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Optimal pre and postfiltering of noisy sampled signals - particular applications to PAM, PCM and DPCM… Chan, Donald 1970

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OPTIMAL PRE AND POSTFILTERING OF NOISY SAMPLED SIGNALS-PARTICULAR APPLICATIONS TO PAM, PCM AND DPCM COMMUNICATION SYSTEMS b y DONALD CHAN B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1964 M.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF - 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 i r e q u i r e d standard Research Supervisor «. Members of the Committee Acting Head of the Department Members of the Department 6f E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In p r e s e n t i n g t h i s t h e s i s in 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 the 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 the 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 . It 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 ELHC.TR IC AL f ^ f f t h i q The 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 V a n c o u v e r 8, Canada Date ~~^>EPT t ^ l f l 6 ABSTRACT In many control, data-processing, and communication systems, sampling i s an inherent part of the system. If the time-continuous input signal is nonbandlimited, and noise i s introduced in the system, an unavoid-able error exists between the actual reconstructed signal and the desired time-continuous output signal. This error can be reduced by the suitable choice of p r e f i l t e r prior to sampling and by the suitable choice of post-f i l t e r for reconstructing the time-continuous signal from the samples. In this thesis, an algorithm for determining the jointly optimal pre and post-f i l t e r s which minimize the frequency weighted mean-integral-squared error of the system is presented, and the valid i t y of the algorithm is proved. In the analysis, no restrictions are placed on the input signal spectrum or the noise spectrum, and the cross-correlation between signal and noise i s taken into account. Applications of the optimization algorithm to M-channel time-multiplexed PAM systems, PCM systems with d i g i t a l channels errors, and DPCM systems are considered. Performance characteristics, showing mean-squared error and inband signal-to-noise ratio versus channel signal-to-noise ratio, are determined expli c i t l y for optimal pre and postfiltered PAM and PCM systems with first-order Butterworth input spectrum. These characteristics are compared with those of PAM and PCM systems which use suboptimal. f i l t e r i n g schemes and with the optimal perfor-mance theoretically attainable. Performance characteristics, showing mean-squared error versus channel capacity, are also determined for PAM, PCM, and DPCM systems when the systems parameters are optimized to yield the least mean-squared error for a given channel capacity. Because of the subjective nature of speech, the effect of pre and postfilters i n PAM, PCM and DPCM communication systems for speech transmission i i i s studied by simulation methods and evaluated with subjective tests. Weak noise pre and postfilters (WNF), which yield v i r t u a l l y the same performance as optimal pre and po s t f i l t e r s , are considered in the subjective evaluation, i n addition to lowpass pre and postfilters (LPF). The di g i t a l simulation f a c i l i t i e s and the subjective testing methods are described, and the sub-jective results interpreted. It was observed that no significant subjective improvement resulted when WNF were used i n place of LPF i n PAM and DPCM systems. In PCM systems, significant differences in WNF and LPF subjective performances could exist. Using the analytical results, an explanation for the subjective behaviour i s presented. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS . i v LIST OF ILLUSTRATIONS v i i LIST OF TABLES x i i ACKNOWLEDGEMENT x i i i 1. INTRODUCTION 1 1.1 System Model 1 1.2 Review of Previous Research 3 1.3 Scope of Thesis 5 2. JOINT OPTIMIZATION OF THE PREFILTER AND THE POSTFILTER 8 2.1 D e r i v a t i o n of Weighted Mean-Integral-Squared Err o r Expression and Some Necessary Conditions 9 2.2 Further Necessary Conditions 13 2.3 Algorithm f o r Determining the J o i n t l y Optimal P r e f i l t e r and P o s t f i l t e r 16 2.4 V a l i d i t y of the Optimization Algorithm 17 3. ANALYSIS AND DISCUSSION OF SOME IMPORTANT FILTERING SCHEMES 21 3.1 A p p l i c a t i o n of Optimal F i l t e r i n g to Some S p e c i f i c Cases 21 3.1.1 High Sampling Rate 21 3.1.2 Lowpass Signals 21 3.1.3 Weak Noise 22 3.1.4 System f o r which p(f)=|A(f)B(f)W(f)K(f) I / « ( £ ) / $ (f) i s constant 23 ' ' x n s i v Page 3.2 Some Suboptimal Filtering Schemes . ... 23 3 .2 .1 Weak Noise F i l t e r s , 23 3 . 2 . 2 Optimal P r e f i l t e r ; Constant Amplitude Pos t f i l t e r 25 3 . 2 . 3 Constant Amplitude Prefilter;. Optimal Pos t f i l t e r 25 4 . OPTIMAL AND SUBOPTIMAL FILTERING IN PAM, PCM, AND DPCM COMMUNICATION SYSTEMS 27 4 .1 Pulse Amplitude Modulation (PAM) 27 4 .2 Pulse Code Modulation (PCM) 34 4 . 2 . 1 Correlation Functions for Quantized Signals Transmitted Over Discrete Memoryless Channels 39 4 . 2 . 2 Pre and Postfiltering in PCM Systems 39 4 . 3 Differential Pulse Code Modulation (DPCM) 46 4.4 System Comparisons 50 5 . COMPUTER SIMULATION OF PRE AND POSTFILTERING PAM, PCM, AND DPCM SPEECH COMMUNICATION SYSTEMS 51 5.1 Introduction 51 5.2 Assumptions and Restrictions 52 5 .3 Digital Computer Simulation 56 5 .3 .1 Digital Recording and Playback System . 58 5 . 3 . 2 Simulation Program 62 6. SUBJECTIVE EVALUATION OF PRE AND POSTFILTERS IN PAM, PCM, AND DPCM SPEECH COMMUNICATION SYSTEM . 66 6.1 Introduction 66 6.2 Subjective Test Method 67 Page 6 .2 .1 Speech Material, Equipment, Listeners, and Further Details on System Simulation 67 6 .2 .2 Determination of Isopreference Contours i 70 6 . 2 . 3 Scaling Isopref erence Contours 79 6 .3 Further Results and Discussion of Subjective Evaluation 81 6.4 Concluding Remarks 90 6 .4 .1 Summary and Comparison with Previous Works ... 90 6 . 4 . 2 Subjective Weighting Function for Speech ... 94 6 . 4 . 3 Application to Television Signals 96 7. CONCLUSIONS 99 APPENDIX CORRELATION FUNCTIONS AND RECONSTRUCTION ERROR FOR QUANTIZED GAUSSIAN SIGNALS TRANSMITTED OVER DISCRETE MEMORYLESS CHANNELS 103 A.l Exact Expressions for Correlation Functions 103 A. 2 Approximation and Bounds 104 A.3 Optimal Postfiltering of Quantized Signals Transmitted over Discrete Memoryless Channels 106 A.4 An Example 109 REFERENCES "... 114 v i LIST OF ILLUSTRATIONS Page F i g . 1 Block diagram of a l i n e a r p r e f i l t e r i n g and p o s t f i l t e r i n g system. Function 6(t) i s the u n i t impulse. Phase angle 9 i s constant. Sampling period T=l/f where f i s the sampling frequency 2 s s F i g . 2 Lowpass equivalent of a l i n e a r i z e d analog modulation system. ... 22 F i g . 3 (a) Multiplexed PAM system. Functions F ± ( f ) , G ^ f ) (i = l , 2 , ... ,M), 0 ( f ) , H(f) , and L ( f ) are t r a n s f e r functions of l i n e a r f i l t e r s . S i g n a l y . ( t ) i s sampled by sampler at t=kT+9.-9, where k i s any i n t e g e r . Impulse t r a i n A.(t) = T. E 6(t-kT-8.+9). .. 28 1 k=-«\ 1 (b) Equivalent representation of a sampler. 28 (c) Channel f i l t e r t r a n s f e r c h a r a c t e r i s t i c f o r Example 2 28 F i g . 4 Normalized e r r o r e / a versus S/N W f o r Example 2. Symbols 0, PR, and PO denote optimal f i l t e r s , optimal p r e f i l t e r - c o n s t a n t amplitude p o s t f i l t e r , and constant amplitude p r e f i l t e r - o p t i m a l p o s t f i l t e r , r e s p e c t i v e l y . For the curves shown the optimal f i l t e r bandwidth W=W 32 c F i g . 5 Signal-to-noise r a t i o SNR f o r Example 2. Symbols 0, PR, and PO are defined i n the caption of F i g . 4. For the curves shown the optimal W=W 33 c F i g . 6 (a) Quantizer and d i s c r e t e memoryless channel 35 (b) System equivalent to the system shown i n F i g . 6a 35 (c) Quantizer c h a r a c t e r i s t i c 35 v i i F i g . 7 (a) A PCM system (b) Quantizer t r a n s f e r c h a r a c t e r i s t i c . Page 40 40 F i g . 8 Normalized e r r o r e/o versus S/N W f o r Example 3. Symbols 0, PR, o c and PO are defined i n the caption of F i g . 4. The number of b i t s of quantization d which minimizes z/a f o r the given values of W^/a and S/N W i s shown on the curves. For the curves shown the optimal o c f i l t e r bandwidth W=W /d 43 c F i g . 9 Signal-to-noise r a t i o SNR f o r Example 3. For the curves shown the optimal f i l t e r bandwidth W=W /d. c (a) W /a=d0. 44 c (b) W /a =100 44 c (c) W /a =1000 44 c F i g . 10 (a) Normalized minimum e r r o r z/a f o r Examples 2, 3, and 4 and optimal channel bandwidth W^/a f o r Examples 2 and 3. The number of b i t s of quantization d which minimizes e/o are shown. Also shown i s the OPTA 45 (b) Err o r e/a and optimum feedback c o e f f i c i e n t f o r DPCM (Example 5). The three f i l t e r i n g schemes, 0, PR, and PO, y i e l d i d e n t i c a l z/a when ct-^  i s chosen to minimize S. Indicated f o r comparison purposes are PAM and PCM r e s u l t s obtained from F i g . 10a: x-PAM 0 and o-PCM 0, optimal code 45 F i g . 11 (a) A DPCM system 47 v i i i Page (b) An equivalent system to F i g . 11a when the d i g i t a l channel CO i s n o i s e l e s s . Impulse t r a i n A(t) = T. Z 6(t-kT+G) 47 k=-°° F i g . 12 T y p i c a l frequency response. (a) Highpass f i l t e r 55 (b) Lowpass f i l t e r 55 F i g . 13 Equivalent r e a l i z a t i o n . (a) P r e f i l t e r 57 (b) P o s t f i l t e r ^ 57 F i g . 14 Block diagram. (a) D i g i t a l recording system 59 (b) D i g i t a l playback system 59 F i g . 15 Simulated communication systems. (a) PAM 63 (b) PCM 63 (c) Previous-sample feedback DPCM 63 F i g . 16 (a) Normalized amplitude p r o b a b i l i t y density of speech. Symmetrical average of p o s i t i v e and negative data 69 (b) Power density spectrum of speech. 69 F i g . 17 PAM isopreference contours. Plus and minus standard deviations of each experimental point are denoted by the bar through the poi n t . Reference points associated with each isopreference contour are ix Page drawn s o l i d . (S/N) , . values are given i n dB and Sc values subj : are enclosed i n brackets. The bandwidth of the pre and post-f i l t e r s equals . (a) Lowpass f i l t e r i n g scheme (LPF). 72 (b) Weak noise f i l t e r i n g scheme (WNF) •. • • • 72 Fig. 18 PCM isopreference contours. See F i g . 17 caption f o r further comments. (a) LPF 73 (b) WNF 73 Fig. 19 DPCM isopreference contours. See F i g . 17 caption f o r furt h e r comments. (a) LPF .... 74 (b) WNF 74 Fig. 20 Psychometric curve f o r obtaining isopreference point B i n F i g . 17a. Reference point i s point A i n F i g . 17a. (a) Ordinate i n l i n e a r preference units 77 (b) Ordinate i n un i t normal deviates 77 Fig. 21 Quality r a t i n g of isopreference contours shown i n Figs. 17, 18, and 19 versus t h e i r corresponding minimum channel capacity. (a) Subjective scale i n ( S / N ) g u b j • 83 (b) Subjective scale i n Sc 83 x Page Fig. 22 R e l a t i o n between s c a l e value Sc and subjective s i g n a l - t o - n o i s e r a t i o (S/N) , 85 subj Fig. 23 Isopreference contours with q u a l i t y ratings derived from the curves f i t t e d to the raw data i n F i g . 21. (S/N) , . values are i n subj dB and Sc values are enclosed i n brackets. (a) PAM 86 (b) PCM and DPCM 86 Fig. 24 Q u a l i t y r a t i n g of isopreference contours versus the isopreference contour satu r a t i n g values of sampling rate f . (a) Subjective scale i n (S/N) , 92 subj (b) Subjective scale i n Sc. 92 Eig. A . l Reconstruction system. Signal v ( t ) i s sampled at t=kT-9 107 Fig. A.2 Functions 2 ( l - a ^ ) , B ^ ( l ) , and B^(3) versus log^N and p when y ( t ) i s a s t a t i o n a r y Gaussian process. The s o l i d curves apply to Max non-uniform quantizers and the dotted curves apply to Max uniform quantizers. (a) Ordinate i s 2 ( l - a 1 ) . 113 (b) Ordinate i s B~(l) 113 (c) Ordinate i s B~(3) 113 x i LIST OF TABLES Page Table .5.1 P r e f i l t e r and p o s t f i l t e r c h a r a c t e r i s t i c s used i n the computer simulation 56 Table A . l Max nonuniform quantizer I l l Table A. 2 Max uniform quantizer 112 x i i ACKNOWLEDGEMENT I am g r a t e f u l to the Defence Research Board of Canada and the Na t i o n a l Research Council of Canada f o r support received under Grants DRB 2801-26 and NRC A-3308, r e s p e c t i v e l y . G r a t e f u l acknowledgement i s also given to the National Research Council of Canada f o r NRC postgraduate scholarships received from 1967 to 1970. I am p a r t i c u l a r l y g r a t e f u l to Dr. R.W. Donaldson f o r h i s valuable suggestions, generous counsel, and constant encouragement which were f r e e l y and r e a d i l y given over the course of the research. I am thankful to Messrs. E. Stanley, W. Dettwiler, and J . Stevens of the U.B.C. Computing Centre and Mr. M. Koombes of the U.B.C. E l e c t r i c a l Engineering Department f o r t h e i r assistance i n the design, construction, and programming of the simulation f a c i l i t i e s . I wish to thank Lenkurt E l e c t r i c Company, Burnaby, Canada, f o r providing the lowpass and highpass f i l t e r s used i n the d i g i t a l recording and playback systems. I also wish to express a my s i n c e r e appreciation to Mr. A. MacKenzie f o r preparing the i l l u s t r a t i o n s , and to Miss B. Harasymchuk and Miss H. DuBois f o r typing the manuscript. F i n a l l y , i n l i e u of the time I might have otherwise spent with them, I would l i k e to thank my wife, P a t r i c i a , and my daughters, Mi c h e l l e , E l y s e , and N i c o l e , f o r t h e i r patience and understanding. x i i i 1. INTRODUCTION 1.1 System Model Widespread use of d i g i t a l computers and the advent of low-priced integrated c i r c u i t s have given great importance to the c h a r a c t e r i z a t i o n of time-continuous signals i n sampled format. In many cont r o l systems, data-processing systems, and communication systems, samples are taken of a time-continuous s i g n a l on input, and on output, a time-continuous s i g n a l i s r e -constructed. I f the time-continuous input s i g n a l i s t a c i t l y assumed to be bandlimited to l e s s than h a l f the sampling frequency of the system, then straight-forward a p p l i c a t i o n of sampled data theory [1-3] can be used i n the analysis and design of the system. Unfortunately, i n many p r a c t i c a l systems the input s i g n a l i s not s t r i c t l y bandlimited. Furthermore, i f noise, n e c e s s a r i l y introduced into the time-discrete s i g n a l , i s also considered, an unavoidable er r o r e x i s t s between the actual reconstructed s i g n a l and the desired continuous output s i g n a l . This e r r o r , however, can be reduced by the su i t a b l e choice of p r e f i l t e r p r i o r to sampling and by the s u i t a b l e choice of p o s t f i l t e r f o r reconstruction from the time samples. A block diagram of the system under consideration i s shown i n F i g . 