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UBC Theses and Dissertations

A study of quadratic processing for passive receiving arrays Turner, Ross Maclean 1972

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\-5JJo7i A STUDY OF QUADRATIC PROCESSING FOR PASSIVE RECEIVING ARRAYS by ROSS MACLEAN TURNER B.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1961 M.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f jr L L~ C TfZ I £ I The U n i v e r s i t y o f B r i t i s h C olumbia V a n c o u v e r 8, Canada A B S T R A C T This work is an investigation and development of theory for the quadratic processing of arrays with Gaussian signals and noise. The quadratic processor is characterized by a weighting matrix, q, in the quadratic form, xtqx, where x is a column vector with its 1^ component the filtered output of the i*"* 1 receiving element. The study is restricted to linear arrays of equally spaced elements. The conditions are established whereby such a quadratic processor has a specified directive pattern. The directive properties of the quadratic pro-cessor are determined and i t is shown that the optimum pattern has side lobes which are 3 dB lower than those of the power pattern of the optimum Dolph-Chebyshev array. The versatility of the quadratic processor for pattern syn-thesis is demonstrated. It is shown that specification of the directive pattern of the processor is not sufficient to specify the elements of the weighting matrix, q, and these additional degrees of freedom can be used to reduce the output noise variance. The performance of the quadratic processor in the presence of noise is studied for a point source model for the background noise. The resulting sig-nal-to-noise ratio is shown to depend on the q matrix through two performance functions, Py(<5) and P^(q). It is shown that /p^ Cq) and p (^<3) become inversely proportional to array directive gain when q is chosen so that the quadratic processor reduces to the square-law-^detected array. However, for the general quadratic processor, i t is shown that these performance functions are not proportional to array directive gain but may be considered as ail extension to the concept of array gain. Two methods for the reduction of the output noise variance are studied, i i based on a minimization of Py(q) and subject to the constraint of specified pattern function and resulting in solution q matrices, q^ . and q . An approximate method for the minimization of the minimum detecta-A ble signal is given based on an interpolation between q^ and q . Numerical results are presented showing how the performance functions vary with array element spacing, steering angle, array size, bandwidth and pattern type. Comparisons are made between the quadratic processor and conventional processors on the basis of the performance functions and signal-to-noise ratio. A q matrix which is maximally uniform subject to the pattern constraints is shown to maximize the signal-to-receivernoise ratio and to maximize the signal-to-background noise ratio for a half-wavelength spaced narrow-band broadside array and to provide performance nearly as good as the optimum quadratic processor for element spacings greater than half-a wavelength. Similar results are established for the endfire array for element spacings greater than or equal to a quarter of a wavelength. i i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v i i LIST OF TABLES .. ix ACKNOWLEDGEMENTS X 1. INTRODUCTION 1 2. QUADRATIC PROCESSING FOR PASSIVE RECEIVING ARRAYS OF EQUALLY SPACED ELEMENTS 8 2.1 Introduction 8 2.2 The General Quadratic Processor of Arques 11 2.3 Quadratic Processing of Linear Arrays with Pattern Constraints 17 2.3.1 Definition of System Structure 17 2.3.2 Optimum Filtering for a Quadratic Processor with Pattern Constraints 19 2.3.3 Directive Properties of a Quadratically Frocessed Linear Array 21 2.3.4 Approaches to the Synthesis of Quadratic Processors with Reduced Noise Output 27 2.4 Pattern Synthesis for the Quadratic Processor 29 2.4.1 General and Optimum Properties of the Pattern Functions 29 2.4.2 Synthesis of Some Classes of Patterns Based on Chebyshev Polynomials . . •. '. 37 2.4.3 Computation of Fourier Coefficients for Chebyshev Pattern Functions 44 2.5 Quadratic Processing for Receiver Noise Limited Passive Arrays 47 2.5.1 The Processing Gain Factor 47 2.5.2 Discussion of Results 55 3. QUADRATIC PROCESSING IN A BACKGROUND NOISE LIMITED ENVIRONMENT ... 64 3.1 Introduction 64 iv Page 3.2 Derivation of Signal-to-Noise Ratio for the Point Source Model of Background Noise . 64 3.3 Relationship of Performance Functions and Array Gains 71 3.4 Matrix Theory for Maximization of Array Gain and Signal-to-Noise Ratio of Square-Law-Detected Arrays 74 3.4.1 Maximization of Narrow Band Array Gain 74 3.4.2 Maximization of Wide Band Gain .. 77 3.4.3 Minimization of the Volume Performance Function for Wide Band Square-Law-Detected Arrays 79 3.5 The Volume Performance Function for the General Quadratic Processor" . 81 3.5.1 Derivation of P y 82 3.5.2 Minimization of P y 83 3.5.3 Symmetry Properties of q^ 86 3.5.4 Formulation of Equations for Computer Solution 90 3.6 The Area Performance Function for the General Quadratic Processor , 97 3.6.1 Minimization of P A 98 3.6.2 Minimization of Py with both Row-Sum and Pattern Constraints 102 3.6.3 Formulation of Equations for Computer Solution 106 3.7 The Minimization of the Minimum Detectable Signal 110 3.7.1 Development of the Minimization Technique ............ I l l 3.7.2 One Parameter Approximation for the F(Py,P^) = 0 Curve 114 3.7.3 The Exact Method 118 3.8 Realization of a Quadratic Processor v?ith a Specified q Matrix 123 4. NUMERICAL RESULTS FOR THE BACKGROUND NOISE LIMITED ENVIRONMENT ... 130 4.1 Introduction 130 4.2 Performance Functions and Signal-to-Noise Ratio 130 4.3 Nomenclature 131 v Page 4.4 Element Spacing 132 4.4.1 Narrow Band Systems 133 4.4.2 Wide Band Systems 144 4.5 The Maximally Uniform q Matrix 147 4.6 The Quadratic Processor Compared to Conventional Array Processors 150 4.6.1 Comparison with the Square-Law-Detected Array 151 4.6.2 Comparison with the Array Correlator 154 4.6.3 Performance Comparisons for Large Arrays 158 4.7 The Minimization of the Minimum Detectable Signal 160 5. SUMMARY AND CONCLUSIONS 163 APPENDIX A GENERATION OF A SET OF FOURIER COEFFICIENTS FOR A CHEBYSHEV POLYNOMIAL 167 A.l Van der Maas' Method 167 A. 2 Extension Method for Large Arrays 169 APPENDIX B DERIVATION OF A RELATIONSHIP BETWEEN VARIANCE AND CONDITIONAL VARIANCE FOR THE POINT SOURCE MODEL OF BACKGROUND NOISE 173 APPENDIX C EQUATIONS REQUIRED FOR WIDE BAND PERFORMANCE FUNCTIONS ... 176 APPENDIX D ONE PARAMETER REPRESENTATION FOR THE PERFORMANCE FUNCTIONS 178 REFERENCES 181 v i LIST OF ILLUSTRATIONS Figure Page 2.1 . Examples of q Matrices for the Square-Law-Detected Array and the Array Correlator 9 2.2 q Matrix for a Three-Element Quadratically Processed Array ... 10 2.3 Symbolic Representation of the Quadratic Processor 12 2.4 Circuit Realization of a General Quadratic Processor 13 2.5 Structure of a Quadratic Processor with Pattern Constraints .. 18 2.6 Quadratically Processed Array 22 2.7 Pre-Detection and Power Patterns for a Dolph-Chebyshev Array 31 2.8 Optimum Squared-Chebyshev Power Pattern and Hypothetical Pattern, ^ ( M - l ) ^ 0 0 3 7 T d u / ^ ) 3 3 2.9 Optimum Post-Detection Patterns for the Dolph-Chebyshev Array and for the Quadratic Processor 36 2.10 Array Correlator with Unequal Subdivisions 39 2.11 Array Correlator with Equal Subdivisions 4l 2.12 Chebyshev-Envelope Patterns for a 30-Element Array 45 2.13 Illustration of Processor Types 48 2.14 Comparison of the Quadratic Processor with the Square-Law-Detected Array for Squared-Chebyshev Patterns 57 2.15 Comparison of the Quadratic Processor with the Square-Law-Detected Array for Chebyshev and Squared-Chebyshev Patterns .. 60 2.16 Comparison of the Quadratic Processor with the Array Correlator 62 3.1 Signal and Noise Spectra 66 3.2 Double Symmetry and the Variable Naming Convention 90 3.3 Graphical Illustration of Minimization of the Minimum Detectable Signal 113 3.4a Symbolic Representation of the Realization of an Arbitrary q Matrix 125 3.4b The Type 1 Multiplicative Processor and Associated Symbolism 125 v i i Figure Page 3.5 Illustration of Multiplier Types and Associated Matrices 126 3.6 The Type 01 Multiplicative Processor 127 4.1 Area Performance Functions for Narrow Band Operation 134 4.2 Performance Functions for a 6-Element Broadside Array 137 4.3 Performance Functions for a 10-Element Broadside Array 138 4.4 Performance Functions for a 14-Element Broadside Array 139 4.5 Effects of Element Spacing on Processing Gain Against Receiver Noise 140 4.6 Effects of Array Steering Angle on Performance Functions 142 4.7 Comparison of Half-Wavelength Spaced with Optimum Spaced Quadratically Processed Arrays 143 4.8 The Effect of Bandwidth on Performance Functions 145 4.9 The Accuracy of the Narrow Band Approximation 146 4.10 Comparison of Maximally Uniform and Optimum Quadratic Processors for Endfire Steering 148 4.11 Comparison of Maximally Uniform and Optimum Quadratic Processors for Broadside Steering 149 4.12 Comparison of the Square-Law-Detected Array with a Quadratically Processed Endfire Steered Array 152 4.13 Comparison of the Square-Law-Detected Array with a Quadratically Processed Broadside Steered Array 153 4.14 Comparison of the Array Correlator with a Quadratically Processed Endfire Steered Array 155 4.15 Comparison of the Array Correlator with a Quadratically Processed Broadside Steered Array 156 4.16 Signal-to-Noise Ratios for the Array Correlator and the Quadratic Processor ; 157 4.17 Estimated F(P^,P.) = 0 Curve for Minimization of the Minimum Detectable Signal 161 v i i i LIST OF TABLES Table Page 2.1 Main Lobe/Envelope Side Lobe Ratios for which the First Zero of the Envelope Coincides with the Second Zero of the Cosine Term 46 i x ACKNOWLEDGEMENT I wish to thank Dr. A.D. Moore, the supervising professor, for his help and encouragement during the course of this project, for suggesting the topic of the research and for arranging for programming assistance in the final stage of the project. Thanks are also due to D. Zilm for providing programming assistance and to Dr. R.L. Olsen for reading the manuscript. Grateful acknowledgement is made to the National Research Council for scholarships held in 1967-1969 and for assistance provided under grant A3357. Thanks are due to the Defence Research Board for assistance received under grant 3801-36 and to the Northern Electric Company for the Northern Electric scholarship held in 1966-1967. Special thanks are due to my wife, Win, for providing encouragement throughout the project, for typing the final draft of the thesis and for expert advice and assistance in drafting the diagrams. x 1 1. INTRODUCTION Multiplicative or correlation processing for receiving arrays is a method for obtaining a narrow beam width directional pattern for an array, at the expense of moderate loss in signal-to-noise ratio or increase in side-lobe heights in order to determine more precisely the location of a radiating source or to resolve two closely spaced sources. The work described in this thesis is a generalization of earlier investigations on correlation processing for arrays. It can also be viewed as a generalization of array directive gain optimization procedures. We shall firs t consider the relationship of this work to correlation processors and review those papers most pertinent to this investigation. Some of the earliest studies in this field consist of analyses of correlation arrays where the array is subdivided into two parts and the summed outputs from each part are multiplied and averaged. Signal-to-noise ratios and beam widths have been determined as functions of the relative sizes of the two subdivisions by Welsby and Tucker.1'2 A Chebyshev synthesis technique for the pattern of the correlation array has been developed by Jacobson.3 A simplification of Jacobson's procedure for finding array weights is given in this thesis. The resolution properties of correlation arrays have been investigated from a theoretical and an experimental viewpoint for a radar receiving array by Shaw and Davies4 and by Ksienski. 5 It was demonstrated that, although superposition does not hold in the response of a correlator to two or more radar returns, the presence and positions of two closely spaced targets may sometimes be determined more accurately than in some conventional systems. S t i l l another approach to multiplicative processing of arrays is the time-averaged-product (TAP) arrays of Berman and Clay.^ Here the outputs of the sub-arrays are averaged before multiplication. The single target 2 response of TAP arrays can be made highly directional; however, there are problems with multiple targets. It has been shown, in a further analysis by Fakley,7 that TAP arrays have severe disadvantages with regard to signal-to-noise ratio and response to distributed sources. Fakley demonstrated that an intra-class correlator (which forms a l l products between array element outputs except for self products and averages the sum of products) gives performance superior to that of a TAP array for a distributed source. The objective of the present investigation is-the study of a general quadratic processor. This general quadratic processor has an output characterized by a quadratic form, xtqx, where x is a column vector and q is the matrix of the quadratic form. The column vector x has components which may be the filtered outputs of the individual array elements. In the most general case, each component of x is a linear combination of a l l the filtered outputs of the array elements. The quadratic processor can be specialized to become the array correl-ator, the square-law-detected array or the intra-class correlator by appropriately choosing the matrix q of the quadratic form which characterizes the processor. Some early analyses of a quadratic processor (not with a general q matrix) are to be found in the reports of Faran and H i l l s . 8 ' 9 Optimum processing for specific quadratic processors has been considered in the papers of Picinbono, 1 0 and of Picinbono and De Suso Barba.11 Analyses of a general quadratic processor with an unrestricted q matrix and most general structure have been presented in papers by Arques 1 2* 1 3 and Bryn.11* The results obtained by Arques and Bryn are equivalent. Arqu£s' analysis is based on maximization of the signal-to-noise ratio. His work is most useful for this investigation because his results are applicable to structures less general than the optimum quadratic processor, 3 that i s , to structures where i t is possible to specify the receiving pattern of the array. Bryn's paper appears to be better known than Arqu£s' and is based on likelihood ratio processing. Bryn has derived a general quadratic processor which determines the likelihood ratio for an array with Gaussian signals and noise. The implementation of the optimum quadratic processor of Arques or Bryn requires a knowledge of the space correlation function of the background noise and the signal. The array beam width is not specified nor are the side lobes. Bryn, however, considered the effect of array element spacing when the background noise was isotropic. He concluded that the optimum processor provided a significant improvement over the conventional uniformly weighted array only when the element spacing became very small and the array became superdirective. In contrast to the approach of Arqu£s and -Bryn, this investigation is concerned with noise reduction in quadratic processors with specified directive pattern functions. For the processor of Arques or Bryn, the x vector has components that are linear combinations of a l l the filtered array element outputs. If a directional pattern is to be specified for the quadratic processor, i t is sufficient to restrict the x vector so that each component consists of a single filtered output of a receiving element with x an M-dimensional vector for an array of M elements. Such is the case in this study, where we are concerned with developing and evaluating techniques for choosing the elements of the q matrix so that certain criteria of performance are satisfied subject to the constraint of specified pattern function. The performance criteria are developed from the concept of a point noise source model for the background noise. The minimization procedures are carried out under the assumption of maximum uncertainty about the positions 4 of the point noise sources and the noise spectra within a specified band. The output signal-to-background noise ratio is derived for a quadratic processor for the point noise source model of background noise. It will be shown that this output signal-to-noise ratio depends on the q matrix through performance functions, Py(q) and P^(q) a s follows: (SNR) Q u t = (BT)Js(SNR).n/(Pv(q) + 2(SNR) .^ ( q ) + (SNR) 2^ where B is the bandwidth and T is an integrating time constant. It will be shown that when the quadratic processor is specialized to a square-law-detected array, "\/ Py (q) and a r e i n v e r s e l y proportional to the directive gain of the array. Thus an appropriate strategy for the square-law-detected array would be the maximization of output signal-to-noise ratio by maximizing the array gain provided the resulting pattern function is acceptable (in some instances the side lobes would be too large or the beam width too wide). In contrast to the square-law-detected array i t will be shown that, for the general quadratic processor,~^/Py (q) and P^(3) are not inversely propor-tional to array gain, are not specified by the processor pattern function and may be minimized subject to the constraint of specified pattern function. Because of the inverse relationship between array gain and the performance functions for a square-law-detected array, the investigation can be viewed as a generalization pf array gain optimization procedures. Parallels will be established and comparisons made to the works of T a i , 1 5 Dfane and Mcllvenna 1 6 and Lo et a l . 1 7 a l l of whom have made contributions to the theory of array gain optimization. Tai has investigated a procedure for choosing the weights of a broadside steered array with cophasal excitation, so that the array gain is a maximum. He has shown that the optimized gain has a local maximum for element spacings of approximately 0.9 wavelengths, t h a t the o p t i m i z e d a r r a y becomes super-d i r e c t i v e f o r s m a l l spacings, and t h a t the o p t i m i z e d a r r a y g i v e s l i t t l e improvement over the u n i f o r m l y weighted a r r a y f o r spacings g r e a t e r than h a l f a wavelength. The phenomenon of an optimum spaci n g at about 0.9 wavelengths appears to have been f i r s t n o t i c e d by K i n g 1 8 f o r a u n i f o r m l y weighted broadside a r r a y . Drane and M c l l v e n n a 1 6 extended T a i ' s a n a l y s i s to o b t a i n a simultaneous gain maximization and p a r t i a l placement of n u l l s of the a r r a y p a t t e r n f u n c t i o n . Lo et a l . 1 7 i n v e s t i g a t e d gain maximization f o r a r r a y s which are not n e c e s s a r i l y l i n e a r a r r a y s and w i t h noncophasal e x c i t a t i o n . They i n v e s t i g a t e d the e f f e c t of o p t i m i z i n g the g a i n on the "Q f a c t o r " which i s a measure of the s e n s i t i v i t y of the a r r a y to s m a l l e r r o r s i n element e x c i t a t i o n s . A l s o considered by Lo et a l . was the maximization of s i g n a l - t o - n o i s e r a t i o f o r known background n o i s e . A method f o r a r r a y g a i n maximization developed by Cheng and Tseng 1 9» 2 £ ) p r o v i d e s the s t a r t i n g p o i n t f o r the s t u d i e s of r e f e r e n c e s 15 to 17. Cheng and Tseng express the a r r a y gain as a r a t i o of two q u a d r a t i c forms w i t h s p e c i a l p r o p e r t i e s which make maximization p a r t i c u l a r l y simple. The a p p l i c a t i o n of the method of Cheng and Tseng to the maximization of s i g n a l - t o - n o i s e r a t i o w i l l be i n v e s t i g a t e d f o r both the narrow band and wide band square-law-detected a r r a y s . T h i s method w i l l be shown to be-p a r t i a l l y a p p l i c a b l e to the general q u a d r a t i c processor because one of the performance f u n c t i o n s , P (q), can be expressed as a q u a d r a t i c form w h i l e the A. o t h e r , P y ( q ) , cannot. The approach taken i n t h i s i n v e s t i g a t i o n most c l o s e l y p a r a l l e l s t h a t of T a i . 1 5 R e s u l t s on optimum spacing s i m i l a r to those of T a i w i l l be obtained f o r the performance f u n c t i o n s of the g e n e r a l q u a d r a t i c processor f o r some 6 pattern functions but not for others. The effects of bandwidth, array steering angle, pattern type, array beam width and side lobe level, and element spacing on the performance functions of a general quadratic processor will be investigated. Two methods of background noise reduction for the quadratic processor will be studied. The first method is based on a minimization of Py(q) subject to pattern constraints. The second method is based on a minimization of and a subsequent minimization of Pv(q) subject.to the constraint of minimum P.(q) and the pattern constraints. A Methods will be investigated for interpolating between these two minimization procedures so as to minimize the minimum detectable signal. Numerical results will be presented to show that in many instances, such an interpolation is unnecessary. The maximization of signal-to-receiver noise ratio of the general quadratic processor is investigated. It will be shown that the q matrix which maximizes the signal-to-receiver noise ratio has the property that its elements are as uniform in size as is possible subject to the pattern constraints. Such a matrix is termed "maximally uniform". Numerical results are presented to show the magnitude of improvement in signal-to-receiver noise ratio for the quadratic processor as compared to conventional array systems. An investigation is made of the pattern synthesis capabilities of the general quadratic processor. It will be shown that the optimum post-detection pattern for a quadratically processed array of M elements is a Chebyshev polynomial of order 2(M - 1 ) . In comparison, the post-detection pattern of the optimum Dolph-Chebyshev array is a squared (M - l ) t n order Chebyshev polynomial. It will be shown that the optimum pattern for the general quadratic processor has side lobes 3.0 dB lower than the pattern of the optimum Dolph-Chebyshev array. The investigations described in the thesis are restricted to linear arrays of equally spaced elements. Applications to other than linear arrays would require a specification of array geometry and spacing. However, the techniques described in the thesis could be applied to various array geometries. New formulations for the performance functions would be required for every type of array geometry considered. 8 2. QUADRATIC PROCESSING FOR PASSIVE RECEIVING ARRAYS OF EQUALLY SPACED ELEMENTS 2.1 Introduction A quadratic processor for a receiving array is characterized by a quadratic form S^qx where q is a square matrix and x is a column vector with components which are filtered linear combinations of the outputs of receiving elements. This investigation is largely concerned with noise reduction in arrays which are constrained to have prescribed directive properties. In this case, x is a column vector with i t n component the filtered output of the i t n receiving element. Illustrations of some array types with their associated q matrices are given in Figures 2.1 and 2.2. Figure 2.1 illustrates the square-law-detected array and the array correlator. Figure 2.2 illustrates the structure of the most general quadratic processor of the type which satisfies the above restriction on x and is thus capable of having a prescribed directive pattern function. In this chapter, the general case where x is a linear combination of a l l the filtered receiver outputs is briefly considered in section 2.2. In section 2.3 the structure of the quadratic processor with pattern constraints is given, as is the optimum filtering. A point source model for the back-ground noise is postulated, and approaches to the synthesis of the q matrix based on this model are also contained in section 2.3. Synthesis techniques for processor pattern functions, analogous to antenna pattern functions, are discussed in section 2.4. The final section in this chapter contains an evaluation of the perform-ance of the quadratic processor with pattern constraints and receiver noise 9 (a) SQUARE-LAW-DETECTED ARRAY e V 2 d ! a 3 V 4 •ft a 2a, 2 a 2 a 2a 3 V 4 V s a 3 a i 3 2 2 OJ 3 V 4 a a 3 5 V l 4 2 a, a 4 3 2 °4 4 5 V . a a 5 2 5 3 2 3 6 ( Z a x ) ( E aix.) i=i 1 1 j=4 J J s-.i Cb) A R R A Y C O R R E L A T O R 0 0 0 a i a 4 a <a 15 0 0 0 V4 a a 2 5 0 0 o 3 4 ct d 3 5 V l ° W 4 3 0 0 O V . a 5 a 3 O o o V . °6a2 0 o o FIGURE 2.1 EXAMPLES OF c[ MATRICES FOR THE SQUARE-LAW-DETECTED ARRAY AND THE ARRAY CORRELATOR 10 FIGURE 2.2 c[ MATRIX FOR A THREE-ELEMENT QUADRATICALLY PROCESSED ARRAY 11 limited operation. 2.2 The General Quadratic Processor of Argues The Arques' 1 2 , 1 3 model for the quadratic processor has a structure sufficiently general to be equivalent to the optimum array processor deter-mined by Bryn,ll+ which determines the likelihood ratio for Gaussian signals and noise. Arques"' analysis is in terms of signal-to-noise ratios. Because his formulation can be specialized to less general processors of the type consid-ered here, his results are summarized in this section. The mathematical operations of the quadratic processor are illustrated in block diagram form in Figure 2.3. Single arrows indicate the trans-mission of scalars and double line arrows, the transmission of vectors. The circuit realization of the quadratic processor is illustrated in Figure 2.4. For the system of Figure 2.3 and Figure 2.4 the following notation will be used. v^ c_i_vr = instantaneous output when signal and noise are present. v^ = instantaneous output when noise only is present. v~ = instantaneous output when signal only is present. Y^ (u)) = matrix of cross-spectral densities at the array input when noise only is present. Yjj+g(o)) = matrix of cross-spectral densities at the array input when signal and noise are present. Yg(to) = matrix of cross-spectral densities at the array input when signal only is present. Expected values are indicated by E(*). 12 A R R A Y E L E M E N T S V V V V V FORMATION OF VECTOR, v Q ) FROM ARRAY INPUTS V V 0 TIME DOMAIN CONVOLUTIONS BY MATRIX OF FILTERS WITH TRANSFORM ckoO V. FORMATION OF QUADRATIC FORM WITH MATRIX c[ _5Z_ A V E R A G E g ^ t ) , Q,Cu>) i FIGURE 2.3 S Y M B O L I C REPRESENTATION OF THE QUADRATIC PROCESSOR 1 3 M RECEIVING ELEMENTS FIGURE 2.4 CIRCUIT REALIZATION OF A GENERAL QUADRATIC PROCESSOR 1 4 The The useful signal output of the array is defined as Su = E<V3,S+N> - E<V3,N> <2-D interfering noise, N?, at the output is the variance of v 3 ^j, i.e. Nf - Var (v 3 j N) = E(v 2 > N) - E 2 ( v 3 j N ) (2.2) The signal-to-noise ratio at the output is defined by SNR = Su/N. = ( E ( v 3 j N + s ) - E ( v 3 ) N ) ) / l / E ( v 2 y - E 2 ( v 3 > N ) (2.3) Arques' formulation for the useful signal output, and the interfering noise variance are as follows: 00 S u = (1/2TT) J tr((Y N + s(a>) - Y N(^))J(w,0)) doj (2.4) —00 where J(to,u)') = Gt(co)qG(oj - a)')G1(co') 00 CO and Var (v 3 N) - (1/2TT2) j / tr(y ( C J ) J ( W , C O * ) ' —OO —CO •Y j^Cu - w'^Cw.w')) da) du' (2.5) J((J3,W'), YM( w) a n d Y^,,o(u) a r e (M x M) matrices where M is the number of " N+o elements in the receiving array. As shown in Figures 2.3 and 2.4, G(o>) is an (m x M) matrix and the matrix of the quadratic form, q, is an (m x m) matrix. The size of m is not specified. In the above equations and those to follow, tr(*) indicates the trace of a matrix. The superscript t indicates the transposed conjugate of a complex matrix. In order to simplify equations (2.4) and (2.5) define CO j I G J C O J ) I 2 du = Gf(0)/2T (2.6) —CO where T is an averaging time constant. With reference to Figure 2.3, note that s v „ (u) = S * (co) + 5(O))E 2(V 9)2TT (2.7) V2 V2 V2 V2 2 where 6(co) is an impulse function, and S y (co) is the power spectral 2 2 density of V2(t) which is the input to the averaging f i l t e r (transfer function G,(co)). S' (to) is a well behaved spectral density function . V2 V2 obtained by eliminating the impulse in S v (to) . 2V2 The following approximation is used: CO CO f |GiOo)|2Sv v > ) dco * S ' (0) j lG1(oJ) |2 dco —CO *~ c. i. —co Th + G|(0)E 2(v 2)2ir (2.8) is approximation implies that v (co) is relatively flat over, the band-width of G1(co) Nov; i t can be shown that CO CO Var (v 3 N) = (1/27T2) J \ tr (yN(co) J(co,co' )YJ(U)-CD' )J^co.co')) dco dco' ' —CO —CO CO (1/2TT2) f |G](a),)|2S' (co') dco' (2.9) J- V2 V2 If the approximation of equation (2.8) is used, one obtains the result CO •Var (v 3 ) = (1/4TT 2)(G 2(0)/T) | tr((YN(co)Q(co))2) dco (2.10) where Q(co) = G*~ (co)qG(co) (2.11) CO and S u = (G1(0)72u) j tr((? N + s(w) - ?N(w))Q(co)) dco (2.12) The signal-to-noise ratio obtained from equations (2.3), (2.10), and (2.12) is 16 (SNR) 2 = S 2/Var ( v 3 ) = oo "7 • / t r ( ( Y N + s ( w ) - ? N(u))Q(u))) dul (2.13) oo J tr((Y N(co)Q(u))) 2) du The condi t i o n f o r (SNR) 2 of equation (2.13) to be a maximum i s obtained by using the Schwarz i n e q u a l i t y , i . e . f b J F(u)H(u) du < J H 2(u) du J F 2(u) du with e q u a l i t y i f H(u) = KF(u) where K i s any constant. A p p l i c a t i o n of the Schwarz i n e q u a l i t y to equation (2.13) e s t a b l i s h e s the co n d i t i o n f o r the sign a l - t o - n o i s e r a t i o to be a maximum as t r ( ( Y N + s ( u ) - y N ( u ) " Y N(w)Q(w)Y N(u))Q(w)) = 0 A s u f f i c i e n t condition f o r equation (2.14) to be true i s (2.14) Y N + s ( U ) - Y N ( w ) = ^ N ( u ) Q ( u ) ^ N ( u ) and thus Q(u) = G t(u)qG(u) = y~x(u)(?„.e(u) - ? H ( u ) ) t M * ( w ) .