1. The research described i n t h i s thesis deals with the optimization of the pre and p o s t f i l t e r s i n F i g . : l and the r e s u l t i n g a p p l i c a t i o n s . A p p l i c a t i o n of F i g . 1 to cont r o l systems r e s u l t s when d i g i t a l c o n t r o l l e r s or d i g i t a l f i l t e r s are employed i n place of continuous networks. I f the computer operations are l i n e a r and time-invariant, then the d i g i t a l computer program can be represented or included i n the p o s t f i l t e r t r a n s f e r function G(f) [4,5]. Noise n(t) can be int e r p r e t e d at the erro r introduced by a n a l o g - t o - d i g i t a l conversion, and t r a n s f e r function A(f) as the continuous network which i s to be simulated by the d i g i t a l computer. For example, A(f) n(t) Linear P r e f i l t e r Transfer Function F(f) Impulse Response f(t) ' y(t) A(t) = T. E 6(t-kT+9) k=-» Sample at t=kT-9 —oo<k<°o Linear Postfilter -Transfer Function G(f) Impulse Response g(t) Linear F i l t e r Transfer Function A(f) Impulse Response a(t) System Error Weighting Function Transfer Function w(f) Impulse Response w(t) t Weighted System Error Fig. 1 Block diagram of a linear prefiltering and postfiltering system. Function 6(t) is the unit impulse. Phase angle 9 is constant. Sampling period T=l/f where f is the sampling frequency. could be the transfer function of a compensating network. Although the advantages and disadvantages of u t i l i z i n g a di g i t a l computer i n control systems for continuous network simulation are dependent upon the particular application, i t suffices to say that d i g i t a l f i l t e r s do not have the problems of d r i f t , sensitivity and component tolerance that analog f i l t e r s have. Furthermore, there are not real bounds on the accuracy that may be achieved in d i g i t a l f i l t e r design. In cases where Fig. 1 models a data-processing system, x(t) may be a noisy version of some random function arising in the course of a measurement or .observation, from which z(t) i s to be obtained by a linear operation. The pr e f i l t e r , sampler, and noise process n(t) may represent the analog-to-digital conversion process, and the p o s t f i l t e r might constitute a d i g i t a l computer program. When Fig. 1 depicts a communication system, x(t) is the input message which is to be prefiltered, transmitted over a pulse modulation system and fi n a l l y , postfiltered to yi e l d the reconstructed time-continuous output x(t). Noise n(t) i s transmission and/or quantization noise arising from such pulse modulation systems as pulse amplitude modulation (PAM), pulse code modulation (PCM), and differential pulse code modulation (DPCM). The p r e f i l t e r and the pos t f i l t e r are chosen to make x(t) approximate the desired output z ( t ) , which is related to input x(t) by a linear operation. 1.2 Review of Previous Research Many significant contributions have been made to the optimal f i l t e r i n g problem since Wiener's original work [6]. The postfiltering problem of reconstructing continuous signal from time-discrete samples has been considered. Stewart [7] obtained the optimal reconstruction f i l t e r for noise-4 l e s s samples, while Ruchkin [8] and Katzenelson [9] have determined the optimal p o s t f i l t e r f o r recovering an input s i g n a l from i t s quantized samples. Various aspects of the j o i n t l y optimal pre and p o s t f i l t e r i n g problem has been examined by numerous i n v e s t i g a t o r s . The f i r s t to examine the problem was Costas [10], who obtained the j o i n t l y optimal f i l t e r s f o r wholly time-continuous systems. Tsybakov [11], Berger and Tufts [12], and Chang and Freeny [13] discussed the j o i n t o p t i m i z a t i o n of transmitter pulse-shaping f i l t e r and l i n e a r r e c e i v e r f i l t e r i n pulse transmission systems where the input and output s i g n a l s are time-d i s c r e t e random sequences. Other i n v e s t i g a t o r s , notably Robbins [14], De Russo [15], and Brown [16], have considered the j o i n t optimization of pre and p o s t f i l t e r s f o r data-processing and c o n t r o l a p p l i c a t i o n s . These systems were modelled as a concatenation of p r e f i l t e r , sampler, and p o s t f i l t e r , i n which the p r e f i l t e r input consisted of s i g n a l plus noise. As a communication system, the pre and p o s t f i l t e r i n g system shown i n F i g . 1 has been analyzed with varying degrees of g e n e r a l i t y and r i g o r . S p i l k e r [17] and Goodman and Drouilhet [18] determined the optimal f i l t e r p a i r when noise n(t) i s white. S p i l k e r considered input s i g n a l s selected from a s p e c i a l class of nonbandlimited power spectra, while Goodman and D r o u i l h e t treated bandlimited spectra. Kimme and Kuo [19], Bruce [20], and Brainard and Candy [21] are among some of the i n v e s t i g a t o r s who have analyzed systems which can e s s e n t i a l l y be reduced to the form shown i n F i g . 1. However, i n t h e i r i n v e s t i g a t i o n s bandlimited input s i g n a l s have been t a c i t l y assumed. Nonbandlimited spectra have been considered by Kellogg [22], who u t i l i z e d Brown's work [16] to obtain approximate optimal pre and p o s t f i l t e r s f o r PCM systems. An unique feature i n Kellogg's work i s the i n c l u s i o n of cross-c o r r e l a t i o n between the p r e f i l t e r output s i g n a l and the quanization noise. 5 1.3 Scope of the Thesis Although the previous works have developed f a i r l y general solutions to the optimal f i l t e r i n g problem, they cannot be applied to nonbandlimited input and noise s i g n a l s with a r b i t r a r y spectra. One of the purposes of t h i s t hesis i s to derive expressions for the j o i n t l y optimal pre and p o s t f i l t e r s under more general conditions. In the analysis of Chapter 2, no r e s t r i c t i o n s are placed on the input s i g n a l spectrum or the noise spectrum, and the cross-c o r r e l a t i o n between s i g n a l and noise i s taken i n t o account. Certain subtle and challenging d i f f i c u l t i e s a r i s e i n solving the necessary equations f o r the f i l t e r s . F i r s t , the equations are nonlinear, and second, the equations along with the associated power constraint can be solved i n an i n f i n i t e number of ways. An algorithm f o r determining the s o l u t i o n which y i e l d s the l e a s t d i s t o r t i o n i s presented and proved to be optimal. The p r i n c i p a l conclusion to be drawn from the algorithm i s that the j o i n t l y optimal pre and p o s t f i l t e r s are band-l i m i t e d to a frequency set of t o t a l measure l e s s than or equal to 1/T, of which no two points coincide under frequency t r a n s l a t i o n k/T f o r any integer k^O. An important p r a c t i c a l consequence of this conclusion i s that the optimal p r e f i l t e r and p o s t f i l t e r can be synthesized by combinations of analog bandpass and d i g i t a l spectral-shaping f i l t e r s . The f i d e l i t y c r i t e r i o n used i n the analysis i s the weighted i n t e g r a l of the system er r o r spectrum. Frequency weighted mean-integral-squared error has been s u c c e s s f u l l y employed by others as a measure of subjective goodness i n t e l e v i s i o n studies [19-21, 23]. In Chapter 3, optimal f i l t e r i n g i s applied to some s p e c i f i c cases and various suboptimal f i l t e r i n g schemes are i n v e s t i g a t e d . One scheme, designated as weak noise f i l t e r i n g , y i e l d s v i r t u a l l y the same performance as optimal f i l t e r i n g i n many cases of i n t e r e s t , and has the p r a c t i c a l advantage that the f i l t e r t r a n s f e r c h a r a c t e r i s t i c s are dependent only on the input s i g n a l spectrum and the relative"spectrum of the noise. Although the contents of the previous Chapters have more general a p p l i c a b i l i t y , only a p p l i c a t i o n s to PAM, PCM, and DPCM communication systems are considered i n Chapter 4. The mean-squared err o r expression f o r a general M-channel time-multiplexed PAM system i s derived. I t i s shown that i f the requirement of d i s t o r t i o n l e s s transmission i s imposed, which means that there i s no intersymbol or interchannel d i s t o r t i o n , then the time-multiplexed PAM system reduces to M independent systems of the form shown i n F i g . 1. I t i s shown that PCM systems with d i g i t a l channel errors can also be modelled by the system of F i g . 1. C o r r e l a t i o n functions f o r quantized s i g n a l s transmitted over d i s c r e t e memoryless channels are derived and included as a necessary part of the f i l t e r o ptimization. F i n a l l y , i t i s shown that DPCM systems can also be reduced to the system depicted i n F i g . 1. Once the major problem of modelling i s solved, the r e s u l t s of Chapters 2 and 3 are e a s i l y applied. Optimal pre and p o s t f i l t e r i n g and some suboptimal f i l t e r i n g schemes which were presented i n Chapter 3 are considered. System errors are evaluated and system parameters optimized to y i e l d the l e a s t d i s t o r t i o n f o r a channel of f i x e d capacity. The optimal performance t h e o r e t i c a l l y a t t a i n a b l e as derived from information theory arguments are computed and compared to the r e s u l t i n g performances achieved by the various f i l t e r i n g schemes u t i l i z e d i n the PAM, PCM, and DPCM communication systems. In Chapter 5, two suboptimal f i l t e r i n g schemes discussed i n Chapter 3, weak noise f i l t e r i n g and optimal p r e f i l t e r i n g with constant amplitude p o s t f i l t e r i n g , are u t i l i z e d i n PAM, PCM, and DPCM speech communication systems which are simulated on a IBM 360/67 d i g i t a l computer. Weak noise f i l t e r s were simulated since they y i e l d v i r t u a l l y the same performance as optimal pre and postfilters and have the practical advantage of relatively simple realization. The optimal prefilter-constant amplitude p o s t f i l t e r scheme was considered since under practical assumptions the f i l t e r s are lowpass. The restrictions and assumptions used in the simulation are tabulated and an explanation of the simulation f a c i l i t i e s , both hardware and software, i s presented. It i s well known that the quality of speech cannot be judged by an objective measure alone. In fact, such a judgment may be quite misleading. The lack of an objective measure for speech quality necessitates the subjec-tive measurements undertaken in Chapter 6, where a subjective testing method i s developed for evaluating the subjective performances of PAM, PCM, and DPCM speech communication systems. The method i s applied and the subjective results interpreted. It was observed that no significant subjective improve-ment resulted when weak noise f i l t e r s (WNF) were used in place of lowpass f i l t e r s (LPF) i n the PAM and the DPCM systems. On the other hand, significant differences in subjective performance can exist between WNF and LPF i n PCM systems. An heuristic explanation for this subjective behaviour i s presented using the objective results of Chapter 4. Finally, a few concluding remarks are presented, including the f e a s i b i l i t y of using a frequency weighted mean-integral-squared error criterion as an objective measure of speech quality, and the po s s i b i l i t i e s of using weak noise f i l t e r s for television signals. 8 2. JOINT OPTIMIZATION OF THE PREFILTER AND THE POSTFILTER In t h i s Chapter the pre and p o s t f i l t e r s shown i n F i g . 1 are j o i n t l y optimized. The sampling operation i s performed by a sampling gate i n s e r i e s with an impulse modulator. This r e s u l t s i n no loss i n g e n e r a l i t y since pulses of f i n i t e amplitude and duration can be converted to impulses by a l i n e a r pulse shaping f i l t e r included i n the p o s t f i l t e r t r a n s f e r function G ( f ) . . F i l t e r s having t r a n s f e r functions A(f) and W(f) may not n e c e s s a r i l y be p h y s i c a l l y r e a l i z e a b l e . In a d d i t i o n , no a p r i o r i r e s t r i c t i o n s are placed on the power spectrum of the input s i g n a l or the noise, and the standard assump-t i o n of zero c r o s s - c o r r e l a t i o n between s i g n a l and noise i s not made. The fi g u r e of merit used f or comparing system performance i s the frequency weighted mean-integral-squared error c r i t e r i o n . In Section 2.1, the err o r expression i s derived and some necessary conditions f o r both r e a l i z e a b l e and unrealizeable optimal f i l t e r s presented. Section 2.2 i s devoted to de r i v i n g further necessary conditions f o r unrealizeable f i l t e r s . Unrealizeable f i l t e r s provide a lower bound on the err o r obtained by l i n e a r time-invariant f i l t e r s , and t h e i r c h a r a c t e r i s t i c s can be approximated a r b i t r a r i l y c l o s e l y by p h y s i c a l l y r e a l i z e a b l e f i l t e r s i f s u f f i c i e n t l a g i s permitted i n the f i l t e r ' s impulse response. For many p r a c t i c a l systems, such i s the case, since reasonable time delay i n the o v e r a l l system response i s usually not c r i t i c a l . In Section 2.3, an algorithm i s presented f o r deter-mining the c h a r a c t e r i s t i c s and passbands of the optimal f i l t e r s and i n Section 2.4, the v a l i d i t y of the algorithm i s est a b l i s h e d . In the following a n a l y s i s , E{ } denotes an ensemble average and T = 1/fg * s t * i e sampling period. Autocorrelation and c r o s s - c o r r e l a t i o n func-tions are defined as 0 u ( t , x ) = E{u(t)u(x)} and 0 u v ( t , x ) =.E{u(t)v(x)} , res-p e c t i v e l y , where u(t) and v( t ) are a r b i t r a r y random processes. I f u(t) and v( t ) are wide-sense s t a t i o n a r y , then 0 (t,x) = 0 (t-x) and 0 (t,x) = 0 ( t - x ) . 9 F o u r i e r t ransform p a i r s are denoted by upper and lower case l e t t e r s . For co • • "~"l2lTfT example, the power spectrum o f u ( t ) i s $ ( f ) = / 0 (x) e J dx and the — CO cross-power spectrum of u ( t ) and v ( t ) i s $ ( f ) = f 0 (x) e dx. uv uv — 0 0 M u l t i p l i c a t i o n , c o n v o l u t i o n and complex conjugate are denoted by "0", "*", r e s p e c t i v e l y . 2.1 D e r i v a t i o n of Weighted Mean-Integral-Squared E r r o r Expression and  Some Necessary Conditions For the system shown i n F i g . 1, the weighted mean-integral-squared e r r o r i s gi v e n by T E = E { i fQ (w(t) 0 [ z ( t ) - x ( t ) ] ) 2 d t } S u b s t i t u t i n g x ( t ) = g ( t ) ® [ A ( t ) . ( y ( t ) + n ( t ) ) ] and z ( t ) = a ( t ) g x ( t ) and i n t e r c h a n g i n g the order of i n t e g r a t i o n and ex p e c t a t i o n y i e l d oo £ = Y / 0 { / / / / w ( e i ) a ( e 2 ) w ( B 3 ) a ( B 4 ) 0 x ( t ~ 6 r 3 2 ' t _ P 3 " B 4 ) d e i d e 2 d B 3 d B 4 + •//// w ( 3 1 ) g ( B 2 ) A ( t - f 3 1 - e 2 ) w ( 6 3 ) g ( B 4 ) A ( t - B 3 - B 4 ) [0y(t-61-B2,t-63-B4) + 0 y n ( t _ 8 r B 2 ' t - B 3 " B 4 ) + 0 n y ( t - B r e 2 ' t - 6 3 - B 4 ) + * n<t-V B2' t _Vty l dM B2 d B3 d?4 -2 / / / / w ( B 1)a ( B 2 ) w ( B 3)g ( 3 4)A ( t - 3 3-.0 4 ) [ 0 x ( t - B ^ , t - B -B .) + 0 ( t - B 1 - B - , t - B , - B / ) ] d B , d B 0 d B , d B . } d t xn 1 2 3 4 1 2 3 4 Now, assume that x ( t ) and n ( t ) are s t a t i o n a r y processes and the c r o s s -c o r r e l a t i o n of y ( t ) and n ( t ) s a t i s f y the r e l a t i o n 0 y n ( T ) = b(x) 0 0 ^ ( x ) , where b(x) i s a r e a l f u n c t i o n . I t then f o l l o w s that ^ ( T ) = b(x)gf(_ T)Q0 ( T ) 10 S u b s t i t u t i n g and i n t e g r a t i n g over t y i e l d oo e .//// W (B 1)a([3 2)w (e 3)a (B 4 )0 x(-3 1-6 2 +e 3+i3 4)dr3 1 dB 2 dB 3 dB 4 CO + //// w(3 1)g (B 2)w (g 3)g(6 4 )A(-6 1-i3 2+i3 3+3 4 -e)[$ n(-f3 1 -B 2+i3 3+6 4) —CO oo + ///f(8 5)f(3 6)c(3 7 ) 0 x(-3 1-3 2+3 3+3 4-3 5+3 6-3 7)d3 5d3 6d3 7 ] dg 1d3 2d3 3d3 4 •—CO CO - 2 ///// w(3 1)a(3 2)w(g 3)g(3 4)f(3 5 ) b(3 6 ) 0 x(-3 1-3 2+3 3+3 4+3 5 - B 6 ) —OO d^dBjde^B^BjdBg (1) where b ( T ) = 1 + b ( T ) and c(x) = 1 + b(x) + b ( - x ) . I n s p e c t i o n of (1) leads to the c o n c l u s i o n that unless some r e s -t r i c t i o n i s placed on the p r e f i l t e r output s i g n a l y ( t ) , e can be made a r b i t r a r i l y s m a l l . In t h i s regard, impose the f o l l o w i n g power c o n s t r a i n t E { ( k ( t ) e y ( t ) ) 2 } = P where k ( t ) i s a r e a l f u n c t i o n . Taking e x p e c t a t i o n and expanding give oo -//// k ( 3 1 ) k ( 3 2 ) f ( 3 3 ) f ( g 4 ) 0 x(-3 i+3 2-3 3+3 4)d3 1d3 2 dg 3d3 4 = P (2) The problem i s to s e l e c t f i l t e r impulse responses f ( t ) and g ( t ) such that e i n (1) i s minimized sub j e c t to c o n s t r a i n t ( 2 ) . Use of v a r i a t i o n a l c a l c u l u s [24-26] shows that CO ////// w(3 1).g(3 2)w(3 3)g(3 4 ) A(-3 1-3 2+3 3+3 4 - e)f(3 6)c(3 7) —oo • 0 x(-3 1-3 2+3 3+3 4-t+3 6-3 7)d3 1d3 2d3 3d3 4d3 6d3 7 CO - ///// w(3 1)a(3 2)w(3 3)g(3 4)b(3 6 ) 0 x(-3 1 - B 2+3 3+3 4+t-3 6)d3 1d3 2d3 d3 d3 -co j H D co + X • /// k ( B 1)k(3 2)f(3 4 ) 0 x(-3 1+3 2-t+3 4)d3 1d3 2d3 4 —CO 11 n - r a l l t, unrealizeable f i l t e r s (3a) = 0 for { ' t > 0, realizeable f i l t e r s . — (3b) and that oo * ~ oo // f(t)f(x){J7./7 w(g 1)g(6 2)w(B 3)g(B 4)A(-g 1-6 2+3 3+g 4-6)c(B 7) • 0X (- 3X- $2+ 33+ B 4 - t + T - B?) d e 1d B2d B3d B4d B ? + X // k(B 1)k(B 2)0 x(-B 1+p -t+ T)dB 1d3 2 }d t t iT > 0 (4) — CO are necessary and sufficient conditions for a pr e f i l t e r impulse response f(t) that minimizes e for a fixed p o s t f i l t e r impulse response g(t). The Lagrange multiplier X must be chosen so that the solution of (3) for f(t) satisfies (2). Assume X >_0; then (4) i s satisfied for a l l possible variations, f ( t ) , of f(t) when C(f) > 0, where C(f) is the Fourier transform of c( t ) . The assumption X _> 0 w i l l be validated in Section 2.2. Similarly, i t can be shown that / / / w(B1)w(B3)g(B4)A(-B1-t+B3+B4-e) ^(-Bj-t+B-j+B^ — 0 0 + ./// f (B 5)f(B 6)c(B 7)0 x(-B 1-t+B 3+B 4-B 5+B 6-B 7)dB 5dB 6dB 7]dB 1dB 3dB 4 - / / / / / w(B1)a(B2)w(B3)f (B 5)b(B6)0 x(-B 1-B 2+B 3+t+B 5-e 6)dB 1dB 2dB 3dB 5dB 6 _ Q ^ a l l t, unrealizeable f i l t e r s (5a) t >_ 0, realizeable f i l t e r s (5b) and //g(t)g(x){// w(B1)w(B3)A(-B1-t+B3+x-e)[0 (-B1-W-B3+T) + /// f(6 5)f(B 6)c(B 7)0 x(-B 1-t+B 3+x-B 5+B 6-B 7)dB 5dB 6dB 7]dB 1dB 3}dtdx > 0. (6) 12 are the necessary and s u f f i c i e n t c o n d i t i o n s f o r a p o s t f i l t e r impulse response g ( t ) that minimizes z f o r a f i x e d p r e f l i t e r . C o n d i t i o n (6) i s s a t i s f i e d f o r a l l p o s s i b l e v a r i a t i o n s , g ( t ) , of g ( t ) when C(f) > 0. Therefore, assume the f i x e d and known r e a l f u n c t i o n C ( f ) = 1 + B ( f ) + B*(f) i s p o s i t i v e . Then (3) and (5) c o n s t i t u t e a set of necessary c o n d i t i o n s t h a t must be s a t i s f i e d by f ( t ) and g ( t ) i n order to be a s o l u t i o n to the j o i n t o p t i m i z a t i o n problem. I f the p h y s i c a l r e a l i z e a b i l i t y c o n s t r a i n t i s not imposed on the pre and p o s t f i l t e r s , then the necessary c o n d i t i o n s f o r the o p t i m a l p r e f i l t e r and p o s t f i l t e r t r a n s f e r f u n c t i o n s can e a s i l y be obtained by t a k i n g the F o u r i e r transforms of (3a) and (5a), r e s p e c t i v e l y . On the other hand, i f the f i l t e r s are c o n s t r a i n e d to be p h y s i c a l l y r e a l i z e a b l e , then the f i l t e r t r a n s f e r f u n c t i o n s can be obtained from Wiener-Hopf equations (3b) and (5b) by the method of s p e c t r a l f a c t o r i z a t i o n [25,26]. In any case, from (3) F ( f ) = A ( f ) B ( f ) G * ( f ) | W ( f ) | 2 $ x ( f ) / D ( f ) (7a) and F ( f ) = [ l / D + ( f ) ] [ A ( f ) B ( f ) G * ( f ) | W ( f ) | 2 $ ( f ) / D " ( f ) l (7b) where 2 ? D(f) = $ ( f ) [ * | K ( f ) | Z + C ( f ) I |G(f+kf )W(f+kf ) T ] X i s s k=-oo are the p r e f i l t e r t r a n s f e r f u n c t i o n s f o r unrealizeab'le and r e a l i z e a b l e optimal f i l t e r s , r e s p e c t i v e l y . S i m i l a r l y , from (5) G(f) = A ( f ) B ( f ) F * ( f ) | W ( f ) | 2 * x ( f ) / E ( f ) (8a) and G(f) = [ l / E + ( f ) ] [ A ( f ) B ( f ) F * ( f ) | W ( f ) | 2 $ ( f ) / E ~ ( f ) l (8b) where (8c) are the t r a n s f e r f u n c t i o n s of the optimal u n r e a l i z e a b l e and r e a l i z e a b l e p o s t f i l t e r s . r e s p e c t i v e l y . Since 13 there may be occasion to c a n c e l the common f a c t o r $ ( f ) from (7a) and the common 2 2 f a c t o r |W(f) | from (8a), * x ( f ) a n < i |W(f)| are assumed p o s i t i v e almost every-where. These assumptions are not c r i t i c a l s i n c e v i r t u a l l y a l l i n p u t s i g n a l power s p e c t r a and weighting f u n c t i o n s of p r a c t i c a l i n t e r e s t s a t i s f y these 2 c o n d i t i o n s . In any case, |W(f)| = 0 i s meaningless s i n c e f o r t h i s c o n d i t i o n F ( f ) and G(f) can take on a r b i t r a r y v a l u e s . A l s o , i t i s obvious that i f $ x ( f ) = 0 for.some frequency f, the o p t i m a l f i l t e r s have the t r i v i a l s o l u t i o n F ( f ) = G(f) = 0 . In both (7b) and (8b), U + ( f ) i s used to denote a l l the l e f t - h a l f plane poles and zeros of any f u n c t i o n U ( f ) , and U ( f ) i s used to denote the r i g h t - h a l f plane poles and zeros. Furthermore, [V(f)/U ( f ) ] + = f Q q ( T ) e J dx where q ( r ) =/°° [V(f )/u"(f) ] e j 2 7 T f T d f . — 0 0 At t h i s p o i n t , i t i s e s s e n t i a l to s t r e s s that simultaneous s a t i s f a c -t i o n of (7) and (8) i s not a s u f f i c i e n t c o n d i t i o n f o r an o p t i m a l system. In f a c t , i t i s the l a c k of a unique s o l u t i o n of the necessary c o n d i t i o n s that n e c e s s i t a t e s the arguments i n the remainder of t h i s Chapter. 2.2 Further Necessary Conditions Since the j o i n t o p t i m i z a t i o n problem i s e a s i e r s o l v e d i n the frequency domain, equations (1) and (2) are expressed i n the f o l l o w i n g e q u i v a l e n t forms by repeated a p p l i c a t i o n of P a r s e v a l ' s theorem, 00 e = fm |A(f)W(f)| 2$ ( f ) d f + r | G ( f ) W ( f ) | 2 I $ (f+kf )df _ o o I I x - o o 1 , n s k=-°° 2 °° ? +/°° |G(f)W(f)| Z C(f+kf )|F(f+kf )| $ (f+kf )df -°° k=-» s s x s -2f°° A * ( f ) B * ( f ) F ( f ) G ( f ) |W(f) | 2$ ( f ) d f (9) o and /" | F ( f ) K ( f ) | 2 $ ( f ) d f = P (10) —co 1 X 14 where B(f) = 1 + B(f) and C(f) = 1 + B(f) + B * ( f ) . For the remainder of t h i s Chapter, the optimal f i l t e r s are assumed 2 unreali z e a b l e and functions |K(f)| , B ( f ) , and C(f) are assumed p e r i o d i c with period f g . Also, C(f) > 0. In a d d i t i o n , we assume i n i t i a l l y that F(a) and G(ct) are non-zero at some frequency a. I t then follows from (7a) and (8a) that fo r any integer k, F(o+kf )/F(ct) = A(cc+kf )G*(o+kf ) |W(a+kf ) | 2/A(a)G*(a) |w(a) | 2 (11) s s s 1 s 1 1 1 G(a+kf )/G(a) = A(a+kf )F*(a+kf )$ (a+kf )/A(a)F*(a)$ (a) (12) S S S x s X S u b s t i t u t i n g (11) in t o (12) gives 6(o+kf )/G(a) = G(a+kf )|A(a+kf )W(o+kf )| 2$ (a+kf )/G(a)IA(a)W(a) I 2$ (a) s s 1 s s ' x s 1 1 x (13) Equation (13) i s s a t i s f i e d i f F(cc+kf ) = G(a+kf ) = 0, or i f s s |A(a+kf )W(a+kf )| 2$ (a+kf )/|A(a)W(a) |2<& (a) = 1 (14) ' S S ' X S ' 1 X In general, (14) i s not s a t i s f i e d f o r a r b i t r a r y integer values of k since 2 |A(f)W(f)| $ x ( f ) Is u s u a l l y aperiodic. Hence, i t may be concluded that the optimal F and G are non-zero f o r at most one frequency i n the set f+kf , where k i s any integer i n c l u d i n g zero. For those frequencies f where F and G are non-zero the optimal F and G are r e l a t e d as follows. F ( f ) = A(f)B(f)G*(f ) | w(f)| 2/[x|K(f ) | 2 + C(f)|G(f)W(f)| 2] (15) G(f) = A ( f ) B ( f ) F * ( f ) * ( f ) / [ * (f ) + C(f)|F(f)| 2$ ( f ) ] . (16a) x n x s CO where « ( £ ) > £ » (f+kf ) (16b) n . n s s k=-°° From (15) and (16) i t follows that X - |G(f)W(f)| 2$ ( f ) / | K ( f ) F ( f ) | 2 $ (f) i 1 n 1 1 x s Thus, X i s r e a l and non-negative as assumed i n Section 2.1 Sub s t i t u t i o n of 2 (16) i n t o (15) y i e l d s a quadratic equation i n |F| , which, when solved gives 15 | F ( f ) | 2 = [ l / C ( f ) ] [ | A ( f ) B ( f ) W ( f ) / K ( f ) | / 4 (f)/X$ ( f ) - * ( f ) / * ( f ) ] (17) 1 1 1 1 n x n x s s S u b s t i t u t i o n of (15) i n t o (16) gives |G(f) | 2 = '[l/C(f)][|A(f)B(f)K(f)/W(f) | A * x ( f ) / * n (f)-x|K(f) | 2/|W(f) | 2] (18) s The p o s i t i v e s i g n s o l u t i o n s of (17) and (18) are chosen f o r the quadratic equations 2 2 because |F| and |G| must be non-negative for a l l f. For t h i s reason, |F| and |G| are n e c e s s a r i l y given by (17) and (18) at only those frequencies f o r which |A(f)B(f)W(f)/K(f)| >/A$n ( f ) / $ x ( f ) (19) s f o r a l l other frequencies, F = G = 0. Let be the s e t of frequencies f o r which F and G are non-zero. S u b s t i t u t i o n of (17) i n t o (10) gives 1//X= (P+/[|K(f) | V ( f ) / C ( f ) ] d f ) / / n |A(f)B(f)K(f)W(f) |/$ (f)$ ( f ) / C ( f ) d f fin- u ' x n s s S u b s t i t u t i o n of G(f) from (16) i n t o (9) y i e l d s ( 2°) ^ A ( f ) W ( f ) | 2 $ (f){$ ( f ) + | F ( f ) | 2 $ ( f ) [ C ( f ) - | B ( f ) j 2 ] } i x n x e =J _ — — ^ - — • — d f -°° $ n (f) + C ( f ) | F ( f ) | 2 $ x ( f ) • s ' I t follows that the phase of F has no e f f e c t on E provided G i s obtained from (16). S i m i l a r l y , the phase of G i s a r b i t r a r y provided F i s obtained from (15). S u b s t i t u t i o n of (20) i n t o (17) and the r e s u l t i n g equation i n t o (21) y i e l d s e - / |A(f)W(f)| 2$ ( f ) d f + / n | A ( f ) W ( f ) | 2 $ ( f ) [ l - | B ( f ) | 2 / C ( f ) ] d f +{/ n|A(f)B(f)K(f)W(f)|/$ (f)$ (f)/C(f)df} 2/{P+ T |K(f)| 2$ (f)/C(f)df> U n x U n S S (22) where f2 contains a l l frequencies not included i n Q. The f i r s t i n t e g r a l i n (22) r e s u l t s from f i l t e r i n g x ( t ) . The other two i n t e g r a l s r e s u l t from inband d i s t o r t i o n . 16 2.3 Algo r i t h m f o r Determining the J o i n t l y Optimal P r e f i l t e r and P o s t f i l t e r I t remains to s e l e c t the frequency set ft which minimizes e i n (22). A c c o r d i n g l y , define the frequency set T ={q:-f _<q<f /2} and l e t T be a subset max s s of frequencies chosen from the s e t T . For each frequency qeT d e f i n e the n . max- max i n t e g e r s e t I which contains e x a c t l y one element chosen from the set of a l l p o s s i b l e i n t e g e r numbers. Frequency s e t ft can now be defined as f f : f = q+kf where q-eT and k e l are chosen so (19) n = \ S q r (23) I i s s a t i s f i e d f o r a l l feft. The problem i s to s e l e c t the s e t s T and I to minimize e i n (22) subje c t to c o n s t r a i n t (19). Based on the r e s u l t s of S e c t i o n 2.4, the f o l l o w i n g a l g o r i t h m i s presented f o r determining the j o i n t l y o p t i m a l f i l t e r s . S u b s c r i p t "o" w i l l designate the o p t i m a l ft, T, I and A. 1. Define f o r a l l |q| < f g / 2 (k:|A(q+kf )W(q+kf )| 2$ (q+kf ) i s maximized \ S S X S J I A > (24) q o /where qe T ( max With T = T and k e l c a l c u l a t e X from (20). max q n o I f (19) i s s a t i s f i e d f o r a l l feft, where ft i s de f i n e d by (23) w i t h T = T and I =1 , s e t X =X, T =T and max q q o o o '«'ft =£2 and go to step 4. Otherwise go to step 2. 2. Define, f o r any X f q:qe T and (19) i s s a t i s f i e d f o r a l l T m ( X , I ) J q m a X ^ (25) q o ) f = q+kf where k el Set X equal to a p o s i t i v e r e a l value. C a l c u l a t e P from A P, - / _ { [ | A ( f ) B ( f ) K ( f ) W ( f ) | / $ ( f U ( f ) / A - | K ( f ) | 2 $ ( f ) ] / C ( f ) } d f s s (26) The i n t e g r a t i o n i n (26), obtained by s o l v i n g (20) f o r P, i s over the set ft c o n s i s t i n g of a l l f=q+kf where qeT and k e l s m q ° 3. Repeat step 2 f o r d i f f e r e n t values of X, thereby obtaining P, vs. X. Choose as X the value of X which makes P = P. X o \ Let the r e s u l t i n g T = T and the r e s u l t i n g ft=ft . m o o 4 . At t h i s point X . T . I , and ft are s p e c i f i e d . For feft , o o q o o n o set F(f)=G(f)=0. For feft , determine | F ( f ) | from (17); the o 1 1 phase of F i s a r b i t r a r y , provided G(f) i s obtained from (16). A l t e r n a t i v e l y , |G(f)| may be obtained from (18) with the phase of G(f) being selected a r b i t r a r i l y , provided F ( f ) i s obtained from (15). A t h i r d a l t e r n a t i v e i s to obtain |F| and |G| from (17) and (18), r e s p e c t i v e l y , and to then s e l e c t the phase of F and G to make FGA*B* r e a l f o r a l l feft . In any case, the o r e s u l t i n g mimimum e i s given by (22) with ft=ftQ 2.4 V a l i d i t y of the Optimization Algorithm*" In t h i s Section, the v a l i d i t y of the preceding algorithm i s established. From among a l l s o l u t i o n s of necessary conditions (17), (18;, and (19) the recom-mended procedure determines the one that minimizes e f o r a given X- However, an optimal system was defined as one that minimizes e f o r a given P, not a given X. Thus, i f we can show that there i s no s o l u t i o n of the necessary conditions, regardless, of the choice of ft and X, which simultaneously possesses power equal to, and e r r o r l e s s than that obtainable using the algorithm, then the v a l i d i t y of the algorithm i s proved. •'. For any p o s i t i v e r e a l X and any integer set I define fq:q i s contained i n a subset of T a n& (19) i s ] f ( x , i ) - ) m a x } q [ ^ s a t i s f i e d f o r a l l f=q+kf g where k e l ^ . j (27) The approach used here i s not unlike that used by others [12,13] i n optimizing the l i n e a r transmitter and r e c e i v e r i n pulse amplitude modulation systems. 