-1 -1 N+S ' N ' ' N (2.15) I f G(u) and q are chosen to s a t i s f y equation (2.15) the maximized si g n a l - t o - n o i s e r a t i o i s CO ( S N R ) m a x = T J t r < ( Y N + S ( t o ) - V<«0 ) Y^ 1 (*>)) 2 du (2.16) — C O The s i g n a l - t o - n o i s e r a t i o of equation (2.16) i s the t h e o r e t i c a l maximum obtainable when both the noise and s i g n a l c r o s s - s p e c t r a l density matrices are known. Such a complete knowledge of the background noise s t a t i s t i c s i s very seldom a v a i l a b l e . Thus (2.16) provides an upper bound to the 17 performance of actual systems. Note that not only the signal-to-noise ratio but also the beam width and side lobe characteristics of an array are of interest. For instance, in radio or radar astronomy, one may be willing to sacrifice a considerable amount of signal-to-noise ratio, which may be compensated for by integrating for a very long time, in order to improve the array resolution. This investigation is restricted to the case where" i t is possible to specify the array pattern. Thus the signal-to-noise ratios obtained are lower than indicated in equation ( 2 , 1 6 ) and, i f optimum filtering is desired, i t is necessary to solve equation ( 2 . 1 4 ) rather than ( 2 . 1 5 ) . We shall be mostly concerned with the case where knowledge of the noise statistics is such that only simple bandpass filters are practical and efforts will be made to choose the elements of the .q matrix to effect some noise reduction. In the following section, the structure of a quadratic processor with a specifiable directive pattern, is given. 2 . 3 Quadratic Processing of Linear Arrays with Pattern Constraints i t 2 . 3 . 1 Definition of System Structure With reference to the general quadratic processor, i t will be shown that, for the application of pattern constraints, a sufficient restriction on the system structure is that the G(to) matrix be diagonal and have the form G(to) •= G(to)I where I is an (M x M) unit matrix. The structure of a quadratic processor with such a diagonal G(to) matrix is illustrated in Figure 2 . 5 . The most general model of this constrained quadratic processor will sometimes be referred to as the general quadratic processor or just the quadratic processor in the remainder of the thesis. In this section the equations for the expected value of the signal 18 LINEAR ARRAY OF EQUALLY SPACED ELEMENTS v OM V I M •G(to) M M v*= y T <i. v, FIGURE 2.5 STRUCTURE OF A QUADRATIC PROCESSOR WITH PATTERN CONSTRAINTS 19 output and the output noise variance are obtained f o r the constrained quadratic processor. These equations are used to obtain the optimum f i l t e r i n g ( s e c t i o n 2.3.2) and equations governing the d i r e c t i v e p roperties of the quadratic processor (section 2.3.3). The a p p l i c a t i o n of pattern constraints forces the expected value of the output response, E(v-j) , to a point target at an angle 6, to vary with 0 i n a prescribed manner. This v a r i a t i o n with 6 i s analogous to the v a r i a t i o n observed i n an antenna pattern. The mean and variance of v-j are obtained from equations (2.10) and (2.12). Assuming independent s i g n a l and a d d i t i v e noise so that Y N + S(w) - Y N(w) = ? s ( w ) • (2.17) and i d e n t i c a l f i l t e r s i n each channel so that G(co) = G(«)I (2.18) and thus Q(co) = Gt(to)qG(co) = [G(to) 12q (2.19) The u s e f u l s i g n a l output (from equation (2.12)) i s CO S u = ( G 1(0)/2T T ) f |G(co)| 2tr(Y s(co)q) dco (2.20) — C O The variance of the output, v^, when noise only i s present, i s obtained from equation (2.10) and i s CO Var ( v 3 N ) = (G 2(0)/4T T 2 T) J |G(co) | 4 t r ( ( Y N ( c o ) q ) 2 ) dco (2.21) ' —CO 2.3.2 Optimum F i l t e r i n g f o r a Quadratic Processor with Pattern Constraints The optimum f i l t e r i n g f o r the quadratic processor with p a t t e r n c o n s t r a i n t s i s obtained from the equations of s e c t i o n 2.2. Because the processor f i l t e r s a t i s f i e s the r e s t r i c t i o n , G(co) = G(co)I, where I i s an (M x M) u n i t matrix, i t i s necessary to use equation (2.14) rather than (2.15). S u b s t i t u t i o n from equations (2.17) and (2.18) into equation (2.14) y i e l d s the r e s u l t tr( Y s(u))q) = |G(a>)| 2tr(( Y N(o))q) 2) (2.22) and thus |G(co) | 2 = tr(Y s(o))q)/tr((Y N(a))q) 2) (2.23) The optimum f i l t e r i n g would almost c e r t a i n l y be designed f o r an array steered d i r e c t l y toward the sig n a l source with the s i g n a l value the same on every receiving element after steering. For t h i s case, where U i s an (M x M) matrix with every element equal to unity. From equation (2.24). M M • tr(Y s(u))q) = Y s sGo) [ [ q i m i = l m=l (2.25) M M M M Similarly t r ( ( Y N ( u ) q ) 2 ) = ^ i ^ m j ^ i k ^ ^ N m l ( ^ i=l k=l m=l j = l J (2.26) • - M M . E E ^ im (2.27) Thus |G(u)| 2 = -i= l m=l M M M M E E E E 4 i k q m j Y N i U ( c o ) Y N m j ( w ) i = l k=l m=l j= l J The maximized signal-to-noise r a t i o for |G(u)| 2 given by equation (2.27) i s obtained by use of the Schwarz inequality as i n section 2.2 and i s ? (tr(Y s(o>)q)) 2 (SNR) 2 = T I du max J ——jz— T—v ~\ ' ) — ~ m t r ( Y N ( u ) q ) / (2.28) or i n expanded form E E E E qik^mjYNikC^YNmjC") 1=1 k=l m=l j=l Note that equation (2.29) is strongly dependent on the q matrix in contrast to equation (2.16). The above expressions for the optimum filtering of a constrained quadratic processor have been given for completeness and are obtained by straightforward application of equations derived by Arques. In the major part of this investigation i t is assumed that one has only a minimal amount of prior information about the signal spectrum and thus a simple bandpass f i l t e r is the logical choice for G(co). The major effort has been directed toward noise.reduction by appropriate choice of the q matrix, subject to pattern constraints, under the assumption that only simple bandpass filtering will be used. 2.3.3 Directive Properties of a Quadratically Processed Linear Array The directive properties of the quadratic processor are determined in this section by reference to a point source model for both the signal or . signals and the background noise. It is assumed that the array (Figure 2.6) is steered by means of ideal delays. For narrow band systems the delay function is adequately approximated by midband phase shifts. Wide band systems require the implementation of true delays. With reference to Figure 2.6, the response at the i receiving element to a plane wave with instantaneous value, s(t), incoming from an angle 6, is v Q i ( t ) = s(t' - A + di(cos 0Q - cos 6)) c or v n - ^ = S ^ ~ - i u ^ (2.30) c FIGURE 2.6 QUADRATICALLY PROCESSED ARRAY where u = cos QQ - cos 0 t = t' - A d = element spacing c = phase velocity and A is a constant delay applied to each input so that the total delay is always positive and realizable. The instantaneous signal value, s(t), is assumed to be a stationary random variable. Let Ri m(f) be the correlation function between the i t n and mtn inputs to the quadratic processor. Then R. (T) = E(v..(t + T)vn(t)) lm Oi O J I K where VQ-^ Ct) and vQ m(t) a r e stationary random variables. .For and V p m ( t ) given by equation (2.30) R I M ( T ) = E(s(t + t - diu)s(t - dmu)) c c or R I M ( T ) = R s s ( T + (m-i)du) (2.31) c • The i,mt n entry of Ys(w) is just Y s i m(w) CO where Y Si m(w) = J R i M ( T ) exp ( - J U T ) dT and thus v . (u) = Y (u) exp (jud(m-i)u) (2.32) sim ss — c ce Y s s(") = J R S S ( T ) e xP ("JWT) d T wher is a purely real, even function of u because R 0 0 ( T ) is even. Thus f (w) is a Hermitian matrix. These results will now be used to obtain an expression for the useful signal output, S . Substitution for q in equation (2.20) yields the result u S u = (02 (0 )7270 J |G(o>) | : M M E E wsira<"> m=l i=l du (2.33) Finally, substitution for Ysim^^ from equation ( 2 . 3 2 ) yields S u = ( G 1 ( 0 ) / 2 T I ) j | G ( c o ) | 2 Y s s ( w ) r M M E E qim e x p (jwd_(m-i)u) i=l m=l c dco ( 2 . 3 4 ) to or, more simply CO S u = ( 0 ^ 0 0 / 2 7 0 J* JG(to)|2YSs(w)P(to,u) d M M where P(to,u) - E E q i m e x p (J w.l ( m _ i ) u ) i=l m=l c Because the q matrix is symmetric, P(to,u) can be expressed as a Fourier series as follows: ( 2 . 3 5 ) ( 2 . 3 6 ) M-l P(co,u) = E c o s ktodu k=0 c (2.37) where M D o = E Hi i=l and for k^ O, M-k M Dk = E ^i.x+k+ E si.i-k i=l i=k+l ( 2 . 3 8 ) Because of the symmetry of q, can be written, for k^ O, M-k M Dk " 2 E <k,i+k " 2 E , <i,i-k i=l ' i=k+l P(io,u) will be called the processor pattern function of the array and differs from the usual array pattern function or space factor which are the corresponding terms in array theory. In the case where the quadratic pro-cessor is a square-law-detected array, the processor pattern function is the square of the array pattern function. No such simple correspondence exists between the standard array pattern function and the processor pattern function for the general quadratic processor. P(to,u) is constrained to have a prescribed form by specifying the set 25 of Fourier c o e f f i c i e n t s {D^} i n equation (2.38). This set of equations (2.38) constitutes a set of c o n s t r a i n t s on the elements of the q matrix when q i s chosen to s a t i s f y some c r i t e r i o n of performance. Performance c r i t e r i a are developed i n terms of the variance of the output of the quadratic processor when operating i n an environment where the background noise i s contributed to by a large number of r a d i a t i n g point sources. Thus we consider the matrix of cross s p e c t r a l d e n s i t i e s , Y^( w)> f ° r a background noise f i e l d a r i s i n g from L point sources. The l o c a t i o n of the h t n point source i s s p e c i f i e d by u^ = cos GQ - cos 0^ , f o r 1 ^ h - L , where 0Q i s the angle to which the array i s steered and 0^ i s the angle at which the h t n source l i e s . We proceed as i n the' determination of Y s(w) f o r a s i n g l e point source s i g n a l . Because the noise sources are uncorrelated, t h e i r c o n t r i b u t i o n s to the matrix y^(o;) can be found by summing c o n t r i b u t i o n s of the i n d i v i d u a l sources. The c o n t r i b u t i o n of each i n d i v i d u a l noise source has the form of equation (2.32) i f one replaces Y s s ( a ) ) with Nj1(co) , the power s p e c t r a l density of the h t n noise source. Thus the i k t n entry of Yv,(u) i s L Y N i k = Y, N h ( ( o ) e xP ( J u ( k - i ) d u h ) (2.39) h= 1 c Because N^(co) i s r e a l , i t follows that Y^j( w) i s a Hermitian matrix. The output noise variance w i l l now be determined f o r Y^C10) with elements given by equation (2.39). The integrand occurring i n equation (2.21) f o r the variance, i s | G ( w ) | " t r ( Y N U ) q ) 2 m=l h=l M M L £ V e xP C-jwdCnu^kuh)) k=l n=l c (2.40) where (2.39) has been substituted i n t o (2.22). We define a f u n c t i o n , F(tou^jtin^) , as follows: F(tou-^ ,tou2) M M E E q k n e x p ( - J u i L ( n u i _ k u 2 ^ k=l n=l c The variance can now be written i n terms of F^u^.tou^) as f o l l o w s : • L L » Var ( v 3 N ) = (G|(0)/4T 7 2 T) E E J I G(to) | 4 N h(to ) N m(co) ' m=l h=l -«> •F(toum,tou^) dto where equations (2.41), (2.40), and (2.21) have been used. Some of the properties of F(tou^,tou2) are F (tou, tou) = P 2(to,u) F(tou^,un^) = F(tou2,tou2) - 0 F(0,0) = P 2(to,0) = P 2(0,0) (2.41) (2.42) • (2.43) where the processor pattern f u n c t i o n , P(to,u), i s given by (2.36) or (2.37). For the s p e c i a l case where the quadratic processor i s a square-law-detected array, the elements of the q matrix s a t i s f y the c o n d i t i o n q i m = q i q m Then F ^ U j . a n ^ ) = P(to,Uj)P(to,U2) and P(to,u) = A 2(to,u) (2.44) (2.45) (2.46) where A(to,u) i s the usual array space f a c t o r or pattern. The expression f o r the output noise variance given i n (2.42) can be rewritten i n s t i l l another form as 27 L " Var (v ) = (G 2 ( 0 ) / 4 T T 2 T ) £ J |G(co) | 4N 2(w)P 2(u),u ) du J ' N m=l -« L L » + ( G 2 ( 0 ) / 4 T T 2 T ) ^ Y, J |G(a))| l tN h(a3)N m(a))F(uu m,a)u h).du (2.47) m=l h=l -°° ntfh The magnitude of the f i r s t term i n (2.47) i s f i x e d once P(w,u) i s p r e s c r i b e d . Any attempt to minimize the v a r i a n c e s u b j e c t to the c o n s t r a i n t of p r e s c r i b e d processor p a t t e r n f u n c t i o n , P(io,u) , must n e c e s s a r i l y be concerned w i t h m i n i m i z a t i o n of the second term i n equation (2.47). 2.3.4 Approaches to the Synthesis of Quadratic P r o c e s s o r s w i t h Reduced  Noise Output There are s e v e r a l p o s s i b l e approaches, based on equation (2.47), which w i l l improve the performance of the q u a d r a t i c processor. I f the p o s i t i o n s of the p o i n t noise sources are known and the system i s narrow band, or i f the s p e c t r a of the p o i n t n o i s e sources are known and the f i l t e r , G ( O J ) , i s chosen f i r s t , the q m a t r i x which .minimizes Var ( V 3 ^) I s e a s i l y found, i n p r i n c i p l e , by d i f f e r e n t i a t i o n w i t h r e s p e c t to M ( M + l ) / 2 independent components, q i m , of the symmetric q m a t r i x . A system of M ( M + l ) / 2 l i n e a r equations i n M ( M + l ) / 2 unknowns i s obtained and should be e a s i l y s o l v a b l e . A maximization of the s i g n a l - t o - n o i s e r a t i o by j o i n t l y choosing G(io) and q s u b j e c t to c o n s t r a i n t s has not been considered. I t i s b e l i e v e d t h a t , f o r p a s s i v e r e c e i v i n g systems f o r which q u a d r a t i c p r o c e s s i n g might be u s e f u l , very seldom are the s i g n a l and n o i s e s p e c t r a known a c c u r a t e l y enough to a l l o w any other s p e c i f i c a t i o n of G(u>) than t h a t i t pass the r e q u i r e d band of frequencies under study. Indeed, the s i g n a l spectrum which i s r e q u i r e d f o r the optimum choice of G(to) may w e l l be what the p r o c e s s o r i s b e i n g used to determine. ' • I f the p a t t e r n i s w h o l l y or p a r t i a l l y s p e c i f i e d , c o n s t r a i n t s may be 28 applied by means of Lagrange multipliers. For the completely specified pattern function, the constraints are obtained from equation (2.38). Partial specification of the pattern function might be effected by choosing the positions of some of the 2(M - 1) zeros of P(co,u) and would require equation (2.38) as constraints with the {D^ } as unknowns and a further set of constraints on the {D^.} obtained by setting P(co,u) = 0 at the various values of u which are chosen to be zeros of the pattern. Before describing the approach taken in this thesis, note that i f there are less than M - 2 interfering noise sources and i f no constraints are placed on P(co,u) with regard to beam width or side lobes, the variance, as given by (2.47), can be made to become essentially zero for very narrow band systems by making the quadratic processor a simple square-law-detected array with pattern zeros at the positions of the interfering noise sources. This follows because, as previously observed, for a square-law-detected array, F(coui , M U 2 ) = P(o),u1)P(w,u2) , and P(COJU) = A2(to,u), where A(co,u), the array pattern function for an M-element array, is a polynomial of order (M - 1) in cos (todu/2c) . Thus, i f zeros of P( IOQ,U) are chosen to occur at u = t^, for m = 1 to L, and L - M — 2, then F(tOQUj1,tOo"m) = 0 and P(co0,u^) = 0 at frequency for 1 - h - L and 1 - m - L. Thus equation (2.47) is zero. The limiting noise becomes the thermal receiver noise, the effect of which is not included in (2.47). The main emphasis in the investigation has been placed on the case of completely specified processor pattern function with minimal knowledge of the directive properties of the background noise. Synthesis of processor pattern functions using Chebyshev polynomials is dealt with in section 2.4 where i t is shown that the quadratic processor can be synthesized to have a narrower beam width and lower side lobes than the square-law-detected array. R e c e i v e r - n o i s e - l i m i t e d q u a d r a t i c a l l y processed arrays are analyzed and evaluated i n s e c t i o n 2.5. In Chapter 3, i n t e g r a l performance functions are developed on the basis of the function, FCwupun^). These performance functions are shown to be generalizations of the concept of d i r e c t i v e gain of a r e c e i v i n g array. Comparisons are made between the performance of the quadratic processor and conventionally processed arrays and numerical r e s u l t s are presented i n Chapter 4. 2.4 Pattern Synthesis f o r the Quadratic Processor 2.4.1 General and Optimum. Properties of the Pattern Functions In t h i s s e c t i o n the r e s u l t s of the i n v e s t i g a t i o n i n t o the d i r e c t i v e properties of the quadratic processor w i l l be presented. The pattern function f o r an M-element q u a d r a t i c a l l y processed l i n e a r array i s given by equation (2.37) which i s re w r i t t e n as fo l l o w s : M-l P ( u , u ) = P(2TTC/X,U) = Y, \ c o s ( 2 k T r d u / A ) (2.48) k=0 k M-l or P ( 2 T T C / X , U ) = Y D k T 2 k ^ c o s m d u / x ) (2.49) k=0 Thus we can express the pattern f u n c t i o n as a polynomial P(2irc/X,u) = Q 2 ( M - l ) ( c o s * d u/*) (2.50) where X i s the wavelength at frequency o)/2ir h e r t z , d i s the element spacing and c i s the wave v e l o c i t y . T2 k(cos T t d u / X ) i s a Chebyshev polynomial of order 2k which s a t i s f i e s the i d e n t i t y T2k(cos irdu/X) = cos 2 k f r d u / X (2.51) The v a r i a b l e u i s u = cos 0 - cos QQ where 0^ i s the s t e e r i n g angle of the array. 3 0 Equation ( 2 . 4 9 ) is obtained from equation ( 2 . 4 8 ) by use of the identity ( 2 . 5 1 ) . Equation ( 2 . 5 0 ) indicates that P ( 2 T T C / X , u) is a polynomial, ^ ^ (cos Trdu/X) , of order 2(M - 1 ) in the variable cos udu/X as can be deduced from equation ( 2 . 4 9 ) . The polynomial, (cos T T d u / A ) , has at most 2(M - 1 ) real zeros. The positions of these zeros can be selected arbitrarily for the general quadratic processor. However, in the case of the square-law-detected array, equation ( 2 . 5 0 ) is the expression for the post-detection pattern function which is proportional to the non-negative power pattern function of the array. Because -Q ( c o s T T d u / A ) is continuous and non-negative, i t f o l -lows that its real zeros must also be zeros of its first derivative. By a 2 2 theorem of algebra , i t follows that i f any zero of a polynomial is also a zero of its first derivative, this zero must be of at least second order. Thus an M-element square-law-detected array has at most (M - 1 ) real, double zeros. The pattern function for the general quadratic processor is a post-detection pattern. However, this pattern is not restricted to be non-negative and hence is not necessarily proportional to the array power pattern The only restriction on the ^ n equations ( 2 . 4 8 ) and ( 2 . 4 9 ) is that they be real. The pattern function of a quadratic processor is a polynomial of order 2(M - 1 ) in cos irdu/X and thus can have 2(M - 1 ) distinct real zeros and M - 1 negative lobes. This allows the quadratic processor to be synthesized with lower side lobes and/or narrower beam widths as compared to the square-law-detected array. The synthesis of optimum patterns for arrays symmetrically weighted about 2 1 the array centre was first described by Dolph. Figure 2 . 7 illustrates the weighting scheme and a typical pattern function for a Dolph-Chebyshev array. Symmetrical weighting about the array centre was assumed for mathematical simplicity. ° 1 0 xrduA (RADIANS') Z ° 3-FIGURE 2.7 PRE-DETECTION AMD POWER PATTERNS FOR A D0LPH-CHE8YSHEV - ARRAY 23 However, recently Schoenberger has shown that, given any asymmetrically weighted array, one can always find a symmetrically weighted array with the same beam width and a smaller maximum side lobe. Thus Dolph's procedure is optimum for a l l linear arrays of equally spaced elements subject to restric-tions on element spacing which will be discussed later in this section. Another method for demonstrating the optimality of symmetrically weighted arrays vis-a-vis asymmetrically weighted arrays will be given here. This method was determined by the author independently of Schoenberger's work. The power pattern of a conventional array is proportional to the post-detection pattern of a square-law-detected array. This pattern was shown to be a polynomial, ( c o s T T d u M ) with at most (M - 1) real, double zeros for an array with M elements and with either symmetrical or asymmetrical weighting about the array center. However, some or a l l of the roots of this polynomial may be complex. Dolph's proof can be applied subject to the condition that the patterns be non-negative to show that the optimum power 2 tli pattern is T^ (Z cos T T d u / A ) where T M ^ (Z cos T T d u / A ) is an (M -1) order Chebyshev polynomial. The proof assumes the existence of a noni-negative pattern, Q.2(^  ^  (cos Trdu/A) 2 which has a beam width the same as that of T^ ^(Z cos T T d u / A ) but lower side lobes. 2 Beam width is defined as shown in Figure 2.8. Since T^ (Z cos T T d u / A ) is equi-ripple, i t follows that O ^ Q ^ (cos T T d u / A ) must intersect i t at at least 2(M - 1) points as shown in Figure 2.8. A polynomial of order 2(M - 1) is completely determined by specifying 2 its value at 2(M-l) points. Thus the two polynomials, T^ ^ (Z cos T T d u / A ) a n d Q2(M-1)^C0S ^ u ^ » m u s t coincide because they have 2(M - 1) points in common and both polynomials are of order 2(M - 1) in the variable cos T T d u / A . Thus we have a contradiction and the assumption that )^ (cos T T d u / A ) can 33 HYPOTHETICAL PATTERN,Q^M-I)*"5 ™*U/0O »S ASSUMED TO HAVE SAME BEAMWIDTH AS T^ ,., (Z cos ndu/A) BUT LOWER SIDE LOBES - BEAM WIDTH FOR BOTH PATTERNS IS 20 B .T^-i CZ cos TTCAU/X") M=7 1Tdu/X (RADIAN6) FIGURE 2.B OPTIMUM SQUARED -CH EBY9H EV POWER PATTERN AND HYPOTHETICAL PATTERN, have the same beam width but lower s i d e lobes than T^_^(Z cos 7Tdu/ A ) i s i n c o r r e c t . Thus T^_^(Z cos udu/X) i s the optimum p a t t e r n . Such an optimum power p a t t e r n i s obtained i f the p r e - d e t e c t i o n p a t t e r n i s the Chebyshev polynomial T^_^(Z cos irdu/X). Such a p r e - d e t e c t i o n p a t t e r n i s obtained by s y n t h e s i z i n g an array which i s symmetrical about i t s centre by u s i n g Dolph's s y n t h e s i s method. This completes the proof. Dolph showed that an a r r a y of 2M - 1 elements w i t h a p r e - d e t e c t i o n array p a t t e r n of the type i n equation (2.49) could be s y n t h e s i z e d , by ap p r o p r i a t e choice of the {D^}, to have a p r e - d e t e c t i o n a r r a y p a t t e r n which was the 2(M - 1) order Chebyshev p o l y n o m i a l , ^2(H-1)^ c o s ^ u / A ) . H e showed th a t t h i s p a t t e r n was optimum i n the sense that i t had minimum beam width f o r a given s i d e lobe l e v e l . However, a p a t t e r n f u n c t i o n of the type i n equation (2.49) i s o b t a i n a b l e w i t h a q u a d r a t i c a l l y processed a r r a y of o n l y M elements. T h i s processor p a t t e r n f u n c t i o n i s a p o s t - d e t e c t i o n p a t t e r n f u n c t i o n whereas the Dolph method was developed f o r the s y n t h e s i s of p r e - d e t e c t i o n p a t t e r n s . The best p r e - d e t e c t i o n Dolph-Chebyshev p a t t e r n o b t a i n a b l e f o r an M-element array i s T^_^(Z cos n d i i / A ) . 2 1 Such an a r r a y has a power p a t t e r n P s o ( 2 T T C / A , U ) = T 2 _ X ( Z cos irdu / A ) (2.52) where Z i s a parameter which s p e c i f i e s the beam width and s i d e lobe l e v e l f o r the p a t t e r n f u n c t i o n . Equation (2.52) g i v e s the p o s t - d e t e c t i o n p a t t e r n f o r a square-law d e t e c t o r . Using the recurrence r e l a t i o n s h i p f o r Chebyshev polynomials f o r n - m, 2 T n ( x ) T m ( x ) = T n + m ( x ) + T n _ m ( x ) (2.53) i t f o l l o w s that equation (2.52) can be expressed as • P s q ( 2 T T C / A , U ) = J 2 ( T 2 ( M _ 1 ) ( Z cos Trdu/X) + 1) (2.54) Consider again the expression for the processor pattern function of the general M-element quadratic processor given by equation (2.49). This has the same form as the pre-detection pattern function for a Dolph-Chebyshev array of 2(M - 1) elements and hence the (D^) can be chosen using Dolph's method to give an optimum pattern function P0(2rrc/A,u) = ?20l-l) ( Z c o s (2.55) Comparing equations (2.54) and (2.55) one sees that the side lobes of the optimum quadratic processor are equi-ripple, alternating in sign and 3.0dB lower than those of the square-law-detected Dolph-Chebyshev array which are non-negative. Figure 2.9 illustrates the optimum post-detection patterns for both the Dolph-Chebyshev array and the quadratic processor. A l l Chebyshev synthesis techniques are based on a Fourier series representation for the array pattern of the type in equations (2.48) and (2.49). Thus a l l Chebyshev synthesis techniques are applicable to the synthesis of processor pattern functions for the quadratic processor. The restriction on the element spacings for Dolph's technique to be optimum is 1/2(1 + |cos eQ|) - d/X ^ (1/TT) cos _ 1(l/Z)/(l + |cos eQ|) (2.56) where Z is a parameter related to beam width as in equation (2.55) and 8Q is the angle to which the array is steered. For spacings less than the lower limit of equation (2.56) i t is necessary to synthesize P Q(2T T C / X , U) by the method of Riblet 2 1'> 2 5 where the {D^ } are chosen so that M-l . T2(M-1)^A C O S ^dn^X + B) = Y \ T 2 k ^ C O S , i r d u/^ (2.. 57) RGUftE 2.9 OPTIMUM POST-DETECTION PATTERNS FOR THE DOLPH-CHE&YSHEV ARRAY AND FOR TH& QUADRATIC PROCESSOR and the A and B are chosen as i n reference 24. The {D, } obtained for spacings tend to alternate i n sign and are dependent on frequency, s t e e r i n g angle and spacing. The patterns thus obtained are " s u p e r d i r e c t i v e " type p a t t e r n s . 2 5 2.4.2 Synthesis of Some Classes of Pattern Functions Based on Chebyshev  Polynomials Several v a r i a t i o n s of the Chebyshev pattern synthesis techniques have been proposed. Because the methods are based on a F o u r i e r s e r i e s represent-ation f o r the pattern function as i n equations (2.48) and (2.49), a l l are a p p l i c a b l e to the general quadratic processor. The a p p l i c a t i o n of these synthesis techniques to the general quadratic processor o f f e r s advantages i n varying degrees. In t h i s s e c t i o n we s h a l l consider two synthesis techniques and s h a l l examine only the advantages and disadvantages of the pattern function i t s e l f . Comparisons of the d i f f e r e n t pattern functions on the basis of s i g n a l - t o - n o i s e r a t i o w i l l be considered i n section 2.5 and again i n Chapter 4. • The f i r s t c l a s s of pattern functions which w i l l be considered i n t h i s s e ction are the modified Chebyshev pattern functions proposed by Rao. 2 5 Rao's method was developed f o r conventional l i n e a r arrays and r e s u l t s i n a pre-detection pattern of type d/X < 1/2(1 + |cos 6 |) (2.58) E(cos T r d u/X) = 2 k ( c o s k T r d u/X)T M-k-1 (Z cos Trdu/X) (2.59) and a- post-detection pattern E 2 ( c o s T r d u/X) = P_ (2nc/X , u ) = 2 2 k ( c o s 2 k T r d u / X ) T 2 M-k-1 (Z cos T r d u/X) or P R(2irc/X,u) = 2 2 k ( c o s 2 k T r d u/X) (T 2(M-k-1) (Z cos T r d u/X) + l ) / 2 (2.60) where the i d e n t i t y (2.53) has been used to get equation (2.60). The integer k i s a design parameter which can be selected i n the range M - 1 - k - 0. When k = 0, the pattern becomes a Dolph-Chebyshev pattern. When k = M - 1, the pattern becomes that of a b i n o m i a l l y weighted a r r a y . 2 6 I t i s evident that the side lobes of the modified Chebyshev patterns are not e q u i - r i p p l e , with f a r side lobes attenuated more than near si d e lobes due to the c o s k Trdu/A f a c t o r . The p r i n c i p a l advantage claimed by Rao i s f o r k = 1 where end element weighting i s reduced at the expense of only moderate beam broadening as compared to the Dolph-Chebyshev array (k = 0). Large Chebyshev arrays often require large end element weighting which i s d i f f i c u l t to achieve i n p r a c t i c e and r e s u l t s i n a degradation of s i g n a l -to-noise r a t i o . Here Rao's patterns may o f f e r some advantage. The quadratic processor can be made to r e a l i z e the pattern of equation (2.60) because the square-law-detected array i s a s p e c i a l i z a t i o n of the quadratic processor. Further f l e x i b i l i t y i n pattern synthesis i s obtained by -equating the F o u r i e r s e r i e s f o r the post-detection pattern of an M-element q u a d r a t i c a l l y processed array to the F o u r i e r s e r i e s f o r the pre-detection pattern f o r a 2M - 1 element array synthesized by Rao's method. In t h i s way one can synthesize post-detection p a t t e r n functions for the quadratic processor of the type P Q(2irc/A,u) = 2 k c o s k (irdu/A) ^2{H-1)-V:(-Z cos Trdu/A) (2.61) A pattern function of the type i n equation (2.61) cannot be r e a l i z e d by a conventional array. Thus equations (2.60) and (2.61) are not d i r e c t l y comparable because of the f a c t o r c o s 2 k (Trdu/A) i n equation (2.60) and the f a c t o r c o s k (Trdu/A) i n equation (2.61). I t can be seen, however, that, the envelope f a c t o r i n equation (2.61) has side lobes one-half the height of 39 those in equation (2.60). Rao showed that the beam broadening for the modified Chebyshev pattern given by equation (2.59) as compared to the Dolph-Chebyshev pattern, was (Beam Broadening) SQ x 100% (2.62) M - k Applying Rao's argument to the quadratic processor with a post-detection pattern of the type in equation (2.61), the beam broadening is (Beam Broadening)Q = k_ x 100% (2.63) 2M - 2 - k a figure considerably smaller than obtainable with a conventionally processed array using Rao's synthesis method. The second class of Chebyshev pattern functions considered in this section is the Chebyshev-envelope pattern function. This pattern function was developed by Jacobson for the array correlator. 3 An array correlator is illustrated in Figure 2.10. ARRAY POINTING DIRECTION ARRAY TT I N| ELEMENTS DIRECTION OF INCIDENCE FOR RECE IVED PLANE. W A V E LOW- - PASS FILTER • Figure 2.10 Array Correlator with Unequal Subdivisions 40 The p o s t - d e t e c t i o n p a t t e r n f u n c t i o n f o r such an a r r a y i s . P (2T T C/X,U ) = % P M (cos 7Tdu/X)P M (cos irdu/X) cos (M-rrdu/X) (2.64) c "2 where P N ^ ( c o s irdu/X) and P N ^ ( c o s irdu/X) are the p r e - d e t e c t i o n p a t t e r n f u n c t i o n s f o r the sub-arrays / / l and #2, w i t h and N 2 elements, r e s p e c t i v e l y . L e t = 1, N 2 = M - 1 and suppose P^^(cos rrdu/X) i s s y n t h e s i z e d to be a Chebyshev polynomial u s i n g Dolph's method. Then the p a t t e r n f u n c t i o n of equation (2.64) becomes P (2irc/A,u) = %T, (Z cos irdu/X) cos (Mirdu/X) (2.65) c M-2 In the above equation, T^ ^ ( Z cos irdu/X) a c t s as an envelope f o r the cosine term, cos Mirdu/X; hence the name, Chebyshev-envelope p a t t e r n . I n t h i s case of m u l t i p l i c a t i o n by an end element, the processor has very poor s i g n a l - t o - n o i s e r a t i o . However, Jacobson 3 has shown t h a t a p a t t e r n of t h i s type can be r e a l i z e d by an a r r a y c o r r e l a t o r w i t h equal s u b d i v i s i o n s (Nj = N 2 = a s i l l u s t r a t e d i n F i g u r e 2.11. He showed th a t the a r r a y weights, {q^}, must s a t i s f y a s e t of n o n - l i n e a r a l g e b r a i c equations which he d e r i v e d . Moreover, Jacobson has shown th a t the envelope, T ^ _ 2 ( Z cos irdu/X) , i s the optimum envelope f o r an M-element a r r a y c o r r e l a t o r , i n the sense that i t has minimum s i d e lobes f o r s p e c i f i e d beam wi d t h and v i c e v e r s a . Since the a r r a y c o r r e l a t o r i s a s p e c i a l case of the q u a d r a t i c p r o c e s s o r , the Chebyshev-envelope p a t t e r n can be expressed as a F o u r i e r s e r i e s as i n equation (2.48). Thus, t h i s p a t t e r n i s r e a l i z a b l e by a g e n e r a l q u a d r a t i c processor. The s i g n a l - t o - n o i s e r a t i o s f o r the g e n e r a l q u a d r a t i c p r o c e s s o r w i t h a Chebyshev-envelope p a t t e r n are s t u d i e d i n d e t a i l i n s e c t i o n 2.5 i n the case of r e c e i v e r n o i s e l i m i t e d o p e r a t i o n and i n Chapter 4 i n the case of back-' ground n o i s e l i m i t e d o p e r a t i o n s . HI 1 < 1 1 1 1 1 i i i i • • i % % ^5 % ^7 %0 • LOW- PASS FILTER Figure 2.11 Array Correlator with Equal Subdivisions In this section we consider briefly the directive properties of the pattern function and give a simple realization of Jacobson's Chebyshev-envelope pattern for the array correlator. In the course of this research a simple solution was found to Jacobson's nonlinear equations. This solution is for the weights, {q^}, which synthesize the Chebyshev-envelope pattern for the array correlator of Figure 2.11. The solution is valid for M even and M/2 odd where M is the number of elements in the array. This solution does not appear to have been reported in the literature. The synthesis is accomplished by choosing the weights, {q^}, to synthesize the two sub-arrays of the array correlator so that they have pattern functions P M (cos Trdu/A) = /2 T. (Z cos Trdu/A) - 1 ®l %M-1 02M-D/2 - E c o s 2k(-rrdu/A) k=0 (2.66) 42 and P„ (cos T r d u / A ) = /2 T,M . (Z cos T r d u / A ) + 1 CsM-D/2 f (2.67) £ F , cos 2 k ( T r d u / A ) J k=0 where the overall pattern function of the array correlator is given by equation (2.64) with = N 2 = M/2, and M/2 is an odd integer. To accomplish the synthesis of the sub-arrays #1 and #2, a set of weights, {H^}, is first chosen using the Dolph-Chebyshev synthesis method to match the^ following pattern: (^M-l)/2 Y Hi cos 2kTrdu/A = T, . (Z cos rrdu/A) (2.68) k=o k ' l M _ 1 The coefficients {E, } and { F . } are obtained from the {H, }.as follows: k k k F K = /2 H k r (2.69) F Q = /2 H Q + 1 k = 0 E k = /2 H k k 4 0 E 0 = /2 H 0 - 1 k = 0 Now substitute equation (2.69) into equations (2.66) and (2.67) which are then substituted into equation (2.64) to obtain the overall pattern function for the array correlator as P.(2TTC/ A , U) = 1"i(2T2 _(Z cos T r d u / A ) - 1) cos Mirdu/X (2.70) L M— 1 Using the identity (2.53) in equation (2.70), one easily shows that 2T2,M_1(Z cos Trdu / A ) - 1 = T M_ 2(Z cos Trdu / A ) (2.71) and hence P c ( 2 T T C / A , U ) = ^ M _ 2 ( z cos T r d u / A ) cos Mrrdu/A which is the optimum Chebyshev-envelope pattern of Jacobson, synthesized without directly solving the set of nonlinear equations. The weights {q^ } of Figure 2.11 are easily obtained from the {E^ } and {F^}. No such simple solution exists i f M/2 is even in which case an iterative solution to Jacobson1s nonlinear equations is necessary. The principal characteristics of the envelope pattern functions are (1) the beam width is determined by the cos M/rrdu/A term and not by the envelope; (2) the pattern has a large negative side lobe directly adjacent to the first null; (3) the envelope controls the side lobe heights for side lobes beyond the second null. The first null of the envelope pattern function occurs at Z E = T r d u E M = TT/2M (2.72) where M is the number of elements in the array. In comparison, the minimum distance to the fir s t null for the square-law-detected .array is Z * udUgp/A = rr/2(M - 1) (2.73) For the general quadratic processor, the minimum distance to the first null is • Z Q = rrdup/A = ir/4(M-l) (2.74) The half beam widths of equations (2.73) and (2.74) are for the limiting cases when only the zeroth and highest order Fourier coefficients are non-zero, in the case of the square-law-detected array and only the highest order Fourier coefficient is non-zero in the case of the quadratic processor. •It will be shown in section 2.5 that the Chebyshev-envelope pattern yields a narrow beam width with a substantially larger signal-to-noise ratio than either the square-law-detected array with squared-Chebyshev pattern or the general quadratic processor with a Chebyshev pattern when these latter two pattern functions are synthesized to have beams of the same width as the former. Thus the Chebyshev-envelope p a t t e r n f u n c t i o n i s worthy of c o n s i d e r a t i o n i n cases where a narrow beam width i s desired, but a l a r g e n e g a t i v e f i r s t lobe can be t o l e r a t e d . Some normalized Chebyshev-envelope p a t t e r n f u n c t i o n s are p l o t t e d i n Fig u r e 2.12 f o r a t h i r t y - e l e m e n t l i n e a r a r r a y . Reasonable f a r lobe a t t e n -u a t i o n can be obtained only i f a l a r g e n e g a t i v e f i r s t s i d e lobe can be t o l e r a t e d . Curves are p l o t t e d w i t h the main lobe/envelope s i d e lobe r a t i o , R, as a parameter. R i s d e f i n e d as R = T^_2(Z) where the p a t t e r n f u n c t i o n i s given by equation (2.65). I n order to attenuate the neg a t i v e f i r s t lobe of the p a t t e r n by means of the envelope term, T M _ 2 ( Z cos rrdu/A) , i t i s necessary to move the f i r s t n u l l of the envelope to l i e between the f i r s t two n u l l s of the co s i n e term of equation (2.65) as i s i l l u s t r a t e d i n F i g u r e 2.12. The border l i n e f o r the r e g i o n where f i r s t lobe a t t e n u a t i o n can be accomplished i s found when the f i r s t n u l l of the envelope i s brought i n t o c o i n c i d e n c e w i t h the second n u l l of the cosine term This c o i n c i d e n c e occurs f o r main lobe/envelope s i d e lobe r a t i o s as given i n Table 2.1, f o r the v a r i o u s a r r a y s i z e s . The r e s u l t s of Table 2.1 i n d i c a t e t h a t the main lobe/envelope s i d e lobe r a t i o must be l e s s than about 16.0 dB i n order to move the envelope n u l l i n t o the r e g i o n between the f i r s t two n u l l s of the cosine term. 2.4.3 Computation of F o u r i e r C o e f f i c i e n t s f o r Chebyshev P a t t e r n Functions A l l p a t t e r n f u n c t i o n s considered i n t h i s t h e s i s r e q u i r e the computation of the F o u r i e r c o e f f i c i e n t s of a Chebyshev p a t t e r n f u n c t i o n . Programs have been developed to generate these c o e f f i c i e n t s i n order to f o r c e a polynomial to have a Chebyshev p a t t e r n f u n c t i o n w i t h a s p e c i f i e d s i d e lobe l e v e l . The b a s i s of these programs i s the s e r i e s r e p r e s e n t a t i o n developed by Van der Maas. 2 7 The s e r i e s r e p r e s e n t a t i o n of r e f e r e n c e 27 was used to develop a program F16URE 2.12 CHEBYSHEV - ENVELOPE PATTERNS FOR A 30-ELEMENT ARRAY TABLE 2 . 1 MAIN LOBE/ENVELOPE SIDE LOBE RATIOS FOR WHICH THE FIRST ZERO OF THE ENVELOPE COINCIDES WITH THE SECOND ZERO OF THE COSINE TERM The pattern function is Tj^CZ cos udu/A) cos Mirdu/A with RDB = 1 0 . 0 Log (T (Z)) M - 2 Array size RDB M : 1 5 ' 1 3 . 6 4 3 0 1 4 . 9 1 5 0 1 5 . 4 4 1 0 0 1 5 . 8 6 2 5 0 1 6 . 1 1 5 0 0 1 6 . 2 0 1 0 0 0 1 6 . 2 4 1 0 , 0 0 0 1 6 . 2 8 47 for generating a Chebyshev pattern with as many as 50 Fourier coefficients for a Chebyshev polynomial of order 98. Later i t was found necessary to generate polynomials with close to 100 Fourier coefficients. This was accomplished by first generating the coefficients for a lower order Chebyshev polynomial and then getting the coefficients of the higher order polynomial by means of identities for Chebyshev polynomials. The use of Van der Maas' method and the method of extension to large arrays is discussed in Appendix A. 2.5 Quadratic Processing for Receiver Noise Limited Passive Arrays In this treatment, i t is assumed that receiver noise on a given array element is independent of noise on any other array element and has a flat power density spectrum over the system bandwidth. The quadratic processor will be shown to be an improvement over conven-tional array processors. It will be shown how the magnitude of the improvement depends on array size, processor pattern type, beam width and side lobe level. Specifically, the quadratic processor, is compared to the square-law-detected array and the array correlator. These processors are illustrated in Figure 2.13. The three pattern functions considered are the Chebyshev, the squared Chebyshev, and the Chebyshev-envelope. A l l three patterns can be realized by the general quadratic processor; the latter two can also be realized by the square-law-detected array and the array correlator, respectively. 2.5.1 The Processing Gain Factor The assumption of independent noise on each element means that the matrix of cross-spectral densities, Y^(w)».is diagonal. If the noise spectra are identical on every element, then YN(w) = S n n ( u ) i (2.75) • QUADRATIC PROCESSOR I 1 1 ! ! ! B2/2 B|/2 B 0 B,/2 B2/2 (a) QUADRATICALLY PROCESSED ARRAY (b) SQUARE-LAW - DETECTED ARRAY SQUARE-LAW DETECTOR I I I Hi H 3 ^4 ARRAY CORRELATOR FIGURE 2.13 ILLUSTRATION OF PROCESSOR TYPES where I is an (M x M) unit matrix and S n n(u) is the power density spectrum of the noise. The substitution of equation (2.75) into equation (2.21) results in the following expression for the variance of the output of the quadratic processor CO Var (v 3 > N) = (1/4TT 2)(G 2(0)/T) J | G(u) | " | S n n(u) | 2 t r q 2 du (2.76) where M M t r 3* = Z. Z q i m % i i=l m=l (2.77) where q • = q .. Thus ^im ^mi Var (v 3 > N) = (l/4rr2)(G2(0)/T) M M ^"*<0>/T £ E ^im J -,i=l m=l J -°° (2.78) The useful signal output with a signal source in the beam centre of the array is obtained from equations (2.35) and (2.36) with u = 0, i.e. 2 im / i G M l ^ O o ) do s u = (1/271 ) 03 (0 ) M M E ' E <*im i=l m=l J l G ^ ) l 2Y s s(w) du (2.79) Finally, from equations (2.3), (2.78), and (2.79) the squared signal-to-noise ratio is (SNR)2 = T M M _i=l m=l im CO f |G(o))|2Ycc(«O) du «/ So (2.80) M M E, E„ ^im i=l m=l J |G(u)|4S2n(u) du Equation (2.80) depends on the q matrix only through a factor, M M E T q-G = t l m^l i m R N (2.81) M M E E <*im _i=l m=l C !RN will be called the processing gain against receiver noise; not to be confused with the array gain, denoted G^. It will be shown in Chapter 3 that Gj^ j is proportional to G ^ R provided the array is square-law-detected, has half-wavelength spacing and the system bandwidth tends to zero. Cauchy's inequality states that k=l ' im and with equality only i f b^ = ca^ where c is constant. If we set a^ b t = 1 , i t is readily seen that G 0 „ - M with equality occurring i f q. = q, a constant. Such a q matrix with a l l elements the same corresponds to a uniformly weighted, square-law-detected array, which has maximum processing gain against receiver noise. This investigation is concerned with the processing of arrays where the constraints of prescribed pattern function preclude the attainment of gains as high as G R ^ = M. Consider the maximization of G ^ subject to the constraints of equation ( 2 , 3 8 ) . From equation ( 2 . 3 8 ) we have M M M-l y y a. = v D . = /Q i - 1 in=l k - 0 Hence the Gn„ can be written RN 'RN M M ZJ qim 1= 1 m=l _i ( 2 . 8 2 ) where Q is fixed. The maximization is carried out using Lagrange's method where the function to be maximized is k=0 ( 2 . 8 3 ) The {Xjj} are Lagrange multipliers and the (Dk) are functions of q as in ( 2 . 3 8 ) and specify the pattern constraints. The solution to the linear set of equations obtained by differentiating equation ( 2 . 8 3 ) with respect to each element of the q matrix and setting 9U/8qrg = 0 for 1 - r, s - M is. 51 q r s = D | r - s | / 2 ( M " l r ~ S , ) f ° r r * S " l and q = D /M rr u (2.84) fo r an array of M elements. Thus G_, i s maximized by making the elements RN of q as uniformly weighted as p o s s i b l e subject to the p a t t e r n c o n s t r a i n t s of equation (2.38). Such a matrix with elements given by equation (2.84) w i l l be c a l l e d a "maximally uniform" q matrix and designated c[ u. I f the independent r e c e i v e r noises are not of the same l e v e l , that i s , i f some of the r e c e i v i n g elements are n o i s i e r than others, G^, w i l l be modified. In t h i s case the cross-power s p e c t r a l density matrix becomes V"0 = S n n ( a ) ) J "(2.85) where J i s an (M x M) diagonal matrix (compare to equation (2.75)). The th power density spectrum of the i r e c e i v i n g element i s J . .S (co) . i i nn The expression for the squared s i g n a l - t o - n o i s e r a t i o becomes (compare to equation (2.80)), (SNR) 2 = ' M M 2 * T E E ^im -i=l m=l J |G((o) |2Yss(w) dco (2.86) M M E E J i i J m m q L / i G M l ^ O o ) do, i = l m=l -«> The processing gain against r e c e i v e r noise (compare to equation (2.81)) i s RN M M E E o i = l m=l im (2.87) ' M M E E t Jii^ m L^ i=l m=l The q matrix which maximizes Gjv^ subject to the c o n s t r a i n t s of equation (2.38) i s D, 'rs r-s f o r r ^ s - (2.88) D, " r r M 4 r £ < 1 / J ? i > The above expressions are given f o r the sake of completeness. In p r a c t i c e , i t i s u s u a l l y necessary to assume = 1 f o r a l l i , i . e . i d e n t i c a l n o i s e on each element. This assumption w i l l be made i n the a n a l y s i s to f o l l o w . I f the q r g are chosen according to equation (2.84), i . e . q = q u , the pr o c e s s i n g gain f a c t o r i s maximized. • From equations (2.81) and (2.84), the maximized i s " RN. M-l E \ k=0 (2.89) max r- M-l (D2/M) + h T D?/(M - k) ° k=l R The s i g n a l - t o - n o i s e r a t i o w i l l now be evaluated f o r the case where both n o i s e and s i g n a l have r e c t a n g u l a r power d e n s i t y s p e c t r a . Thus f o r centre frequency, U Q , h a l f bandwidth b, and T s s ( a j ) = Yss | G ( u ) | 2 = l S n n(w) = S n n f o r a) 0 - b - 1 to I ^ co 0 + b |G ( co ) | 2 = Y s s(w) = S n n(a)) = 0 otherwise Y (2.90) Then CO a 2 = f IG(OJ) 1 2S (to) du = 4bS n J ' ' nn' nn (2.91) and a | = J | G ( u ) | 2 y s s ( u ) du = 4by ss (2.92) Thus, f o r t h e c o n d i t i o n s o f e q u a t i o n s (2.90) t o ( 2 . 9 2 ) , CO / i G C u O l ^ C c o ) du, = a£/4b (2.93) The s i g n a l - t o - n o i s e r a t i o o f t h e q u a d r a t i c p r o c e s s o r w i t h i d e n t i c a l n o i s e on each r e c e i v i n g e l e ment and w i t h s i g n a l and n o i s e s a t i s f y i n g e q u a t i o n s (2.92) and (2.93) i s ( S N R ) 2 = 4 b T G 2 N ( a g / a n ) I t (2.94) E q u a t i o n s ( 2 . 9 3 ) , ( 2 . 9 2 ) , and (2.81) have been s u b s t i t u t e d i n e q u a t i o n (2.80) t o o b t a i n t h e above e q u a t i o n . I f q i s c h o s e n t o be ;q„ so t h a t G_., i s m a x i m i z e d , t h e s i g n a l - t o - n o i s e r a t i o f o r t h e q u a d r a t i c p r o c e s s o r i s o b t a i n e d f r o m e q u a t i o n s (2.89) and (2.94) and i s ' 4bT (SNR)2 = M - l .k=0 M - l (D 2/M) + T D 2/2(M - k) k = l k fa I" )k (2.95) T h i s e q u a t i o n w i l l be used t o compare t h e q u a d r a t i c p r o c e s s o r w i t h c o n v e n -t i o n a l l y p r o c e s s e d a r r a y s u s i n g a s q u a r e - l a w - d e t e c t o r and w i t h t h e a r r a y c o r r e l a t o r . The s q u a r e - l a w - d e t e c t e d a r r a y i s a s p e c i a l c a s e o f t h e q u a d r a t i c p r o c e s s o r , as i s t h e a r r a y c o r r e l a t o r . The e l e m e n t s o f t h e q m a t r i x f o r t h e s q u a r e - l a w - d e t e c t e d a r r a y s a t i s f y t h e e q u a t i o n and t h u s q i m = q i q m M M E E « (2.96) i = l m=l M M im and E E <?. -i = l m=l im M E H i = l * M .(2.97) The pre-detection array pattern for a symmetrically weighted array with an odd number of elements as shown in Figure 2.13b is N P^ CtOjU) = E Bk c o s ktoclu/c k=0 (2.98) where N = (M - l)/2 and M is odd. If the number of elements, M, is even the array pattern is N PL(u,u) = YJ B c o s C( 2 k " Dwdu/2c) (2.99) k=l where N = M/2, d is the element spacing and u = cos 6 - cos 0Q. The post-detection pattern for the square-law-detected array is P S Q ( a ) , u ) = % P j ( u , u ) For P (w,u) given by either equation (2.98) or (2.99),-equation (2.97) ti becomes M N i=l k=0 >• (2.100) and N N E <i| = Bo + * E Bk i=l k=l where B Q = 0 i f M is even and N = M/2, and B Q 4 0 for M odd and N = (M - l)/2. The processing gain against receiver noise of (2.81) becomes = Ggg, where ' S O r N i E \ _k=0 (2.101) B§ + h £ » k=l The signal-to-noise ratio for the square-law-detected array, obtained by substituting equation (2.101) for G^ ., in (2.94), is (SNR)2 Q -r N 4 4bT y B , .k=0 . (2.102) N E k=l Bl + h E B 2 Finally we consider the array correlator illustrated in Figure 2.13c. The array correlator is a special case of the quadratic processor with q matrix satisfying for or and for qim = q i q m / 2 1 - i - %M, ^ M + l -m-M 1 - m -%M, m + 1 - i - M 1 - i , ra - %M and + 1 - i , m - M r (2.103) where M is the array size and an even integer. Thus and M M E <C qim i=l m=l M M E E i L ' h i=l m=l M E i m^ M+1 m M V „2 T q 2 A q i 1=1 m=^M+l r (2.104) The processing gain against receiver noise, Gc, for the array correlator is obtained from equations (2.81) and (2.104) and is (2)' G = c E i i i=l M £ qm Lm=J£M+l . (2.105) V q? M E m=%M+l *m The maximum value of G occurs when a l l the q. are the same and is c 1 (Gp) = M//2 c max 2.5.2 Discussion of Results The maximum value of the processing gain, G^, for a quadratic processor of M receiving elements is G ^ m a x = M. Processing gains are normalized with respect to M in this discussion and converted to decibels as follows: (Normalized Processing Gain)dP> = 10 Log^ (Processing Gain/M) 56 The quadratic processor is compared with the square-law-detected array and with the array correlator, on the basis of two criteria. The first criterion specifies that the processors being compared have the same processor pattern function. The second criterion specifies that the processors have the same beamwidth but not the same pattern function. Under the f i r s t criterion, the quadratic processor is compared with the square-law-detected array with both processors having a squared-Chebyshev pattern function . ' P q ( 2 T T C A , U ) = P s q ( 2 T T C / X , U ) = T 2_ X(Z cos Trdu/X) (2.106) Equation (2.106) is the optimum pattern for the square-law-detected array as discussed in section 2.4.1. Comparison of the quadratic processor and the array correlator, under the first criterion, is carried out for the Chebyshev-envelope pattern P^(2!rc/X,u) = P c ( 2 T T C / X , U ) = %T M_ 2(Z cos Trdu/X) cos M-rrdu/X (2.107) Equation (2.107) is the optimum pattern for the array correlator as discussed in section 2.4.2. • -In a l l cases the elements of the q matrix for the quadratic processor are chosen so that the processing gain, G ^ , is a maximum, subject to the pattern constraints of equation (2.38). The appropriate expression for GQ is given in equation (2.89). The processing gains for the square-law-detected array and the array correlator are given by equations (2.101) and (2.105), respectively. Processing gains for the quadratic processor and the square-law-detected array are plotted in Figure 2.14. Plots are given for array sizes of M = 30, 50, and 99. The quadratic processor shows a superiority of about 1.4dB maximum for M = 30, 1.6dB maximum for M = 50, and 1.8dB maximum for M = 99. 57 SIDE. LOBE LEVEL CdB) SIDE LOBE LEVEL CdtQ 0 -6 -12 -Ift -24 -30 FIGURE 2.14 COMPARISON OF THE QUADRATIC PROCESSOR WITH THE SQUARE-LAW.-DETECTED ARR&Y FOR SQUARED CHEBYSHEV PATTERNS 58 As the side lobe levels decrease the curves for (Gp/MjdB and (Ggq/M)dB approach closely but do not converge. Although the improvements are not large they may be quite worthwhile depending on the application and cost of implementation. The maximum improvement appears to be obtained for the limiting condition of an edge-type pattern when the side lobe level tends to OdB. The term edge-type pattern refers to an array with very small element weighting for a l l but the two end elements. As the beamwidth of a Chebyshev pattern is decreased, the main lobe/side lobe ratio also decreases until the limiting condition of an edge pattern is reached. The edge pattern for a square-law-detected array has a q matrix with elements q ^ = q ^ = h, q ^ = Ij,^ = ^ and a l l other elements equal to zero.' In contrast, the same edge-type pattern can be realized by a quadratic processor with maximally uniform matrix, q u, with elements q-j^ = q = h, q^i = 1/2M for 1 - i - M and a l l other elements equal to zero. This spreading out of the weights among the diagonal elements results in a greater processing gain than is obtainable with the square-law-detected array. The Fourier coefficients for the edge pattern function are D k •> 0 for k 5s 1 for k ^ M - 1 D0 - DM-1 * °'5 (2.108) For this case G Q ~\fst>l/ (M+2) and for large M , G ^ ->• /8. The corresponding value for the square-law-detected array is GgQ -> 2 and thus 10 Log 1 0(G Q/G S Q) m a x - 2.4dB (2.109) It.can be argued that, for sufficiently large array size, a performance improvement of the order of 2.4dB can be obtained for the squared Chebyshev pattern. This happens because, i f the side lobe level is fixed at some value, and the array size is increased indefinitely, the limiting condition of equation (2.108) is approached. The second criterion requires that the processors being compared have the same half-power beam width. Comparison of the quadratic processor and the square-law-detected array according to this second criterion is carried out for the pattern functions P 0(2T T C / A ,U) = T 2 ( M - 1 ) ^ Z Q C O S irdu/X) and PgQ(27rc/X ,u) = T ^ _ ^ ( Z g ^ cos irdu/X) r (2.110) That i s , in equation (2.110), the Chebyshev pattern is chosen for the quadratic processor and the squared-Chebyshev pattern for.the square-law-detected array. These patterns are optimum in the sense of minimum beam, width for specified side lobe height, for the respective system structures. Under this second criterion for comparison, the quadratic processor has the advantage of a side lobe level 3dB lower than the square-law-detected array (section 2.4.1) as well as advantages with regard to processing gains. Normalized processing gains are plotted as functions of half-power beam width in Figure 2.15 for the patterns of equation (2.110) and array sizes of M = 30, 50, and 99. The quadratic processor processing gain, and hence the signal-to-noise ratio, is .between 2 and 3dB greater than the square-law-detected array over much of the range of beam widths in Figure 2.15. The 3dB gain is only obtained for the 99-element array. However, the trend of the plots indicates that, as the array size is increased beyond 99 elements, one should expect improvements of the order of 3dB in signal-to-noise ratio over an increasingly larger portion of the range of useable beam width/side lobe ratios. This last comparison is a better indication of the relative worth of the quadratic processor as compared to the square-law-detected array because the 60 2.0 HALF-POWER BEAMWIDTH (DEGREES) 2.0 AO A N 3 2 z < -4 cr Z -6 lf> ID o v a o. -10 ~ ~ — PoC2nc A,u) = r 5 a (2c .06 noli ARRAY SIZE M=30 d/x= i/z BROADSIDE STEERING. 10 L OS (Gc \ Ps?< 10 LC 2-rrc/A )G C6 5 ,t0 -T 29 ("zS i FOR — BEAMWIDTH FOR CHEBYSHEV-ENVELOPE PATTERN HALF-POWER BEAM WIDTH (DEGREES) •z < -4 CD O 5 Cfl ui U <x 1.0 2.0 3.0 PQ(2TTC u/X> ARRAY SIZE M=50 'd/x--l/2 BROADSIDE STEERING 10 LO 'MI / > \ 10 L O G ( \ u % Q / M T&CZ i i ) FOR rr> COS nrf.u/.V *) / -10 BEAMWIDTH FOR C H EBYSH EV - EUVE LOPE PATTERN o o -2 -4 - 6 -8 -10 -!2 HALF-POWER BEAMWITTTH (DEGREES) .5 1-0 ARRAY SIZE M^99 C*A=l/2 8R0ADSIDE STEERING P5Q(2r 10 LOC ifefiQ/M) FC R >s rrdu A) 10 LOG (Gfi/M) FOR TJQ(2nc/A,u')-T|qfa(ZQco«. ridd.ti / / r «JVJ / — BE/1 iMVMIDTH FOR CHEBYSHEV-ENVELOPE PATTERN (0 S z < Z 5> cO Ui o o or a -14 FIGURE ?.I5 COMPARISON OF THE QUADRATIC PROCESSOR WITH THE SQUARE-LAW- -DETECTED ARRAY FOR CHEBYSHEV AND SQUARED - CH EBYSH EV PATTERNS 0 1 quadratic processor has pattern synthesis capabilities not possible for the square-law-detected arrays. The gain in using the quadratic processor is a fairly significant improvement in signal-to-noise ratio and, in addition, a reduction in side lobe level of 3dB vis-a-vis the square-law-detected array. Beam widths attainable with a Chebyshev-envelope pattern function are marked on Figure 2.15 in order to assist in the evaluation of the performance . of the quadratic processor and the array correlator using this pattern function. Processing gains for the array correlator and quadratic processor, using the Chebyshev-envelope class of pattern functions, are plotted against the envelope side lobe level in Figure 2.16 for array sizes of M = 30, 50, and 98. The results plotted in Figure 2.16 indicate a substantially better performance for the quadratic processor in comparison to the array correlator than were obtained x^ith Chebyshev and squared-Chebyshev pattern functions in comparison to the square-law-detected array. For example, for the 50-element array, the signal-to-noise ratio of the quadratic processor is larger than that of array correlator by amounts of - 3dB for envelope side lobe levels - -17dB and - 2dB for envelope side lobe levels - -19dB. The corresponding figures for the 98-element array are signal-to-noise ratio improvement - 3.5dB for envelope side lobes - -17dB, - 3dB for envelope side lobes - -20dB and - 5dB for envelope side lobes - -15dB. The salient features of the array correlator and the quadratic processor with the Chebyshev-envelope pattern are a very narrow beam, a large negative first side lobe, and relatively small far side lobes. Ignoring the d i f f i c u l -ties and costs of instrumentation, the choice between a quadratic processor and an array correlator depends on the amount of first lobe suppression required and on the array size. . If the pattern is synthesized to have some significant attenuation of D Z ENVELOPE SIDE-LOBE LEVEL CdEO -12 -10 -24 -30 s z (9 Z <I> UJ o O a HM.F-POWER SEAMWDTH-2.55 CHEBYSHEV-ENVELOPE PATTERN ARRAY SIZE M-30 ENVELOPE 61 D E - LOBE LEVEL (d&> -fe -12 -I© -24 10 L O G ( G Q / M ) H A L F - P O W E R B E A M W I O T H - 1-53° d/x= «/?-C H E B Y S H E V - E N V E L O P E P A T T E R N A R R A Y S I Z E M = 5 0 -2 CO . z < -6 o iii o o c: a. 10 -12 ENVELOPE SIDE-LOSE LEVEL CelB) - 6 -12 - 1 6 -24 10 L O G (GQ/VO 10 LOG (QC/M") / HALF - POWEft BEAMWItJTH - .70° d/X= l/2 CHEBYSHEV-ENVELOPE PATTERN ARRAY SIZE M=93 3 < o & z tn u> U l o O a: a - 2 -4 -6 -8 -10 -12. -14 -16 FIGURE 2.16 COMPARISON OP THE QUADRATIC PROCESSOR WITH THE ARRAY CORRELATOR 63 the first negative lobe by the envelope, then as shown in section 2.4.2, the envelope side lobe level must be - -16dB. This is the region where the quadratic processor shows the greatest increase in signal-to-noise ratio as compared to the array correlator. On the other hand, i f a large negative lobe can be tolerated and the envelope pattern is used largely to control the far side lobes, and i f the array size is sufficiently large, a substantial increase in signal-to-noise ratio may s t i l l be had by using the quadratic processor. For example, for the 99-element array of Figure 2.16, the quadratic processor signal-to-noise ratio is s t i l l 3dB greater than that of the array correlator at an envelope side lobe level of - 20dB. The choice between a Chebyshev-envelope pattern or a Chebyshev pattern depends on (1) whether the large negative lobe of the Chebyshev-envelope pattern can be tolerated and (2) whether the beam width of this pattern is sufficiently narrow for the particular application. Chebyshev pattern functions with beam widths as narrow as Chebyshev-envelope patterns have signal-to-noise ratios significantly lower than realizable using the latter pattern as can be seen by comparing Figures 2.15 and 2.16. The difference can be as much as 6dB for a large array depending on how large a first lobe of the Chebyshev-envelope pattern can be tolerated. 