18 Note that ¥(X,I ) C T ) where I and T (A,I ) are de f i n e d by (24) and q m q q m q n n o n o ^o (25), r e s p e c t i v e l y . I t w i l l now be shown that f o r any p o s i t i v e r e a l X, there e x i s t no p o s i t i v e r e a l X^ X and no set s I ^1 and ¥(X,I ) such that q q o q P[X,V(X,IJ, I J = P[X, Tm(x,I„ ), I„ ] (28) and q q q o q Q e[X,V(X,I ), I ] .< e[X,T (X , I ) , I ] (29) q q - m ' q Q q Q where the dependence of e i n (22) and P i n (26) on X, I , T(x» I ) > and ) q q q has been made e x p l i c i t . Define f o r a l l qe T ' max p(q) = |A(q+kf )B(q+kf )W(q+kf )| /$ (q+kf )/C(q+kf) k e I (30) S S S X S S CJ o p(q)-Ap(q) = |A(q+kf )B(q+kf )W(q+kf ) | •* (q+kf )/C(q+kf ) k E I (31) S S S X S S C[ X ( q ) = |K(q)|/* n (q)/C(q) (32a) s Since |K|, $ , and C are assumed p e r i o d i c w i t h p e r i o d f s x(q+kf s) = x(q) (32b) From the d e f i n i t i o n of I , Ap(q) > 0 f o r a l l q e > • Define J=T (X.,I ) . » _ 1^. xuax m q Then any frequency set f(X , I ^ ) i n (27) can be expressed as ¥(X , I^) = J-AJ^+A^* where AJ^ c J i s a frequency set d e l e t e d from J and AJ^'P J i s a frequency set added to J i n order to compose ¥(X , I ^ ) . Let j = fx and Y+Ay = / x \ Use of (30) , (31) , (32) , the c o n s t r a i n t that P i n (26) remain constant, as i n (28), and the change of v a r i a b l e f=q+kf g y i e l d s Y ( Y + A Y ) A P = Y ( Y + A Y ) [ P C Y + A Y . J - A J ^ A J , I ) - P ( Y , J , I ) ] 2 2 = / [ Y ( P _ A P ) X _ Y ( Y + A Y ) X ]dq - / [ ( Y + A Y ) X P - Y ( Y + A Y ) X ] d q J-AJ 1+AJ 2 J • • , = 0 (33) 19 From (20) and (22) i t follows that e=/2jA(f)w(f)| 2$ x(f)df - f {[fA(f)B(f)W(f)| 2$ (f)-|A(f)B(f)K(f)W(f)| A* (f)$ (f)J/C(f)}df • (34) Q x x n^ Use of (30), (31), (32), (34) and the substitution f=q+kfg gives the following increment i n e . Ae = e [ y + A y , J - A J ^ + A J ^ I ] - e [ y , J , I 1 q o = - J" ( p - A p ) [ ( p - A p ) - (y+A Y )x]dq + / p[p- Y x]<iq (35) J - A J 1 + A J 2 We now show that A e > 0 independent of whether A Y > 0 or A Y < 0. Consider f i r s t the case A Y >_ 0. From (19), (24), (25), and (27) i t follows that J - A J . . + A J , , = ¥ [ y + A y , I ] c T ( y , I ) = J . Therefore, A J ~ is an empty frequency ± z q m q z . o set. Since (19) requires that both terms in square brackets in (35) be non-negative, and because Ap>_ 0, A e >_ 0 for A Y > 0. Finally, assume that A Y < 0. Addition of (33) and (35) gives Ae = A e + Y ( Y + A Y ) P 2 2 = /jt - ( p - A p ) + ( Y + A Y ) (p-Ap )x + p - YPX+ Y(p-Ap)x 2 2 2 - y ( Y + A Y ) x ~ ( Y +A Y )px + Y ( Y + A Y ) x l d q + / [ ( p - A p ) - A J 1 + A J 2 2 + ( Y + A Y ) ( p - A p )x + Y ( p - A p ) x - Y(Y+AY)X ldq (36) Rearrangement of (36) gives 2 Ae= / Ap (p-yx )dq + / Ap [ (p-Ap ) - (y+Ay )x ]dq + / (p-yx ) dq J - A J ^ J - A J ^ + A ^ A J 1 + / ( - A y ) x ( p - A y ) d q + / [ - ( p - y X ) [ ( p - A p ) - ( y + A y ^ l d q (37) A J 1 A J 2 From (19), (25), (27), the periodicity of yx and the definitions of J , A J ^ and J 2 : 20 1. P - YX >0 V q e J and, therefore, v q e A J ^ and v q e J - A J ^ 2. ( p - A p ) - ( Y + A y ) x > 0 V q e J - A J 1 + A J 2 and, therefore, V q e A J " 2 3. p - Y X < 0 V q e A J 0 i f J is a proper subset of T . — ^ 2 max 4. A J _ is an empty set i f J = T 2 r J max It follows that Ae > 0 as claimed, and that with I as given by qo (24), T is obtained by varying X in (26) unti l a value X = X i s found such o o that P[X ,T (X ,1 ) , I ] = P . The optimal frequency set i s then ° m o <30 q Q •T = T [X ,1 ] c T o m o q max o 21 3. ANALYSIS AND DISCUSSION OF SOME IMPORTANT FILTERING SCHEMES 3.1 A p p l i c a t i o n of Optimal F i l t e r i n g to S ome S p e c i f i c Cases In t h i s Section the optimal frequency set £2q i s determined f o r some s p e c i f i c cases. 3.1.1 High Sampling Rate In p r a c t i c e $ x ( f ) 0 as f •>• », i n which case ftQ contains only frequen-cies f o r which | f | < f g / 2 i f f i s s u f f i c i e n t l y l arge. I f A = B = C = | K | = |w| = 1, for a l l f e f t Q , then the optimal f i l t e r s and the r e s u l t i n g e are i d e n t i c a l to the r e s u l t s f o r unsampled systems [10] . The lowpass equivalent of the l i n e a r i z e d analog modulation system i n F i g . 2 can be represented as i n F i g . 1 i f G and H are combined and T = 0 i n Fi g . 1. For amplitude modulation P represents the power i n the modulated s i g n a l , 2 2 2 provided |K(f)| =1. For angle modulation |K(f)| = f , i n which case P equals the mean square bandwidth [27]-3.1.2 Lowpass Signals A s i g n a l u(t) with power spectrum U'('f) i s lowpass i f dU/df<0 for f>0. I f w(t)©a(t)9x(t) i s lowpass, then from (24) i t follows that f o r a l l |q.|<f /2, I contains the integer zero. Also, i f d{ | B ( f ) / K ( f ) I//$ ( f ) } / df < 0 S q o n s for f>0 a p p l i c a t i o n of the algorithm shows that Q q contains a l l frequencies |f|<W. I f the s o l u t i o n f o r V i n the following equation, obtained by s u b s t i t u t i n g (20) i n t o (19), i s l e s s than fg/2, then W=V. V V ) K(V) [ | A ( f ) B ( f ) K ( f ) W ( f ) | / C ( f ) ] / 5 ^ ( r 7 $ x W df -# 00 = U(V)B(V)W(V) I " ' ( 3 8 ) "s P + /.,[|K(f)| 2$ ( f ) / C ( f ) ] df —V ' ' n s I f V >f /2 then W = f /2. — s s 22 x(t) F ( f ) 1 V H(f) | G ( f ) i ! xft) CHANNEL F i g . 2. Lowpass equivalent of a l i n e a r i z e d analog-modulation-system. 3.1.3 Weak Noise For $ (f) s u f f i c i e n t l y small i n e q u a l i t y (19) i s s a t i s f i e d f o r a l l s f=q+kf where |q|<f /2 and k e l . Hence, ft w i l l contain a frequency band s s q o whose t o t a l width equals f . The smaller $ ( f ) , the greater IA(f)B(f)W(f)/K(f) s n 1 s i s , compared to /x$ ( f ) / $ ( f ) , and the more accurate are the following approx-, xi x s imate equations, obtained from (16), (17), (20), and (22). 1 / / 5 T = P// n [|A(f)B(f)K(f)W(f)|/$ (f)$ ( f ) / C ( f ) ] df o ft n x o s (39a) | F ( f ) | = |A(f)B(f)W(f)/K(f)|/$ (f)/X <& ( f ) / C ( f ) 1 1 1 1 n o x s (39b) e = / |A(f)W(f)| 2$ ( f ) d f +/ |A(f)W(f)| 2$ (f) [ l - | B ( f ) | 2 / C ( f ) . ] d f ft X ft X o o +{/ [|A(f)B(f)K(f)W(f)|/C(f)]/$ (f)$ (f) df}^/P o n x (39c) G(f) = A ( f ) B ( f ) / C ( f ) F ( f ) (39d) - I f A ( f ) B ( f ) / C ( f ) i s constant, F and G become r e c i p r o c a l f i l t e r s . . 2 2 Also, i f $ ( f ) | w ( f ) / K ( f ) I i s constant, then | F ( f ) | i s p r o p o r t i o n a l to n s 23 l / / $ x ( f ) , i n which case F becomes a " h a l f - w h i t e n i n g " f i l t e r [14] 3.1.4 Systems f o r which p(f) = IA(f)B(f)W(f)/K(f ) I/$ ( f ) / $ (?) x n s i s Constant Let T denote the set of frequencies f o r which |A(f )W(f) |'"$ ( f ) > 0, and l e t p ( f ) = p Q f o r a l l feT, where p^ i s any constant. S u b s t i t u t i o n of p Q i n t o (20) shows that ( 1 9 ) i s always s a t i s f i e d f o r any Q which contains at most one frequency i n the set f=q+kf g where feT, |q| < a n d ^ ^ s anY i n t e g e r . I t f o l l o w s that i f W < f where W i s the t o t a l width of the band of frequencies r s r M . i n T, and i f no frequencies i n r c o i n c i d e under t r a n s l a t i o n by k f g where k i s any non-zero i n t e g e r , then ^ 0 = r . With p ( f ) = p Q f o r a l l non-zero p ( f ) , ( 1 7 ) and ( 1 8 ) give | F ( f ) | 2 = [*n ( f ) / $ x ( f ) C ( f ) ] [ p o / A ~ o - l ] (40a) s | G ( f ) | 2 = [ X O | K ( f ) | 2 / C ( f ) | W ( f ) | 2 [ p o / A " o - l ] (40b) 2 2 f o r a l l fen . I f $ /C$ and IKI / C l w l are constant f o r a l l fefi , then (40) o n x 1 1 1 1 o s shows th a t |F| and |G| are constant f o r a l l f e ^ . 3.2 Some Suboptimal F i l t e r i n g Schemes In the f o l l o w i n g d i s c u s s i o n on suboptimal f i l t e r i n g schemes the frequency set 9. has the property that i f feft then f + k f g e f i f o r any non-zero i n t e g e r k, where fi i s as de f i n e d i n (23). 3.2.1- Weak Noise F i l t e r s When | A ( f ) B ( f ) W ( f ) / K ( f ) | i s s u f f i c i e n t l y l a r g e r than A~$ ( f ) / $ ( f ) n x s the o p t i m a l F and G are r e l a t e d approximately by (39d), i n which case they w i l l be c a l l e d weak n o i s e f i l t e r s . With F and G c o n s t r a i n e d f o r feQ, and w i t h 24 F=G=0 f o r feft , i t f o l l o w s from s u b s t i t u t i n g (39d) i n t o (9) tha t e - / |A ( f )W( f ) | 2 $ ( f ) d f + T |A( f )W( f ) |2<5 ( f ) [ l - | B ( f ) | 2 / C ( f ) ] d f 5 " X + / f i [ | A ( f ) B ( f ) W ( f ) | 2 $ n ( f ) / C 2 ( f ) | F ( f ) | 2 ] d f (41) s From S c h w a r t z ' s i n e q u a l i t y / f i [ | A ( f ) B ( f ) W ( f ) | 2 $ n ( f ) / C 2 ( f ) | F ( f ) | 2 ] d f / f l | F ( f ) K ( f ) | 2 * ( f ) d f s i { / 0 [ | A ( f ) B ( f ) K ( f ) W ( f ) | / $ ( f ) * ( f ) / C ( f ) ] d f } 2 s E q u a l i t y h o l d s i f , and on l y i f u | F ( f ) | 2 = | A ( f ) B ( f ) W ( f ) / K ( f ) | / $ ( f ) / $ ( f ) / C ( f ) (42) s where u i s any c o n s t a n t . Thus, to m in im ize (41) sub jec t to (10) r e q u i r e s tha t 2 | F | s a t i s f y (42) and , from (10 ) , tha t u = / n [ | A ( f ) B ( f ) K ( f ) | / $ ( f ) $ ( f ) / C ( f ) ] d f / P (43) a n x s The r e s u l t i n g mean-square e r r o r i s g i ven by e - / | A ( f ) W ( f ) | \ ( f ) d f + / j A ( f ) W ( f ) | 2 $ ( f ) [ l - | B ( f ) | 2 / C ( f ) ] d f a + {/ [ | A ( f ) B ( f ) K ( f ) W ( f ) | / * n ( f ) $ x ( f ) / C ( f ) ] d f } 2 / P (44) s If f requency s e t s Q i n (44) and (22) are i d e n t i c a l then the two equat ions d i f f e r 2 o n l y tha t J"_IK(f)I $ ( f ) / C ( f ) ] d f i s m i s s i n g from the denominator i n the l a s t Sl1 ' n s 2 term of ( 22 ) . Whenever P >> / f i [ | K ( f ) | $ n ( f ) / C ( f ) ] d f , as i s n e a r l y always the s c a s e , the performance o b t a i n a b l e u s i n g weak n o i s e f i l t e r s i s e s s e n t i a l l y e q u i v a l e n t to tha t o b t a i n a b l e u s i n g o p t i m a l f i l t e r s . Weak n o i s e f i l t e r s have a p r a c t i c a l advantage wh ich o p t i m a l f i l t e r s do not have ; t h e i r t r a n s f e r c h a r a c t e r i s t i c s are e s s e n t i a l l y dependent on l y on the r e l a t i v e power s p e c t r a o f x ( t ) and n ( t ) . I n s p e c t i o n of ( 17 ) , ( 18 ) , (39d) , and (42) r e v e a l s tha t f o r op t ima l f i l t e r s , p r e c i s e knowledge of s p e c t r a $ x ( f ) and 25 $ (f) i s required, whereas for weak noise f i l t e r s , only knowledge of their s relative values i s necessary to .determine the f i l t e r s to within a gain factor. Adjustment of the f i l t e r gains to satisfy power constraint can easily be accomplished during installation. 3.2.2 Optimal P r e f i l t e r ; Constant Amplitude Po s t f i l t e r Let A=A1, C=Clf |K|=K , |G| =G , and |w|=l for a l l feft, where A^, B^, C^, and are constants. From.(7) i t follows that the optimal | F | = F 1 for a l l fen, and from (10 )that F 1 = (P/KJ / n $ x ( f ) d f ) 1 / 2 (45) Let the phase of F and G are such that FG=F^ G^  for a l l fen. Substitu-tion of | F | = F 1 , |G|=G1 and FG=F^ G^  into (9) yields the optimal as follows G 1 = (A 1B 1P/K 2)/F 1(PC 1/K 2 + /^$ n (f)df) (46) s The following equation gives the resulting error. B 2 A 2B 2 " * n ( f ) e - .AJ/ • (f)df + A 2 ( l - ^ ) V x ( f ) d f + ( f ) d f V n ( f ) d f / ( ^ + / n - g ^ d f ) " 1 C. s K 1 1 1 (A 7) If the sets £2 in (22) and (47) are identical, e i n (47) exceeds e in (22) because the numerator of the last term in (47) exceeds the numerator in the last term of (22), unless $ (f)/$ (f) i s constant for a l l fen, in which x n s case (22) and (47) give the same e . 3.2.3 Constant Amplitude P r e f i l t e r ; Optimal P o s t f i l t e r Let A=A1, B=B1, C=C1, •|K|=K , |F| = F^ and |w|=l for a l l f e n , where A^, B^, C^, and F^  are constants. From (10) F^  i s given by (45). Substitu-tion of |F|=F^ in (16) yields the optimal G as follows G(f) = A.B F*(f)« ( f ) / ( * (f) + C nF 2$ (f)) (48) 1 1 x n 1 1 x s 26 S u b s t i t u t i o n of (45) and (48) i n t o (9) g i v e s B 2 A 2 B 2 $ n ( f ) e=A2/ * ( f ) d f + A 2 ( l - A/o* ( f ) d f + -M/0[* (f)* (f)/(-7 + F2$ ( f ) ) ] d f 1 ft x . 1 • • C x fl x c 2 ft n s x C 1 1 x (49) 27 4 OPTIMAL AND SUBOPTIMAL FILTERING IN PAM, PCM, AND DPCM COMMUNICATION SYSTEMS . • In t h i s Chapter, optimal and suboptimal f i l t e r i n g schemes are applied to PAM, PCM and DPCM communication systems. System errors i n PAM, PCM, and DPCM are evaluated f o r optimal pre and p o s t f i l t e r i n g , optimal pre-f i l t e r i n g only, and optimal p o s t f i l t e r i n g only schemes when the input power spectrum i s f i r s t - o r d e r Butterworth. The r e s u l t i n g performances are compared with the optimum t h e o r e t i c a l l y a t t a i n a b l e as cal c u l a t e d from information theory. 4.1 Pulse Amplitude Modulation (PAM) Fi g . 3 depicts, a multiplexed pulse amplitude modulation (PAM) system. Signals x ^ ( t ) , x 2 ( t ) , •.'.. ,x^(t)'and n (t). are assumed to be s t a t i s t i c a l l y independent random processes with power s p e c t r a l d e n s i t i e s $ (f) (i=l,2,...,M) X i and $ ( f ) ; Refe r r i n g to the procedure used i n Section 2.1 the mean-squared n '.'.'..;. ^ ^. • ' . •' ^.Yi e r r o r (no weighting between x^(t) and the desired i output z^(t)=a^(t)8x^(t) i s given by ' 9 e i = E { T f0 ( z i ( t ) " X i ( t ) ) d t } In F i g . 3a l e t f ± ( t ) , g ± ( t ) , o ( t ) , h(t) and l ( t ) denote, r e s p e c t i v e l y the impulse responses of f i l t e r s having t r a n s f e r functions F ^ ( f ) , G ^ ( f ) , 0 ( f ) , H(f) and L ( f ) . Since x. (t) = g ^ t ^ b (t) i t follows that e ^ f / J [ z j c t ) - 2 z i ( t ) x i ( t ) . + ( X i ( t ) ) 2 ] d t } - Y ' X ( 0 ) - 2 O i ^ ^ b 2 ( t - 8 r t ) d B + O ^ . g 1 - ( B 1 ) g 1 ( B 2 ) 0 b . ( t - e 1 , t - B >d B ldB 2}dt i i i i Since b,(t)=A ( t ) . [ J l ( t ) 8 n ( t ) + E (y (t) • A . (t) )0u(t) ] where u(t)=o(t)8h(t)8Jl(t)/M, and since the y.'s and n are uncorrelated 1 c < 8 b i Z i ( t ^ ) = A 1 ( t ) O ^ u ( 8 2 ) a i ( B 3 ) A . ( t - B 2 ) 0 y i X i ( t - B 2 , T - B 3 ) d B 2 d B 3 CHANNEL r LINEAR TRANSMITTER s(t), CHANNEL FILTER H(t) -4+ L I (a) LINEAR RECEIVER ft]/M)L(f) T Z {(t-kT+8) SAMPLE AT t = kT-6 -*>((( <• SAMPLE AT t;kT+9-8 (b) H(f) 1 •MWC 0 MWC (c) . F i g . 