64 3. QUADRATIC PROCESSING IN A.BACKGROUND NOISE LIMITED ENVIRONMENT 3.1 Introduction This chapter contains a derivation of the signal-to-background noise ratio in terms of performance functions. Methods of noise reduction based on minimization of these performance functions will be developed. The performance functions will be related to the familiar concept of array gain or directivity. It will be shown that, in general, these new performance functions are more accurate measures of system performance against noise than is the array gain. The signal-to-noise ratio is derived for a point noise source model of background noise. The:model is manipulated to represent background noise under conditions of maximum uncertainty about the positions of the various noise sources. Individual noise sources in the model a l l have the same power as received at the array. Thus the model represents anisotropic noise by a grouping of the noise sources in space in some nonuniform manner. The objective is a model which is capable of giving a realistic representation of fields as would be received by an array operating in an environment where both the noise and signal are Gaussian random processes. 3.2 Derivation of Signal-to-Noise Ratio for the Point Noise Source Model  of Background Noise The equations for the mean output signal (2.35) and noise variance (2.47) of the quadratic processor were derived in section 2.3 for a system of L point noise sources. The mean output signal is fixed and not a function of the q* matrix for signals in the array boresight except for an overall scaling factor. Thus the signal-to-noise ratio is effectively determined when the noise variance 65 has b e e n e v a l u a t e d as a f u n c t i o n o f q . E q u a t i o n ( 2 . 47 ) f o r t h e n o i s e v a r i a n c e w i l l be g i v e n a g a i n f o r c o n v e n i e n t r e f e r e n c e . L « V a r ( v 3 N ) = ( G 2 ( 0 ) / 4 T T 2 T ) £ J |G((o) | 4 N 2 ( u ) P 2 ( u , u m ) du ' m=l - o o L L °°' + ( G2 ( 0 ) / 4 T T 2 T ) £ £ J* | G ( o J ) | 1 + N h ( u ) N m ( c o ) F ( a ) u m , c o u h ) du ( 2 . 47 ) m=l h=l -» itr/h L i s t h e t o t a l number o f n o i s e s o u r c e s , N j 1 (u ) i s t h e power s p e c t r a l d e n s i t y o f t h e r a d i a t i o n o f t h e h n o i s e s o u r c e as r e c e i v e d a t t h e a r r a y and u^ d e f i n e s t h e p o s i t i o n o f t h e n o i s e s o u r c e , i . e . u ^ = co s 6Q - c o s 9^  w h e r e t h e a r r a y i s s t e e r e d t o an a n g l e 8Q. I t i s assumed t h a t t h e a r r a y i s a p a s s i v e r e c e i v i n g d e v i c e and t h a t t h e s i g n a l and n o i s e have f l a t s p e c t r a o v e r t h e b a n d w i d t h o f i n t e r e s t . T h i s a s s u m p t i o n i s f a i r l y r e a l i s t i c f o r most p a s s i v e s o n a r and r a d i o a s t r o n o m y a p p l i c a t i o n s . More i m p o r t a n t , i n t h e o p t i m i z a t i o n p r o c e d u r e s , t h e a s s u m p t i o n o f f l a t n o i s e and s i g n a l s p e c t r a has t h e e f f e c t o f p l a c i n g e q u a l e m p h a s i s on a l l f r e q u e n c i e s i n t h e s y s t e m b a n d w i d t h . T h i s i s l o g i c a l l y c o n s i s t e n t w i t h t h e c o n c e p t o f maximum u n c e r t a i n t y a b o u t t h e b a c k g r o u n d n o i s e . The a s s u m p t i o n o f maximum u n c e r t a i n t y w i t h r e g a r d t o t h e d i r e c t i v e p r o p e r t i e s and s p e c t r u m shape ( w i t h i n a s p e c i f i e d b a n d w i d t h ) o f t h e n o i s e i s t h e g o v e r n i n g p h i l o s o p h y i n t h i s i n v e s t i g a t i o n i n t o methods o f n o i s e r e d u c t i o n . The assumed n o i s e and s i g n a l s p e c t r a a r e i l l u s t r a t e d i n F i g u r e 3.1. F o r t h e s i g n a l and n o i s e s p e c t r a o f F i g u r e 3.1, e q u a t i o n ( 2 . 4 7 ) f o r t h e n o i s e v a r i a n c e can be r e w r i t t e n as 66 Var ( v 3 y = (G 2(0)/4bT) L - l uQ+b L - l L - l u0+b E E (^5/2b) J F("u m,a,u h) dco h=l m=l co + 2 a 2 £ (o-2/2b) Uf F(0,uu h) du + op(0, h=l u n-b 0) (3.1) where a 2 i s the mean squared signal' power i n the system bandwidth and o^ th i s the mean squared power of the h noise source. In equation (3.1) i t has been assumed that one of the L point sources i s the s i g n a l source; hence summations run from 1 to L - 1. In the equations to follow, L - l w i l l be replaced by L where L i s the number of noise sources. NOISE. SPECTRAL D E N S I T Y , N^(tx)) K h-ttcr^/2b SIGNAL SPECTRAL DENSITY, Y^Cw) FREQUENCY (RADIANS/SEC.) K =TTcr|/2b / ~ i o 0 - b oo, »0-b W0+b FREQUENCY (RADlANs/sEC.) Figure 3.1 Sign a l and Noise Spectra Inherent i n the model i s the assumption that a l l p o i n t sources have equal strength. Thus = a 2 f o r a l l 1 - h - L, and the variance of equation (3.1) becomes 67 Var (v 3 j N) = F(0,0)(G2(0)/4bT) < i ta/2b) 7 w0+ b F( du h=l m=l to0-b F(0,0) + 2afc 2 V (l/2b) U/ h-1 c^-V bF(0,uu h) „ s b F(0,0) (3.2) Equation (3.2) was derived under the assumption of noise sources in the far field of the array. The defining characteristic of the far field region is that, i f the sources are in the far field, waves received from these sources by the array are very nearly plane waves. Thus the model can represent any statistical wave field which can be represented as a sum of random plane waves provided no restrictions are placed on the number of sources L, or their angular position.. If, in the model, angular positions are unrestricted, noise sources may actually be superimposed at least in their effect. Therefore, specifying a l l sources to have the same strength is really not a restriction. It appears that the model may not be able to give an accurate represent-ation of noise fields made up of the superposition of non plane waves due to noise sources in the near field of the array. Thus conclusions drawn from the model should probably be restricted to situations where the back-ground noise has its origin primarily in the far field region of the array. The effect of allowing maximum uncertainty in the positions of the noise sources will now be discussed. The positions of the noise sources will be considered to be random variables. The assumption of maximum uncertainty in a source angular position is interpreted to mean that the random source position is uniformly distributed in solid angle. That i s , the probability of a source occurring in a differential solid angle, A f t , is directly proportional to A f t . The differential solid angle surrounding a linear array at an angle 0 68 to the array is dft = 2TT sin Gd6. But u = cos QQ - cos 6 and thus du = sin 6d6 = dft/27T. The probability of a source lying in dfi is Pfi(ft)-dft with P^ (£2) = 1/4TT. Thus noting that Pu(u) du = P^(ft) dfi i t follows that Pu(u) = h (3.3) for cos 0 Q - 1 - U - 1 + cos 8Q. When the noise source positions are allowed to become random variables, the noise variance of equation (3.2) becomes a conditional variance as follows: Var (v 3 j N|u) = E((v 3 j N |n) 2 ) - E 2(v 3 ) N|u) (3.4) where u is an L-vector with components which are the independent random positions of the noise sources. Thus, defining Pu^(u^) = P^(u^) with Pu^(u^) given, in equation (3.3), the L-fold probability density for the position vector, u, describing the positions of a l l L noise sources, is L L p(u) = n p.(u.) = (hr (3.5) i=l It is shown in Appendix B, that for the point noise source model with P(u) given by equation (3.5) and the conditional variance given by (3.4), the following relationship is true. Var (v 3 N) - J Var (v 3 N|H)P(u) du (3.6) ' —CO ' Note that equation (3.6) is not generally true but holds for this particular model of background noise. If the integral of equation (3.6) is evaluated with the conditional variance given by equation (3.2) the result is Var (v 3 j [ j) = F(0,0)(G|(0)/4bT) < E E '<"a> h=l m=l 0,0+b cos 90+l p ( } L • f // v l ' 2 dux du2 du + 2a 2a 2 ( l/2b) U Q - b cos 0O-1 F(0,0) h = l ton+b C O S 6 n + l • / J u ^ - b cos e Q - l F(0,0) / / F(0,uu) du du + a£ (3.7) It is now possible to identify the performance functions in the above equation. Two measures of performance are defined, namely, a volume perform-ance function Py and an area performance function, P^ . Specifically, U Q + b cos P v = (l/2b) f J J F(uu l tuu 2) d U ] L d u 2 du. ug-b cos GQ-1 . F(0,0) (3.8) and P = toQ+b cos 8o+l (l/2b) J J F(0,uu) du du uQ-b cos 6 0-l F(0,0) (3.9) The final expression for the noise variance will now be written in terms of the total interfering noise power, a 2^ = La 2 and the performance functions: Var (v 3 N) = F(0,0) Gf (0)aj T["(P vM) + 2(a 2/a 2 T)P A + aj/arf (3.10) • ' 4bT L J Notice that the number of noise sources, L, has been absorbed in the total interfering noise term 0 2^. Thus, the form of the variance and the signal-to-noise ratio, is independent of the number of sources used to make up the model. The output signal from a signal source in the array boresight, is obtained by substituting the signal spectrum of Figure 3.1 into equation (2.35) and is S u = G1(0)P(0,0)o (3.11) where P 2(0,0) = F(0,0) (equation 2.43). The system s i g n a l - t o - n o i s e r a t i o can now be w r i t t e n i n terms of the performance f u n c t i o n s u s i n g equations (3.11), (3.10), and (2.3) to get (SNR) 2 = S 2/ Var ( v ^ ) = 4bT (0|/a 2 T) 2/ (P v/4) + 2(a|/a 2 T)P A + a£/a. NT (3.12) For v a n i s h i n g l y s m a l l s i g n a l s , the system s i g n a l - t o - n o i s e r a t i o i s governed by the volume performance f u n c t i o n , Py. For very l a r g e s i g n a l s , Py and P^ have l i t t l e i n f l u e n c e on system performance which i s a f u n c t i o n only of bT and the in p u t s i g n a l - t o - n o i s e r a t i o , a ^ 0 ^ ' At some i n t e r m e d i a t e s i g n a l s i z e , P^ may become more important than Py i n determining output SNR. Much of t h i s chapter i s concerned w i t h techniques f o r i n c r e a s i n g the s i g n a l -t o - n o i s e r a t i o by choosing q so that Py or P^ are minimized. I t i s of i n t e r e s t to determine the t h r e s h o l d l e v e l which separates the regions where i t i s more advantageous to minimize e i t h e r Py or P^. In order to determine t h i s t h r e s h o l d , i t i s necessary to d e f i n e the f o l l o w i n g terms: Pyy = minimum valu e of the volume performance f u n c t i o n , Py. T?AA = minimum valu e of the area performance f u n c t i o n , P^. P^y = volume performance f u n c t i o n obtained when P^ i s minimized. Py^ = area performance f u n c t i o n obtained when Py i s minimized. In terms of the above q u a n t i t i e s , the t h r e s h o l d Th i s Th - ( 1 / 8 ) ( P A V - P V V ) / ( P V A - P ^ ) . (3.13) I f > T h , i t i s advantageous to minimize P^. I f a g / ° ^ T < Th, i t i s advantageous to minimize Py. The above i s only a rough d i v i s i o n . More p r e c i s e methods f o r choosing the q m a t r i x w i l l be di s c u s s e d i n l a t e r p o r t i o n s 71 of this chapter. 3.3 Relationship of Performance Functions and Array Gains In order to demonstrate the relationship of the performance functions and array gain, the quadratic processor is f i r s t restricted to be a square-law-detected array. It was shown in section 2.3.3, that, for the square-law-detected array, F(wu^ ,tou2) = P(io,ui)P(co,U2) and P(w,u) = A2(to,u) where A(u),u) is the pre-detection pattern of the array and P(io,u) is the power pattern or post-detection pattern of the square-law-detected array. Substitution of F(wui ,u)u2) into the expressions for the performance functions gives ton+b cos 6 n+l , . . . p v q = ci/2b) r tr PKui)p(u,u 2) d d U o d M ( 3 > u ) V S Q tonJ-b cos 6 n-l Pz(0,0) 2 and UQ—U LUB OQ-tOQ+b COS 0Q+1 P = (l/2b) f f P((o,u) du du (3.15) A S Q to0J-b cJs 9 0 - l P(0,0) The subscript, SQ, indicates the restriction to the square-law-detected array. Let the system bandwidth tend to zero. Then equations (3.14) and (3.15) reduce to 'cos 0^ +1 p VSQ f 0 PCCOQ.U) du cos 0O-1 P(0,0) 2 (3.16) cos 0+1 a n d P A Q n = f P U )0> u ) du (3.17) A S Q cos e Q - l P(0,0) and thus P V S Q = p2gQ (3.18) Because P(tog,u) is the power pattern at frequency tog, equations (3.16) and (3.17) are recognizable as PASQ = 2'GAR^ (3^19) and P V S Q = 4/G^(co 0) (3.20) where G A R(cog) is the array directive gain at a frequency COQ. G a r(C O Q) is defined as the ratio of power per unit solid angle in the direction of the maximum to the average received power per unit solid angle. GAR((0g) m a ^ ^ e written G A R(co 0) = 4TTP(Q,0) (3.21) Total Received Power The narrow-band signal-to-noise ratio of equation (3.12) can now be written (SNR)2Q = 4bT(o-2/o-2T)/ (l/G2 R(to 0)) + (cf f / a ^ G ^ e O g ) ) + o^/ofo. (3.22) From this equation i t is apparent that the best strategy for improving the signal-to-noise ratio of a narrow band square-law-detected array is to maximize the array gain. The wide band expressions for the performance functions were given in equations (3.14) and (3.15). Equation (3.14) may be written as tOQ+b P V S Q - (1/Zb) f co0J-b cos OQ+1 j* " P(co,u) du cos 6 0-l P(0,0) dco (3.23) Now from equations (3.17) and (3.19) i t follows that cos en+i f U P(co,u) du = 2/G (co) cos 6 0-l P(0,0) **• (3.24) The wide band performance functions can now be expressed in terms of the narrowband gain. Substituting equation (3.24) into equations (3.23) and (3.15) gives C0g+b P V S P = (l/2b) f (4/G|R(co)) dco coQ-b wQ+b and P ^ Q = (l/2b) / (2/GAR(co)) dco co0-b (3.25) (3.26) 73 A wide band array gain, G^ g, is defined using the usual definition of ' array gain (equation (3.21)) in terms of radiated powers under £he assumption of flat signal spectrum over the system bandwidth. Gyg satisfies the equation ojQ+b 2 / P(w,0) do; GITR = "O-b (3.27) WB — r—7-; UiQ+b cos QQ+L J J P(o),u) du do) tog-b cos 6Q-1 Now P(to,0) =T(0,0) and thus WQ+b cos 6Q+1 G^ = 2P(0,0)/(l/2b) J- j P(to,u) du dto (3.28) COg-b COS 6g _l From equations (3.15) and (3.28), i t can be seen that the-area performance function is inversely proportional to the wide band gain, PASQ " 2/GWB <3'29> Thus, from equations (3.29) and (3.26), the vide band gain is written in terms of the narrow band gain. OJg+b 1/G^ = (l/2b) f (l/GAR(co)) dco , (3.30) u0-b • • i As a consequence of equations (3.25) and (3.12), the small-signal signal-to-noise ratio is not maximized by maximizing the wide band array gain. Instead, the wide band small-signal (SNR)2 is maximized by minimizing Py which is proportional to the inverse square of the narrow band gain averaged over the system bandwidth as shown in equation (3.25). Only when the signal becomes large enough so that 2(o s/o^)P A is the dominant term in equation (3.12) is the SNR maximized by maximizing the array gain for wide band systems. • 3.4 M a t r i x T h e o r y f o r M a x i m i z a t i o n o f A r r a y G a i n and S i g n a l - t o - N o i s e R a t i o  o f S q u a r e - L a w - D e t e c t e d A r r a y s Cheng and T s e n g 1 9 ' 2 0 h a v e d e v e l o p e d an e l e g a n t m a t r i x t e c h n i q u e f o r t h e m a x i m i z a t i o n o f a r r a y g a i n s . T h e i r t e c h n i q u e i s a s t a r t i n g p o i n t f o r t h e w o r k p r e v i o u s l y c i t e d i n r e f e r e n c e s 15 t o 17. An o u t l i n e o f t h e t h e o r y o f Cheng and T seng w i l l be g i v e n i n s e c t i o n 3 . 4 . 1 . E x t e n s i o n s t o t h e i r w o r k w i l l be p r e s e n t e d i n s e c t i o n 3 .4 .2 w h e r e w i d e band s q u a r e - l a w - d e t e c t e d a r r a y s a r e c o n s i d e r e d . M i n i m i z a t i o n o f t h e v o l ume p e r f o r m a n c e f u n c t i o n , P y , as a means o f s m a l l - s i g n a l SNR m a x i m i z a t i o n w i l l be c o n s i d e r e d i n s e c t i o n 3 . 4 . 3 . I t i s shown t h a t t h e m a t r i x methods o f Cheng and T seng c a n n o t be u sed f o r m i n i -m i z a t i o n o f P y when t h e n a r r o w band a p p r o x i m a t i o n i s n o t a p p l i c a b l e . Me thod s a p p l i c a b l e t o t h e g e n e r a l q u a d r a t i c p r o c e s s o r w i l l be i n t r o d u c e d . 3.4.1 M a x i m i z a t i o n o f t h e N a r r o w Band A r r a y G a i n The a r r a y g a i n i s e x p r e s s e d as a r a t i o o f two q u a d r a t i c f o r m s . 1 9 ' 2 0 M a x i m i z a t i o n i s a c c o m p l i s h e d by s i m u l t a n e o u s d i a g o n a l i z a t i o n o f t h e two q u a d r a t i c f o r m s . The a r r a y g a i n f o r n a r r o w band s y s t e m s , o b t a i n e d f r o m e q u a t i o n s ( 3 . 17 ) and ( 3 . 1 9 ) , i s whe re P(to,u) i s t h e a r r a y power p a t t e r n . I f t h e q u a d r a t i c p r o c e s s o r i s s p e c i a l i z e d t o a s q u a r e - l a w - d e t e c t e d a r r a y , t h e q m a t r i x becomes t h e o u t e r p r o d u c t d u ( 3 . 3 1 ) ( 3 . 3 2 ) where 5 i s a co lumn v e c t o r o f t h e a r r a y e l e m e n t w e i g h t s . Thus ( 3 . 3 3 ) From equations (3.33) and (2.36), the power pattern for the square-law-detected array is M M '(co,u) = yT Y\ a a exp (j (m - i)codu/c) i=l m=l 1 m (3.34) Equation (3.34) is readily integrated to obtain cos S Q + I cos 6Q-1 M M P(to,u) du = 2 Y, Y a , i i=l m=l m •sin (m-i)iod/c cos (cod/c)(m-i) cos 8^ (m-i)cod/c \ j (3.35) The numerator of equation (3.31) for the narrow band gain, is obtained by setting u = 0 in equation (3.34) to get M M P(0,0) = Y Y ai am ( 3 ' 3 6 ) i=l m=l The above two equations may be written in terms of quadratic forms as and P(0,0) = aZ~A& cos J " P(co,u) du = 2atBa cos 9Q-1 (3.37) (3.38) where and lm sin (m-i)cod/c cos (m-i) cod/c (cod/c)(m-i) cos 6 (3.39) (3.40) Thus the array gain may be written G A R = a^a/a^a (3.41) A and B are real symmetric matrices and atBa is positive definite. Thus the quadratic forms in the numerator and denominator of equation (3.41) 27 may be simultaneously diagonalized so that the array gain becomes M E Ki<4>2 M E Cap 2 i=l where a' is the transformed version of the weight vector a. The {K^ } are a set of eigenvalues satisfying the equation det (A - KB) = 0 (3.43) It is evident that GA1J - (K.) (3.44) AR n/max v ' It also follows that i f a is chosen to be the eigenvector,..^, corresponding to the eigenvalue, K^ , i.e., i f a = J. where AJ. = K,BJ. l 1 l r (3.45) then the corresponding gain is GAR = K i ( 3 * 4 6 ) as can be verified by substituting equation (3.45) into equation ( 3 . 4 1 ) . Thus the gain is maximized i f a is chosen to be the eigenvector corresponding to the largest eigenvalue. It was shown by Cheng and Tseng 2 0 that i f A satisfies A = ee1- (3.47) where t is a column vector, a l l but one of the eigenvalues are zero. The nonzero eigenvalue is given by K = ^B^e (3.48) and thus ( G A R ) m a x = K = (3.49) The weight vector, a, which maximizes i s the ei g e n v e c t o r corresponding to the eigenvalue, K, of equation (3.48). From r e f e r e n c e 20 ( a>opt = B ~ l g ( 3 ' 5 0 ) I n the paper of Cheng and Tseng, a r r a y s t e e r i n g i s not i n c l u d e d i n the f o r m u l a t i o n of the equations, so A i s complex and A i m = exp ( j ( i - m)todu 0/c) (3.51) where UQ i s the d i r e c t i o n i n which the g a i n i s to be maximized. For A w i t h elements s a t i s f y i n g equation (3.51) the optimum weight v e c t o r a w i l l be complex i n d i c a t i n g a phase s h i f t i n g o p e r a t i o n i s r e q u i r e d to maximize the ga i n . In t h i s i n v e s t i g a t i o n the equations are formulated under the assumption of i d e a l delays and only r e a l a v e c t o r s are obtained. The assumption of i d e a l delays w i l l be shown to be advantageous when wide band gains are considered i n the f o l l o w i n g s e c t i o n . 3.4.2 Maximization of Wide Band Gain In t h i s s e c t i o n i t w i l l be shown th a t the wide band g a i n can be maximized using the method of Cheng and Tseng. I t w i l l be shown t h a t t h i s maximization presents l i t t l e mathematical d i f f i c u l t y . I n c o n t r a s t , i t i s shown i n s e c t i o n 3.4.3, that c o n s i d e r a b l e d i f f i c u l t y i s encountered i n the maximization of the s m a l l - s i g n a l s i g n a l - t o - n o i s e r a t i o by means of a m i n i m i z a t i o n of Py. .The wide band a r r a y g a i n , G^g, i s given by equation (3.28). S u b s t i t u t i o n of equations (3.37) and (3.38) i n t o (3.28) y i e l d s I0g+b G ^ = 2a tAa/(l/2b) f 2a tB(to)a dco (3.52) co0-b where A^ m = 1 and B^ m i s given by equation (3.40). Thus i t i s apparent that Gyg can be w r i t t e n as the r a t i o of two q u a d r a t i c forms Gm = a ^ a / a t C a (3.53) where A s a t i s f i e s the c o n d i t i o n of equation (3.47) i . e . A = e£ t and where C = (l/2b) J B(u) du (3.54) V b The s o l u t i o n s developed i n s e c t i o n 3.4.1 are now a p p l i c a b l e i f B i s r e p l a c e d by C w i t h C given by equation (3.54). The advantage of f o r m u l a t i n g the equations to i n c l u d e i d e a l s t e e r i n g delays becomes evident here. Suppose t h a t the equations had been formulated u s i n g the method of Cheng,and Tseng w i t h P ( O J , U 0 ) = a ^ a (3.55) and w i t h the elements of A given by equation (3.51). Then tOQ+b ( l / 2 b ) f P(co,u 0) du = atDa u Q-b cog+b w i t h . D i m = (l/2b) f A i m ( l o ) du (3.56) u 0-b and Gyg = atDa/atCa (3.57) The elements D^m of equation (3.56) are obtained by s u b s t i t u t i n g equation (3.51) i n t o (3.56) and performing the i n d i c a t e d i n t e g r a t i o n . The r e s u l t i s Uim = exp ( j ( m - i ) u d u 0 / c ) s i n ((i-m)bdu n/c) (3.58) ((i-m)bdu 0/c) I f the D^m are given by equation (3.58) w i t h b ^ 0, i t i s e v i d e n t t h a t D ^ e£ t (3.59) except f o r ug = 0 corresponding to gain maximization i n the broadside d i r e c t i o n . Hence, the simple s o l u t i o n of Cheng and Tseng cannot be found un l e s s U Q = 0 i f b 4 0. The p h y s i c a l e x p l a n a t i o n of t h i s i s t h a t the weight v e c t o r . a obtained by u s i n g the method of r e f e r e n c e 20 accomplishes beam s t e e r i n g by frequency independent phase s h i f t s . T his i s adequate f o r a s i n g l e frequency but not f o r a band of f r e q u e n c i e s . In f a c t one could maximize (3.57) w i t h the D^m given by (3.58); however the eigenvalue equation det (D - KC) = 0 (3.60) would have more than one nonzero eigenvalue. The s o l u t i o n corresponding to the l a r g e s t eigenvalue would maximize the g a i n under the c o n s t r a i n t of beam s t e e r i n g by a simple frequency, independent phase s h i f t . The l a r g e r the bandwidth, the g r e a t e r the d i f f e r e n c e one would expect between maximized gains obtained by u s i n g equation (3.57) i n s t e a d of equation (3.53). 3.4.3 M i n i m i z a t i o n of the Volume Performance F u n c t i o n f o r Wide Band  Square-Law-Detected Arrays The s m a l l - s i g n a l s i g n a l - t o - n o i s e r a t i o i s maximized by, m i n i m i z i n g the volume performance f u n c t i o n , Py, as can be seen by l e t t i n g o|/o§j -> 0 i n equation (3.12). In the case of wide band systems, Py i s not i n v e r s e l y p r o p o r t i o n a l to the square of a r r a y g a i n as shown i n s e c t i o n 3.3. The purpose of t h i s s e c t i o n i s to demonstrate that the methods of Cheng and Tseng 1. 9' 2 0 are not a p p l i c a b l e to the m i n i m i z a t i o n of Py f o r wide band systems. From equation (3.38) one o b t a i n s cos 6(3+1 J P(to,u) du = 2a tB(co)a (3.61) cos 6Q-1 where B(to) i s a f u n c t i o n of frequency as can be seen from equation (3.40); 80 r M n PVSQ " 2/ Equations (3.61) and (3.37) are s u b s t i t u t e d i n t o equation (3.23) to o b t a i n 1 a)Q+b PVSQ = U/bC^Aa)2! J" a^Cu)aa^Cu)a do (3.62) u0-b This e x p r e s s i o n f o r PysQ cannot be reduced to the r a t i o of two q u a d r a t i c forms or to the square of such a r a t i o unless .the bandwidth, b, i s allowed to go to zero. Hence the method of ga i n maximization of Cheng and Tseng i s not a p p l i c a b l e i n t h i s case. Equation (3.62), expressed i n terms of summations, i s M M M M E E E E a i a n a m a k i = l n=l m=l k=l cog+b •(l / 2 b ) . J B i n(u))Bm k(u)) dco (3.62a) to 0-b The i n t e g r a t i o n of equation (3.62a) must be c a r r i e d out n u m e r i c a l l y f o r B-j_n(co) given by equation (3.40). A numerical i n t e g r a t i o n of t h i s type has been programmed f o r the general q u a d r a t i c processor to be d i s c u s s e d i n l a t e r s e c t i o n s of t h i s chapter. However, i f the weight v e c t o r , a, i s chosen to M minimize Pysq u s i n g the method of Lagrange m u l t i p l i e r s w i t h E a i = c o n s t a n t i = l as a c o n s t r a i n t , a h i g h l y n o n l i n e a r s e t of equations i s obtained. No attempt has been made to s o l v e such a s e t of equations. The main e f f o r t has been concentrated on the gene r a l q u a d r a t i c processor where q ^ n ^  a ± a n - When Py i s expressed i n terms of { q ^ n } 5 o n l y second order products of the q ^ n occur and the d e r i v a t i v e i s l i n e a r . This can be i n f e r r e d by r e p l a c i n g the products a i a n a m a k ^ ^ i n ^ mk * n equation (3.62a) which i s a l l t h a t i s necessary to ob t a i n expressions f o r the volume performance f u n c t i o n of the gene r a l q u a d r a t i c processor. The r e s u l t s obtained by m i n i m i z a t i o n of Py f o r the gene r a l q u a d r a t i c processor provide an upper bound to the improvement t h a t might be obtained by choosing the weight vector, a, of the square-law-detected array so that Pygq of equation (3.62a) is minimized. 3.5 The Volume Performance Function for the General Quadratic Processor The volume performance function, Py, will now be derived in a form suitable for computation. A set of-linear equations will be obtained for the elements of the q matrix which minimizes Py subject to a set of constraints obtained from the prescribed pattern function. The solution matrix, q, obtained by solution of these equations, will be denoted qy. It will be shown that the q matrix of the quadratic processor is not completely determined by a complete specification of the pattern function of the array. This is in contrast to.the conventional array where the pattern function determines the weight vector, a, the array gain, and the performance functions, Py and P^ . The set of equations for the elements of q^ r will be shown to be particu-larly simple when the array elements are half-wavelength spaced and the narrow band approximation is valid. This solution will be denoted qy It will be shown that qy 1^ also maximizes the signal-to-receiver noise ratio, qy 1^ will be shown to have elements as uniform in size as the constraints will allow. Such a matrix will also be referred to as "maximally uniform" and denoted q u. Symmetry properties of qy will be deduced from the set of equations for minimizing P ^ . It will be shown the qy is symmetrical about both diagonals. This symmetry property will be used to obtain a significant reduction in the dimension of the set of equations which must be solved. The equations for the elements of qy will be formulated for computer solution. These equations are i n i t i a l l y set up to minimize Py, subject to the constraints, by Lagrange's method. A further reduction in the dimension of the equation is obtained by using a substitution method. This substitu-tion method is shown to be computationally efficient. This efficiency is a result of the sparse nature of certain submatrices in the matrix of co-efficients for the linear equations. 3.5.1 Derivation of P. -V The volume performance function, P^ , has been defined as o)rj+b cos 6A+1 P = ( i / 2 b ) J ff •FCam^MUz) ^ ^ tOQ-b cos 0O-1 F(0,0) dto (3.63) Py is a triple integral of the function F(O ) U ^ , O J U2) and is an average over the frequency band of the volume under F(wu^,wu2). This is the origin of the name "volume performance function". The function Ftwu^on^) of equation (2^41) can be written M M M M . F(oju1,Wu2) = E E E E q k nq i n i cos ud((m-n)u1 + (k-i)u 2) k=l n=l i=l m=l c (3.64) The integration of equation (3.63) with respect to u^ and U 2 is easily carried out and a closed form expression obtained. The integration over GJ must be carried out numerically. For narrow band systems, frequency averaging is unnecessary. The narrow band expression for P^  is cos 8^ +1 V<°0> = / / F("0U1'M0U2> du. du, V cos 8 0-l F(0,0) 1 1 (3.65) A general expression for P^. has been obtained. This equation is useful for computations for both the narrow and wide band cases. 1/ E E q m ; L 2 m 1 * E E <± + 8 E E E q m l q j ± G < j , i n ) - J i j m j^ m m 1 + 8 E E E E q m iq j kE(m,i,k 5j) i m j k i^k j^ m (3.66) A l l summations run from 1 to M in equation (3.66). If the narrow band approximation of equation (3.65) is valid, sin tOQd(j - m) G(j,m) = c cos (COQCKJ - m) cos 0Q ) ( 3 . 6 7 ) WQCUJ - m) c c sin C0gd_(j-m) sin u)Qd(i-k) and E(m,i,k,j) = c c cos (cogd(i-k+j-m) cos 0Q ) U)Qd(j-m) co-^ cKi-k) c c c ( 3 . 6 8 ) The corresponding wide band expressions are frequency averages of equations (3.67) and (3.68). These averages must be evaluated numerically. The wide band versions of G(j,m) and E(m,i,k,j) are given in Appendix C. 3.5.2 Minimization of Py Py is minimized subject to the constraint of specified array pattern function. This pattern function is given as a terminated Fourier series in equation (2.37). The set of coefficients of the Fourier series, (O^K completely specify the pattern function. The Fourier coefficients, {D^ .