3 (a) M u l t i p l e x e d PAM system. Functions F ^ f ) , G ± ( f ) , ( i = l , 2 , ... ,M) , 0 ( f ) , H ( f ) , and L ( f ) are t r a n s f e r functions of l i n e a r f i l t e r s . S i g n a l y ^ ( t ) i s sampled by sampler at t=kT+9.-G, where k i s any i n t e g e r . Impulse t r a i n A. (t)=T. Z°° 5(t-RT-G.+G). 1 I k=-°° 1 (b) E q u i v a l e n t r e p r e s e n t a t i o n of a sampler. (c) Channel f i l t e r t r a n s f e r c h a r a c t e r i s t i c f o r Example 2. oo 29 0 B ( t,T ) = A i ( t ) A i ( x ) { / _ o o / _ o o i l ( B 3 ) ) l ( B 4 ) 0 N ( t - 6 3 » T - g 4 ) d B 3 d 3 4 i c-M + / " o o / " o o o ( g 3 ) o ( g 4 ) [ z Aj ( t - 6 3 ) A j ( T - 6 4 ) 0 y ^ ( t - B 3 , T - B 4 ) d B 3 d B 4 } S u b s t i t u t i n g and i n t e g r a t i n g over t , g i ves e±-0 ( O ) - 2 / V ^ 8 1 ( B 1 ) u ( B 2 ) a 1 ( B 3 ) A 1 ( B 2 ^ 1 - e ) 0 y i X i ( - B 2 + e 1 - 0 , B 1 - B 3 + e 1 - e ) d B 1 d B 2 d B 3 + ////8±( B^ 8. < B2> A (B3> % ( B4) A ± ( B ^ 6 2 + 6 i " e ) 0 n ( " 3 3 + 9 i ~ 9 ' h~62~S4+ 9 i " 9 ) d 3 l d B 2 d 3 3 d 3 4 -oo 1 C M + / / / / 8 i ( B 1 ) 8 i ( B 2 ) 0 ( B 3 ) 0 ( B 4 ) A i ( 6 1 - B 2 + e i - e ) E A . ' ( - 8 3 + 8 ^ - 8 ) A. ( B 1 - B 2 - B 3 + 9 ± - 6 ) -oo j = l J J ^ ( - B 3 + e ± - e » B ^ B 2 _ 6 3 + e ± - e ) d B ^ B 2d g 3d g 4 S ince x . ( t ) and n ( t ) a re s t a t i o n a r y , 0 ( t , x ) , 0 ( t , x ) and ^ i 1 c 0 ( t , x ) depend on l y on t - x . A p p l y i n g P a r s e v a l ' s theorem repea ted l y and us ing i 2 $ ( f ) = | F . ( f ) l $ ( f ) and $ ( f ) = F . ( f ) $ ( f ) , e . can be expressed as y ± 1 i 1 x ± y ^ 1 x ± ' 1 £, = • / " | A . ( f ) | 2 $ ( f ) d f + ( i ) / ° ° | G . ( f ) | E |L( f+kf ) U (f+kf )d f i -oo 1 i 1 x . M - 0 0 1 1 1 1 s 1 n s 1 k=-«° c M CO +/°° | G , ( f ) | E | U . ( f ) | 2 E | F . ( f +k f ) | 2 f ( f+kf )d f -2 / ° ° F . ( f ) G . ( f ) A * ( f ) U ( f ) $ ( f ) d f —  1 j = 1' 1 k = _ i 3 S 1 X j S -co 1 1 1 x ± (50a) CO X U(f ) = E O ( f + k f )H(f+kf )L ( f+k f ) /M . (50b) S S S k=-°° O  U m ( f )= [ E 0 ( f+k f )H( f+kf )L ( f+k f )exp{ -2JTT ( f+kf )(e -e:)}]/M (50c) *** t s s s s xn x The t r a n s m i t t e r power S^ i s g i ven by ST = E { | / J s 2 ( t ) d t } (51a) M ', . - I S (51b) i = l where 30 S 4 = (1/M)/ 0 0 | F . ( f ) | V (f) Z 10 (f+kf ) | 2 d f (51c) l k=-°° i s the transmitter power associated with the i * " * 1 s i g n a l x ^ ( t ) . The requirement of d i s t o r t i o n l e s s channel transmission i s often imposed, which means that there i s no intersymbol or interchannel d i s t o r t i o n at sampling times t=kT+ei-9 (i=l,.2,.... ,M) . Replacement of the samplers i n F i g . 3a by t h e i r equivalent representations i n F i g . 3b[28], followed by a p p l i c a t i o n of Smith's [24] r e s u l t s shows that d i s t o r t i o n l e s s transmission occurs i f and only i f 0 0 (1/M) E 0(f+kf )H(f+kf )L(f+kf )exp{-j2Tr(f+kf )(e.-6.)}= 6. .A (52) , s s s s i i i i k=-oo where A i s any r e a l constant and <5^ j=0 f o r i ^ j and 5^=1 f o r i = j . S u b s t i t u t i o n of (52) i n t o (50a) gives an equation i d e n t i c a l to (9), provided the following s u b s t i t u t i o n s are made: * x ( f ) = * x (f)» F(f)=F ( f ) , G(f)=G ±(f) ,'-B(f)=A, C(f)=A 2, |W(f)| 2=l,$ (f)=$ ( f ) | L ( f ) | 2 / M , and e=e.. The f i l t e r s F .(f) and G.(f) c which minimize subject to the.constraint (51c) can now be obtained using the 2 2 2 method of Chapter 2, with P=S ± and |K(f)| =|K ( f ) | = |0 ( f ) | /M. In many cases of i n t e r e s t , $ (f)=$.(f) and S =S=S /M, i n which case the optimal f i l t e r s and x i x . i T the r e s u l t i n g are independent of i . Example 1: Let 0(f)=L(f)=H(f) , where H(f) i s as shown i n F i g . 3c. Let Q^Qj 3 1 ( i - j ) T / M , f =2W ,$ (f)=N 12, S.=S=SjM and A. (f)=1 f o r a l l i . Let. s c n o i l l c ^a/2a f <_ a $ x (f) .- $ x ( f ) =1 (i=l,2,...,M) 1 [ 0 f > a The system i s d i s t o r t i o n l e s s , and (9) and (10) apply with $ n(f)=N q/(2M) v|.f|<MW , A(f)=B(f)=C(f)=|K(f)| 2=|W(f)| 2=1, «' (f)=N /2, and P=S. The optimal F and 1 , 1 1 n o s G are given i n Section 3.1.4. I f a < Wc then F and G are i d e a l lowpass f i l t e r s of bandwidth a, and E / O = (1+S/aN ) ^ . I f a > W then n i s not unique but contains o — c any frequency set of t o t a l bandwidth 2W^  chosen from the set | f | < a i n such 31 a way t h a t i t s elements do not c o i n c i d e under frequency t r a n s l a t i o n k f , where k i s any non-zero i n t e g e r . In t h i s case e/a=(l-W /a)+(W /a)(l+S/W N ) ^, c c c o Example 2: Consider the same d i s t o r t i o n l e s s PAM system i n Example 1, w i t h A ± ( f ) = l , S±=S and .« ( f ) = $ ( f ) = aa/Tr(f 2+a 2) (1=1,2,... ,M) X . X 1 S e c t i o n s 2.3 and 3.1.2 show that f o r o p t i m a l pre and p o s t f i l t e r i n g ft contains a l l f r e q u encies If I < W, where W < f /2. F i g . 4 shows e/o vs. S/N W f o r v a r i o u s s o c Wc f o r the o p t i m a l pre and p o s t f i l t e r case ( 0 ) , as w e l l as f o r the o p t i m a l pre-f i l t e r o n l y case (PR) i n S e c t i o n 3.2.2 and the o p t i m a l p o s t f i l t e r only case (PO) i n S e c t i o n 3.2.3. The frequency s e t s ft i n a l l three cases are assumed i d e n t i c a l to the set determined f o r the optimal pre and p o s t f i l t e r case. F i g . 5 shows the s i g n a l - t o - n o i s e r a t i o SNR = / * ( f ) d f / [ e - /.* ( f ) d f ] (53) »u X X f o r each of the above f i l t e r i n g schemes. The d i f f e r e n c e i n both E and SNR f o r the three cases i s seen to be s i g n i f i c a n t f o r W^ /a > 10. I f the channel n o i s e n c ( t ) i n Examples 1 and 2 i s white Gaussian, then the c a p a c i t y per message i s C=W l o g 0 ( l 4 S / N W ) (54) c 2 o c I f x_^(t) i s a Gaussian process w i t h $ ( f ) as i n Example 2, then the r a t e d i s t o r -i t i o n f u n c t i o n i s expressed p a r a m e t r i c a l l y i n 0 as [25] R(0) = ( 2 a / l n 2 ) ( 0 - t a n - 1 0 ) (55a) e ( 0 ) / a = l+(2/Tr ) ( [ 0/(l+ 0 2)]-tan - 1 ( 0 ) ) (55b) The optimum performance t h e o r e t i c a l l y o b t a i n a b l e (OPTA) by a communication system w i t h c a p a c i t y C i s obtained when R(0)=C. Using (54) and (55), the OPTA curves shown i n F i g . 4 were obtained. 32 Fig. 4 Normalized error e/a versus S/N W for Example 2. Symbols 0, PR, and PO denote optimal f i l t e r s , optimal prefilter-constant amplitude po s t f i l t e r , and constant amplitude prefilter-optimal p o s t f i l t e r , respectively. For the curves shown the optimal f i l t e r bandwidth W=W . c 10,000 SNR l.OOOl 1 0 0 1 S/N0WC 1000 Fig. 5 Signal-to-noise ratio SNR for Example 2. PO are defined in the caption of Fig. 4. the op timal W=W . Symbols 0, PR, and For the curves shown 34 For any given C one p a i r of values o f S/N W and W w i l l minimize £ o c c and the r e s u l t i n g e represents the minimum d i s t o r t i o n o b t a i n a b l e by a channel w i t h the corresponding C f o r the PAM system under c o n s i d e r a t i o n . F i g . 10a shows f o r cases (0) and (PR) the minimum e/a and the corresponding W /a f o r Example 2. F i g . 10a can a l s o be regarded as a p l o t of the minimum C vs E. I n g e n e r a l , e w i l l exceed t h a t shown i n F i g . 10a s i n c e the values of- S/NQ and W"c f o r the given channel w i l l not be those which minimize e f o r the r e s u l t i n g C. The a n a l y s i s r e l a t i v e to Examples 1 and 2 and the r e s u l t s i n F i g s . 4,5» and 10a a l s o apply when an M-channel s i n g l e sideband, s u p p r e s s e d - c a r r i e r , amplitude modulated (SSB AM/SC) system i s used on the channel described i n Example 2, provided S i s i n t e r p r e t e d as the power i n the tr a n s m i t t e d s i g n a l . The a n a l y s i s a l s o a p p l i e s to an M-channel double sideband (DSB) AM/SC system provided S again i s i n t e r p r e t e d as the power i n the tr a n s m i t t e d s i g n a l , except that the bandwidth per message f o r an DSB AM/SC system i s twice that r e q u i r e d by an SSB AM/SC system. 4.2 Pul s e Code Modulation fPCM) , I n t h i s S e c t i o n the PCM system shown i n F i g . 7 i s analyzed assuming x ( t ) , and t h e r e f o r e , y ( t ) i s Gaussian, and the mapping of the q u a n t i z e r output samples v i n t o v i s by a memoryless d i g i t a l channel. C o r r e l a t i o n funcations are obtained using the Hermite polynomial expansion of the Gaussian p r o b a b i l i t y d e n s i t y f u n c t i o n i n S e c t i o n 4.2.1 and the pre and p o s t f i l t e r i n g problem i s examined i n S e c t i o n 4.2.2. 4.2.1 C o r r e l a t i o n Functions f o r Quantized S i g n a l s Transmitted Over  D i s c r e t e Memoryless Channels F i g . 6a shows a s i g n a l y ( t ) which i s sampled, qua n t i z e d , and t r a n s -DISCRETE MEMORY-LESS CHANNEL TRANSITION PROBABILITY Pjj= P(v=Vj \ v = v . ) (a) QUANTIZER v = Q(y) v(t) TIME CONTINUOUS MEMORYL ESS CHA NNEL TRANSITION PROBABILITY f ) j = - P ( v ( . t ) = V j . : \ v ( t ) = vf) v(t) (b) y 0 = - m y1 N V N-1 y y y2 3 '=Q[yJ mm ma m* m mym " n s I N-1 v2 (c) yN=<° F i g . 6 (a) Quantizer and discrete memoryless channel. (b) System equivalent to the system shown in Fig. (c) Quantizer characteristic. 36 m i t t e d over a d i s c r e t e memoryless communication channe l . In any r e a l i s t i c sys tem, the sampl ing o p e r a t i o n must be performed be fo re q u a n t i z a t i o n and d i g i t a l t r a n s -m i s s i o n . However, s i n c e the q u a n t i z e r and d i g i t a l channel a re assumed to be memoryless, F i g . 6a can be rep resen ted f o r a n a l y s i s purposes by the a n a l y t i c a l l y e q u i v a l e n t system o f F i g . 6b. In F i g . 6b, l e t 0 ~(T) = E [ y ( t ) v ( t - x ) ] denote the c r o s s c o r r e l a t i o n f u n c t i o n o f s t a t i o n a r y random processes y ( t ) and v ( t ) , 0^(x) denote the a u t o c o r r e l a t i o n f u n c t i o n of v ( t ) , and l e t n ( t ) = v ( x ) - y ( t ) . A l though 0 *(T ) , - 0 ( x ) , and 0 (x) have been ob ta ined when the channel i n F i g . 6 i s yv n yn n o i s e l e s s [ 31 ,33 ] , t ha t i s , when 1 i= j P i i = { the purpose o f t h i s S e c t i o n i s to o b t a i n 0~(x) , 0 " (x) , 0 ( x ) , and 0 (x) when , v yv n yn y ( t ) i s a Gauss ian p r o c e s s and d i g i t a l t r a n s m i s s i o n i s not e r r o r - f r e e . L e t P ( A , B ) and P ( A / B ) denote , r e s p e c t i v e l y , the j o i n t p r o b a b i l i t y o f events A and B and the p r o b a b i l i t y o f event A g i v e n B. I f Py(c t ,3 ;x) denotes the second o r d e r amp l i t ude p r o b a b i l i t y d e n s i t y o f y ( t ) a t t imes t and t - x , then 0; (T ) = E [ v ( t ) v ( t - x ) ] N N N N „ \ ' = E Z v . v . E I P [ v ( t ) = v . | v ( t - x ) = v , v ( t )=v , v ( t - x )=v ] i - 1 1=1 1 j k= l m=l 1 j k m P[v ( t - x )=v^ . / v ( t ) =v k , v ( t - x ) = v m ] P [ v ( t ) = v k , v ( t - x ) = v m ] S ince 1 " ^Equa t ion (56) f o l l o w s f rom P[v(mT)=v /v(mT-nT)=v. ]=• 1 - i o n*0 where m and n are any i n t e g e r s , and T i s the sampl ing p e r i o d o f the PCM system shown i n F i g . 6 a . ' 37 P.. T=0 P[v(t)=v./v(t - r)=v, ] ={ K l (56) 1 K 0 x^O then N N N N- y y 0- (T)= E E V.V. E E P, .P . / k / m p (O:,8;T) dadg (57) i = l j = l . k=l m=l J k-1 7m-l 3 2 I f y ( t ) i s a Gaussian process with mean u , variance a , a u t o c o r r e l a t i o n 2 2 function 0^(T) and c o r r e l a t i o n c o e f f i c i e n t 6^(T)= [0^(x)-y ]/a then [29,30] oo g n ( x ) p v(a,B ; T ) = [ ^ - T ] [ e x p { - [ ( a - y ) 2 + ( 6 - y ) 2 ] / 2 a 2 } ] E -X r H ( ^ ) H (-^) (58) y 2 ™ 2 n=0 n! n a n a H (r) = ( - l ) n e r 2 / 2 d n ( e _ r 2 / 2 ) / d r n (59a) n We note here the orthogonality property of the Hermite polynomials ( r ) , ' °° 1 - r 2 / 2 n! m=n -°° 0 mfn N S u b s t i t u t i o n of (58) i n t o (57) followed by a p p l i c a t i o n of . (59b) and E P,.=l 1=1 k l y i e l d s 0-X T ) = o2 E a 2 [ 6 v ( T ) ] n (60) . n=0 y where N v. N y a = ( 1 / v ^ ) E . — E P.,/* e x p [ - ( r - u ) 2 / 2 a 2 ] d r - 1=1 0 k=l K 1 y k - l = E ( v ) / a (61a) and, f o r n > 1 N v.-c N an =• [ ( - l ) n / v & n E ( 1 N ) E P, . r (k) (61b) n i = l ° k=l k l n F n ( k ) = ? " 1 [ ( y k - u ) / a ] - ; n - 1 [ ( y k _ i - M ) / a ] (61c) $ n ( r ) = d n [ e " r 2 / 2 ] / d r n (61d) where c i s any constant, n 3 A s i m i l a r approach gives 38 ' 0^(T) = E [ y ( t ) v ( t - T ) ] N N oo y, = E E v.P / a t / 3 p (a ,B;x)dg]da i = l j = l 1 J 1 -=o y. y = a 2ta 16 y(x)+(y/a)a ( )] <62) Equations (61) and (62) can be used with n ( t ) = v ( t ) - y ( t ) , 6 (x) = 6 (-T) and 0~ (T) = 0 *(-x) to obtain 0 (x) and 0 (x) i n y y vy yv n yn F i g . 6 as follows. 0 n(x) = 0;(x) + 0 y(x) - 0 y;(x) - 0 y ; ( - t ) = a 2 [ ( p / o - a 0 ) 2 + ( 1 - a ^ 2 6 y(x) + E a 2 6 y U ( x ) ] (63) 0 y n ( r ) = a2 [ ( y / a ) a 0 - ( y / a ) 2 ] + ( a 1 - l ) 6 y ( x ) (64) 2 Let R (x)=0 (x)-[E(u)] and R (x)=0 (x)-E(u)E(x) , where u and x are any XI Xi UX XIX stationary random processes. I t follows from (61a) and E(y)=y that 2 °° R~(x) = a E 2. n . . . v i a (5 (x) (65a) n=l n y R y v ( x ) = ^ i ^ y ^ ) ( 6 5 b ^ OO R (x) = o 2 [ ( l - a 1 ) 2 5 (x) + E aV(x)] (65c) 1 Y n = 2 n y . R y n ( x ) = o 2 ( a 1 - l ) 6 y ( x ) (65d) In the Appendix, a l t e r n a t i v e exact expressions, which are sometimes more a t t r a c t i v e than the Hermite s e r i e s expansions, are presented, as w e l l as some useful approximations and bounds f o r the c o r r e l a t i o n functions. Also included i n the Appendix are the r e s u l t s of t h i s Section applied to the p o s t f i l t e r i n g problem. I t i s shown that the e f f e c t of c r o s s c o r r e l a t i o n 0 (x) yn on both the optimal reconstruction p o s t f i l t e r and the reconstruction e r r o r i s small i f the channel i s s u f f i c i e n t l y good and i f the number of l e v e l s N i n both Max and optimal quantizers i s s u f f i c i e n t l y l a r g e . In general, the same conclusion did not apply f o r small N or f o r poor channels. 39 4.2.2 Pre and P o s t f i l t e r i n g i n PCM Systems I t f o l l o w s from the a n a l y s i s i n the previous S e c t i o n and Appendix A that the PCM system i n F i g . 7 can be represented as i n F i g . 1. Without l o s s i n g e n e r a l i t y l e t E[x(t)]=E[y(t) ] = y =0, then from (61), (63) and (64) 0 ( T) = (a - 1 ) 0 (T) (66a) yn 1 y OO 0 (T ) = P { a 2 + ( l - a . ) 2 [ 0 (T)/P]•+ Z a f [ 0 (x)/P] } (66b) n 0 1 y k=2 y N v. N „ - i a = [ ( - l ) n / 2 i m ! ] Z (-%) Z P . . { $ n l [ (y .) / v^"]-$ n _ i [y ._ . ) / & ] } (66c) n i = l A j=l  J 1 2 3 x $ n ( r ) = d n [ e x p ( - r 2 / 2 ) ] / d r n (66d) 2 where P i s given by (10) w i t h | K ( f ) | = 1,. 