}, are expressed in terms of the elements of the q matrix in equation (2.38). The constraint equations obtained from (2.38) are M D6tf) = ~ D0 + E I i i i=l M-k M and V (3.69) = - Dk + E <*i,i+k + [ ^ i _ k for k ^ 0 i=1 i=k+l ' The constraints are applied by setting ^(q) = 0 f° r 1 - k - M - 1. Mini-mization subject to these constraints is accomplished by Lagrange's method. In the formulation of the equations to follow, the dependence of Py on the q matrix is indicated by writing P^  as Py(q). Let M-l U = P (q) + £ p D (q) (3.70) v k= o k k and r e q u i r e that 3q 3U_ = 0 f o r 1 - r , s - M r s D k(q) = 0 f o r 0 - k - M - 1 (3.71) .th Here, i s the Lagrange m u l t i p l i e r corresponding to the k c o n s t r a i n t equation. I t can be seen i n equation (3.66), that o n l y second order products of the elements of q occur i n Py(c[). Thus 3 P y ( q ) / 3 q r s i s a l i n e a r combin-a t i o n of the elements of q. The s e t of l i n e a r equations which must be s o l v e d i s ( 3 P v / 3 q r s ) + p | r _ s | = 0 f o r 1 - r , s ^ M D k(q) = 0 f o r 0 - k - M-l " (3.72) In t h i s l i n e a r set of equations there are M 2 + M unknowns; t h a t i s , the M 2 elements of c[ and the M Lagrange m u l t i p l i e r s . Because q i s symmetric, the dimension of the equations i s r e d u c i b l e to M(M + l ) / 2 . We s h a l l see that f u r t h e r r e d u c t i o n i s p o s s i b l e . Consider the case of an a r r a y w i t h h a l f - w a v e l e n g t h spacing between elements. Let the a r r a y system be narrow band so t h a t equations (3.67) and (3.68) are v a l i d approximations. Then f o r h a l f - w a v e l e n g t h s p a c i n g , COQC! = ir c and equation (3.66) reduces to M M i=1 m=l M M E Z i = l m=l •I 2 (3.73) The denominator of the above equation i s f i x e d by the c o n s t r a i n t s , i . e . M M M-l E E ^mi = E \ i = l m=l k=0 K (3.74) and thus, M M ' i=l m=l M-l • E \ k=0 . (3.75) From equations (2.81) and (3.75) we see that P i s proportional to the inverse square of the processing gain against receiver noise. That i s , P V ^ A " A / G 2 (3.76) where G i s given by equation (2.81). Thus minimization of Py ^ also maximizes the signal-to-receiver noise ratio. For Py given by equation (3.75) the set of linear equations (3.72) becomes qrs + p | r - s | = ° f o r 1 ~ r> s ~ M M i = l Y (3.77) M-k M £ q± i + k + E ^ i , i - k = \ f o r 1 - k -i=l ' K i=k+l M - l Note that the factor l/(EDj c) 2 has been dropped. The solution to the above set of equations is easily found to be q r s ** D | r - s | / 2 ( M " ' r _ S ' ) f o r r •' S q r r = DQ/M (3.78) which i s the same solution obtained in equation (2.84). A (4 x 4) matrix q j with elements given by equation (3.78) is 4 D l 6 D 2 4 D 3 2 D l 6 Do 4 D l 6 D 2 4 D 2 4 D l 6 D 0 4 D l 6 » 3 2 c 2 4 D l 6 D 0 4 ( 3 . 7 9 ) A q matrix with elements as in equations ( 3 . 7 8 ) and ( 3 . 7 9 ) will be called a "maximally uniform" q matrix. Such a matrix has the property that its elements are as uniform in size as is possible while s t i l l satisfying the pattern constraints. The maximally uniform q matrix was designated . t J u in section 2 . 5 . 3 . 5 . 3 Symmetry Properties of q^ In this section the symmetry properties of cjy will be established. These properties will be used to reduce the dimension of the set of linear equations in ( 3 . 7 2 ) . The q matrix has been assumed to be symmetrical about the principal diagonal throughout the thesis. It will be shown that qy is also symmetrical about the diagonal running from the bottom left hand corner to the upper right hand corner of the matrix. This symmetry condition will be called "double symmetry". The two symmetry conditions are <*rs = ^sr and i f c( = q*. ( 3 . 8 0 ) Irs = ^ M-r+l.M-s+l ( 3 ' 8 1 ) for 1 - r , s - M. If equation ( 3 . 8 1 ) is true i t is possible to make a reduction in the dimension of the set of linear equations of ( 3 . 7 2 ) . Equation ( 3 . 8 1 ) is a solution which can be verified by substitution. If, on substitution of equation (3.81) into (3.72), i t can be shown that two equations become identical, term for terra, then one of the equations may be eliminated and the reduced set solved. If the reduced set has a solution, then equation (3.81) is verified. Consider the two equations and (3Pv(q)/3qrs) + p| r_ s| = 0 (3P v(q)/3q M_ r + 1 > M_ s + 1) + P|M-r+l-(M-s+1)| = 0 (3.82) (3.83) It will be shown that equation (3.82) can be made identical, term for term, to equation (3.83) under the assumption of equations (3.80) and (3.81). The last terms of equations (3.82) and (3.83) are obviously identical, i.e., P |M_ r +i - (M-s+l)| = p | s " r l = p | r-s | " T h u s i t : i s o n l y necessary to show that the derivatives are identical, i.e. , (3Pv(q)/3qrs) = ( 9 ? V ( ^ 7 9 V r + 1 .M-s+P (3.84) The equations for the derivatives in (3.82) and (3.83) are obtained by differentiating equation (3.66) for the volume performance function (omitting the l/(ZD k) 2 factor). Thus M •OP„(3)/8q r s) = 8q + 16 £ q .G(j,r) V rs rs ^ sj 3Z M M +16 E [ q , k E(r,s,k,j) j=l k=l 3 * r (3.85) and 88 ( 9 P v ( q ) / 9 q M _ r + 1 > M _ s + 1 ) = 8 q M _ r + 1 > M _ g + 1 + 1 6 t V s + l , j G ( ^ M - r + 1 ) M M  + 1 6 £ E q. kE(M-r+l,M-s+l,k,j) j = l k=l j^M-r+1 MM-s+1 (3.86) The three terms i n the above equation w i l l now reduce to the three terms i n equation (3.85) under the symmetry c o n d i t i o n s of equations (3.80) and (3.81). From equation (3.81) the f i r s t term i s S q ^ _ r + ^ j^-s+l = ^ q r s - * n t* i e s e c o n d term we make a change of v a r i a b l e s . L e t j = M - k + 1. Then the second term becomes M 1 6 E q M - s + l , j G ^ ' M - r + 1 > j = l j^M-r+1 Now from equation (3.67) i t i s evident t h a t G(j+a, k4a) = G(j,k) = G(k,j) ~ G ( - j , -k) and hence G(M-kt-l, M-r+1) = G(k,r) (3.88) Thus from equations (3.88) and (3.81) the second term i s M M 16 E %-s+i ^(j,M-r+n = i 6 Y qSkG(k>r> j = l , J k=l j=frl-r+l k^ r I n the t h i r d term of equation (3.86) make the s u b s t i t u t i o n n = M - j + 1 m = M - k + 1 (3.89) (3.90) 89 Then the t h i r d term becomes M M 16 T V q.,E(M-r+l,M-s+l,k,j) j = l k=l J k j^M-r+1 k^M-s+1 M M = 16 [ [ q M_ n + 1 } M_ n i + 1E(M-H-l )M-s+l,M-m+l,M-n+l) (3.91) n=l m=l n^r m^s From equation (3.68) i t can be shown th a t E ( n , i , k , j ) = E(n+a,i+b,k+b,j+a) (3.92) where a and b are a r b i t r a r y . A l s o E ( n , i , k , j ) = E ( - n , - i , - k , - j ) . ' (3,93) Now, from equations (3.92) and (3.93), E(M-r+l,M-s+l,M-rc+l,M-n+l) = E(r,s,m,n) (3.94) Thus, from equations (3.94) and (3.81), the t h i r d term becomes M M 16 Y. L q j kE(M-r+l,M-s+l,k,j) j = l k=l j^M-r+1 k^M-s+1 M M = 1 6 Y Y q n m E ( r ' s ' m ' n ) ( 3 - 9 5 ) n=l m=l n^r m^s •It has t h e r e f o r e been shown that equations (3.85) and (3.86) are i d e n t i c a l , term f o r term, under the assumption of double symmetry of the c[ m a t r i x . Hence, the number of equations may be reduced to take advantage of t h i s symmetry c o n d i t i o n . 90 3.5.4 Formulation of Equations f o r Computer S o l u t i o n The double symmetry p r o p e r t i e s of the qy m a t r i x w i l l now be used to e s t a b l i s h a v a r i a b l e naming convention. T h i s w i l l a l l o w a r e d u c t i o n i n the dimension of the equations to be s o l v e d . The v a r i a b l e naming convention i s XjCr.s) = q r g - q g r = V r + 1 ,M-s+l = qM-s+l ,M-r+l (3.96) This v a r i a b l e naming convention i s i l l u s t r a t e d i n F i g u r e 3.2 f o r a 4-element a r r a y . N" 71 q n q 1 2 q i 3 qu q 2 1 q 2 2 q 2 3 i q 2 4 q 3 1 q 3 2 q 3 3 : q 3 4 <tttl %2 %3 q44 \ axes of symmetry x 2 x 5 , x6 x 3 x 3 x 6 X 5 . x 2 x 4 x 3 x 2 x l Fi g u r e 3.2 Double Symmetry and the V a r i a b l e Naming Convention ' The s e t of l i n e a r equations (3.72) i s formulated i n terms of the x^ v a r i a b l e s of equation (3.96) w i t h r,s dependence suppressed. As a consequence of equation (3.96) the number of unknown elements i n the q m a t r i x becomes S Q e where the "o" s u b s c r i p t denotes odd M and the "e" s u b s c r i p t denotes even M. S i s given by o, e S e = M(M + 2) S = 1(M + l ) 2 ° 4 (3.97) where M i s the number of r e c e i v i n g elements i n the a r r a y . The numbers S of equation (3.97) are to be compared to M2 unknowns i f no symmetry i s used or to M(M+1) unknowns i f only symmetry about the 2 principal diagonal is assumed. In terms of the x (r,s) variables, the partial derivative of P v becomes j OP v(q)/3x j(r,s)) = (9P v(q)/8q rs) + (3P y(q)/3q sr) + <V3>/^M-r+l,M-s+l> + ( 3 PV ( q ) / 3 q»-s+l il*.r+l ) ( 3 ' 9 8 ) Now, from this expression, a set of linear equations equivalent to equation (3.72) is %K(r,s)(3Pv(x)/3x:j(r,s)) + P|r_g| = 0 for 1 ^  j - S Q > e M E x i ( r > r ) = Do f (3.99) M-k •2 Y, x.(r,r+k) .= Dk for 1 - k - M - 1 r=l J J The values for K(r,s) are K(r,s) =1 for r ± s, M - r + 1 i s, M K(r,s) =2 for r i s, M - r + 1 = s, M K(r,r) =2 for M - r + W r K(r,r) =4 for M - r + 1 = r s + 1 i r s + 1 = r y ( 3 . 1 0 0 ) The p| r_ sj are the Lagrange multipliers. The K(r,s) factor has been chosen for simplicity in programming. The same q solution matrix would be obtained with any nonzero set of K(r,s) values. Specifically, the K(r,s) values of equation (3.100) have been chosen so that JSK(r,s)(3Pv(x)/3x:J(r,s)) = (3P y(q)/3q r g) + (3P v(q)/3q s r) (3.101) The minimized value of Py(q) (i.e., P v(q v) = P y v) will now be derived in terms of the Lagrange multipliers and the Fourier coefficients, {D,}. The equation for Py in (3.66) is rewritten as M M Py ~ .£ £. ^ m i i=l m=l M M £ £ j#n k^i M 4 cW + 8 E <li,G(J>m) jVm - "M-l q^kE(m,i,k,j) / E \ _ (3.102) where the expression in the brackets is identifiable as the derivative, ^3Py/3q^m, by comparison with equation (3.85). Thus, P^v becomes PW = E E ^ i ^ V ^ 7 i=l m=l M-l E \ k=0 (3.103) But 3Py/3qm-L = 9 Pv 7 9 clim a n d h e n c e f r o m equations (3.99) and (3.101) M M p = - v y o - o i . i ik V V &?1 mil ' U U' | m _ 1 1 M-l E B> k=0 From the constraint equations i t follows that P can be written H W M-l PW = - E V k / 4 k=0 M-l E \ k=0 (3.104) This equation is useful for computational purposes both as a check on the program and as an efficient formula for the evaluation of Pyy A check on the program is provided by computing P from equation (3.104) and also by substituting q^ in equation (3.66). If the results agree one can be certain the program is correct. The Lagrange multipliers, p^ ., can be determined in the solution to the linear equations of equation (3.99). Only one program loop is required to evaluate the numerator of equation (3.104) with M iterations. In contrast, i f Pyy is evaluated by substitution of q^ into equation (3.66),.the program must c o n t a i n four nested loops r e q u i r i n g M 4 i t e r a t i o n s . Thus, equation (3.104) i s computationally much more e f f i c i e n t than equation (3.66). A s u b s t i t u t i o n method was used to reduce the dimension of the l i n e a r equations (3.99) which must be s o l v e d . T h i s method i s e f f i c i e n t because i t makes use of the sparseness of the m a t r i x of c o e f f i c i e n t s f o r the s e t of equations i n (3.99). Equation (3.99) i s expressed as a m a t r i x equation as CZ = B (3.105) where C i s a square m a t r i x of c o e f f i c i e n t s . Z and B are column v e c t o r s which can be p a r t i t i o n e d as (3.106) (3.107) X i s a column v e c t o r w i t h values given by equation (3.96). The components of x s p e c i f y the elements of the q m a t r i x , p i s a column v e c t o r of the M Lagrange m u l t i p l i e r s . D i s a column v e c t o r of the M F o u r i e r c o e f f i c i e n t s which s p e c i f y the p a t t e r n f u n c t i o n and the p a t t e r n c o n s t r a i n t s . The m a t r i x C can be p a r t i t i o n e d as Z = o,e M I and. B = D o,e M 94 C = A : E I F J 0 ) ISo,e | M (3.108) So,e M Now C i s sparse i n the r e g i o n of the lower r i g h t hand M x M m a t r i x which i s i d e n t i c a l l y zero. The submatrix E i s a l s o sparse c o n t a i n i n g o n l y one element per row. A i s not sparse i n general except f o r the s p e c i a l case of a h a l f -wavelength spaced narrow band a r r a y . The m a t r i x F i s a l s o sparse and has s p e c i a l p r o p e r t i e s which w i l l be used to advantage. The s u b s t i t u t i o n method e l i m i n a t e s the unknown Lagrange m u l t i p l i e r s and reduces the dimension of equation (3.105) by M . We s t a r t w i t h the c o n s t r a i n t equations . Fx = D • (3.109) In these equations there are S Q e unknowns and only M equations. Thus a s o l u t i o n to equation (3.109) has S c e - M a r b i t r a r y v a r i a b l e s . Let us denote t h i s set of S„ - M a r b i t r a r y v a r i a b l e s as {xv } w i t h 1 - j - S „ - M . o, e J K j J o, e We s u b s t i t u t e f o r x i n terms of the { x k . } . i n the e x p r e s s i o n f o r P^. Then i f 9P /9x, = 0 f o r a l l 1 - j - S - M (3.110) V k j o,e , Py w i l l be minimized s u b j e c t to the c o n s t r a i n t s . That i s , any x which s a t i s f i e s equation (3.110) must a l s o s a t i s f y the c o n s t r a i n t s of equation (3.109). . Consider now the f o l l o w i n g set of equations: So,e 9P„/9xk = V (9P v/3x.)(dx./dx ) = 0 f o r l - j - S o e - M J i = l 1 J ' Fx = D (3.111) T h i s set of equations has dimension S Q e which i s M l e s s than the dimension of the o r i g i n a l set of equations (3.105). In f a c t i t i s necessary to s o l v e an M x M set of equations for the derivatives, dx^/dxk^ . as will be shown. The set of unknowns, dx./dxv is determined by differentiating the constraint equations (3.109) as follows: or d(Fx)/dxk_ = d(D)/dxk. F(dS/dxkj) = FY = 0 (3.112) Equation (3.112) is a homogeneous set of M equations in S D > e unknowns and hence has SQ e - M independent solutions. These S 0 j 6 - M solutions constitute the derivatives of x with respect to the S 0 j 6 - M arbitrary variables xk_. That is. f j = K j ( d x / d x k ) (3.113) where Kj is an arbitrary constant of proportionality which will be set to K- = 1 for a l l j . Y J is the j*"*1 independent solution to equation (3.112) The entire solution space of equation (3.112) is V " [ ? i Y 2 Y 3 . . . Y S O ) e_ M] The set of equations which must be solved now reduces to (3.114) Gx = H (3.115) where G = Jo,e - M (3.116) M < > S. o.e and H = r* 9 0 D 4-So,e - M M (3.117) The elements of L are obtained from A and V as or in matrix form L = AV (3.119) The Lagrange multipliers, p, are then obtained by substituting the x solution of equation (3.115) into a subset of the original set of linear equations in equation (3.105). The first M rows and columns of the sparse E submatrix of equation (3.108) constitute a unit matrix Ij^. Thus, i f we form a matrix from the first M rows and S Q e columns of C, i t follows that p = - Cix (3.120) ..Determination of the solution space for a homogeneous set of equations is not generally an easy matter. In general, i t is necessary to determine a set of M independent columns in the matrix of coefficients. However, the special nature of the constraint equations simplify the problem in this case. If the constraint equations (3.109) are written out in detail, i t becomes apparent that F may be written as the partitioned matrix F = [ l M j j ] (3.121) where 1^ is the M x M unit matrix. Thus, the first M columns of F are independent by inspection. The method of solution of the homogeneous set of equations is carried out as in reference 28. Let Y be partitioned as 9 7 W, t M Then Y = (3.122) J $ o ,e - M FY = i M : J = 0 and thus W, = - JW- (3.123) The v e c t o r W2 may be chosen a r b i t r a r i l y , each cho i c e g i v i n g a v a l i d s o l u t i o n f o r W^. The . W]_ and W2 v e c t o r s are combined to form a s o l u t i o n v e c t o r Y. as i n equation (3.122). An' independent s e t of Y. v e c t o r s i s J J obtained i f an independent s e t of a r b i t r a r y v e c t o r s , W2 i s chosen. The o v e r a l l s o l u t i o n space, V, made up of a l l the Y j v e c t o r s , i s V = -J <- So,e- M - M t M I T s r (3.124) o ,e M The s u b s t i t u t i o n procedure which has j u s t been o u t l i n e d i s mathematically complex w i t h c o n s i d e r a b l e programming e f f o r t r e q u i r e d to s e t up the equations. However, because of the nature of the e q u a t i o n s , a gain i n computational e f f i c i e n c y was r e a l i z e d . Numerical r e s u l t s obtained by u s i n g t h i s computa-t i o n procedure, are given i n Chapter 4, 3.6 The Area Performance F u n c t i o n f o r the General Quadratic P r o c e s s o r The area performance f u n c t i o n , P^, w i l l now be d e r i v e d i n a form s u i t a b l e f o r computation. I t w i l l be shown that P^ depends on the q m a t r i x only through a set of v a r i a b l e s which are the row sums of the q m a t r i x . I t w i l l a l s o be shown th a t m i n i m i z a t i o n of P A i s not s u f f i c i e n t to s p e c i f y a l l the elements of the q m a t r i x . A method f o r the subsequent m i n i m i z a t i o n of P v s u b j e c t to the p a t t e r n c o n s t r a i n t s and the c o n s t r a i n t of minimum P A w i l l be given. The s o l u t i o n m a t r i x f o r t h i s m i n i m i z a t i o n procedure w i l l be denoted q^. The double symmetry p r o p e r t i e s of the c[y m a t r i x , d e s c r i b e d i n s e c t i o n 3.5.3, w i l l be shown to be a p p l i c a b l e to the q A matrix. A s u b s t i t u t i o n method s i m i l a r to t h a t of s e c t i o n 3.5.4 w i l l be developed. This s u b s t i t u t i o n method w i l l be shown to be c o m p u t a t i o n a l l y e f f i c i e n t . 3.6.1 M i n i m i z a t i o n of P^ P A i s the double i n t e g r a l of the f u n c t i o n , F(0,wu) as shown i n equation (3.9). This equation i s r e w r i t t e n and renumbered below f o r convenient r e f e r e n c e coQ+b cos 6o+l P A = ( l / 2 b ) f f F(0,uu) du do) (3.125) u>0-b cos GQ-1 F(0,0) Thus, P A i s the average over frequency of the area of a c r o s s - s e c t i o n of the F(coui,wu2) f u n c t i o n along one or the other of the u^,u2 axes. Hence, P has been named the "area performance f u n c t i o n " . The f u n c t i o n F(0,wu) i s obtained from equation (2.41) as F(0,cou) = £ E £ E qni^mk c o s (wd(n-m)u) (3.126) n i k m c w i t h summations running from 1 to M. A new v a r i a b l e w i l l now be d e f i n e d as the row sum *n= E q n i < 3- 1 2 7> i = l . • F(0,tou) can be w r i t t e n i n terms of these row sums as M M F(0,iou) = [ [ XnXm cos (tod(n-m)u) (3.128) n=1 m= 1 c The f i r s t i n t e g r a t i o n of equation (3.125) w i t h respect to u i s e a s i l y c a r r i e d out. I f the narrow band approximation cannot be used, the second i n t e g r a t i o n w i t h respect to to must be done n u m e r i c a l l y . E v a l u a t i o n of the i n t e g r a l of equation (3.125) r e s u l t s i n P A = 2 M M M E ^ + 2 [ £ V m G ( n , m ) n=l n=l m=l n^m / M n=l (3.129) where G(n,m) i s given i n equation (3.67) f o r the narrow band case or i n Appendix C f o r the wide band case. Because P^ can be w r i t t e n as a q u a d r a t i c form the method of Cheng and T s e n g , 1 9 ' 2 0 d e s c r i b e d i n s e c t i o n 3.A, could be used. However, the Lagrange m i n i m i z a t i o n technique i s not much more complex. Thus, the Lagrange method has been chosen i n order to make use of r e s u l t s and techniques developed i n s e c t i o n 3.5. I t i s evident from an examination of equation (3.129) that the m i n i -m i z a t i o n of P^ cannot uniquely s p e c i f y the q m a t r i x . M i n i m i z a t i o n of P^ leads to the s p e c i f i c a t i o n of a s e t of row sums {X n ) which form the components of a row-sum column v e c t o r , X^. These row sums form a s e t of c o n s t r a i n t s on the q m a t r i x . P^ i s minimized s u b j e c t to a c o n s t r a i n t which s p e c i f i e s t h a t the sum of a l l the elements i n the q m a t r i x must be a constant. T h i s c o n s t r a i n t can be expressed i n terms of the row sum v a r i a b l e s as M C U ) = £ X n - Q = 0 (3.130) n=l where Q i s a constant. I f p a t t e r n c o n s t r a i n t s are to be a p p l i e d as w e l l , then M-l Q - E Dk k=0 We define a function U ( X ) = P A ( X ) + 6 C ( X ) ( 3 . 1 3 1 ) where 6 is a Lagrange multiplier. Then P ^ ( X ) is minimized subject to the constraints of equation ( 3 . 1 3 0 ) i f 3 U ( X ) / 3 X r = 0 for 1 - r - M. The resulting set of linear equations is ( 3 P A ( X ) / 3 X r ) + 6 = 0 for 1 - r - M C ( X ) = 0 ( 3 . 1 3 2 ) The solution to equation ( 3 . 1 3 2 ) for a narrow band half-wavelength spaced array is of interest. For half-wavelength spacing P A becomes '*A,hX = 2 E # n=l M T. >-n n=l ( 3 . 1 3 3 ) Equation ( 3 . 1 3 2 ) reduces to (neglecting the constant term) 4 X r + 6 = 0 for 1 * r - M C ( X ) = 0 ( 3 . 1 3 4 ) Substitution of X r = - 6 / 4 in equation ( 3 . 1 3 3 ) yields the result P A A , ^ X - 2 / M ( 3 . 1 3 5 ) where M is the array size. The symmetry properties of the solution vector X A of equation ( 3 . 1 3 2 ) will now be examined. It will be shown that Xr - *M-r+l ( 3 . 1 3 6 ) 101 i n the same manner as double symmetry was demonstrated i n s e c t i o n 3.5.3. That i s , i t w i l l be shown th a t 9U(X)/cU r = 9 U ( X ) / 9 X M _ r + 1 f o r 1 ^  r ^ M (3.137) under the assumption of equation (3.136). Equation (3.137) i s s a t i s f i e d i f the d e r i v a t i v e s of P A(A) w i t h r e s p e c t to X r and A^-r+l a r e I d e n t i c a l , i . e . , 3P A(X)/9X r = 9 P A ( X ) / 9 X M _ r + 1 (3.138) The d e r i v a t i v e s i n equation (3.138) are obtained by d i f f e r e n t i a t i n g e q uation (3.129). That i s , M 3P A(X)/3X r = 4X r + 4 Y, \G(r,m) (3.139) m=l rn^-r M and 3 ^ / 3 W l = V + l ' 4 E XffiG(M-r+l,m) (3.140) m=l m^M-r+1 Equation (3.140) w i l l now be shown to reduce to equation (3.139) under the assumption of the symmetry c o n d i t i o n of equation (3.136). The f i r s t term of equation (3.140) s a t i s f i e s 4X M_ r +-^ = 4X r from equation (3.136). I n the second term of equation (3.140) make the s u b s t i t u t i o n k = M - m + 1 to o b t a i n M M 4 Y \nG(M-r+l,m) = 4 Y X M - k + l G ( M - r + 1 » M ~ k + 1 ) (3.141) m=l k=l m^M-r+1 k^r From equation (3.136) A^-kM = X k and from equation (3.88) G(M-r+l,M-k+l) = G(r,k) = G(k,r) and t h e r e f o r e M M A E XmG(M-r+l,m) = 4 Y \ G ( r , k ) .(3.142) m=l k=l m^M-r+1 kfr 102 Thus i t has been shown that equations (3.139) and (3.140) are identical, term for term, under the assumption of the symmetry conditions of equation (3.136). This allows a reduction in the number of equations to be solved. A computer program has been written for the solution of the linear equations of equation (3.132). The solution is a set of row-sum constraints on the q matrix which are used in a subsequent minimization of Py. The row-sum constraints are specified by the column vector, X^ , which is the solution to equation (3.132). An equation for P^, the minimized value of P^ , will now be derived in terms of the constraint and the Lagrange multiplier 6. Writing P^ as M n=l M 2Xn + 2 £ XmG(n,m) m=l m^ n " M £ *n n=l (3.143) and using equation (3.139), we obtain P^ as PAA= 2 £ V 9 PA(>>< / S-V/[£ *n n=l Ln=l (3.144) Substituting from equation (3.132) into the above equation yields M *AA = - M £ *n Ln=l = - 6/Q (3.145) This equation is useful for checking the program operation and is an efficient formula for evaluating P^ because 6 is obtained as part of the solution to the set of equations of (3.132). 3.6.2 Minimization of Py With Both Row-Sum and Pattern Constraints In the previous section i t was demonstrated that minimization of P^ was not sufficient to uniquely define the q matrix. Instead a set of row sums of the q matrix were determined. These row sums form the components of a column vector, XA> Since the q matrix is not uniquely defined by specifying a set of row sums i t i s p o s s i b l e to c a r r y out a subsequent m i n i m i z a t i o n of Py. T h i s m i n i m i z a t i o n i s sub j e c t to the c o n s t r a i n t of s p e c i f i e d row sums and the p a t t e r n f u n c t i o n c o n s t r a i n t s of equation (3.69). The row-sum c o n s t r a i n t equations are M R r a(q) - 0 - £ q m i - X m f o r 1 * m * M i = l (3.146) where X i s the m component of X . The p a t t e r n c o n s t r a i n t s are m A D. (q) = 0 f o r 0 - k - M - 1 (3.147) The e x p l i c i t form of the D^(q) equations i s given i n equation (3.69). The f u n c t i o n to be minimized i s M M-l U(q) = P v ( q ) + h £ 6 nR n(q) + h £ P kD k(q) n=l k=0 (3.148) where g n and p k are. Lagrange m u l t i p l i e r s . The f a c t o r s of % are used f o r convenience i n what f o l l o w s . • Since i t i s a c t u a l l y the sum, q r g + q s r , that has a p h y s i c a l e f f e c t on system performance, the equations can be formulated as (3U(q)/3q r s) + (3U(q)/3q s r) = 0 f o r 1 ^ r , s ^ M (3.149) In a d d i t i o n to equation (3.149), the c o n s t r a i n t equations must be s a t i s f i e d . The complete set of equations obtained by s u b s t i t u t i n g f o r . U(q) i n equation (3.149) and i n c l u d i n g the c o n s t r a i n t s i s ( 3 P v ( q ) / 3 q r s ) + ( 3 P y ( q ) / 3 q s r ) + h&r + % B S + P | r _ g j = 0 f o r 1 - r , s - M R^q) = 0 f o r 1 ^ n < M D, (q) = 0 f o r 0 ^ k - M - 1 >• (3.150) It will now be shown that the double symmetry conditions of equations (3.80) and (3.81) are satisfied by the q A matrix which is the solution to equation (3.150). The method used in section 3.5.3 will be adapted to show that (3Pv(q)/3qrs) + (3Pv(q)/3qsr) + %B r + h&a + P| s_ rJ = OPv(q)/3qM-r+l,M-s+l) + OPvW/^M-s+l.M-r+P + ^M-r+1 . + ^ M - s + l •+ P|M-r+l - (M-s+1) | (3.151) under the assumption of equations (3.80) and (3.81). In section 3.5.3 i t was shown that (3Pv(q)/3qrg) = (3P y(q)/3q M_ r + 1 y M_ s + 1) under the assumption of equations (3.80) and (3.81). It is obvious that P|r-s| = P|M-r+l - (M-s+1)I' T n u s> i t i s only necessary to show that B M_ r + 1 = 6r for 1 - r - M (3.152) This will be true i f the corresponding row-constraint equations are identical, i.e., i f V q ) ~ RM-n+l ( q ) for 1 - n - M (3.153) It will now be shown that equation (3.153) holds under the assumption of the double symmetry condition of equations (3.80) and (3.81). Consider the r row sum X r= t q r i (3.154) i=l From equation (3.81) q r i = ^M-r+l,M-i+l a n d t h u s 105 M X r " qM-r+l,M-i+l (3.155) Provided equation (3.81) holds, equations (3.154) and (3.155) are i d e n t i c a l , terra f o r term. A change i n the index v a r i a b l e i s now made i n equation (3.155). L e t t i n g i = M - k + 1, equation (3.155) becomes M x r - E 3M-r+l,k k=l (3.156) Therefore, X r = X M_ r +^ f o r 1 - r - M s a t i s f y i n g the requirement on the X^ vector. Also, R r(q) = % _ r - f i ( 3 ) f o r 1 ~ r ~ M * T n i s i s t n e c o n d i t i o n of equation (3.153) which i s necessary to show that the double symmetry c o n d i t i o n i s v a l i d . Therefore, the v a r i a b l e naming convention of equation (3.96) can be used to reduce the number of unknowns i n the set of l i n e a r equations i n (3.150). The formula f o r the minimized volume performance f u n c t i o n w i l l now be derived i n terms of the Lagrange m u l t i p l i e r s and the pattern c o n s t r a i n t s . The value of Py obtained by minimization subject to both row and pattern co n s t r a i n t s w i l l be designated P^y In s e c t i o n 3.5.4 i t was shown that the minimized value of Py i s given by CAV M M - E E 1=1 n=l ' n i 1 "M-l /4 Y D k .k=0 J (3.157) where equation (3.103) has been used. S u b s t i t u t i n g f o r (9Py(q)/3q i n) + (3Py(q)/3q n^) from equation (3.150) y i e l d s the r e s u l t -M-l M PAV = <•*) £ Pk Dk + E v n Lk=0 n=l -M-l 2 / E D k (3.158) .k=0 . This equation i s u s e f u l f o r computational purposes both as a programming check and as an e f f i c i e n t formula f o r evaluation of P^y. The Lagrange m u l t i p l i e r s necessary to evaluate equation (3.158) are obtained as p a r t of the s o l u t i o n to equation (3.150). 3.6.3 Formulation of Equations f o r Computer S o l u t i o n The developments of t h i s s e c t i o n p a r a l l e l s the development of s e c t i o n 3.5.4. " T h e r e f o r e , e x p l a n a t i o n of the approach w i l l be v e r y b r i e f . The performance f u n c t i o n , Py, i s to be minimized s u b j e c t to a s e t of p a t t e r n c o n s t r a i n t s and a'set of row c o n s t r a i n t s on the q m a t r i x . The a d d i t i o n of the row c o n s t r a i n t s c o n s t i t u t e the only d i f f e r e n c e between t h i s s e c t i o n and s e c t i o n 3.5.4. The set of equations to be solved i s J 2 K ( r , s ) ( 3 P v ( x ) / 9 x j ( r , s ) ) + % 6 r + h&8 + P | r - S | = -° f o r 1 * j - S o,e w i t h row c o n s t r a i n t s M Y x . ( n , i ) = X n f o r 1 ^ n ^ A i = l and p a t t e r n c o n s t r a i n t s M E X j ( r . r ) = D 0 r = l M-k 2 Y, x,(r,r+k) = D, f o r 1 - k * M - 1 r= l > (3.159) Th i s set of equations should be compared to equation (3.99). The x column v e c t o r i s cons t r u c t e d from the elements of the q m a t r i x as i n equation (3.96) w i t h the values of K ( r , s ) given i n equation (3.100). The number of row c o n s t r a i n t equations, A Q e , i s given by and A„ = *5(M - 1) f o r M odd A e = ^M - 1 f o r M even - (3.160) 107 Here, the double symmetry c o n d i t i o n has been used to reduce the number of ro w - c o n s t r a i n t equations from M to A Q e . The number of row c o n s t r a i n t s s p e c i f i e d i s one l e s s than the number of independent row sum equations. T h i s i s because the p a t t e r n c o n s t r a i n t s plus the f u l l s e t of row-sum c o n s t r a i n t equations are not independent. In m a t r i x form, the s e t of equations given i n equation (3.159) becomes C a Z a ~ ^a (3.161) where and x B = 0 X . D t t ° > e o,e 1-t ° ' e f M (3.162) (3.163) The column v e c t o r , B , has components which are the A Q £ Lagrange m u l t i p l i e r s corresponding to the A Q e r o w - c o n s t r a i n t equations. A l s o , p i s a column v e c t o r of Lagrange m u l t i p l i e r s corresponding to the M p a t t e r n c o n s t r a i n t s ; X_ c o n s i s t s of'the f i r s t A _ elements of the XA v e c t o r which minimizes P A; and D i s the column v e c t o r of F o u r i e r c o e f f i c i e n t s which s p e c i f y the array p a t t e r n . The m a t r i x C a i n equation (3.161) i s (3.164) A0,e+M So,e A o , e + M The sparseness of the C a matrix will be used to develop a computationally efficient substitution method as was done in section 3.5,4. Because the approach parallels that of section 3.5.4 almost exactly, the equations will be given with a minimum of explanation. The constraint equations F ax = D (3.165) are solved in terms of the SQ e - M - A Q^ e arbitrary variables. Denoting this set of arbitrary variables, ^ x k j ^ w i t n 1 - j - s 0 > e ~ M ~ Ao,e> t n e n as before in equation (3.111), the reduced set of equations to be solved is S (9P v(q)/8x k j) = (3Pv(q)/3xi)(dxi/dxk.) = 0 i = l for 1 * j * S 0 j e - M - A 0 j ( and F ax = D (3.166) The derivatives (dx^/dx^ ) are determined by finding the solution space of the homogeneous set of equations FaY = 0 (3.167) which has S 0 } 6 unknowns. Thus, the solution space has dimension S 0 j 6 - M - A o,e" The j*"*1 solution vector is Yj = Kj(dx/dxkj) (3.168) The constant K. will be set to L = 1 in what follows. The entire solution 3 J space is V a " • i o,e * "o,e (3.169) I f the c o n s t r a i n t equations are w r i t t e n out i n d e t a i l i t can be seen that F a can be p a r t i t i o n e d as F a " V , J a where k& i s a no n s i n g u l a r ((M + A Q e ) x (M + A 0 ? e ) ) m a t r i x , space can be shown to be The s o l u t i o n V a = A a J a s o , e - M - A o , e <- So,e- M- Ao,e~> f M t S 0 e-M-A_ . , e o, e The reduced set of equations now becomes (3.170) where and G a x = H a H a = 0 L 5 (3.171) (3.172) (3,173) The m a t r i x L„ i s given by K = A V a a a (3.174) Equation (3.171) i s the ma t r i x equation which must be solved to o b t a i n the elements of the q m a t r i x . The Lagrange m u l t i p l i e r s , 3 and p, are obtained by s u b s t i t u t i n g the x s o l u t i o n i n t o a subset of equation (3.159). A m a t r i x P a i s c o n s t r u c t e d from the f i r s t M + 1 rows of E (equation (3.164)) and then every 2(M - k ) t b row a f o r k running from 1 to A o,e 1. I f the submatrix, E a , i s w r i t t e n out i n d e t a i l , i t can be seen t h a t such a s e l e c t i o n r e s u l t s i n an independent set of equations f o r the Lagrange m u l t i p l i e r s . The corresponding rows of the A m a t r i x are s e l e c t e d to form a m a t r i x N a. Then, 3 N ax = - P £ (3.175) I f x i s the s o l u t i o n v e c t o r which i s s u b s t i t u t e d i n t o equation (3.175), the Lagrange m u l t i p l i e r s are obtained from (3.176) These Lagrange m u l t i p l i e r s are used to e v a l u a t e P^y from equation (3.158). T h i s s u b s t i t u t i o n method des c r i b e d here has been programmed to o b t a i n the numerical r e s u l t s of Chapter 4. 