0 ( T) = E [ y ( t ) y ( t - t ) ] , 0 n ( x ) = E [ n ( t ) n ( t — r ) ] and 0 y n ( T ) = E [ y ( t ) n ( t - x ) ] . T r a n s i t i o n p r o b a b i l i t y P =P(v=v./v=v.) depends on the encoding and decoding scheme as w e l l as on the l j 3 i modulator, demodulator and p h y s i c a l channel. From (66a) i t f o l l o w s t h a t B ( f ) = a 1 - 1 . Therefore B(f)=1+B(f ) = a ± and C(f)=1+B(f)+B*(f ) =2a -1. Although ^ n ( f ) c a n D e obtained by F o u r i e r transforming 0 n ( T ) i n (66b) c a l c u l a t i o n i s t e d i o u s , p a r t i c u l a r l y when i t has tc be performed f o r s e v e r a l values of T. Ruchkin [34] and Robertson [35] have shown th a t f o r (67) $ ( f ) i n (16b) i s approximately constant f o r a l a r g e c l a s s of $ ( f ) and a l l but very coarse q u a n t i z i n g and very s m a l l T. One would expect t h i s c o n c l u s i o n to h o l d f o r any P..t i n which case $ ( f ) = T ' 0 (n) where IT n n s t The statement i s supported by the r e s u l t s i n Appendix A.4 when a Max uniform or nonuniform q u a n t i z e r i s used and P i s g i v e n by (71a). DIGITAL CHANNEL xft) LINEAR PREFILTER F(f) y(t) ENCODER MODULATOR CHANNEL DEMODULATOR DECODER 1 LINEAR POSTFILTER G(f) 1 (a) y a- . VN-l (b) Fig. 7 (a) A PCM system. (b) Quantizer transfer characteristic. 41 0 n(O) = E(n 2) N N y 2 2 - Z E P. . / (v.-a) p (a) p (a)d a (68) i=l j=l 1 3 y i - l 3 y y and py(ct) Is the amplitude probablity density of y(t) . If the quantizer output levels are optimally spaced [36] then y y v = / 1 ctp (a)da/ /- 1 p (a)d a (69) 1 y i - l y y i - l y i n which case the crosscorrelation between quantization noise and noise resulting from channel transmission errors i s zero and N N y. 0 ( 0 ) - Z f Y ± (v.-a) 2p y(a)da + X r V V V ^ p ( a ) d a (70) , " i=i v Z X J J . i J - y y i - l The y^'s and v j ' s °f t n e quantizer are usually proportional to /P~, in which case $^ (f) i s proportional to P. It follows that e in Chapters 2 s 2 —2 and 3 i s independent of P while |F| and |G| are proportional to P. In fact, substitution of $ (f)=TP0 (0) into (9) where 0 (0)=E(n2) when P=l, followed n n n §> . ,2 by replacement of P by /_ |K(f)F(f)| $ (f)df and subsequent minimization of e co . X with respect to F and G i s an alternative way to obtain the optimal F, G and e for PCM without use of the power constraint (10), Example 3: For the PCM system i n Fig. 7 l e t d be such that d=log2N takes on only integer values, and let d . d-d V P ± j = p 1 3(1-p) 1 3 (71a) p = Q[/(2S/N of gd)] (71b) 2 ' •> Q(x) = (1/^7) e" r 1 1 dr (71c) where d ^ i s the Hamming distance between the binary numbers i and j . 42 Equation (71a) gives P which r e s u l t s when the quantizer output amplitudes are n a t u r a l binary coded and each d i g i t i s then transmitted over a white Guassian channel of bandwidth W^  using optimum demodulation and binary antipodal modu-la t e d signals whose average power equals S and whose t o t a l energy l i e s i n the frequency band of the channel. Fi g . 8 shows e/a vs. S/N W fo r optimal pre and p o s t f i l t e r s when o c $^(f) and the modulator, demodulator, and channel are as i n Example 2 with M=l and f d=2W , i n which case P.. i s given by (71). Max [37] nonuniform s c i j quantizers were used, with d chosen to minimize e/a. The e f f e c t of a non-optimal d on e/a was not c r i t i c a l . For example, with S/N W = 10 and o c W^/a = 1000, the optimal d=3 y i e l d s e/a = .00385 f o r the (0) case, as opposed to e/a=.00490 f o r the non-optimal value of d=6. Also shown i s 0PTA as given by (55), and e/a f o r the (PR) and (PO) cases. F i g . 9 shows SNR as given by (53). We note here that a furthe r ( s l i g h t ) improvement i n both e and SNR 2 i s p o s s i b l e i f the quantizer i s chosen to minimize E(n ) i n (68) [38], since Max nonuniform quantizers are optimal i f and only i f P i s given by (67). As i n the PAM case, one S/N^ W^ , W^  p a i r minimizes e f o r a given value of C i n (54). F i g . 10a shows t h i s minimum e vs. C/a f o r the (0) and (PR) cases i n Example 3. For these two curves and f o r the curves i n F i g s . 8 and 9 the optimal f i l t e r bandwidth W=W^ /d. Example 4 If f d <_ C, where C i s the capacity of the p h y s i c a l channel,, then optimal encoding and decoding makes -P^  as given by (67)^' Although optimal codes are unrealizable because of. t h e i r i n f i n i t e block/length, they provide a bound on the performance obtainable using any encoding scheme. The e and SNR tFor optimal codes d i s constrained to assume values f o r which 2^ i s a p o s i t i v e integer. . 1 : I : : J JO wo woo S/N0 Wc Fig. 8 Normalized e r r o r e/o versus S/N W f o r Example 3. Symbols 0. PR, and PO are defined i n the caption of F i g . 4. The number of b i t s of quantization d which minimizes e/o for the given values of W /a and S/N W i s shown on the curves. For the O c curves shown the optimal f i l t e r bandwidth W=W /d. 1000 Fig. 9 Signal-to-noise ratio SNR for Example 3. For the curves shown the optimal f i l t e r bandwidth W=W /d. (a) W /a=10. c (b) WC/a=100. (c) WC/a=1000. c i 1000 10,000 0.01 0.001 100 1000 (b) C/a 10,000 F i g . 10 (a) Normalized minimum e r r o r e / a f o r Examples•2,3, and 4 and optimal channel bandwidth W /a f o r Examples 2 and 3. The number of b i t s of q u a n t i z a t i o n d which minimizes e / a are shown. Also shown i s the OPTA. (b) E r r o r e / a and optimum feedback c o e f f i c i e n t ct- f o r DPCM (Example 5). The three f i l t e r i n g schemes, 0, PR, and PO, y i e l d i d e n t i c a l e / a when i s chosen to minimize e . I n d i c a t e d f o r comparison purposes are PAM and PCM r e s u l t s obtained from F i g . 10a: x-PAM 0 and o-PCM 0, optimal code. 46 obtained using optimal codes with f gd=C equals that which r e s u l t s when an analog s i g n a l x(t) i s p r e f i l t e r e d , sampled and quantized, stored d i g i t a l l y i n a memory having no read and write errors at a rate of C b i t s per second, and l a t e r p o s t f i l t e r e d . Let P.. be given by (67), l e t f d=C and l e t the modulator, demodulator, i j s channel and $ x ( f ) be as given i n Example 2 with M=l, i n which case C=W l o g (1+S/N W ). F i g . 10a shows e/a and the optimal d vs. C/a f o r the f i l t e r s c 2 o c discussed i n Section 3.2. Max nonuniform quantizers were used. When x(t) i s non-Gaussian exact t h e o r e t i c a l optimization of the system i n F i g . 7 i s impossible, since the second order amplitude p r o b a b i l i t y density of y ( t ) at times t and T cannot be c a l c u l a t e d from F ( f ) and the s t a t i s t i c s of x ( t ) . Consequently, 0 ( T) and $ (f) cannot be c a l c u l a t e d exactly and an y n n s approach using approximations i s required. 4.3 D i f f e r e n t i a l Pulse Code Modulation (DPCM) A DPCM system i s shown i n F i g . 11. Input x ( t ) i s p r e f i l t e r e d , sampled and compared with a l i n e a r p r e d i c t i o n y, of y, based on i t s past. The dif f e r e n c e e between y and the predicted value y i s quantized and transmitted over a d i g i t a l channel. At the re c e i v e r a p r e d i c t i o n of y i s also made based k on previously received samples and added to the present received sample g i v i n g r ^ . The sequence {r^} i s then p o s t f i l t e r e d to y i e l d x ( t ) . I f d i g i t a l transmission i s e r r o r - f r e e , or i f the d i g i t a l channel represents a d i g i t a l memory having no read-write e r r o r s , then the reconstructed samples r^=y^+q^, and the systems i n Fi g s . 11a and l i b y i e l d the same x ( t ) . The system i n F i g . l i b can be represented as shown i n F i g . 1 provided n(t)=q(t) i s the noise r e s u l t i n g from quantization of e(t>. I f one makes the standard assumption v a l i d f o r N > 8, [39-42] that the feedback quantization noise i s small i n comparison with e(t) then x(t) LINEAR PREFILTER SAMPLE AT tsT-kT+8,-*°<l< <~* 7 ^ QUANTIZER •'•k+*kl DIGITAL CHANNEL (see FIG 7a) + LINEAR FEEDBACK FILTER H(f) (a) y(t) s e(t) QUANTIZER ERROR -FREE DIGITAL CHANNEL ) H(f) e(t)+q(t)-(b) to) LINEAR POSTFILTER + G(f) LINEAR FEEDBACK -FILTER H(f) A(t) yfthQ(t) H(f) LT Fig. 11 (a) A DPCM system. (b) An equivalent system to Fig. 11a when the di g i t a l channel is noiseless. Impulse train A(t)=T. E 6 (t-kT+0) -k=-°° 48 E [ e 2 ] = /™Jl-H(f)|2 | F ( f ) | 2 $ x ( f ) d f # Assuming that the spectrum of the sampled q u a n t i z a t i o n n o i s e i s f l a t , $ ( f ) = s T0 ( 0 ) where 0 ( 0 ) i s given by (68) w i t h p (a) r e p l a c i n g p ( a ) , n n e y I f x ( t ) i s Gaussian, then y ( t ) and e ( t ) are Gaussian, provided the feedback q u a n t i z a t i o n n o i s e i s neg l e c t e d . From (66) i t f o l l o w s that $ (f)= eq ( a ^ - l ) * ( f ) , from which one obtains $ n(f)=(a±-l) [1-H(f) ]*| F ( f ) | 2 $ x ( f ) where 2 i s obtained from (66c) w i t h P.. defined by (67) and P = E[e ]. Thus, B ( f ) = ( a i - l ) [ l - H ( f ) ] * . The input and output of the l i n e a r f i l t e r s H(f) i n F i g . 11a are s i g n a l s sampled every T seconds. I t f o l l o w s that f o r any r e a l i z a b l e H(f) no ^ - i 2 i T k f T l o s s i n g e n e r a l i t y r e s u l t s i f H(f) = Z a, e where L i s any p o s i t i v e k=l k i n t e g e r ; i f a j c = 0 ( k = l , 2 , . . . , L ) then a PCM system r e s u l t s . With t h i s H(f) and 2 x ( t ) Gaussian the a n a l y s i s i n Chapters 2 and 3 a p p l i e s w i t h |W(f)| =1, | K ( f ) | = 1 -j2TrkfT 1 - Z a k e k=l k (72a) B ( f ) = 1 + ( a i - l ) [ l - z a , e j 2 7 f k f T ] (72b) 1 k=l K C ( f ) = 1 + 2(a -1) [ 1 -Z a cos 2ivkfT] (72c) 1 k=l k provided C ( f ) > 0 . Functions K, B and C are p e r i o d i c i n 1/T as r e q u i r e d i n Chapter 2. We note here that the paragraph which immediately preceeds Example 3 i n S e c t i o n 4.2 a l s o a p p l i e s here. Example 5 For the DPCM system described above, l e t the channel c a p a c i t y C=f gd and the modulator, demodulator, channel and $ x ( f ) be as given i n Example 2 w i t h M=l, i n which case C=W log„(l+S/N W ). I f the p r e f i l t e r i s an i d e a l low-c z o c pass f i l t e r w i t h c u t o f f frequency W=f /d>>a, then {y } approaches a f i r s t - o r d e r S K Markov sequence and the l i n e a r p r e d i c t i o n of y^ which minimizes E[q^] i s obtained by c o n s i d e r i n g the most recent a v a i l a b l e sample y^_^ [391 Based on t h i s f a c t , i t i s a n t i c i p a t e d that u s i n g one sample of feedback (L=l) to minimize e w i l l be almost as good as using many samples, i n which case |K(f) | B ( f ) , and C(f) are given by (72) w i t h L=l. In order to ensure that C(f)>0, one must r e s t r i c t ]a^| < 1 and the q u a n t i z e r c h a r a c t e r i s t i c such that a^>0.75. Let the q u a n t i z e r be Max nonuniform [37]; then the r e s t r i c t i o n a^>0.75 i s s a t i s f i e d f o r d >_ 2. Sections 2.3 and 3.1.2 show that the optimal pre and p o s t f i l t e r c ontains a l l frequencies | f | < W where W <_ f g / 2 . F i g . 10b shows e / a vs. C/a f o r the three f i l t e r i n g schemes, 0, PR and PO when d=3,4 and 5. Also shown i s the value of c t ^ which minimizes e f o r the designated C/a f o r the 0 and PR cases. For the curves shown the o p t i m a l f i l t e r band-width i s W=f /2d. The f a c t that e / a vs. C/a i s the same f o r a l l three s f i l t e r i n g schemes i n d i c a t e s that whatever redundancy r e d u c t i o n i s not done by the p r e f i l t e r s w i l l be done by H ( f ) , provided c t ^ i s optimized. Note th a t the s m a l l e r d the c l o s e r i s e / a to OPTA. The reason f o r t h i s behaviour i s t hat f o r a f i x e d C, T decreases as d decreases, w i t h the r e s u l t that y^ becomes a more accurate p r e d i c t i o n of y^ and e / a decreases. Curves of e / a vs. C/a f o r d=2 and d=l are not shown because the v a l i d i t y of the assumption t h a t the feedback q u a n t i z a t i o n n o i s e i s s m a l l i n comparison w i t h e ( t ) becomes d o u b t f u l . When x ( t ) i n F i g . 11 i s non-Gaussian the above a n a l y s i s i s not n e c e s s a r i l y a p p l i c a b l e , i n which case an approach using approximations i s r e q u i r e d f o r o p t i m i z a t i o n . When the d i g i t a l channel i s n o i s y , an approach to o p t i m i z a t i o n using approximations i s a l s o necessary even when x ( t ) i s Gaussian. 50 4.4 System Comparisons F i g . 10a shows the advantage of optimal encoding r e l a t i v e to that of PCM n a t u r a l coding when S / N W and W are such t h a t , f o r n a t u r a l coding, e o c c i s minimized f o r a given c a p a c i t y C. This advantage i s even more pronounced when S / N Q W c and W"c are not chosen to minimize e f o r n a t u r a l coding. For given f i l t e r s of bandwidth W the d i f f e r e n c e s (not shown) i n S N R f o r the two types of coding are more pronounced than are the d i f f e r e n c e s i n e. The d i f f e r e n c e s i n e f o r PCM and DPCM are evident from F i g . 10b. I t f o l l o w s from (53) that f o r any given d and f i l t e r p a i r of bandwidth W the advantage i n DPCM over PCM w i t h respect to S N R i s even more pronounced than i s the advantage w i t h respect to e. The d i f f e r e n c e i n S N R between PCM and DPCM appears elsewhere f o r i d e a l lowpass f i l t e r s [40] • Comparison of F i g . 4 w i t h F i g . 8 shows that when S / N W i s s m a l l o c and W"c i s l a r g e , PCM tends to y i e l d lower values of e than does PAM, w h i l e the converse i s tru e when S / N W i s l a r g e and W i s s m a l l . F i g . 10a shows o c 0 c 0 t h a t f o r a l l values of C/a shown the best o b t a i n a b l e performance f o r PAM i s s u p e r i o r to that f o r PCM f o r both the 0 and PR cases. F i g . 10b shows the best PAM to be s u p e r i o r to optimum DPCM only f o r C/a s 100. F i n a l l y , we note that when x ( t ) and the channel are as described i n Example 2 w i t h M=l, e o b t a i n a b l e using o p t i m a l f i l t e r s , o p t i m a l q u a n t i z a t i o n and o p t i m a l encoding i s approximately twice OPTA f o r PCM and 1.3 times OPTA f o r DPCM w i t h d=3. A f u r t h e r ( s l i g h t ) decrease i n e f o r o p t i m a l coded PCM and DPCM i s p o s s i b l e i f the q u a n t i z e r outputs are entropyt coded p r i o r to being o p t i m a l l y channel-encoded [40,41]. tEntropy coding i s the same as Huffman coding or Shannon-Fano coding. 5. COMPUTER SIMULATION OF PRE AND POSTFILTERING IN PAM, PCM, AND DPCM SPEECH COMMUNICATION SYSTEMS 5.1 I n t r o d u c t i o n The purpose of the f o l l o w i n g Chapters i s to r e l a t e the r a t h e r general t h e o r e t i c a l study of the previous Chapters to the p r a c t i c a l design of speech communication systems. At t h i s time i t i s not p o s s i b l e to formulate a mathematically t r a c t a b l e e v a l u a t i o n c r i t e r i o n f o r speech that agrees com-p l e t e l y w i t h the s u b j e c t i v e judgments made by human a u d i t o r s . I t i s the l a c k of such a d e s i r a b l e mathematical c r i t e r i o n that n e c e s s i t a t e s s u b j e c t i v e measurements before more conclusions regarding the p o s s i b l e b e n e f i t s of pre and p o s t f i l t e r i n g i n speech communication systems can be reached. f o r PAM, PCM, and DPCM communication systems are simulated and s u b j e c t i v e l y evaluated when the i n p u t s i g n a l i s speech. Because of the e m p i r i c a l nature of the problem, the f i l t e r i n g schemes described i n the previous Chapters are not s t r i c t l y adhered t o , but are i n c o r p o r a t e d i n t o the s i m u l a t i o n w i t h p r a c t i c a l m o d i f i c a t i o n s . The f i l t e r i n g schemes considered are the unweighted (W(f) —1 Vf i n F i g . 1) weak n o i s e f i l t e r i n g scheme and the o p t i m a l p r e f i l t e r - c o n s t a n t a m p l i -tude p o s t f i l t e r scheme. Weak n o i s e f i l t e r s are used i n s t e a d of opt i m a l pre and p o s t f i l t e r s s i n c e weak noise f i l t e r s y i e l d v i r t u a l l y the same performance and have the p r a c t i c a l advantage that the f i l t e r t r a n s f e r c h a r a c t e r i s t i c s are e s s e n t i a l l y dependent only on the r e l a t i v e s p e c t r a of the i n p u t s i g n a l and the n o i s e . The opt i m a l p r e f i l t e r - c o n s t a n t amplitude p o s t f i l t e r scheme (see S e c t i o n 3.2.2) i s considered s i n c e under c e r t a i n p r a c t i c a l assumptions the I n the f o l l o w i n g Chapters, v a r i o u s pre and p o s t f i l t e r i n g schemes [|K(f)| 23> n ( f ) / C ( f ) ] df»l (see S e c t i o n 3.2.1) "s 52 f i l t e r s are lowpass. As lowpass f i l t e r s are commonly employed i n speech communication systems, t h i s scheme w i l l serve as a u s e f u l comparison f o r the weak noise f i l t e r i n g scheme. I n S e c t i o n 5.2, the assumptions and r e s t r i c t i o n s imposed i n the i n v e s t i g a t i o n are ta b u l a t e d . A l s o i n c l u d e d i s a Table g i v i n g the f i l t e r c h a r a c t e r i s t i c s used i n the s i m u l a t i o n of the PAM, PCM, and DPCM systems. S e c t i o n 5.3 i s devoted to a b r i e f e x p l a n a t i o n of the d i g i t a l s i m u l a t i o n f a c i l i t i e s . Both hardware and software are discussed. I n Chapter 6, the s u b j e c t i v e t e s t i n g method i s e x p l a i n e d and the s u b j e c t i v e r e s u l t s i n t e r p r e t e d . Included i s a h e u r i s t i c e x p l a n a t i o n f o r the s u b j e c t i v e behaviour of lowpass and weak n o i s e f i l t e r i n g schemes used i n PAM, PCM, and DPCM communication systems. F i n a l l y , a few concluding remarks are presented, i n c l u d i n g the f e a s i b i l i t y of using a frequency weighted mean-integral-squared e r r o r c r i t e r i o n as an o b j e c t i v e measure of speech q u a l i t y and the p o s s i b i l i t i e s of using weak no i s e f i l t e r s f o r PCM t e l e v i s i o n systems. 5.2 Assumptions and R e s t r i c t i o n s The f o l l o w i n g assumptions and r e s t r i c t i o n s are used i n the i n v e s -t i g a t i o n : 1) No inte r s y m b o l or i n t e r c h a n n e l d i s t o r t i o n i s assumed i n the PAM system shown i n F i g . 3. 2) E r r o r - f r e e t r a n s m i s s i o n i s assumed i n the d i g i t a l channel of the PCM and DPCM systems shown i n F i g s . 7 and 11, r e s p e c t i v e l y . 3) The PCM and DPCM qu a n t i z e r s are assumed to have nonuniform steps conforming to the y=100 l o g a r i t h m i c n o n l i n e a r i t y of Smith [45]. Smith has shown that such a q u a n t i z e r c h a r a c t e r i s t i c has the 53 p r a c t i c a l advantage t h a t when the q u a n t i z e r i n p u t s i g n a l i s speech the s i g n a l - t o - q u a n t i z a t i o n n o i s e r a t i o i s r e l a t i v e l y i n s e n s i t i v e to t a l k e r volumes. Log a r i t h m i c (u=100) q u a n t i z e r s w i t h 1,2,...7 b i t s of q u a n t i z a t i o n are sim u l a t e d . 4) The q u a n t i z e r overload v o l t a g e i s s e t at 4 times the RMS value of the q u a n t i z e r i n p u t s i g n a l . P u b l i s h e d r e s u l t s [46] show th a t f o r a wide c l a s s of f i l t e r s the instantaneous amplitude of f i l t e r e d speech has l e s s than a 1 percent p r o b a b i l i t y of exceeding 4X the RMS v a l u e , and s u b j e c t i v e t e s t s have confirmed that a 1 percent p r o b a b i l i t y of peak c l i p p i n g i s v i r t u a l l y undetectable [47]. For DPCM the RMS value i s evaluated by n e g l e c t i n g the q u a n t i z e r i n the feedback loop. 5) Previous-sample feedback i s used i n the DPCM s i m u l a t i o n . There-f o r e , from (72a), |K(f) | = |l - c t ^ e " ^ 2 7 ^ T |. The p r e d i c t i o n c o e f f i c i e n t ct^ i s s e t to the normalized a u t o c o r r e l a t i o n f u n c t i o n of the p r e . f i l t e r e d speech evaluated at the sampling p e r i o d T [39-41]. 6) The a d d i t i v e noise i n the PAM systems and the q u a n t i z a t i o n noise i n the PCM and DPCM systems are assumed to be u n c o r r e l a t e d w i t h the s i g n a l . Hence, B ( f ) = C(f) = 1. In a d d i t i o n , the sampled 00 q u a n t i z a t i o n n o i s e spectrum $ ( f ) = E $ (f+kf ) i s assumed to a3 k=-°° n s be constant f o r a l l frequency f . Since the systems are simulated on a d i g i t a l computer and are not operated i n r e a l - t i m e , A ( f ) i s a r b i t r a r i l y s e t to u n i t y f o r a l l f. I f r e a l - t i m e o p e r a t i o n was performed, A ( f ) would be the t r a n s f e r f u n c t i o n o f the system time-delay. 7) The systems t e s t e d are assumed to be of s u f f i c i e n t l y h i g h q u a l i t y 54 so that the frequency s e t ft, over which the p r e f i l t e r and po s t -f i l t e r t r a n s f e r f u n c t i o n s are non-zero, i s given by f :f = q + k f where qe [- 7 7 f , -i- f ] and k e l Q = J S N • • 2 s 2 s q r Integer s e t I i s defined by (24). In most cases of i n t e r e s t , ft w i l l be equal to the optimal s e t ft derived i n S e c t i o n 2.3. o 8) On the b a s i s of assumptions 5 ) , 6 ) , and 7 ) , and the r e s u l t s of Sections 3.1.3 and 3.2.1, p r e f i l t e r and p o s t f i l t e r c h a r a c t e r i s -t i c s are presented i n Table 5.1 f o r PAM, PCM, and DPCM. 9) From assumption 5 ) , ct^ i s adjusted i n the DPCM weak no i s e f i l t e r s i m u l a t i o n s so / | F ( f ) | 2 * ( f ) e " j 2 T r f T d f a _ -eo X  1 ~ oo / | F ( f ) [ 2 $ ( f ) d f X —co where | F ( f ) | 2 =/* (f ) / | l - c ^ e " ^ 7 1 * T | i s given i n Table 5.1. 10) Bandpass f i l t e r i n g , i f r e q u i r e d , i s achieved by concatenating highpass f i l t e r s (HPF) and lowpass f i l t e r s (LPF). The HPF has an a t t e n u a t i o n of 45 dB or more at 1/1.07 X the 3 dB c u t - o f f frequency and the LPF has an a t t e n u a t i o n of 45 dB o r more at 1 .07 X the 3 dB c u t - o f f (see F i g . 12). Frequency s e t ft i s p h y s i c a l l y r e a l i z e d by s e t t i n g the 3 db c u t - o f f frequencies of the HPF at 1.07f„ and those of the LPF at f /1.07, where f„ X U X, and f are the lower and upper c u t - o f f frequencies of the frequency bands comprising ft. Such frequency s c a l i n g i s expected to render n e g l i g i b l e the d i s t o r t i o n caused by a l i a s i n g . (73) 501 401 ^ 30[ g | ^ 101 0./ SCALE A SCALE B _ l L_ -I 1 I I I I _J I I I I I l _ 1.0 FREQUENCY (kHz) f a ) 10.0 tj o Lu |2<a Ui o LO 60r 501 S 401 § UJ —j 2 to 101 111/ 03 T3 2: o UJ 'SCALE' B 0.1 10 FREQUENCY (kHz) (b) Fig. 12 Typical frequency response. (a) Highpass f i l t e r . (b) Lowpass f i l t e r . 100 56 PAM PCM DPCM Unweighted Weak Noise Fi l t e r s | K f ) | 2 'V ( f ) / * (f) (f) ( f ) / | l - a e " j 2 1 T f T | , X . A. G(f) 1/F(f) 1/F(f) 1/F(f) ft from (73) with. A(f)-W(f)=l from (73) with A(f)=W(f)=l from (73) with A(f)=W(f)=l Optimal P r e f i l t e r -Constant Amplitude Po s t f i l t e r I.F(f)|2 1 1 1 G(f) 1/F(f) 1/F(f) 1/F(f) ft from (73) with A(f)=W(f)=l from (73) with A(f)=W(f)=l from (73) with A(f)=W(f)=l Table 5.1 P r e f i l t e r and Post f i l t e r Characteristics used i n the Computer Simulation 5.3 Di g i t a l Computer Simulation Application of d i g i t a l simulation to the subjective study of speech communication systems has been described by previous investigators [48-51]. It was demonstrated that savings i n time and money could be achieved through the elimination of extensive hardware construction. Additional benefits include ease and f l e x i b i l i t y in modifying system parameters, exact reproduction of data (stored on d i g i t a l tape), and precise control of the simulated system. In this Section, hardware and programming requirements peculiar to this simulation are discussed. The key to the simulations considered here i s found i n the realization of the pre and postfilters as analog bandpass f i l t e r s i n series with d i g i t a l f i l t e r s . Some of the advantages of using d i g i t a l f i l t e r s include very predict-able stable performance of ar b i t r a r i l y high precision and great ease i n changing f i l t e r response [52,53]. The equivalent realizations are shown i n Fig. 13. x(t) X(f)' Analog Pr e f i l t e r y(t) F(f), ft Y(f) / Y (f) • f s s x(t) X(f)' Analog Bandpass F i l t e r H(f), ft | Digital P r e f i l t e r ,V£) (a) • Digital Postfilter I r(t) R(f) R s ( f ) G(f), 'ft G s(f) Analog Bandpass F i l t e r H(f), ft x(ty x ( f ) (b) Fig. 13 Equivalent realization. (a) P r e f i l t e r . (b) P o s t f i l t e r . 58 The passbands occupied by the f i l t e r s are denoted as s e t ft and are con s t r a i n e d such t h a t there i s no overlapping' of passbands when t r a n s l a t e d by i n t e g e r oo m u l t i p l e s of the sampling frequency (no a l i a s i n g ) . I f U ( f ) = [ U ( f ) ] =Z U(f+kf ) , s s . s k=-°° then Y ( f ) = [ F ( f ) X ( f ) ] HF ( f ) [ H ( f ) X ( f ) ] (74a) s s s s and X(f)=G(f)R (f)=H(f)G ( f ) R ( f ) (74b) s s s Equivalence r e l a t i o n (74a) f o l l o w s from the f a c t t h a t [ F ( f ) X ( f ) ] = [ F ( f ) H ( f ) X ( f ) ] = F ( f ) [ H ( f ) X ( f ) L and (74b), from s s s s G(f) = H ( f ) G g ( f ) . 5.3.1 D i g i t a l Recording and Playback System A b l o c k diagram of the d i g i t a l r e c o r d i n g system used i n the s i m u l a t i o n i s shown i n F i g . 14a. The analog s i g n a l i s f i l t e r e d i n t o passbands which prevent a l i a s i n g f o r a gi v e n sampling r a t e (see (73) and assumption 10) i n S e c t i o n 4.2) and then sampled-and-held f o r a n a l o g - t o - d i g i t a l conversion. The a n a l o g - t o - d i g i t a l converter (ADC) codes each sample i n t o 10 b i t s and t r a n s f e r s i t to the b u f f e r and c o n t r o l u n i t (BCU). Noise i n t r o d u c e d by the conversion process i s ignored s i n c e i t s l e v e l i s s u f f i c i e n t l y low to be masked out by the noise i n t r o d u c e d i n the s i m u l a t i o n s . The b u f f e r c o n s i s t s of two b l o c k s , each capable of s t o r i n g 508 t e n - b i t c h a r a c t e r s . The b l o c k s are u t i l i z e d i n a double-buffered manner so that data i s being w r i t t e n out of one bl o c k onto tape w h i l e a t the same time data i s being read i n t o the other b l o c k from the ADC. The sequence of operations i s t h a t when a b l o c k becomes f u l l , the tape t r a n s p o r t i s s t a r t e d and w r i t i n g commences. A f t e r the b l o c k i s w r i t t e n the tape t r a n s p o r t i s stopped to await f i l l i n g of the other b l o c k . The procedure i s repeated u n t i l an e n d - o f - f i l e (EOF) p u l s e i s encountered on the BCU. Analog Tape Recorder S c u l l y 280 E q u a l i z e r Network * Bandpass F i l t e r Sample-and-Hold Analogic "250-01 Magnetic Tape Transport CD 601 E x t e r n a l Timing and Co n t r o l Unit Magnetic Tape Synchronizer CD 8093 B u f f e r and Co n t r o l CD 8092 Analog-to D i g i t a l Converter. Analogic 2200 (a) E x t e r n a l Timing and Co n t r o l U n i t (b) Fig. 14 Block diagram. (a) D i g i t a l Recording System. (b) D i g i t a l Playback System. 60 The tape synchronizer f u r n i s h e s timing and c o n t r o l pulses to i n s u r e t h a t the data are w r i t t e n on d i g i t a l tape i n proper format. Since the CD 601 i s a 7 t r a c k tape d r i v e , the f i v e most s i g n i f i c a n t b i t s of each sample are w r i t t e n i n one tape byte and the remaining f i v e l e a s t s i g n i f i c a n t b i t s i n the f o l l o w i n g b y t e . The s i x t h tape t r a c k i s used to i d e n t i f y the most or l e a s t s i g n i f i c a n t byte and the seventh t r a c k i s reserved f o r a p a r i t y check b i t . The tape speed and d e n s i t y are s e t at 37.5 inches/second and 556 b y t e s / i n c h , r e s p e c t i v e l y . • Due to the r e l a t i v e l y long memory c y c l e time of the CD 8092 (approx-i m a t e l y 4 usee), the maximum sampling r a t e i s approximately 6500 samples per second. Speech experiments [41,42] have shown that such a l i m i t imposes a c o n s t r a i n t on the q u a l i t y of speech that can be analyzed. Since a maximum sampling r a t e of 12 kHz i s r e q u i r e d f o r the s i m u l a t i o n , the analog s i g n a l was recorded on the S c u l l y 280 at 15 inches/second and played back at 7.5 inches/ second when a sampling r a t e of greater than 6.5 kHz i s r e q u i r e d . This e f f e c t i v e l y doubles the a c t u a l sampling r a t e . To compensate mismatching of r e c o r d and playback e q u a l i z e r s r e s u l t i n g from the d i f f e r e n c e i n re c o r d and playback tape speeds, an e x t e r n a l e q u a l i z e r i s used to smooth out the tape recorder frequency response. The e x t e r n a l t i m i n g and c o n t r o l u n i t (ETCU) s t a r t s and stops the analog tape r e c o r d e r , s u p p l i e s command pulses to the sample-and-hold (SH) and ADC, and EOF pulses to the BCU. Timing i s under c o n t r o l of a Wavetek Model 111 s i g n a l generator whose frequency i s adjusted to the d e s i r e d sampling r a t e . P r i o r to d i g i t a l r e c o r d i n g , the analog tape i s p o s i t i o n e d so that the playback head on the tape recorder c o i n c i d e w i t h a v i s u a l mark on the tape. The mark i s placed f a r enough ahead of the pre-recorded analog s i g n a l to a l l o w the tape recorder to come up to f u l l speed before the s i g n a l passes over the playback 61 head. Once the analog tape i s p o s i t i o n e d , the tape recorder i s s t a r t e d manually by op e r a t i n g a push button on the ETCU. D i g i t a l r e c o r d i n g i s delayed u n t i l the analog tape recorder reaches f u l l speed. At the end of the delay, command pulses are a u t o m a t i c a l l y s u p p l i e d to the SH and ADC to t r i g g e r a n a l o g - t o - d i g i t a l conversion. A f t e r a predetermined span of time corresponding to the l e n g t h of the analog sample, an EOF pulse i s a u t o m a t i c a l l y s u p p l i e d to the BCU to terminate d i g i t a l r e c o r d i n g and a pulse a p p l i e d to the analog tape recorder to stop playback. Analog r e c o n s t r u c t i o n of the d i g i t a l samples i s accomplished by the d i g i t a l playback system shown i n F i g . 