3.7 The M i n i m i z a t i o n of the Minimum D e t e c t a b l e S i g n a l The miiiimum d e t e c t a b l e s i g n a l i s d e f i n e d as the s m a l l e s t s i g n a l which w i l l g i v e ah output s i g n a l - t o - n o i s e r a t i o of a s p e c i f i e d s i z e . The s i z e of the minimum output s i g n a l - t o - n o i s e r a t i o necessary i s chosen by the system designer and depends on the p a r t i c u l a r a p p l i c a t i o n . For example, a minimum output s i g n a l - t o - n o i s e r a t i o of 3dB might be the minimum acc e p t a b l e output s i g n a l - t o - n o i s e r a t i o . Input s i g n a l s r e s u l t i n g i n an output s i g n a l - t o - n o i s e r a t i o of l e s s than 3dB would be considered u n d e t e c t a b l e . A system design s t r a t e g y based on m i n i m i z a t i o n of the minimum d e t e c t a b l e s i g n a l , removes the emphasis from the very s m a l l s i g n a l s which are undetectable on any account. In s e c t i o n s 3.5 and 3.6, methods were given f o r choosing the q m a t r i x to s a t i s f y s p e c i f i e d c r i t e r i a f o r system performance. The f i r s t method minimized Py with resulting matrix q^. This method is optimum for vanishingly small signals. The second method minimized P^ , and then Py, with minimum P A as a constraint. The q matrix obtained from this minimization is q A. The second method becomes approximately optimum for systems with signals of some intermediate size. A method of combining the matrices, q and q , in a linear combination, V A will be given. This linear combination, q = (qy + Lq^)/(1 + L) will be shown to approximately minimize the minimum detectable signal i f L is appropriately chosen. The problem of an exact minimization of the minimum detectable signal will be briefly considered. It will be shown that the exact approach leads to a set of nonlinear equations. The solution of these nonlinear equations does not appear necessary for the system study described in this thesis. However, the nonlinear equations are partially formulated to serve as a guide for further research. 3.7.1 Development of the Minimization Technique The starting point for the development is equation (3.12) which relates the output signal-to-noise ratio to the input signal-to-noise ratio. Let /y m denote the minimum acceptable output signal-to-noise ratio. Let ^ denote the minimum detectable input power signal-to-noise ratio. From equation (3.12), and y m are related as JfiPv(q) + 2xmPA(q) + x 2 (3.177) The first stage in the development of the method is to define a curve in Py,P^ space which is described by the equation F(Py,PA) = 0. This curve is defined so that, for any point (Pvj'^Ap l y l n 8 o n t n e curve, q is such that Py^ = Py(q) is a minimum subject to the constraint, 1*^(3) = P A j ' " Depending on the application, pattei"n constraints or any other auxiliary constraints may be applied to the minimization of Py(q) values lying on the curve. The method is independent of the auxiliary constraints applied, except that the same constraints must be applied for determination of a l l values lying on the curve. The minimization procedures of sections 3.5 and 3.6 provide a set of end points for the curve F(Py,PA) = 0. These end points are {VqV> = P W W " PVA} and • (P v(q A) = P A V, P ^ ) = P M> (3.178) The concept of the F(Py,PA) = 0 curve will be used to develop a method for interpolating between the two minimization procedures of sections 3.5 and 3.6. The equation for a straight line in (Py,PA) space which specifies the minimum detectable signal, x^ is obtained from equation (3.177). This equation is hVv + 2x mP A = x£(4bT - 1) (3.179) Equation (3.179) is plotted in Figure 3.3 together with a representation of the curve, F(Py,PA) = 0. The intersections of this line with the curve, F(Py,PA) = 0, specify the Py*^ v a l u e s f o r a particular set of x m, y m, and bT values. .In Figure 3.3 the straight line of equation (3.179) intersects the Py and P A axes at pV n = A ( ^ - " 1 ) x £ (3.180) ym and 113 (3.181) Figure 3.3 Graphical Illustration of Minimization of the Minimum Detectable Signal The intersection points on the axes, Py and ?AQ> s P e c i f y the minimum detectable signal. Py^ and P^  are minimized for given y m i f x m is made as small as possible. Conversely, x is minimized i f PT T and P. are made m v 0 A Q as small as possible. The smallest values of P«„ and PA and hence, of x , r v0 A Q ' m' which will allow a point in common for the F(Py,PA) = 0 curve and the straight line of equation (3.179), are obtained when the two curves are tangent provided F(Py,PA) is a continuous function. The point of tangency specifies the values Py(qm) = Py m and P^(qm) = P^ which minimize x m for given bT/ym. The objective of the following section is to develop an approximate method for determining q . 3.7.2 One Parameter Approximation f o r the F ( P V , P A ) = 0 Curve Exact determination of the curve, F ( P y , P A ) = 0, w i l l be shown, i n a l a t e r s e c t i o n of the t h e s i s , to r e q u i r e the s o l u t i o n of a s e t of n o n l i n e a r equations. An approximate method f o r determining t h i s curve w i l l be developed i n t h i s s e c t i o n . The approximation c o n s i s t s of r e q u i r i n g the row-sum v e c t o r , X, of the q m a t r i x to s a t i s f y the l i n e a r r e l a t i o n s h i p X = ( X A + K X V ) / ( 1 + K) (3.182) where X A and Xy are the row-sum v e c t o r s corresponding to the q A and qy m a t r i c e s . Equation (3.182) must be s a t i s f i e d f o r a l l p o i n t s Py(q) and P^(q) l y i n g on the curve, F ^ ( P y , P A ) =• 0, which i s an approximation to the t r u e curve, F ( P Y , P A ) = 0.• As shown i n equation (3.129), P A i s completely s p e c i f i e d by the X v e c t o r . Thus, the v a l u e of Py necessary to s p e c i f y a p o i n t on F (Py,P A) = 0 i s obtained by m i n i m i z i n g Py(q) s u b j e c t to a s e t of row c o n s t r a i n t s obtained from equation (3.182). I t w i l l be shown t h a t the r e s u l t a n t q m a t r i x a l s o s a t i s f i e s a l i n e a r r e l a t i o n s h i p , q = ( q A + Kqy)/(1 + K) 1 > (3.183) or q = ( q y + L q A ) / ( l + L) where L = 1/K. The m a t r i x equations f o r the m i n i m i z a t i o n of Py(q) w i t h both row and p a t t e r n c o n s t r a i n t s are obtained from equations (3.171) and (3.173). The x v e c t o r from which q i s obtained s a t i s f i e s 115 G x = a 0 x\ D (3.184) The X a v e c t o r i n equation (3.184) i s a c t u a l l y a trun c a t e d v e r s i o n of the X v e c t o r obtained by a p p l i c a t i o n of the symmetry c o n d i t i o n , X^  = X^_^^, as i n equation (3.136). The v e c t o r x i s obtained from equation (3.184) as [— —T A 0 x 0 X, D S o , e - M - A o , e o.e (3.185) A r M 4-Let j^Gaj 1 be p a r t i t i o n e d as " [ E l ! E 2 j£ 3 ] (3.186) where Ei i s a (S„ ~ x (S n „-M-A^ „)) m a t r i x , E 0 i s a (S„ 0 x A„ „) m a t r i x i , c w , c o,e /. O j c o,e and E-j i s a ( S 0 > e x M) m a t r i x . Then x = E 0X„ + EoD z a 3 (3.187) F u r t h e r , l e t *a = <*aA + K X a V ) / ( l + K) (3.188) where X^ and Xay are made up of the f i r s t A Q e components of X^  and Xy r e s p e c t i v e l y w i t h A given i n equation (3.160). I t was shown i n s e c t i o n o, e 3.6 that a reduced s e t of row c o n s t r a i n t s s p e c i f i e d by Xa could be used because of the double symmetry p r o p e r t i e s of the q m a t r i x . l i b Let X y and x A be the solution vectors to equation (3,187) from which the qy and q A matrices can be constructed, i.e., x v = E~A y + E->D J - (3.189) *A = £2*aA + E 3 6 -Substitute Aa from equation (3.188) into equation (3.187) to obtain x = 1 (ES . + E-D) + K ( E 9 X y + EoD) (3.190) T+K 1 a A J 1+K a V J Then, from equation (3.189), x = (x A + Kxv)/(1 + K) (3.191) The q matrix which can be constructed from the x vector of equation (3.191) is q = (q + Kqv)/(1 + K) L (3.192) or q = (qy + Lq A)/(l + L) where L = 1/K, which was to be shown. The approximate curve, F^(Py,PA) = 0, can be specified completely by using the relationship of equation (3.192). It is shown in Appendix D that • Py, P A values lying on the Fj/PyjPA) = ^ curve satisfy the parametric equations Py = Pyy(l + 2L + L 2P Ay/Pyy)/(l + L ) 2 (3.193) P A = P M ( L 2 + 2L + P V A/P M)/(1 + L ) 2 (3.194) with the end-point quantities, PAy, Pyy> PvA' aiK* *AA defined in equation (3.178). The above equations completely specify the curve, F^(Py,PA) = 0, which will be called the "L-parameter" curve. The derivative along the L-parameter curve is easily determined from 117 (dPv/dPA) = ( d p v / d L ) (dPA/dL) with (dPv/dL) = 2L(P A V - Py^/Cl + L ) 3 (3.195) and (dPA/dL) = 2 ( P M - P V A)/(1 + L ) 3 (3.196) Therefore, (dPv/dPA) = -2L(P A V - P W ) / ( P V A - P M ) (3.197) Now PAy - Pyy and Py A - P^ and hence (dPy/dPA) - 0 for L - 0 (3.198) Thus, the F^(Py,PA) = 0 curve is concave downward for a l l non-negative values of L. Moreover, (dPv/dPA) •+ 0 as L -»- 0 and (dPy/dPA) -> -«> as L -> 0 0 Therefore, the L-parameter curve and the exact F(Py,PA) = u curve have the same end points and the same derivatives at the end points.' The L-parameter approximation is thus most accurate in the regions of the end points of the curve. The yalue of L which minimizes the minimum detectable signal on the L-parameter curve is found as the solution to a quartic equation in L. From equation (3.177) and the expressions of equations (3.193) and (3.194), the minimum detectable signal can be written as a function of L as ^ 2(l-*W/y n) + 2xmPA(L) + I PV(L) - 0 (3.199) This equation is differentiated with respect to L and (dxm/dL) is set to 118 zero to o b t a i n ^ ( L ) = - ( l / 8 ) ( d P v / d L ) = _ ( 1 / 8 ) ( d P v / d L ) (dP A/dL) Thus, from equation (3.197), x ^ L ) = J«L(PAV - P V V ) / ( p V A - P A A } ( 3 ' 2 0 0 ) Equation (3.200) give s the v a l u e of x m i f L i s known. The v a l u e of L depends on the square of the s p e c i f i e d output s i g n a l - t o - n o i s e r a t i o , y m , and the time bandwidth product, bT. To o b t a i n L f o r s p e c i f i e d y m , equations (3.200), (3.194), and (3.193) are s u b s t i t u t e d i n t o equation (3.199) t o o b t a i n the q u a r t i c L V ( l - 4 b T / y m ) - + L 3 ( 2 a P A A + 2 a 2 ( l - 4 b T / y m ) ) + L 2 < 4 ( * M + 4 PAV + « 2(l-4bT/ y m)) + L ( 2 a P V A + | P w ) + \ P w - 0 ( 3 > 2 0 1 ) where a = ? s ( P A V - P W ) / ( P V A - P M ) (3.202) and X j n ( L ) = ctL. Equation (3.201) must be solved n u m e r i c a l l y u s i n g one of the many algorithms or l i b r a r y programs now a v a i l a b l e . The s m a l l e s t p o s i t i v e r o o t i s the d e s i r e d v a l u e of L. The q m a t r i x which approximately minimizes x^ i s then obtained from q = ( q v + L q A ) / ( l + L ) . The minimum d e t e c t a b l e s i g n a l i s then obtained as yi^ = <xL. 3.7.3 The Exact Method In t h i s s e c t i o n , a method w i l l be o u t l i n e d f o r e v a l u a t i n g a s e r i e s of p o i n t s on the F(Py,P A) = 0 curve. I t w i l l be shown th a t t h i s method r e q u i r e s the s o l u t i o n of a s e t of n o n l i n e a r a l g e b r a i c equations. The method i s based 119 on the second minimization procedure of section 3.6.3, which gives the mini-mum Py for a set of pattern constraints and a set of row constraints. It was shown in section 3.6.3 that Py can be expressed as a function of the pattern-constraint vector, D, and the row-constraint vector, A. The row-constraint vector, A, will then be allowed to vary and Py will be mini-mized as a function of A subject to'the pattern constraints and the constraint of a specified value for P . The solution is a A vector which minimizes Py for given P^ . The resulting q matrix obtained gives Py and P A for a point on the curve, F(Py,PA) =0. A series of points on this curve is generated by specifying a series of P A values and repeating the minimization procedure for each P A value. The constraint equations will now be formulated as a function of the A vector. From equation (3.129), P A can be expressed as a quadratic form P A = XtGjX (3.203) -M-l i 2 where Gj = 2G(n,m)/ k=0 Thus the constraint of specified P A value is expressed as Cr(A) = A ^ A - P A = 0 (3.204) The row-constraint equation is M M- l G2(A) = Y. \ i " T. Dk = 0 (3.205) n=l k=0 In equation (3.158) Py is expressed as a function of the pattern con-straints, the row-sum constraints and the Lagrange multipliers. This relationship can be written as 120 P y = -h p f c D + (3.206) 'M-l lk=0 where p and g are the column vectors of the pattern constraint and row-constraint Lagrange multipliers, respectively. It was shown in equation (3.176) that (3 and p could be expressed in terms of x which specifies the elements of the q matrix. The x solution vector is in turn dependent on the Aa vector and the D vector through either equation (3.171) or (3.187). Equation (3.187) expresses x in the form most useful to this approach. The X vector is made up of the first A components 3. O j 6 of X. The complete X vector is obtainable from the elements of Xa by applying the symmetry condition X^  = X^_ +^^ . The details of the manipulations will be omitted. If the above procedure is followed i t is possible to write the p and B vectors as p = F,X + F9D (3.207) and' B = FoX + F/D (3.208) The exact form of the F^, F 2, F^, and F^ matrices will not be given. If 6 and p of equations (3.207) and (3.208) are substituted into the expression for in equation (3.206) an expression of the following type is obtained: ?V(X) = -kCX'fjD + DtF2D + XCF3X + D CF 4X)/ M-l Lk=0 (3.209) From equation (3.209) one sees that the general form of Py(X) on the F(PV,PA) = 0 curve is P,7(X) = X^X + JCX + K t r (3.210) 121 where H i s a symmetric (M x M) matrix, J is an M-element column vector and K is a constant. The minimization of Py will be carried out for Py(X) given by equation (3.210) with constraints given by equations (3.204) and (3.205). The function to be minimized is u(x) = P v ( X ) + s c i ( x ) + r c 2 ( x ) (3.211) Minimization is carried out by requiring that (9U(X)/9Xr) = 0 for 1 - r - M. This leads to the set of equations (9Pv(X)/8Xr) + 6(3C1(X)/3Xr) + r(9C2(X)/9Ar) = 0 for 1 ^  r ^  M and Cx(X) = 0 C2(X) = Q The above equations may be represented in partial matrix form as y (3.212) 2(H + SG^ ) X + TU + J = 0 M n=l M-l E Dk k=0 y (3.213) X^A = P A where U is an M-element column vector with every element equal to one. The H matrix and J column vector occur in equation (3.210) for Py(X). The set of equations of (3.213) consists of M + 2 equations in M + 2 unknowns where M is the receiving array size. The unknowns are the M com-ponents of X and the two Lagrange multipliers, 6 and T. The equations are nonlinear because of the product <5X and the nonlinear constraint, P A = XtG^X. The dimension of this set of equations can be reduced by the application of the double symmetry condition, X^  = X^_ +^^ . This reduction would be very worthwhile in this case because of the nonlinear nature of the equations.' Many algorithms and library programs are available for the solution of sets of nonlinear equations. Often there is difficulty in obtaining a good starting point for these algorithms. The L-parameter approximation may be quite useful for obtaining the i n i t i a l t r i a l solution. As stated previously, i t is not intended to obtain a solution to the nonlinear set of equations. The L-parameter approximation is quite adequate for the system study of this thesis. However, formulation of the equations constitutes part of the contribution of this investigation and provides a starting point for future work. A similar but more complex set of equations can be formulated for mini-mizing the minimum detectable signal when the acceptable output signal-to-noise ratio, ^ym> is specified. The minimum detectable signal, x^, is expressed as a function of Py, P^ , ym., and bT as in equation (3.199), i.e., x*(l-4bT/y ) + 2 X P. (A) +xV^ = 0 (3.214) m m m A 4 v Here, x is minimized subject to the constraints of equations (3.204) and (3.205). Thus, O ^ / a x , . ) + 60c1(x)/axr) + r(ac 2(x)/3X R) for 1 - r - M and CL(X) c 2 ( x ) where 6 and T are Lagrange multipliers. The partial derivative, Sx^ /SA.^ ., is obtained by differentiating equation (3.214) and substituting into equation (3.215). The resulting set of equations is much more nonlinear than obtained for the F(Py,PA) = 0 curve. Generally, one would expect that the bT/ym ratio would not be well specified. Thus a solution for the entire F(Py,PA) = 0 curve appears more useful from a system study viewpoint. 0 0 0 (3.215) 123 3.8 Realization of a. Quadratic Processor with a Specified q Matrix This investigation has been primarily concerned with the numerical determination of the element values of the q matrix so that certain criteria of performance are satisfied. However, in this section one of the problems of implementation will be considered. One of the difficulties in realizing a real-time quadratic processor is the large number of multipliers required. At audio frequencies, hardware multipliers are feasible but i t is desirable to keep their numbers as small as possible. If processing is to be performed off-line on recorded data, the large number of multiplications is not such a problem. A method of realizing the q matrix with a reduced number of multipliers will be given. If the q matrix is (M x M), then M(M + l)/2 multipliers appear necessary in the formation of the quadratic form M M = E I W±* (3.216) i=l n=l and M(M + l)/2 weightings of the product x^x by the factor q^n- A method will be given for the realization of the q matrix which requires at most (M + 1) multiplications of the x^xn type and at most M(M + 3)/2 weighting multiplications. The method is best demonstrated by means of an example. Consider the (6 x 6) q matrix which is necessary to quadratically process a six element receiving array. q = Jqyj for 1 * i , j * 6 (3.217) The off-diagonal elements of this q matrix can be realized by a set of five multiplication operations. In general, for an M-element array, the off-diagonal elements are realizable by M - 1 multiplicative operations. 124 Realization of the complete matrix is accomplished by taking the M outputs of the M receiving elements after filtering, and feeding these in parallel to the M+1 multiplicative processors as illustrated in Figure 3.4a for M = 6. The double width arrows in Figure 3.4a indicate the transmission of the vector x1- = (x^ ,x2, • • *x ) where x^ is the filtered output of the i receiving, element. Blocks labelled 1, 2, 3, etc. are multiplicative processors denoted Type 1, Type 2, etc. Blocks labelled 01 and 02 are square-law processors, denoted Type 01 and Type 02, respectively. Consider the multiplicative processor of Type 1 which is illustrated in Figure 3.4b. This Figure illustrates the circuit layout, symbolic represent-ation and associated q^ matrix for the Type 1 multiplicative processor. Other processor types are illustrated in Figure 3.5 by symbolic represent-ations only. The circuit layouts for the various types can be inferred by analogy to the Type 1 processor. Type 01 and Type 02 processors are square-law-detected processors, as illustrated in Figure 3.6, and are necessary to realize the diagonal elements-of the q matrix. Figure 3.6 illustrates only the Type 01 processor. The Type 02 processor is obtained in the same manner as the Type 01, except that a different set of weights, p^, (i = 1 to 6), is used instead of the q^, (i = 1 to 6). On the basis of the various types of multiplicative processors illustrated in Figures 3.4, 3.5, and 3.6, a given q matrix can be realized as M-l q °° £ ^ i + 901 ~ . (3.218) i=l u z If the q matrix has zero diagonal elements, i t is realized by M -plicative processors as 1 multi-FIGURE 3.4a SYMBOUC REPRESENTATION OF THE REALIZATION OF AN ARBITRARY MATRIX m i n 12 13 14 15 16 CIRCUIT LAYOUT 0 a 0 0 0 0 ePA 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SYMBOLIC REPRESENTATION A l l | A I 2 A I3 A I 4 A I 5 A I 6 ^ 12^ 1 j *| 0 0 0 0 FIGURE 3.4b THE TYPE I MULTIPLICATIVE PROCESSOR AND ASSOCIATED SYMBOLISM SYMBOLIC REPRESENTATION OF MULTIPLICATIVE PROCESSORS A l l A 12 . A , 3 A , 4 A I 5 A , 6 x l 0 0 0 0 T Y P E I q,: MATRICES (i*|+o5) 1 i B 0 el2 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 o 0 0 0 0 o 0 0 0 0 0 o 0 A2I A 2 2 I A23 A 2 4 A 2 5 A2fc 2e l 5/x 2 0 O O T Y P E 2 0 0 ei3 0 0 0 0 0 0 0 0 e32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A3I A 3 2 A 3 S | A 3 4 A 3 5 A 3 b Z«W*3 2 a24/ X3 &34/ K3 I *3 0 0 T Y P E 3 /v 0 0 o e l 4 0 0 0 o 0 e 24 0 0 0 0 0 e 34 0 0 e 4. G42 e 43 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A 4 . A 4 2 A 4S A 4 4 I A 4 5 A 4 6 2 e i s / x 4 2 e 2 5 / x 4 & 4 5 / x 4 x 4 O T Y P E 4 0 0 o 0 e!5 o o 0 0 0 0 0 0 o 0 Ss 0 0 0 0 0 645 0 e5l e53 G54 o 0 0 0 0 o 0 0 A 5 I A 5 2 A 5 5 A 5 4 A 5 5 I A 5 f c 2 <W X5 2W KS 2 e3(,/* 5 T Y P E 5 0 o 0 0 0 SI6 0 0 0 0 0 0 0 0 0 0 e3& 0 o o 0 0 E46 0 0 0 0 0 % hi % e64 e65 O FIGURE 3.5 ILLUSTRATION OF MULTIPLIER TYPES AND ASSOCIATED MATRICES 01 % % %% %% \% \% %% H 5 FIGURE 3.b THE TYPE 01 MULTIPLICATIVE PROCESSOR M-l q = E q ± (3.219) i = l by choosing e.,-. = q.., where e.. i s a weight f a c t o r as shown i n F i g u r e 3.5. I f the q m a t r i x has non-negative d i a g o n a l elements, i t can be r e a l i z e d as M-l q" - £ q ± + q where CJQ^  i s s p e c i f i e d by q i =/q7I f o r 1 - 1 " M (3.220) and the q^ are s p e c i f i e d by e i j = l i j - l i l j <3'221> Thus, a q m a t r i x w i t h a l l non-negative or a l l n o n - p o s i t i v e d i a g o n a l elements can be r e a l i z e d by M m u l t i p l i c a t i v e p r o c e s s o r s . I f the d i a g o n a l elements are both p o s i t i v e and n e g a t i v e , the q m a t r i x i s - r e a l i z e d as M-l q = E q± + q Q 1 - q 0 2 ' (3.222) i = l The set of weights q^ and p^ are s e l e c t e d to r e a l i z e the d i a g o n a l elements, of t h i s q m a t r i x as q i : L = q 2 - p j f o r 1 - i - M (3.223) There are 2M unknowns and o n l y M equations i n (3.223) and thus t h e r e i s a l a r g e measure of a r b i t r a r i n e s s i n the s e l e c t i o n of q^ and p^. A method which reduces the number of weighting o p e r a t i o n s r e q u i r e d to a minimum i s to choose q^ and p^ so t h a t , i f q ^ - 0 then p^ = 0 and i f q ^ < 0 then q^ = 0. This means that M weighting o p e r a t i o n s i n t o t a l are r e q u i r e d i n the 129 formation of q g i and qg2* '^ne w e i 8 n t s V± a n < i q^ are P i = " V | q ± i l " q u < 0 p ± = 0 i f q.. - 0 1 1 (3.224) q i - " ^ q i i l f q u - 0 •I < . < q = 0 i f q i ± < 0 f o r 1 - i - M. Once the weights and q^ have been s e l e c t e d , the weights e^j are found to be 6 i j = q i j + p i p j " q i q J (3.225) Therefore, a q u a d r a t i c processor w i t h an (M x M) square q m a t r i x w i t h both p o s i t i v e and negative d i a g o n a l elements, i s r e a l i z e d by means of M + 1 m u l t i p l i e r s , not counting the m u l t i p l i c a t i o n s which are r e q u i r e d to produce weighting f a c t o r s . 130 4. NUMERICAL RESULTS FOR THE BACKGROUND NOISE LIMITED ENVIRONMENT 4.1 Introduction The numerical results of this chapter are obtained from computer programs based on the theoretical developments, of Chapter 3. The results are arranged to show how the system performance is affected by variations in the following parameters: (i) element spacing, (ii) array steering angle, ( i i i ) bandwidth, (iv) array size, (v) pattern type, (vi) beam width and side lobe level of pattern. The three pattern types to be. considered, a l l variations of the Chebyshev type, are (a) Chebyshev, (b) Squared Chebyshev, and (c) Chebyshev-Envelope. The characteristics of these pattern types were fully discussed in section 2.4. The method of generation of Fourier coefficients for these patterns is described in Appendix A. The performance of the quadratic processor is compared to the array correlator and the square-law-detected array. This comparison is made on the basis of the performance functions, P A and Py. Some of the performance function data is selected to illustrate the one-parameter approximation for the minimization of the minimum detectable signal. 4.2 Performance Functions and Signal-to-Noise Ratio The numerical results to be presented consist largely of graphs showing how the Py and P^ performance functions are affected by the parameter variations listed in the previous section. 1 3 1 The performance functions are r e l a t e d to the output s i g n a l - t o - n o i s e r a t i o by equation (3.12) which can be w r i t t e n as SNR = 2<bT) J s(o-2/ a§ T)/ kP v(q) + 2(a s/a2 T)P A(q) + ( a s / a § T ) 2 * ( 4 . 1 ) The performance functions, rather than the s i g n a l - t o - n o i s e r a t i o , are p l o t t e d . i n order to i s o l a t e the e f f e c t of the q matrix from the time band-width product, bT, and the input s i g n a l - t o - n o i s e r a t i o , c f / a 2 ^ . Also, the performance functions more c l e a r l y show the e f f e c t s of v a r i a t i o n s i n the parameters of s e c t i o n 4.1. However, equation (4.1) w i l l be used to p l o t SNR/(bT)^ versus input s i g n a l - t o - n o i s e r a t i o f o r selected Py and P A values. 4.3 Nomenclature In order to provide a convenient reference, a l i s t i n g of the various terms used i n t h i s chapter w i l l be .given as follows: p w = = the volume performance function obtained when Py i s minimized subject to pattern c o n s t r a i n t s only PVA = PA^V> = area performance f u n c t i o n obtained when q i s chosen so that pv = pvv PAV = PV<3A> = minimum Py obtained subject to pattern c o n s t r a i n t s and the con s t r a i n t P A = P ^ PAA = P A ( q A > = mxnimum area performance f u n c t i o n . Gy = processing gain against r e c e i v e r noise when q = qy GDBy = 1 0 log 1 0(Gy/M) G A = processing gain against r e c e i v e r noise when q = q A GDBA = 1 0 log 1 0(G A/M) Pgy = Py for a square-law-detected array P„. = P. for a square-law-detected array SA A 132 Gg = processing gain against receiver noise for square-law-detected array G D B S = 10 log10(Gs/M) P Q Y = P V for an array correlator P Q A = P^ for an array correlator GQ = processing gain against receiver noise for an array correlator . G D B C = 10 log 1 Q(G c/M) Vjpj = P v(q u) where q u is a "maximally uniform" q matrix P T T A = P.(q ) UA A Mu Gy = processing gain against receiver noise for q = q u GDBTJ = 10 log^CGxj/M) PNVV PNVA Performance functions corresponding to Pyy, ^vA' PAV» a n c* - respectively, when the narrow-band approximation for N^AV ° r A^ """S u s e c* "*"n a wlde-band system P NAA These terms may be redefined on the f i r s t occurrence when,'in the author's opinion, this appears necessary. 4.4 Element Spacing It has been known for some time that there is an optimum spacing for the array gain of broadside arrays. King 1 8 found that the optimum spacing was at about 0.9 wavelengths for a uniformly weighted array. T a i 1 5 found that the optimized array gain also had a maximum at about 0.9 wavelengths. A similar result was recently obtained by Hansen29 for a Dolph-Chebyshev array. It was shown in Chapter 3 that the performance functions, Py and P^ , are generalizations of the concept of array gain. This generalization, was developed to deal with the general quadratic processor where the array gain concept d i d not seem a p p l i c a b l e . I t w i l l be shown, by means of computed r e s u l t s , that these performance f u n c t i o n s a l s o l e a d to an optimum spacing f o r most p a t t e r n c o n s t r a i n t s . However, p a t t e r n c o n s t r a i n t s can be s e l e c t e d which r e s u l t i n no optimum spacing. The numerical r e s u l t s obtained are more e x t e n s i v e than i n the c i t e d r e f e r e n c e s . 1 5 ' 1 8 ' 2 9 P a t t e r n f u n c t i o n s are considered which have not been p r e v i o u s l y considered. The e f f e c t s of s i d e - l o b e l e v e l and beam widths on the optimum spacing are demonstrated. The l i m i t i n g case of an edge-type p a t t e r n i s used to i n f e r the e f f e c t s of v e r y l a r g e a r r a y s i z e s . R e s u l t s are l a t e r obtained f o r half-wavelength spaced a r r a y s w i t h a l a r g e number of elements. These r e s u l t s are used to strengthen the i n f e r e n c e s made on the b a s i s of edge-type p a t t e r n s . The e f f e c t of a r r a y s t e e r i n g angle and band-width on the p o s i t i o n of the optimum spacing i s demonstrated. 4.4.1 Narrow Band Systems The performance f u n c t i o n data of t h i s s e c t i o n were computed u s i n g a narrow band approximation. These performance f u n c t i o n s e x h i b i t an optimum spacing except f o r edge-type p a t t e r n c o n s t r a i n t s . The minimized area performance f u n c t i o n , P ^ , i s p l o t t e d i n F i g u r e 4.1 f o r three s t e e r i n g angles, broadside, e n d f i r e , and TT/4. The optimum spaci n g f o r P ^ i s approximately 0.9 wavelengths f o r the b r o a d s i d e , 0.6 wavelengths f o r the ir/4, and 0.44 wavelengths f o r the e n d f i r e s t e e r e d a r r a y . The v a l u e s of P ^ at the optimum spacing are almost i d e n t i c a l f o r the e n d f i r e and broadside a r r a y s but are about 50% h i g h e r f o r the TT/4 s t e e r e d a r r a y . The p o s i t i o n of the minimum of P ^ i s s h i f t e d w i t h s t e e r i n g angle so t h a t the optimum spacing i s s l i g h t l y l e s s than the spacing which b r i n g s the f i r s t g r a t i n g lobe of the p a t t e r n i n t o the range of r e a l angles. Since P i s i n v e r s e l y p r o p o r t i o n a l to a r r a y g a i n , G^, F i g u r e 4.1 i s i n d i c a t i v e of the behaviour of optimized a r r a y g a i n . The data of F i g u r e 4.1 P A A KAA .6 .5 .4 .S .2 .1 0 .7 .6 .5 .4 .3 .2 .1 0 .5 .4 .3 .2 .1 M»6 .25 .25 .25 .5 .75 • 5 _ > / 7 5 .5 , .75 d/x 1.0 l.O (a) BROADSIDE STEERED ARRAY M« NO. OF ELEMENTS 1.25 M=6 M-lO M-14 .._ — . (b) ARRAY STEERED I T A F R O M . B R O A D S I D E M > M O . O F E L E M E N T S 1.23 M=6 M^io M=I4 (c) ENDFIRE STEERED ARRAY M= NO. OF ELEMENTS l.O 1.25 FI6URE 4.1 AREA PERFORMANCE FUNCTIONS FOR NARROW BAND OPERATION are e q u i v a l e n t to the curves obtained by T a i . 1 5 As was shown i n s e c t i o n 3.6, ]>AA i s the only one of the performance f u n c t i o n s which i s completely indepen-dent of the p a t t e r n c o n s t r a i n t s . The e f f e c t s of a r r a y s i z e and s i d e lobe l e v e l are i l l u s t r a t e d i n F i g u r e s 4.2, 4.3, and 4.4. Performance f u n c t i o n s Pyy> PyA> PAV> a n d PAA c o r r e s P o n d i n g to m a t r i c e s , qy and q A , are p l o t t e d ' v e r s u s spacing i n wavelengths f o r broad-s i d e a r r a y s of M = 6, 10, and 14 elements, r e s p e c t i v e l y . Data f o r these curves are computed by u s i n g the m i n i m i z a t i o n procedures of s e c t i o n 3.5 and 3.6. The c o n s t r a i n t s f o r the m i n i m i z a t i o n are obtained by s p e c i f y i n g a Chebyshev p o s t - d e t e c t i o n p a t t e r n f o r the q u a d r a t i c processor. The curves of F i g u r e s 4.2, 4.3, and 4.4 each c o n s i s t of m u l t i p l e p l o t s f o r main l o b e / s i d e lobe r a t i o s of R =• 1.0, 5.0, 45.0, and 500.0. The R = 1.0 and R = 5.0 r a t i o s are u n r e a l i s t i c but are i n c l u d e d to i n d i c a t e l i m i t i n g behaviour. A r a t i o of R =1.0 corresponds to what w i l l be c a l l e d an "edge-type p a t t e r n " . This term comes from c o n v e n t i o n a l arrays where the weights or gains f o r a l l but the two end elements tend to zero. T h i s c o n d i t i o n i s a l s o approached i n very l a r g e Chebyshev array s where the main l o b e / s i d e lobe r a t i o , R, i s h e l d f i x e d at some acceptably l a r g e v a l u e and the a r r a y s i z e i s increased to get a narrow beam. For t h i s reason, the R = 1.0 and R = 5.0 data have been given to i n d i c a t e the type of behaviour one i s l i k e l y to get w i t h very l a r g e a r r a y s . The performance f u n c t i o n s f o r the edge-type p a t t e r n s are markedly d i f f e r e n t from non edge-type p a t t e r n s . These performance f u n c t i o n s do not have an optimum spacing except f o r P ^ which i s independent of the p a t t e r n c o n s t r a i n t s . I n s t e a d , a l l the performance f u n c t i o n s are r e l a t i v e l y constant as f u n c t i o n s of spacing u n t i l the r e g i o n of s m a l l spacings i s reached.. A' r a p i d drop i n the magnitude of these performance f u n c t i o n s occurs s t a r t i n g at a spacing of h a l f a wavelength. P l o t data were obtained f o r spacings down to 0.25 wavelengths f o r the s i x -element arra y but only to 0.375 wavelengths f o r the 14-element a r r a y . P l o t data f o r s m a l l e r spacings were not obtained because the l i n e a r equations became extremely i l l - c o n d i t i o n e d . T h i s i l l - c o n d i t i o n i n g a t s m a l l spacings i n d i c a t e s that the a r r a y i s approaching a super d i r e c t i v e c o n d i t i o n . The t r a n s i t i o n from an edge-type p a t t e r n to a r e g u l a r p a t t e r n i s i n d i -cated by the R = 5.0 curves. I t can be seen i n F i g u r e s 4.2, 4.3, and 4.4 that the l a r g e r the a r r a y s i z e , the l e s s complete i s the t r a n s i t i o n f o r the R = 5.0 curve. I t i s only f o r the edge-type and t r a n s i t i o n a l p a t t e r n f u n c t i o n s that the d i f f e r e n c e s , P - Pyy, and P ^ - P • are s i g n i f i c a n t . The non edge-type p a t t e r n s w i t h a w e l l d e f i n e d optimum spacing a t 0.9 wavelengths have -P and P ^ - P ^ d i f f e r e n c e s so s m a l l as to be b a r e l y d i s c e r n i b l e on the p l o t s . This means th a t the two m i n i m i z a t i o n procedures of s e c t i o n s 3.5 and 3.6 converge f o r non edge-type p a t t e r n s and no i n t e r p o l a t i o n between them i s -necessary. I t i s only f o r the edge-type p a t t e r n s that the techniques f o r m i n i m i z a t i o n of minimum d e t e c t a b l e s i g n a l of s e c t i o n 3.8 are r e q u i r e d . The magnitude of the performance f u n c t i o n s f o r non edge-type p a t t e r n c o n s t r a i n t s appear almost independent of the main l o b e / s i d e lobe r a t i o as i n d i c a t e d by the R =45.0 and R = 500.0 data. In Figure 4.5 are p l o t t e d the p r o c e s s i n g gains a g a i n s t r e c e i v e r n o i s e f o r a six-element a r r a y u s i n g the q^ and q^ m a t r i c e s which y i e l d e d the performance f u n c t i o n data of F i g u r e 4.2. The normalized p r o c e s s i n g g a i n s , corresponding to m a t r i c e s q^ and q^, are denoted GDB^ and GDB^ r e s p e c t i v e l y . The n o r m a l i z a t i o n i s w i t h respect to M, the a r r a y s i z e as i n s e c t i o n 2.5.2. Thus the maximum p o s s i b l e v a l u e f o r GDBy or GDB i s O.OdB. From F i g u r e 4.5, GDBy and GDBA are l a r g e l y f u n c t i o n s of the p a t t e r n tf> z 2 2.0 C J Z u-Z < s: cc o ti-cs: U i a z o B z 3 z < O u. or u l o. <s> z o z => u. o < s: o: o U . a in a 1.0 PA\/ P W L Pv/A PAA (a) T2&a)*R= i.o .25 d/A .75 1.0 .2 PAA PvA PM/ Cb) T26(Z) = -R-B.O .25 .5 d / x .75 1.0 Pv'A, PAA P A V Pw T26(Z)=R=45.0 << 3 F-4 o u N vO 3^ O CM z o 5 z ZD U-z cc ui 5 a. o .25 .75 I.O in z o o Z . o 5 s: cc o IL C£ l u a * V A ?AA PAV ^ (d) T26(Z)*R = 5oo.o .25 .S d A .75 ,.0 FIGURE 4 . 4 PERFORMANCE FUNCTIONS FOR A 14-ELEMENT BROADSIDE ARRAY FIGURE 4.5 EFFECTS OF ELEMENT SPACING ON QAIN AGAINST RECEIVER NOISE PROCESSING 141 constraints and almost constant for spacings in the range 1.0 - d/A - 0.5. For spacings less than 0.5 wavelengths, GDBy and GDB^  drop off sharply as the array tends towards a super directive condition. Thus, i f the q^ . and q^ matrices are determined for small spacings and used to implement the quad-ratic processor, an improvement in signal-to-background noise ratio is accompanied by a much larger degradation in signal-to-receiver noise ratio. This is a well known condition for super directive arrays. The effect of steering angle is illustrated in Figure 4.6. Plots are given for P^ and Pyy for steering angles of broadside, TT/4, TT/8 and endfire for a six-element array with a Chebyshev pattern. The effect of steering angle, 8g, is clearly indicated as a shift of the minimum of the performance functions so that the optimum spacing- is less than the minimum spacing where grating lobes first appear at d/A ~ 1/(1 + [cos 6Q|). The optimum spacings occur at 0.9, .63, 0.5, and 0.4 wavelengths respectively. It was also found that the region where the equations start to become ill-conditioned, indicating the onset of super directivity, is shifted with steering angle. This boundary was found to be at d/A = 1/2(1 + |cos 6Q|). The effect of optimum spacing has been demonstrated for the performance functions. However, the best measure of system performance is the signal-to-noise ratio. The signal-to-noise ratio is calculated on the basis of equation (4.1) and plotted in Figure 4.7 for half-wavelength and optimum spacing. The results apply to both the broadside and endfire arrays which have almost identical Py and P^  values at the optimum spacing. Both Py and P^ are independent of steering angle for half-wavelength spacing. The array chosen for the example is a 14-element array with a Chebyshev pattern and a main lobe/side lobe ratio of R = 500.0. It was found for this array that P^ y - Pyy and Py^ = * A A ' S° t* i a t ^ t doesn't matter whether q^ or q is used. F16URE 4.6 EFFECTS OF ARRAY STEERING ANGLE OM PERFORMANCE FUNCTIONS 143 S N R = [ 4 b T / C P v / 4 + 2 j T ) P A + frj/ar^ ) ) J 2 Cff*/crN2T ) O P T I M U M S P A C E D B R O A D S I D E A R R A Y P v = . 0 0 7 3 6 P A * . O 0 5 O S d / A • % H A L F - W A V E L E N & T H S P A C E D A R R A Y P A = P V A * . I 4 3 7 4 d/K => 1/2 O P T I M U M S P A C E D E N D F I R E A R R A Y P v » . 0 0 7 3 3 P A - . 0 0 7 0 2 d / X = 7 / l 6 I N P U T SIGNAL-TO-NOISE RAT IO 10 LOS ( e r | /<T^ T ) o!S • 6 0 - 5 0 - 4 0 - 3 0 - 2 0 -10 <Q o Q IO I cc z S •8 P A T T E R N F U N C T I O N P ( 2 T T C A , L O = T 2 & ( Z COSTICSU/X) W I T H T 2 F A ( Z ) « R = 5 O O . 0 A R R A Y S I Z E M = M 4 F I 6 U R E 4 . 7 C O M P A R I S O N O F H A L F - W A V E L E N 6 T H S P A C E D W I T H O P T I M U M S P A C E D Q U A D R A T I C A L L Y P R O C E S S E D A R R A Y S 1 4 4 The difference in output signal-to-noise ratio due to optimum spacing was found to be about 2.3dB for vanishingly small signals, tapering to zero for very large signals. 4 . 4 . 2 Wide Band Systems Wide band systems of the type investigated here are likely to occur only i f the quadratic processor is used for studying very low frequency signals. Such low frequency signals are encountered in passive sonar and geophysical studies. The performance functions, Pyy and P^> are plotted as functions of d/A in Figure 4.8, where d is the element spacing and X is the wavelength for the centre frequency, co. The computed results of Figure 4 . 8 are obtained for a six-element array with a Chebyshev pattern and a main lobe/side lobe ratio of 5 0 0 . 0 . The effect of increasing bandwidth on the results of Figure 4 . 8 is an increase in the values of the performance functions. This is due in part to the greater inter-element noise correlation because of the noise energy at the low end of the band. Also observed is a gradual elimination of the fir s t minimum and a shifting to the left of the position of this minimum with a progressive increase in bandwidth. Also considered is a type of narrow band approximation where q^ and are determined using the assumption of zero bandwidth. The performance functions obtained by operating at some non-zero bandwidth with these q matrices are denoted P^yy, ^j^y^' PNAV a n c* PNAA* ^ e subscript N denotes that the q matrix was determined using the narrow band approximation. In Figure 4 . 9 are plotted the minimized performance functions, Pyy and ]?AA' together with the narrow band approximations, P^ jyy and PJJAA> f° r a six-element broadside steered array. The results indicate that the narrow.band approximation is adequate for half bandwidth, b - 0.4OJ, and spacings d/X - 0 . 5 1 4 5 •7 .6 .5 P A A .4 .3 .2 .1 0 b=..5> b=.6 b=.4 b=.2 b=o BROADSIDE STEERED ARRAY WITH CHEBYSHEV PATTERN b= HALF BANDWIDTH/ CENTRE FREQUENCY X= WAVELENGTH AT CENTRE FREQUENCY d= ELEMENT SPACING R= 500.0 ARRAY SIZE Ki-6 (b) .25 d/x .75 l.O FIGURE 4.9 THE EFFECT OF BANDWIDTH ON PERFORMANCE FUNCTIONS 14 b 0) z o 5 z Z3 u-u) u z < s. o u. °? U l 0. ul J o > o z o p o z U . Ul o z < a o u. Ul a. < Ul < .7 .6 .5 .3 .2 .1 0 9 .8 .7 .6 .5 .4 .3 b».8 • PNVV b».4- *• \ \ \ \ \ \ \ \ PNVV b».6 b».4 PNVV b».2 Pvv r b*.2 .23 d/A .75 1-0 CcO pHAf* b '.B 1 \ P^  A b».& \ PHAA b»-4 ?AA b«.4 PNAA' b*-2 PAA b*.2 B R O A D S I D E S T E E R E D ARRAY V M t T H C H E B Y S H E V P A T T E R N b= H A L F B A N D W V D T H / C E N T R E F R E Q U E N C Y X- W A V E L E N G T H A T C E N T R E F R E Q U E N C Y d= ELEMENT SPACING R=5.0 ARRAY fciZE 14=6 Cb) .25 d/A .75 \.0 FIGURE 4.9 THE ACCURACY OF THE MARROW BAND APPROXIMATION 147 where OJ i s t h e c e n t r e f r e q u e n c y and X i s t h e w a v e l e n g t h a t t h a t f r e q u e n c y . The d a t a o f F i g u r e 4 .9 a r e computed f o r a Cheby shev p a t t e r n f u n c t i o n w i t h a m a i n l o b e / s i d e l o b e r a t i o o f R = 5 . 0 . T h i s v a l u e o f R i s u n r e a l i s t i c f o r a p r a c t i c a l a r r a y . Howeve r , R = 5.0 p r o v i d e s a good t e s t o f t h e n a r r o w band a p p r o x i m a t i o n b e c a u s e t h e p e r f o r m a n c e f u n c t i o n s become more s e n s i t i v e t o v a r i a t i o n s i n t h e q m a t r i x as t h e p a t t e r n c o n s t r a i n t s become more " e d g e " l i k e . 4 . 5 The M a x i m a l l y U n i f o r m q M a t r i x The m a x i m a l l y u n i f o r m q m a t r i x , q ^ , was d e f i n e d i n s e c t i o n 2 .5 t o have t h e p r o p e r t y t h a t i t s e l e m e n t s a r e as u n i f o r m i n s i z e as p o s s i b l e , s u b j e c t t o t h e p a t t e r n c o n s t r a i n t s . Such a m a t r i x was shown t o m a x i m i z e t h e s i g n a l -t o - r e c e i v e r n o i s e r a t i o i n s e c t i o n 2 . 5 . I t was shown i n C h a p t e r 3 , s e c t i o n 3 . 5 . 2 , t h a t q u m i n i m i z e s t h e vo lume p e r f o r m a n c e f u n c t i o n , P y , f o r h a l f -w a v e l e n g t h e l e m e n t s p a c i n g p r o v i d e d t h e n a r r o w - b a n d a p p r o x i m a t i o n i s v a l i d . I n t h i s s e c t i o n a c o m p a r i s o n w i l l be made b e t w e e n t h e p e r f o r m a n c e o f a q u a d r a t i c p r o c e s s o r w i t h m a t r i x q u and a p r o c e s s o r w i t h m a t r i x qy o v e r a r a n ge o f e l e m e n t s p a c i n g s . The p e r f o r m a n c e f u n c t i o n s f o r t h e m a x i m a l l y u n i f o r m q u a d r a t i c p r o c e s s o r a r e P y ( q u ) = P y y and ^ ( q u ) = P U A " T h e s e p e r f o r -mance f u n c t i o n s a r e t o be compared w i t h t h o s e w h i c h a r e o b t a i n e d w i t h m a t r i x qy u s i n g t h e n a r r o w band a p p r o x i m a t i o n , t h a t i s , w i t h Py (c jy ) = P and P A^v^ = P V A w n e r e P V V i s t * i e minimum v a l u e o f P y ( q ) s u b j e c t t o t h e p a t t e r n c o n s t r a i n t s . •Data a r e p l o t t e d f o r a t e n - e l e m e n t b r o a d s i d e - s t e e r e d a r r a y . Two p a t t e r n t y p e s a r e c o n s i d e r e d , t h e Cheby shev and t h e C h e b y s h e v - e n v e l o p e t y p e s , w i t h m a i n l o b e / s i d e l o b e r a t i o and m a i n l o b e / e n v e l o p e s i d e l o b e r a t i o o f 4 5 . 0 , r e s p e c t i v e l y . D a t a f o r t h e e n d f i r e and b r o a d s i d e - s t e e r e d a r r a y s a r e p l o t t e d i n F i g u r e s 4 .10 and 4 . 1 1 , r e s p e c t i v e l y . The r e s u l t s i n d i c a t e t h a t q y i e l d s p e r f o r m a n c e f u n c t i o n s v e r y c l o s e t o 148 •3 •z o h z -3 UJ .2 <J Z z cc o & .1 ui CL PUA PVA Puv .25 1 a o H N > Ui X <n CQ UI X u to F 3 O Ui m y c 11 > ^ a: CO ci h < d/A .3 tn z o b •z a .2 o T cc o u. or ui a PVA p v v y y .25 1 o o Z § S I ul Ou o z ,oo Ui \r I I to X 0 3 IO it II ui n 5> O Q. 5 d/A FIGURE 4.10 COMPARISON OF MAXIMALLY UNIFORM AND OPTIMUM QUADRATIC PROCESSORS FOR ENDFIRE STEERING .149 .4 to z o E z z> u Ui z < cc £ OS Ul a PUA Pw (a) .25 d / A .75 1.0 z Ci Ul \r % > Ul I > oq ui % I m o u fJ & f l» /-^ 3 0 t u 0. u m N •t 55 ii ^ > or or i- < CO to z o 5 2 IL Ui CJ Z < or o u. .oc ul a PAA puv A Pw "sag—^ y .25 d/A .75 1.0 Z or Ul 3 O O o u Ul tf> Ou o 3 ° ui N > ^ ui h I ti Ul Kj 3, in II Ui W X 5 or (0 oc Q. H < FIGURE 4.11 COMPARISON OF MAXIMALLY UNIFORM AND OPTIMUM QUADRATIC PROCESSORS FOR BROAD91DE STEERING 150 qy f o r e l e m e n t s p a c i n g s i n t h e r a n g e 1/2(1 + |cos 0 O | ) ^ d/X ^ 1/(1 + |cos 9 0 | ) where 0Q i s t h e a r r a y s t e e r i n g a n g l e . F o r s m a l l s p a c i n g s d/X < 1/2(1 + |cos 6 0|) t h e p e r f o r m a n c e f u n c t i o n c u r v e s f o r q u and qy d i v e r g e as t h e a r r a y becomes p r o g r e s s i v e l y more s u p e r d i r e c t i v e . I t was shown i n F i g u r e 4 .5 t h a t , a s t h e s p a c i n g becomes s m a l l , t h e p r o c e s s i n g g a i n a g a i n s t r e c e i v e r n o i s e f o r qy d r o p s o f f r a p i d l y . I n c o n t r a s t t h e p r o c e s s i n g g a i n a g a i n s t r e c e i v e r n o i s e f o r q u i s a maximum and i n d e p e n -d e n t o f s p a c i n g . The se r e s u l t s i n d i c a t e t h a t i f a q u a d r a t i c p r o c e s s o r i s t o be i m p l e m e n t e d , s e r i o u s c o n s i d e r a t i o n s h o u l d be g i v e n t o u s i n g q u u n l e s s t h e e l e m e n t s p a c i n g s a r e s m a l l and t h e d e s i g n e r w i s h e s t o a c h i e v e some measu re o f s u p e r d i r e c t i v i t y . 4 . 6 The Q u a d r a t i c P r o c e s s o r Compared t o C o n v e n t i o n a l A r r a y P r o c e s s o r s The two c o n v e n t i o n a l p r o c e s s o r s t o be u s e d as s t a n d a r d s o f c o m p a r i s o n a r e t h e s q u a r e - l a w - d e t e c t e d a r r a y and t h e a r r a y c o r r e l a t o r . The p o s t - d e t e c -t i o n p a t t e r n s assumed f o r t h e s e c o n v e n t i o n a l p r o c e s s o r s a r e - t h e s q u a r e d -Cheby shev p a t t e r n f o r t h e s q u a r e - l a w - d e t e c t e d a r r a y and t h e C h e b y s h e v - e n v e l o p e p a t t e r n f o r t h e a r r a y c o r r e l a t o r . •The q u a d r a t i c p r o c e s s o r i s assumed t o have t h e same p a t t e r n as t h e c o n v e n t i o n a l a r r a y w i t h w h i c h i t i s b e i n g compared i n t h e f i r s t t y p e o f c o m p a r i s o n . A s e c o n d t y p e o f c o m p a r i s o n w h e r e t h e q u a d r a t i c p r o c e s s o r ha s a d i f f e r e n t p a t t e r n w i l l be d i s c u s s e d w i t h r e f e r e n c e t o t h e s i g n a l - t o - n o i s e r a t i o d a t a f o r l a r g e a r r a y s g i v e n i n s e c t i o n 2 . 5 . 2 ( s e e F i g u r e s 2 . 1 4 , .2 .15 , and 2 . 1 6 ) . 151 Data are p l o t t e d f o r a 14-element a r r a y w i t h both e n d f i r e and broadside s t e e r i n g . The 14-element array i s the l a r g e s t a r r a y f o r which a complete set of performance f u n c t i o n data was obtained f o r a wide range of element spacings. Data f o r the half-wavelength spaced a r r a y s were obtained f o r a r r a y s w i t h up to 99 elements. 4.6.1 Comparison w i t h the Square-Law-Detected Array The volume and area performance f u n c t i o n s f o r the square-law-detected a r r a y are denoted Pgy and Pg^, r e s p e c t i v e l y . The comparison i s c a r r i e d out by p l o t t i n g the narrow band v e r s i o n of Pgy, PgA' *Vv a n ^ PAA "*"n f i g u r e 4.12 f o r the e n d f i r e - s t e e r e d a r r a y and i n F i g u r e 4.13 f o r the b r o a d s i d e - s t e e r e d a r r a y . The number of ar r a y elements i s M = 14. In s e c t i o n 4.4.1 i t was shown th a t Pyy = P^y and P ^ - Py^ except f o r edge-type p a t t e r n s . Thus, i n order to avoid a c l u t t e r e d diagram, o n l y Pyy and P.. have been p l o t t e d . A l s o , PT„, and P are the lowest v a l u e s of Vv and AA W AA v P^ and thus represent the best performance a c h i e v a b l e w i t h the q u a d r a t i c processor. The p a t t e r n f u n c t i o n i s squared-Chebyshev f o r both the q u a d r a t i c p r o c e s s o r and the square-law-detected a r r a y . The three s i d e lobe l e v e l s s e l e c t e d are R = 5.0, R = 15.0, and R = 45.0. The s i d e lobe l e v e l s of R = 5.0 and R = 15.0 are u n r e a l i s t i c a l l y low but are used to represent the t r a n s i t i o n between edge-type and non edge-type p a t t e r n s . Thus r e s u l t s f o r R = 5.0 and R = 15.0 are r e p r e s e n t a t i v e of the k i n d of behaviour to be expected i n l a r g e a r r a y s . The curves f o r R = 45.0 are r e p r e s e n t a t i v e of the non edge-type p a t t e r n s . I t was found that the performance f u n c t i o n data f o r R = 500.0 (not given) f a l l almost on top of the data f o r R = 45.0. The data of F i g u r e s 4.12 and 4.13 i n d i c a t e that the q u a d r a t i c p r o c e s s o r , w h i l e always s u p e r i o r to the square-law-detected a r r a y , shows a worthwhile improvement only i f the p a t t e r n has an edge-type c h a r a c t e r i s t i c ID .3 z o O z .2 z oc 1 o > 1 LL Q. Ul Cu 0 .3 (D Z o f-o z .2 U. o z < oc o u* Cxf UJ ( X (ft z o i— o z => ui o z' < u. or Ul a .2 ... T i 1 i | y j ! i PAA / Psv Py/V .25 d/x .25 dA .5 .25 d/A 152 (a) T,| (Z) = R= 5 0 i [ %A • i • t i J Pvv i i r i Cb> T ( | ( Z ) = R = I 5 0 P A A / ( S ! f V Psv Cc) T , | ( Z ) » R > 4 & 0 1 u M MS u o a z o O Z Z OC Ul Ou FIGURE 4.12 COMPARISON. OF A SQUARt-LAVJ-DETECTED A R R A Y WITH A QUADRATICALLY PROCESSED E N D F I R E - STEERED A R R A Y N a or < FIGURE 4.13 COMPARISON OF THE SQUARE-LAW-DETECTED ARRAY WITH A QUADRATICALLY PROCESSED BROADSIDE-STEERED ARRAY From the data of F i g u r e 4.12 f o r the e n d f i r e a r r a y , the maximum PgV^vV r a t i o i s approximately 3.0, which would y i e l d a maximum d i f f e r e n c e of 2.4dB i n the sm a l l s i g n a l s i g n a l - t o - n o i s e r a t i o computed from equation (4.1). The improvement i s l e s s f o r a broadside a r r a y where the maximum Pgy/Pyy r a t i o was about 2.0 which y i e l d s a d i f f e r e n c e of 1.5dB i n the s m a l l s i g n a l s i g n a l -t o - n o i s e r a t i o . 4.6.2 Comparison w i t h the Array C o r r e l a t o r The volume and area performance f u n c t i o n s f o r the a r r a y c o r r e l a t o r are denoted P^y and P ^ , r e s p e c t i v e l y . The ar r a y c o r r e l a t o r i s compared w i t h the qu a d r a t i c processor by means of p l o t s of the performance f u n c t i o n s PQV> ^ C A . ' Pyy and P^. The narrow band approximation i s used i n the computations. Data f o r 14-^element e n d f i r e and broadside s t e e r e d a r r a y s are p l o t t e d i n Fi g u r e s 4.14 and 4.15, r e s p e c t i v e l y . A Chebyshev-envelope p a t t e r n i s used f o r both the q u a d r a t i c processor and the ar r a y c o r r e l a t o r . The s e l e c t e d . envelope s i d e lobe l e v e l s are = 5.0, 15.0, and 45.0. The q u a d r a t i c processor compares much more f a v o u r a b l y w i t h the ar r a y c o r r e l a t o r than w i t h the square-law-detected a r r a y . For Rg = 15.0 and an e n d f i r e a r r a y , the maximum P^y/Pyy r a t i o , f o r s p a c i n g d/X ^ 0 . 2 5 , i s approx-i m a t e l y 7.0.which corresponds to a d i f f e r e n c e i n s m a l l s i g n a l s i g n a l - t o - n o i s e r a t i o of 4.2dB. For Rg = 5.0, the corresponding maximum Pcv^vV r a t*° approximately 20.0, f o r a d i f f e r e n c e i n sm a l l s i g n a l s i g n a l - t o - n o i s e r a t i o of about 5.7dB. The corresponding improvements f o r the broadside a r r a y are not q u i t e so l a r g e . Performance f u n c t i o n s f o r p a t t e r n s w i t h low envelope s i d e l o b e s , f o r example R^ , = 45.0, are l i t t l e d i f f e r e n t f o r e i t h e r the a r r a y c o r r e l a t o r or the q u a d r a t i c processor. P r o p e r t i e s of the Chebyshev-envelope p a t t e r n were di s c u s s e d i n s e c t i o n 2.4.2. Th i s p a t t e r n f u n c t i o n has the p r o p e r t y of y i e l d i n g a narrow beamwidth p a t t e r n w i t h a l a r g e negative f i r s t lobe. The performance f u n c t i o n s are such F I G U R E 4 . 1 4 COMPARISON OF T H E A R R A Y CORRELATOR WITH A -QUADRATICALLY PROCESSED ENDF IRE - STEERED ARRAY 156 FIGURE 4.15 COMPARISON OF THE ARRAY CORRELATOR WITH A QUADRATICALLY PROCESSED BROADSIDE-STEERED ARRAY 157 INPUT SIGNAL-TO-NOISE RATIO 10 LOG (0"|/o T^) dB P A T T E R N F U N C T I O N P(2T T C / X , U0 = T , E (Z cos -ndu/x) co3 (\4--nduA) W I T H T , 2 C Z ) » I 5 . 0 E N D F I R E — S T E E R E D A R R A Y , S I Z E M = 1 4 E L E M E N T S P A C I N G d/ A =T/l6 F I G U R E 4 . 1 6 S I G N A L - T O - N O I S E R A T I O S F O R T H E A R R A Y C O R R E L A T O R A N D T H E Q U A D R A T I C P R O C E S S O R that good signal-to-noise ratios are obtained for narrow beamwidths provided the negative lobe can be tolerated. Signal-to-noise ratios are computed on the basis of equation (4.1). Figure 4.16 contains a plot of output signal-to-noise ratio as a function of input signal-to-noise ratio for the array correlator and the quadratic processor for a selected set of performance function data. The quadratic processor shows an improvement of about 4.2dB over the array correlator for small input signal-to-noise ratios. The amount of the improvement tapers to zero as the input signal-to-noise ratio becomes very large. 4.6.3 Performance Comparisons for Large Arrays The results of this chapter have so far indicated that the quadratic processor shows to best advantage when used with very large arrays. This is subject to the proviso that the large array size is used principally to achieve narrow beamwidth, thus resulting in post-detection patterns with edge-type characteristics. The results of section 2.5.2 for processing gain against receiver noise, Gj^-, were obtained for moderately large arrays with up to 99 elements. These results will form the basis for the arguments presented in this section. The relationship between the minimum volume performance function, Pyy, and the maximum processing gain against receiver noise, G^, was established in equation (3.76) for an array with half-wavelength spacing and narrow band operation. This relationship is PW,^ = 4 / GRN (4-2> Thus the curves of Figures 2.14, 2.15, and 2.16 for can be used together with a knowledge of the general shape of the performance function curves to strengthen the projections made for large arrays on the basis of edge-type patterns. It was shown in sections 2.5.2 and 3 . 5 . 2 that the maximally uniform q matrix, q u, maximizes and minimizes Pyy i ^ . The results of section 4 . 5 indicate that Py(qv) is approximated closely by Py(3u) over the range of spacings 1 / 2 ( 1 + |cos 9 0 | ) ^ d/X ^ 1 / ( 1 + |cos 0 O | ) ( 4 . 3 ) The signal-to-receiver noise ratio is proportional to G^ as shown in equation (2.94). From equation (4 . 1 ) i t can be seen that the small signal signal-to-background noise ratio is proportional to 1//Py. Therefore, for half wavelength spacing, the signal-to-background noise ratio is proportional to the signal-to-receiver ;noise ratio. It follows that the curves of Figures 2 . 1 4 , 2 . 1 5 , and 2 . 1 6 , for Gj^, provide an exact comparison, on the basis of signal-to-background noise ratio, between the quadratic processor, the square-law-detected array and the array correlator for half-wavelength spacing. The relative performance indicated by this data is strongly indicative of the relative performance of the quadratic processor for spacings lying in the range of equation (4.3). The discussion of results given in section 2.5.2 regarding the signal-to-receiver noise ratio is applicable to the signal-to-background noise ratio. Some of the pertinent results of section 2.5.2 are that the quadratic processor shows maximum improvements over some conventional array processors by 'amounts of (i) 1.8dB over a 99-element square-law-detected array when both processors have the squared Chebyshev pattern, (ii) 3.0dB over a 99-element square-law-detected array when the quadratic processor has an optimum Chebyshev pattern, ( i i i ) 5.0dB over a 98-element array correlator when both processors have a Chebyshev-envelope pattern. 1UU 4.7 The M i n i m i z a t i o n of Minimum Detectable S i g n a l A theory f o r the m i n i m i z a t i o n of the minimum d e t e c t a b l e s i g n a l was developed i n s e c t i o n 3.7. There, i t was shown that m i n i m i z a t i o n of the minimum d e t e c t a b l e s i g n a l r e q u i r e s an i n t e r p o l a t i o n between the two m i n i -m i z a t i o n procedures de s c r i b e d i n s e c t i o n s 3.5 and 3.6. These two m i n i m i -z a t i o n methods w i l l be reviewed f o r the purpose of t h i s d i s c u s s i o n . The f i r s t method ( s e c t i o n 3.5) minimizes Py(q) s u b j e c t to a s e t of p a t t e r n c o n s t r a i n t s . A m a t r i x qy i s found such t h a t Py(q^) = Pyy i s a minimum. The r e s u l t i n g area performance f u n c t i o n i s P^(qy) = PyA> which i s not the minimum of P^(<i) • T h i s procedure has the e f f e c t of maximizing the s i g n a l - t o - n o i s e r a t i o f o r ; v a n i s h i n g l y s m a l l s i g n a l s . The second method ( s e c t i o n 3.6) i s a two step procedure where P ^ C Q ) I S f i r s t minimized to get P^(q) = ^ A A ' T h i s s p e c i f i e s a s e t of row c o n s t r a i n t s on q which are used together w i t h the p a t t e r n c o n s t r a i n t s i n a subsequent m i n i m i z a t i o n of Py(q). The s o l u t i o n m a t r i x f o r t h i s procedure i s q^ w i t h W = P A V a n d P A « A > " P A A ' The b a s i s of the method f o r the m i n i m i z a t i o n of the minimum d e t e c t a b l e s i g n a l i s the determination of the curve, F(Py,P^) = 0, on which Py i s a minimum f o r . s p e c i f i e d P^. I t was shown th a t the ( P ^ J P A ) p a i r which minimize the minimum d e t e c t a b l e s i g n a l l i e on t h i s curve. The end p o i n t s of the curve are (Py = P Ay, P^ = P ^ ) and (Py = Pyy, P^ = P y A ) • The n u m e r i c a l r e s u l t s of t h i s chapter i n d i c a t e that these end p o i n t s are so c l o s e together t h a t i t i s not worthwhile to i n t e r p o l a t e between qy and q^. The l a r g e s t spread i n the end p o i n t s occurs f o r edge-type p a t t e r n s . F i g u r e 4.17 c o n t a i n s p l o t s of the estimated F(Py,P^) = 0 curve obtained by u s i n g the one parameter approximation of s e c t i o n 3.7.2. The data f o r F i g u r e 4.17 are obtained f o r a 14-element b r o a d s i d e - s t e e r e d a r r a y w i t h a spacing of 7/8 wavelengths and a Chebyshev p a t t e r n w i t h main l o b e / P A V F L ( P V T P AV-0 — pvv ! i j i i • P A A j 0 - Z . - 4 6 . 8 1.0 1.2 1.4 1. P A <"c0 T26(Z)M.O . 0 8 .06 . 0 4 .02 P A V F L ( P V L P A > 0 "i i ! j | ; i ; P A A l P V A 0 . 0 2 . 0 4 .Ob . 0 6 T 2 6 ( Z ) = 5 . 0 . 1 2 . 1 4 . 1 6 3 t « o II SI /~- r 3 . Z a z o o V u. ui Z a: 5 P A F I G U R E 4 . 1 7 E S T I M A T E D F ( P V , P A ; = 0 C U R V E F O R M I N I M I Z A T I O N O F T H E M I N I M U M D E T E C T A B L E S I G N A L s i d e lobe r a t i o s of R.= 1.0 and R = 5.0. Values of R = 1.0 and R = 5.0 correspond to edge-type and t r a n s i t i o n - t y p e p a t t e r n s , r e s p e c t i v e l y . Non edge-type p a t t e r n s y i e l d e d a n e g l i g i b l e spread between the end p o i n t s of the F ( P V , P A ) = 0 curve. The estimated F(Py,P^) = 0 curves of F i g u r e 4.17 show a f a i r l y s m a l l v a r i a t i o n i n values of Py compared w i t h the v a r i a t i o n i n P^. Because the output s i g n a l - t o - h o i s e r a t i o of equation (4.1) i s most s t r o n g l y i n f l u e n c e d by the Py v a l u e s , i t f o l l o w s that there i s l i t t l e v a r i a t i o n i n output s i g n a l -t o - n o i s e r a t i o f o r the (Py,P ) p o i n t s l y i n g on the F(Py,P^) = 0 curves. Thus i t appears that i n t e r p o l a t i o n between qy and q^ i s not necessary f o r systems of the type considered i n the t h e s i s where ( i ) knowledge of the s p a t i a l d i s t r i b u t i o n of background n o i s e i s minimal and the p o i n t n o i s e source model of i s o t r o p i c n o i s e i s used, ( i i ) the r e c e i v i n g elements are o m n i d i r e c t i o n a l , ( i i i ) the array elements are e q u a l l y spaced i n s t r a i g h t l i n e s . I f the equations f o r the performance f u n c t i o n s are formulated to i n c l u d e element p a t t e r n s and/or d i f f e r e n t a r r a y geometries and/or a d i f f e r e n t model f o r background n o i s e r e p r e s e n t i n g a g r e a t e r knowledge of the n o i s e s p a t i a l d i s t r i b u t i o n , such an i n t e r p o l a t i o n may w e l l be u s e f u l and necessary. 5. SUMMARY AND CONCLUSIONS The general quadratic processor for a linear array has been analyzed and compared to conventional processors from the point of view of pattern synthesis and signal-to-noise ratio for both a background noise and a receiver noise limited environment. It has been shown that the quadratic processor has pattern synthesis capabilities not shared by conventional array processors. The optimum pattern has been derived for the quadratic processor and shown to have side lobes 3.0dB lower than the pattern of the Dolph-Chebyshev array. 2 1 A model for the background noise has been developed. The model is based on a system of independent, equal strength, radiating sources with positions that are assumed to be independent random variables. This model for the background noise has been'shown to represent the case where there is maximum uncertainty about the directive properties of the background noise. The signal-to-backgrcund noise ratio derived for this model of background noise has been shown to depend on the q matrix through performance functions Py(q) and P^(q). It has been shown that the performance functions, Py(q) and P^(q) become inversely proportional to array directive gain when the quad-ratic processor is specialized to become the square-law-detected-array. Thus, for the background noise model, the signal-to-background noise ratio is maximized by maximizing the array gain. The performance functions, P (q) and P«(q) have been shown to serve as V A generalizations of the concept of array gain. They are useful for evaluation of the general quadratic processor where i t is not possible to define the conventional array gain as a function of the q matrix. It has been shown, for the quadratic processor, that specification of the pattern function does not completely specify the performance functions, Pv(q) and P^(^)» which may be reduced according to some criteria of optimality 164 subject to the constraints of specified pattern function. In contrast the specification of the pattern function determines the array gain for a con-ventional array. Two methods of reducing the output noise have been proposed and evaluated numerically. The first method is a minimization of Py(q) subject to pattern constraints with solution matrix, q^. The second method is a two step pro-cedure where Py(q) is minimized to find a set of row constraints for the q matrix. Py(<?) is subsequently minimized subject to these row and pattern constraints to obtain the solution matrix, q^. An investigation has been made of methods for interpolating between q^ and q^ so as to minimize the minimum detectable signal. The numerical results obtained indicate that in many instances, such an interpolation is unnecessary. These numerical results were obtained for linear arrays of omnidirectional receiving elements with isotropic background noise. Thus i t may be necessary to perform an interpolation between q^ and qy i f other than linear array geometries are considered and/or i f element directivity is taken into account and/or i f knowledge is available about the spatial distribution of anisotropic background noise. The two minimization methods provide a theoretical found-ation for further investigations. Numerical results were obtained showing the effects of element spacing, array size and steering angle, bandwidth, pattern type, and array beamwidth and side lobe height on the performance functions of the quadratic processor. The performance functions are shown to lead to an optimum array element spacing similar to the optimum spacings for array gain found by Tai.* 5 Comparisons were made of the quadratic processor and conventional array processors on the basis of signal-to-background noise ratio and signal-to-receiver noise ratio for element spacings in the range 1/2(1 + |cos 9 0 | ) ^ d/X ^  1/(1 + |cos 80|) 165 The c o n v e n t i o n a l processors which were the standards, of comparison are the square-law-detected array and the a r r a y c o r r e l a t o r . The numerical r e s u l t s of Chapter 4 and s e c t i o n 2.5.2 i n Chapter 2, i n d i c a t e that improvements o b t a i n a b l e by u s i n g a q u a d r a t i c processor aire dependent on the type of p a t t e r n f u n c t i o n and the array s i z e . R e s u l t s were obtained u s i n g the general computer' program f o r a r r a y s i z e s of up to 14 elements. For a 14-element a r r a y , the q u a d r a t i c processor showed l i t t l e improvement i n signal-to-background n o i s e r a t i o when compared w i t h the square-law-detected a r r a y when both processors had a squared-Chebyshev p a t t e r n . The q u a d r a t i c processor compares more f a v o u r a b l y w i t h the a r r a y c o r r e l a t o r . About a 4.0dB improvement.in signal-to-background n o i s e r a t i o was obtained f o r a 14-element q u a d r a t i c a l l y processed array as compared to an a r r a y c o r -r e l a t o r when both had a Chebyshev-envelope p a t t e r n w i t h main lobe/envelope s i d e lobe r a t i o of 15.0. Large arra y s were s t u d i e d only f o r half-wavelength spacing and only Py(q) was minimized. I t was shown that the q m a t r i x which minimizes Py(q) f o r a half-wavelength spaced narrow band a r r a y a l s o maximizes the s i g n a l - t o -r e c e i v e r noise r a t i o . Numerical r e s u l t s were obtained f o r a r r a y s of a max-imum s i z e of 99 elements i n the case of the Chebyshev and squared-Chebyshev p a t t e r n s and 98 elements i n the case of the Chebyshev-envelope p a t t e r n . Maximum improvements i n s i g n a l - t o - n o i s e r a t i o f o r the q u a d r a t i c processor were 3.0dB over the square-law-detected a r r a y and 5.0dB over the a r r a y c o r r e l a t o r . These gains occurred f o r narrow beamwidths and decreased as the beamwidth of the p a t t e r n was i n c r e a s e d . The h a l f x^avelength s o l u t i o n f o r the q m a t r i x t h a t minimized Py(q) has been c a l l e d the "maximally uniform" m a t r i x , q u , because i t s elements are as uniform i n s i z e as i s p o s s i b l e w h i l e s t i l l s a t i s f y i n g the p a t t e r n c o n s t r a i n t s . Such a q ma t r i x was found to provide performance a g a i n s t background n o i s e 166 very nearly equal to the optimized system for element spacings in the range 1/2(1 + |cos 60|) ^ d/X and to provide optimum performance against receiver noise. Once the Fourier coefficients of the pattern are known, the elements of the maximally uniform q matrix can be written down immediately. Thus q u is important for large arrays where i t would not be feasible to solve the linear equations required to determine q. and q for an arbitrary spacing, steering angle and bandwidth. 167 APPENDIX A GENERATION OF A SET OF FOURIER COEFFICIENTS FOR A CHEBYSHEV POLYNOMIAL A.1 Van der Maas' Method The method of Van der Maas27 is used for the generation of the coefficients of a Chebyshev polynomial. These coefficients are used to specify the pattern constraints for the various minimization procedures developed in the thesis. The formulation is in terms of currents, i H , where 1^  is the relative N N current in the k radiator for an N-element array with elements numbered from left to right. Van der Maas obtained the following expression for I M : j k = ( N - D y N ( N - k) s e 0 k - s s N~V + 1 " (A.l) s + 1 for k 4 1 and k + N, and I* = I N = 1 N N N where L = k - 2 for k ^ %(N +1) and L = N - k - 1 for k - h (N + 1). The value of a is a = 1 - (1/Z)2 (A.2) where Z is the parameter which specifies the beam width and side lobe level for the Chebyshev pattern T (Z cos irdu/X) which is an (N - l)*-* 1 order N-1 Chebyshev polynomial. The variable u is, in the notation of this thesis, u = cos 6Q - cos 8, where 9 Q is the steering angle of the array and 6 is the angle of incidence of the plane wave on the array. In many practical cases, Z is only slightly larger than 1 and thus a is quite small. The maximum value of a = 1 corresponds to a binomially weighted array while the minimum value of a = 0 corresponds to an edge-type array. These extreme cases have main lobe/side lobe ratios of 0.0 and 1.0 respectively. 2 7 It is necessary to determine Z in order to determine a in equation (A.2). Z is related to the main lobe/side lobe ratio, R, by T N-1 ( Z ) = R <A-3> A well known expression for Chebyshev polynomials is T N_ X(Z). = cosh {(N - 1)(cosh - 1 Z)} (A.4) Equations (A.3) and (A.4) may be combined to get Z as a complicated function of R and thus to get a as a function of R from equation (A.2). Fourier coefficients are required for the synthesis of three types of patterns. These are the Chebyshev, squared-Chebyshev and Chebyshev-envelope patterns. The M Fourier coefficients of an M-element Chebyshev array are obtained from equation (A.l) as D l = ^M-l' D k = ^ " l 1 f° r 1 - k - M - 1 (A.5) • Patterns for the squared-Chebyshev pattern can be obtained from equation (A.5) and the identity 2l2(x) = T 2 N(x) + 1 (A.6) That i s , the M Fourier coefficients for the squared-Chebyshev pattern are El » k(T>1 + 1) , E k = J5D for 1 - k ± M - 1 (A. 7) The generation of the Fourier coefficients for the Chebyshev-envelope pattern is explained in section 2.4.2 of Chapter 2. Programs using Van der Maas' series of equation (A.l) were developed for arrays with up to M = 50 elements. It was later found necessary to obtain synthesis for arrays of approximately 100 elements. 169 A. 2 Extension Method for Large Arrays The identity for Chebyshev polynomials T 2 N(x) = 2T2(X) - 1 (A.8) will be used to obtain the Fourier coefficients of a large array from those of a smaller array. The patterns to be synthesized are the Chebyshev, squared-Chebyshev and Chebyshev-envelope. Consider first the Chebyshev pattern. A program has been written, using the techniques described in section A.l, to synthesize a pattern of type N-1 T 2 ( N_ 1 }(Zx) = F kT 2 k(x) • (A.9) k=0 by determining the F k > Here x = cos irdu/X. The program is limited to values of N - 50. It will be shown how the identity of equation (A.8) allows a pattern of the type in equation (A.9) with N = 99'to be obtained from a pattern with N = 50. Let us start with the formulation of equation (A.9) and obtain the indiv-idual weights or relative currents for a conventional Dolph-Chebyshev array having a pattern of ^ (Zx) . Denote these weights by q^ with 1 - i - 2N - 1 and numbering of array elements running from left to right. Thus '% = F0' qk = h FN-k f ° r k < N ' qk = ^Fk-N f° r k > N (A. 10) The post-detection pattern for a square-law-detected array with element weights, {qk}^of equation (A.10) is (N-1)^ Z x)' T ^ e s e r ^ e s representation for this pattern is 2(N-1) T2(N-l)<Zx> = T Hk T2k ( x ) ( A ' U ) k=0 J-/U w i t h and 2N-1 T q ? h 1 2 N - l - k 1=1 i+k f o r 1 - k ^ 2N - 1 (A.12) S u b s t i t u t i o n of equation (A.11) i n t o the i d e n t i t y of equation (A.8) y i e l d s (A.13) 2(N-1) T 4 ( N - l ) ( Z x ) " 2 En Hk T2k< x> - 1 k=0 The above equation can be w r i t t e n i n the d e s i r e d form as M-l T 2 ( M - l ) ( Z x ) - ^ E k T 2 k ( K ) where M = 2N - 1 (A.14) (A.15) The { E , } are given by , E Q = 2 H 0 - 1, .E F C = 2Hfc f o r 1 - k - M - 1 (A.16) I f N = 50 i s the l a r g e s t v a l u e of N f o r which the method of Van der Maas has been programmed, the l a r g e s t v a l u e of M obtained i s M = 99 from equation (A.15). The method then c o n s i s t s of s y n t h e s i z i n g the p a t t e r n of equation (A. 9) f o r a v a l u e of N w i t h T 2 ( N - 1 ) ( Z > - (A.17) The {q k) weights are co n s t r u c t e d from the c o e f f i c i e n t s of equation (A. 9). The (H k) and ( E K ) c o e f f i c i e n t s are obtained from the {qj-} weights as i n equations (A.12) and (A.16) to accomplish the s y n t h e s i s of the p a t t e r n of equation (A.14) f o r an M-element a r r a y w i t h M = 2N - 1 and a s p e c i f i e d main l o b e / s i d e lobe r a t i o of T 2 ( M - 1 ) ( Z ) " R (A,18) Pattern synthesis for a 99-element array with a squared-Chebyshev pattern is accomplished by using the pattern of equation (A.14) and the identity of equation (A.8) with M = 99 to get TJ^CZx) = %{T 2 ( M_ 1 }(Zx) + 1} (A. 19) This pattern function can be synthesized by determining the coefficients in TM-l<Zx> ' % G k T 2 k « <A'20> k=0 such that the main lobe/side lobe ratio specified is T2_ X(Z) = R (A.21) The {G^ } which synthesize this pattern are given by Gk = %\ f o r 1 ~ k ~ M " l f G0 = ^ ( E0 + ^ (A.22) where, the 0^} are chosen so that the pattern of equation (A. 14) is synthe-sized with main lobe/side lobe ratio of T2(M-1) ( Z ) = 2 R " 1 (k.23) The cycle can be repeated indefinitely with the extended array size patterns used as a basis for the above method to obtain synthesis for array sizes of 2N - 1, 4N - 3, 8N - 7, 16N -15, etc., from the Fourier coefficients for an N-element quadratically processed array. The last type of pattern for which the extended size synthesis will be given is the Chebyshev-envelope pattern. This pattern is synthesized only for array sizes M where M is even and M/2 is odd. This restriction is imposed so that the synthesis method of section 2.4.2 of Chapter 2 can be used for the array correlator. The pattern to be synthesized was derived in section 2.4.2.. This pattern 172 is P(2T T C/X,U) =JgP (2tfc/X,u)P2(2Trc/A,u) cos (Mirdu/X) (A.24) or P(2irc/X,u) = 1 & K _ 2 ( Z c o s ™ W A ) cos Mirdu/X (A.25) with P 1(2TTC/X,U) = /2 T ^ j C Z cos irdu/X) - 1 (A.26) and P 0(2TTC/X,U) = /2 T l M . (Z cos irdu/X) + 1 (A.27) In order to apply the extended size synthesis technique to equation (A.25) for a 98-element array, i t is only necessary to determine the Fourier coeffi-cients, {H^ }, for a conventional array with a pattern of type Tj^_^(Z cos -rrdu/X) for M = 98. This is accomplished by determining the Fourier coefficients, {H^ }, for the pattern of equation (A.9), T2(N-1)^Z c o s ^u/X) » using Van der Maas1 method. (The name of the set of coefficients in equation (A.9) has been changed from {F^ .} to {H^ } to conform with some results in Chapter 2.) N is •chosen so that 2 (N - 1 ) = %M - 1 = 48 and thus N = 25. The {lly} coefficients are found such that T_,„ 1 N(Z) = T i „ , (Z) =~\J(R + l)/2 where i t is required that the overall pattern of equation (A.25) have a main lobe/envelope side lobe ratio of T M_ 2(Z) = R. Al l that is required now to accomplish the synthesis of equation (A.25) is to substitute the {H } into equation (2.69) in section 2.4.2 in the main k body of the thesis. APPENDIX B DERIVATION OF A RELATIONSHIP BETWEEN VARIANCE AND CONDITIONAL VARIANCE FOR THE POINT SOURCE MODEL OF BACKGROUND NOISE The c o n d i t i o n a l variance f o r the point source model i s conditioned with respect to the p o s i t i o n s of the sources as foll o w s : Var (V |u) = E ( ( V 3 | u ) 2 ) - E 2 ( V 3 | u ) (B.l) where u i s an N-vector of the p o s i t i o n s of the N sources. That i s , u = (u^ ,U2>.. .Ujq) and u_^  = cos 9 ^ - cos 9 Q . We assume the p o s i t i o n of every noise source i s independent of the p o s i t i o n of every other noise source. Thus the p r o b a b i l i t y density f o r u i s ' N P(u) = n P.(u-) ' (B.2) i = l 1 . N and du = H .du. (B.3) 1=1 1 I t i s e s s e n t i a l to the d e r i v a t i o n of the r e l a t i o n s h i p between variance and c o n d i t i o n a l variance to show that the c o n d i t i o n a l mean can be w r i t t e n E< v 3i u> - £ w • ( b - 4 ) J h=l The s t a r t i n g point f o r the d e r i v a t i o n of E(V.j|u) i s equation (2.12) where Ys(u>) i s interchanged with Y^( w) t 0 obtain CO E(V 3|u) = ( G ^ O ^ T T ) J tr{y (O ) ) Q (O ) ) } du) (B.5) — CO Equation (2.19) i s substituted i n t o equation (B.5) to get CO E(V 3|u) = i _ | | G(o))| 2tr{Y N(w)q} du (B.6) 2ir -°° M But tr(Y N(u))q) = Y. Y N i k ( w ) q k i '(B.7) k=l 174 The elements of YM(u) » i.e. y (oi), for the point source model, are w Nik obtained from equation (2.39). Thus from equations (2.39), (B.6), and (B.7) N E(V3|u) = (G^CO^TT) j |G(o))| 2 £ V W ) -co h=l M M • V V q, . exp ((k - i)uduL/c) du> (B.8) k=l i=l Now equation (B.8) is simplified by using equation (2.36) to obtain N ~ E(V |u) = ^ ( 0 / 2 7 0 Y J |G(oj)|2N (a))P(o),uh) doj (B.9) h=l -» If we define CO • f h(u h) = ( G ^ O ) ^ * ) J |G( U)| 2N h(a J )P(co,u h) dto (B.10) — CO i t is evident that equation (B.4) is satisfied. It will now be shown that equations (B.4) and (B.2) are sufficient conditions for the following equation to be true: CO J Var (V3|G)P(u) du = Var (V3) (B.ll) — CO where the integration is an N-fold integration over a l l the components of Now i t is apparent that oo J E((V |u ) 2 )P(u) du = E(V2) — C O In order to demonstrate the truth of equation (B.ll) i t is necessary to show that oo J E 2(V 3|u)P(G) du = E 2(V 3) (B.12) oo where E(V3) = J E(V 3|u)P(u) du (B.13) E(V^) w i l l f i r s t be e v a l u a t e d f r o m e q u a t i o n s ( B . 13 ) and ( B . 4 ) . The r e s u l t w i l l be s q u a r e d and t h e v a l u e o b t a i n e d w i l l be compared t o t h e e x p r e s -s i o n o b t a i n e d by e v a l u a t i n g e q u a t i o n ( B . 1 2 ) . S u b s t i t u t i o n o f e q u a t i o n s ( B . 2 ) , ( B . 3 ) and (B . 4 ) i n t o e q u a t i o n ( B . 13 ) y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r E ( V ^ ) : N <? E ( V 3 ) N-1 » n J P i ( u . ) d u . 1=1 -°° £ J W W d u h h=l -<» ( B . 1 4 ) N and t h u s E ( V 3 ) = ]T E ( f ( u h ) ) h= l and E 2 ( V 3 ) = f f E ( f h ( u h ) ) E ( f m ( u ) ) h= l m=l ( B . 1 5 ) The i n t e g r a l o f e q u a t i o n (B .12 ) " w i l l now be e v a l u a t e d by s u b s t i t u t i o n f o r E ( V 3 | u ) and P ( u ) f r o m e q u a t i o n s (B .2 ) and ( B . 4 ) . CO Thus j E ? ( V 3 | u ) P ( u ) du CO CO N N - — I I W W _ o o -co m=l h=l N - f o l d N n P , ( U ± ) du . i=l N-2 » n f P . ( u . ) d u . _ i L i i i i=l -«> £ / W .^ m=l -°° m N ~ £ J f h K ) p h H > d u l h=l -« - £ £ ^ h ^ ^ S i " m=l h=l ( B . 1 6 ) E q u a t i o n s (B .15 ) and (B .16 ) a r e i d e n t i c a l . H e n c e , i f t h e c o n d i t i o n s o f e q u a t i o n s (B .2 ) and (B .4 ) a r e s a t i s f i e d , i t f o l l o w s t h a t J E 2 ( V 3 | { I ) P ( u ) du = E 2 ( V 3 ) and J V a r (V 3 |u " )P (u ) du = V a r ( V 3 ) 176 APPENDIX C EQUATIONS REQUIRED FOR WIDE BAND PERFORMANCE FUNCTIONS The wide band versions of G(j ,m) and E(m,i,k,j) are given by the average over frequency of expressions (3.67) and (3.68). Thus iOQ+b 1 (wd/c)(j-m) fe J (OQ-U • cos ((ood/c) (j-m) cos 6Q) dto (C.l) o ) 0 +b / J .x _ J L f sin (ood/c) (j-m) and E(m,i,k,j) - 2b j (wd/c)(j-m) ( 0 Q - b • sin (ud/c) (i-k) cos ((cod/c) (i-k+j-m) cos 0Q) dw (C.2) (oxi/c)(i-k) . Evaluation of equation (C.l) for G(j,m) yields the result G(j,m) = S i((u 0 +b)d(j - i n)(l + cos 0Q)) • \ c / + SjL|(co0+b)d(j-m)(l - cos 8Q)J - S i ^ o - b ^ a ^ d + cos.60)| - S i ^ W g - b J d a - n O d - cos G0)j ' (C.3) x where SjL(x) = J (sin y)/y dy (C.4) 0 Evaluation of the integral of equation (C.2) for E(m,i,k,j) yields the result E(m,i,k,j) = ktt (i-k)(l + cos 60) + (j-m)(cos e 0 - 1)] + I|(i-k)(cos 60 - 1) + (j-m)(cos 0 Q + 1) - I (i-k+j-m)(1 + cos 0O) - I (i-k+j-m)(cos 0 O - 1) } (C.5) v/here 177 I ( K ) = C O S K^ u0 ~ b)d/c _ cos K((I)Q + b)d/c co„ - b 0 w0 + b + Kd c S.(Kd(w0 - b)) - Si(Kd(a)0 + b)) (C.6) The evaluation of the wide band versions of G(j,m) and E(m,i,k,j) requires the numerical determination of the function of equation (C.4). 178 APPENDIX D ONE PARAMETER REPRESENTATION FOR THE PERFORMANCE FUNCTIONS The volume and area performance f u n c t i o n s , Py, and P^, are d e r i v e d f o r a q matri x given by q L = ( q y + L q A ) / ( l + L) (D.l) Py and P^ are given i n equations (3.8) and (3.9) as wg+b COS6Q+1 P v = (l/2b) J Jj F ( O J U 1,03U 2)/F(0,0) d u x d u 2 du (D.2) ajQ-b COSGQ-1 u)Q+b COS8Q+1 and P = (l/2b) f f F(0,aiu)/F(0,0) du da) (D.3)  ,O J U ) / F(0,0)  A J -i J Q 1 0)g-b COS9Q-1 Now F(CJU^,WU2) i s obtained from equation (3.64) by s u b s t i t u t i n g from equation ( D . l ) . Thus F(a)u 1,cou 2) = I E E E E ( q v k n + L q A k n ) (1 + L ) z k n i m " ^ V i m + L q A i m ^ C ° S — ^ m ~ n ) u i + (k - l ) u 2 ^ (D.4) c where a l l summations run from 1 to M i n any summation where index l i m i t s are not e x p l i c i t l y s t a t e d . F(tuupU)u2) can be w r i t t e n as F(iou^ ,wu2) Fy(coUj ,om2) + L 2 F A ( t o u ^ ,wu2) (D.5) ( T + L F + L F A y ( c o u 1 ,uu 2) + L F V A ( ( O U 1 , C J U 2 ) l -F^(wuj ,tou2) and F A(o)u^ ,CJU 2) are the expr e s s i o n s o b t a i n a b l e by subst t u t i n g q„ and c[ , r e s p e c t i v e l y , i n t o equation (3.64). F (ii)u,,wuo) i s given by V A AV F A V((ou 1,o Ju 2) = E E E E ^vkn^Aim k n l m • cos wd((n - m)u + ( i - k ) u 2 ) (D.6) c • I t can be e a s i l y shown that (D.7) by a simple change of index variables in equation (D.6). Thus FCojUjjtou,) 1 F (OJU. ,wu„) (1 + L F L + 2LF^(uUj , U)U 2 ) + L2FA(wu^ ,um2) (D.8) The parametric expression for Py will be fi r s t obtained by substituting equation (D.8) into equation (D.2). Thus we obtain (D.9) where UiQ+b COS6Q+1 and and P W = p v ( ( V = ( 1 / 2 b> / If toQ-b C O S O Q - I • Fv(wu]L,a)U2)/F(0,0) du^ du 2 dai wn+b cos8 +1 a)Q-b COS8Q-1 • FA(cju^ ,cou2) /F(0,0) du^ du2 da) a)Q+b COS8Q+1 W = (l/2b) J JJ F ((ou1,a)u2)/F(0,0) duL du 2 du (On-b cos9 n-l A V (D.10) (D.ll) (D.12) ;0 Now PTTTT is the minimum value of P (q) . Also P v(q L) L=0 W Thus dPy/dL = 0 L=0 (D.13) Differentiating equation (D.9) and setting L = 0 yields the result dPv/dL JL=0 = 2W - 2 P W = 0 (D.1A) Thus W = P and the final result is W = P w ( 1 + 2 L + L 2 PAV/ PVV ) / ( 1 + L ) 2 (D.15) A similar representation for P.(qT) will now be obtained. We start by A Li 180 substituting K = 1/L in equation (D.l) to obtain q K = (q + Kqv)/(1 + K) (D.16) F(0,oju) is then obtained by substituting equation (D.16) into equation (D.4) to obtain F(0,cou) = {F (O.wu) + K2Fv(0,uu) + 2KF^ (0, uu) } / (1 + K) 2 (D.17) where F^(0,wu) and F^(0,uu) are the expressions obtained by substituting q^ and qy, respectively, into equation (3.64). F^y(0,0)u) is obtainable from equation (D.6). The above equation for F(0,cou) is substituted in equation (D.3) to obtain P A(q K) = ( P M + K 2P V A + 2KZ)/(1 + K) 2 (D.18) (OQ+b COS8Q+1 where Z = (l/2b) f f F (0,wu)/F(0,0) du dco (D.19) <o*-b cos0 0-l A V Now i f K = 0, p A ( q 0 ) = P^, the minimum value for ?^(q). Thus i t follows that dP/dK A = 0 = 2PA. - 2Z (D.20) K=0 Substituting Z = P^ into equation (D.18) yields the result V V = (PAA + K % A + 2KPAA> / ( 1 + K> 2 CD. 21) The final result is obtained by setting K = 1/L to get V V = PAA ( L 2 + 2 L + TVA/?AA)/a + L ) 2 (D'22) 181 REFERENCES 1. D.G. Tucker, "Multiplicative Arrays in Radio Astronomy and Sonar Systems," The Radio and Electronic Engineer, vol.25, p.113, Feb. 1963. 2. V.G. Welsby and D.G. Tucker, "Multiplicative Receiving Arrays," The  Radio and Electronic Engineer, vol.19, p.369, June 1959. 3. M.J.Jacobson, "Optimum Envelope Resolution in an Array Correlator," J.Acoust. Soc. Am., vol.33, p.1055, Aug. 1961. 4. E. Shaw and D.E.N. Davis, "Theoretical and Experimental Studies of the Resolution Properties of Multiplicative and Additive Aerial Arrays," in Proc. of a Symp. on Signal Processing in Radar and Sonar Directional  Systems, Birmingham, July 1964. 5. A. Ksienski, "Multiplicative Processing Antennas for Radar Applications," in Proc. of a Symp. on Signal Processing in Radar and Sonar Directional  Systems, Birmingham, July 1964. 6. A. Berman and C.S. Clay, "Theory of Time-Averaged-Product Arrays," J, Acoust. Soc. Am.,. vol.29, p.805, July 1957. 7. D.C. Fakley, "Comparison Between the Performance of a Time-Averaged- •..• Product Array and an Intra-Class Correlator," J. Acoust. Soc. Am., vol.31, p.1307, Oct. 1959. 8. J.J. Faran and R.H. Hills Jr., Correlation for Signal Reception, Acoustic Research Lab., Harvard Univ., Tech. Memo.27, Sept.15, 1952. 9., J.J. Faran and R.H. Hills Jr., The Application of Correlation Techniques  to Acoustic Receiving Systems, Acoustic Research Lab., Harvard Univ., Tech. Memo.28, Nov.l, 1952. 10. B. Picinbono, "Optimum Filtering and Multi-Channel Receivers," IEEE  Trans, on Information Theory, vol.IT-12, April 1966. 11. B. Picinbono, and J. de Suso Barba, "Filtrage Optimal et Correlation," C.R. Acad. Sc., Paris, p.290, Oct.19, 1964. 12. P.Y. Arques, "Optimalisation des Systemes de Detection a n Entrees par Traitement Quadratique et Integration Forte," C.R. Acad. Sc., Paris, p.260, April 26, 1965. 13. P.Y. Arques, "Optimalisation des Systemes de Detection a n Entrees de Signaux Aleatoire Comportant un Traitement Quadratique et Integration Forte," Annales des Telecommunications, vol.20, p.119, May 1965. 14. F. Bryn, "Optimum Signal Processing of Three-Dimensional Arrays Operating on Gaussian Signals and Noise," J. Acoust. Soc. Am., vol.34, p.289, March 1962. 182 15. CT. Tai, "The Optimum Directivity of the Uniformly Spaced Broadside Array of Dipoles," IEEE Trans, on Antennas and Propagation, vol.AP-12, p.447, July"1964. 16. C. Drane Jr. and J. Mcllvenna, "Gain Maximization and Controlled Null Placement Simultaneously Achieved in Aerial Array Patterns," The Radio  and Electronic Engineer, vol.39, p.49, Jan. 1970. 17. Y.L. Lo, S.W. Lee, and O.H. Lee, "Optimization of Directivity and Signal-to-Noise Ratio of an Arbitrary Antenna Array," Proc. IEEE, vol.54, p.1033, Aug. 1966. 18. H.E, King, "Directivity of a Broadside Array of Isotropic Radiators," IRE Trans, on Antennas and Propagation, vol.AP-7, p.197, April 1959. 19. D.K. Cheng and F.I. Tseng, "Gain Optimization for Arbitrary Antenna Arrays," IEEE Trans, on Antennas and Propagation, vol.AP-13, p.973, Nov. 1965. 20. D.K. Cheng and F.I. Tseng, "Maximization of Directive Gain for Circular and Elliptical Arrays," Proc. IEE, vol.114, p.589, May 1967. 21. C.L. Dolph, "Current Distribution of Broadside Arrays," Proc. IRE, vol.34, p.335, June 1946. 22. K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, Cambridge, Mass. , 1969), p.60. 23. M. Schoenberger, "Symmetry Properties of Linear Arrays," Proc. IEEE, vol.59, p.287, Feb/ 1971. 24. H.J. Riblet, "Discussion on "A current Distribution for Broadside Arrays" . by C.L. Dolph," Proc. IRE, vol.35, p.489, May 1947. 25. R.L. Pritchard, "Optimum Directivity Patterns for Linear Point Arrays," J. Acoust. Soc. Am., vol.25, p.879, Sept. 1953. 26. B.L.J. Rao, "Modified Dolph-Chebyshev Arrays," Radio Science, vol.3 (New Series), p.459, May 1968. 27. G.J. Van der Maas, "A Simplified Calculation for Dolph-Tchebyscheff Arrays," J. Appl. Physics, vol.25, p.121, June 1954. 28. D.C. Murdoch, Linear Algebra for Undergraduates (John Wiley and Sons, .New York, 1957), pp.165-166. 29. D.C. Murdoch, Linear Algebra for Undergraduates (John Wiley and Sons, New York, 1957), p.49. 30. R.C. Hansen, "Directivity of Chebyshev Arrays," IEEE Trans, oh Antennas  and Propagation, vol.AP-18, p.815, Nov. 1970. 

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