14b. The f i l t e r passbands are the same as those used during d i g i t a l r e c o r d i n g . The r a t i o of sampling p e r i o d to sampling i n t e r v a l i s c o n s t r a i n e d to be grea t e r than 10 i n order to approximate the c h a r a c t e r i s t i c s of impulse sampling. I f the d e s i r e d sampling r a t e exceeds 6.5 kHz, the a c t u a l sampling r a t e i s s e t at one-half t h i s value and the analog s i g n a l recorded at 7.5 inches/second and played back at 15 inches/second. The e x t e r n a l e q u a l i z e r i s s u p p l i e d to compensate mismatching of the tape recorder.!s i n t e r n a l r ecord and playback e q u a l i z e r s . P r i o r to d i g i t a l playback, the 7 t r a c k d i g i t a l tape i s blocked i n t o p h y s i c a l records c o n s i s t i n g of 508 data samples and and EOF mark w r i t t e n to s i g n i f y the end of the f i l e . The blocks are separated by an i n t e r r e c o r d gap of 0.75 inches to a l l o w s t o p p i n g and s t a r t i n g of the tape t r a n s p o r t . The sequence of o p e r a t i o n i s i n i t i a t e d by l o a d i n g the BCU w i t h the f i r s t two data b l o c k s read from d i g i t a l tape. The analog tape recorder i s then s t a r t e d manually by depressing a push button on the ETCU. A f t e r a s u f f i c i e n t time delay to allow the tape recorder to.reach f u l l speed, command pulses are s u p p l i e d at the sampling r a t e to the BCU, d i g i t a l - t o - a n a l o g converter (DAC), and sampler to t r i g g e r the conversion of d i g i t a l samples from the b u f f e r i n t o 62 analog samples. Each time a b u f f e r b l o c k i s emptied, the tape t r a n s p o r t i s s t a r t e d and another data b l o c k i s read from tape. When an EOF mark i s en-countered on tape, the e n t i r e b u f f e r i s emptied and an EOF pu l s e sent to the ETCU. The ETCU then stops the analog tape recorder. 5.3.2 S i m u l a t i o n Program The PAM, PCM, and DPCM communication systems are simulated on a IBM System/360 Model 67 data p r o c e s s i n g system. Since the b i n a r y tape produced by the d i g i t a l r e c o r d i n g system i s w r i t t e n i n format which cannot be accessed d i r e c t l y by FORTRAN input/output statements, a p r e l i m i n a r y conversion stage i s r e q u i r e d . A FORTRAN-callable subroutine i n assembly language reads the 7 tr a c k tape and returns to the c a l l i n g program a f i x e d - p o i n t number r e p r e s e n t i n g the decimal e q u i v a l e n t of the b i n a r y number that was w r i t t e n onto d i g i t a l tape. The c a l l i n g program blo c k s the data i n t o p h y s i c a l records each c o n t a i n i n g 768 data samples, and w r i t e s the data onto 9 t r a c k tape using an unformatted WRITE statement. The 9 t r a c k tape i s then used as the i n p u t tape f o r the s i m u l a t i o n program w r i t t e n e n t i r e l y i n FORTRAN IV. The communication systems simulated on the IBM/360 are shown i n F i g . 15. Noise data p r e v i o u s l y recorded onto d i g i t a l tape from a Grason-S t r a d l e r Model 4550 noise generator connected to a lowpass f i l t e r w i t h c u t - o f f frequency s e t at h a l f the sampling r a t e i s used to form the n o i s e sequence {n^} i n the PAM system of F i g . 15a. M u l t i p l y i n g the sequence by a s c a l e f a c t o r gives the d e s i r e d channel s i g n a l - t o - n o i s e r a t i o , S/N W , at the o c p o s t f i l t e r i n p u t . S i g n a l power S i s the p r e f i l t e r output power and N 0 W C i s the n o i s e power. The l o g a r i t h m i c q u a n t i z e r s and l i n e a r p r e d i c t o r used i n the PCM and DPCM s i m u l a t i o n s shown i n F i g s . 15b and c are designed i n accordance w i t h assumptions 3) to 5) of S e c t i o n 5.2. Mapping of the q u a n t i z e r . Digital P r e f i l t e r F(f) m ( (a) " Scale Factor D i g i t a l P o s t f i l t e r G ( f ) / m D i g i t a l P r e f i l t e r F ( f ) D i g i t a l P r e f i l t e r F ( f ) Logarithmic Quantizer d b i t s D i g i t a l P o s t f i l t e r G(f) (b) Logarithmic Quantizer d b i t s Linear Predictor a. 0 Digital Postfilter G(f) (c) F i g . 15 Simulated communication systems. (a) PAM. (b) PCM. (c) Previous-sample feedback DPCM. 64 i n p u t samples i n t o the 2 p o s s i b l e output l e v e l s i s performed by a d-step s u c c e s s i v e approximation procedure. E x t e n s i v e l i t e r a t u r e has been p u b l i s h e d on the design of d i g i t a l f i l t e r s [53-57]. In t h i s t h e s i s , the method proposed by Helms [57] i s f o l l o w e d f o r d e s i g n i n g the d i g i t a l pre and p o s t f i l t e r s shown i n F i g . 15. I n the method, c o n v o l u t i o n of the d e s i r e d frequency response w i t h the Dolph-Chebyshev f u n c t i o n i s used to achieve nonrecursive ( t r a n s v e r s a l ) d i g i t a l f i l t e r s w i t h the f o l l o w i n g d e s i r a b l e p r o p e r t i e s : 1) The d e s i r e d frequency response can be s p e c i f i e d n u m e r i c a l l y , g r a p h i c a l l y , or a n a l y t i c a l l y . 2) The method allows n o n r e c u r s i v e f i l t e r s to be designed to a s p e c i f i c r e s o l u t i o n , defined as the bandwidth of the t r a n s i t i o n s between d i s c o n t i n u i t i e s i n the d e s i r e d frequency response, and to a s p e c i f i c r i p p l e , d efined as the maximum d e v i a t i o n from the d e s i r e d frequency -response f o r frequencies o u t s i d e the t r a n s i t i o n r e g i o n s . 3) The method tends to produce r e l a t i v e l y good r e s o l u t i o n f o r a given r i p p l e and f o r a given number of c o e f f i c i e n t s used i n s p e c i f y i n g the impulse response on the nonrecursive f i l t e r . 4) The nonrecursive f i l t e r i s e a s i l y implemented by u s i n g the f a s t c o n v o l u t i o n a p p l i c a t i o n of the f a s t F o u r i e r transform [53,57-59]. The above method i s a p p l i e d to the design of nonrecu r s i v e f i l t e r s which s i m u l a t e the pre and p o s t f i l t e r s s p e c i f i e d i n Table 5.1. Exception to the Table i s made by modifying the frequency s e t ft so f o r any band contained i n a, the lower and upper c u t - o f f f r e q u e n c i e s , f and f , are s c a l e d to 1.07f . i u a and f^/1.07, r e s p e c t i v e l y . The s c a l e d values correspond to the 3 dB c u t - o f f frequencies used i n the analog bandpass f i l t e r s during d i g i t a l r e c o r d i n g (see assumption, (10). i n S e c t i o n 5.2). S p e c i f i c a t i o n s used i n the design were r i p p l e equalto 0.1 percent and number of coefficients equal to 256. This resulted in a resolution of (1.46/256)f slog^Q(Q/.001) where f is the sampling rate and Q is the number of discontinuities i n the desired frequency response. Implementation of the nonrecursive f i l t e r s so designed is by the select-saving method of fast convolution [57,58]. Extensive literature on power spectra estimation has also been pub-lished [60-62], The method used here for the estimation of power spectra follows closely that proposed by Welch [62]. The method involves sectioning the time series, taking modified periodograms of these sections, and averaging these modified periodograms. Dolph-Chebyshev window coefficients [57] and non-overlapping segments consisting of 256 samples each are used i n the ca l -culation of the modified periodograms. The autocorrelation function is estimated by computing the inverse Fourier transform of the power spectrum. Computation of discrete Fourier transforms (DFT's) and inverse DFT's (IDFT's) in the above design, implementation, and estimation methods is performed using Cooley's FORTRAN IV subroutine [63] based on the fast Fourier transform (FFT) algorithm [59,61,64]. The FFT allows computationally efficient design and implementation of nonrecursive d i g i t a l pre and postfilters, and provides computationally ef f i c i e n t estimation of power spectra and auto-correlation functions. 66 6 . SUBJECTIVE EVALUATION OF PRE AND POSTFILTERS IN PAM, PCM, AND DPCM SPEECH COMMUNICATION SYSTEMS 6.1 Introduction During the design, development, and testing of speech communication systems, there i s a need for evaluation and for an optimization criterion. In the past, i n t e l l i g i b i l i t y has been u t i l i z e d as the main criterion for the subjective evaluation of speech communication systems. However, since the i n t e l l i g i b i l i t y of speech output signals from modern communication systems is close to 100 percent, i n t e l l i g i b i l i t y alone as a measure of speech quality cannot suffice as a design criterion. The concept of speech quality encompasses the total auditory impression of speech on a listener and not just i t s i n t e l l i g i b i l i t y aspect. Speech quality includes additional factors such as loudness, naturalness, c l a r i t y , speaker i d e n t i f i a b i l i t y , timbre and rhythmic character, amplitude or time distortions, and many others. In general, quantitative evaluation of a l l these factors may be d i f f i c u l t or impossible. However, i f certain assumptions on the psychological dimensions which characterize speech quality are made, speech quality can be described on the unidimensional scale, preference [65]. Preference as a parameter of speech quality i s the attitude of a listener towards a speech signal when he compares i t with a second speech signal and i s , therefore, a relative measure of quality. The aspect of preference becomes dominant with respect to over-all speech quality when the following conditions are f u l f i l l e d , which is often true in many practical cases: a) The i n t e l l i g i b i l i t y of speech is high. b) The level of the speech signals i s presented at optimum loudness, which i s defined as the speech level at which a listener prefers 67 to hear speech. c) The recognizability of the speaker i s of minor interest to the l i s tener. A method of subjective preference testing i s described i n Section 6.2 and the results of the subjective evaluation using the method i s discussed i n Sections 6.3 and 6.4. 6.2 Subjective Test Method The subjective test method ut i l i z e d i n this study is similar to the isop-reference method proposed by numerous investigators [65-68]. In the method, preference i s evaluated by a forced pair-comparison test, and the results shown as isopreference (equal preference) contours on a sampling rate versus channel signal-to-noise ratio, or number of quantization b i t s , diagram. The quality rating assigned to each isopreference contour is the signal-to-noise ratio of a degraded speech signal which is subjectively equivalent to the reference signal associated with the isopreference contour [69]. An alternative quality scale obtained by the subjective estimate method [70] i s also presented. 6.2.1 Speech Material, Equipment, Listeners, and Further Details  on System Simulation The speech material used throughout the study consists of the two sentences, "Joe took father's shoe bench out. She was waiting at my lawn". These sentences contain most of the phonemes found in English and have a power spectrum typical of conversational speech [71]. The sentences, spoken by a 31 year old male university professor with a western Canadian accent, were recorded on a single-track Scully 280 tape recorder at 15 ips using an 68 AKG D-200E low impedance, c a r d i o i d microphone. The r e c o r d i n g was performed i n an I n d u s t r i a l A c o u s t i c s Company Model 1205-A q u i e t room. A d d i t i o n a l speech m a t e r i a l was not considered due to the p r o h i b i t e d amount of data p r o c e s s i n g i n v o l v e d . S t a t i s t i c s on the spoken sentences were obtained u s i n g the d i g i t a l s i m u l a t i o n f a c i l i t i e s d e s c r i b e d i n S e c t i o n 5.3. The e f f e c t i v e bandwidth of the d i g i t i z e d spoken sentences was l i m i t e d to 6 kHz. This was accomplished by reducing the S c u l l y tape recorder to 7.5 i p s , lowpass f i l t e r i n g a t 3 kHz, and sampling at 6 kHz. F i g . 16a shows the amplitude p r o b a b i l i t y d e n s i t y of the speech samples normalized r e l a t i v e to t h e i r RMS v a l u e . A l s o shown f o r comparison are the L a p l a c i a n d i s t r i b u t i o n [45] and the Gamma d i s t r i b u t i o n [72], which are o f t e n used as speech models. I n F i g . 16b, the r e l a t i v e power spectrum of the d i g i t i z e d speech i s presented. The method used f o r . t h e power spectrum e s t i m a t i o n has been described i n S e c t i o n 5.3.2. A l s o shown f o r comparison purposes i s a r e l a t i v e speech spectrum from Benson and H i r s h [71]. T h e i r spectrum represents the long-time average s p e c t r a of 90-second samples of t e c h n i c a l and news m a t e r i a l f o r f i v e male speakers. Although the sentences used i n t h i s study represent only approximately 5 seconds of speech i n r e a l -time, F i g . 16 shows there i s c l o s e enough agreement w i t h p u b l i s h e d r e s u l t s to p r o v i d e a r e p r e s e n t a t i v e sample which i s t y p i c a l of c o n v e r s a t i o n a l speech. Speech samples f o r the l i s t e n i n g t e s t s were obtained by: a) d i g i t a l l y r e c o r d i n g the analog recorder sentences (see F i g . 14a) b) d i g i t a l l y s i m u l a t i n g PCM, PCM, or DPCM communication systems {see F i g . 15), and c) analog r e c o r d i n g the processed sentences (see F i g . 14b). Since the o r i g i n a l speech s i g n a l i s approximately lowpass w i t h power d e n s i t y spectrum shown i n F i g . 16b, the passband f o r the pre and p o s t -Fig. 16 (a) Normalized amplitude probability density of speech. Symmetrical average of positive and negative data, (b) Power density spectrum of speech. 70 f i l t e r s given i n Table 5.1 i s -ft= {f: I f I <-rf }, where f i s the sampling f r e -Z s s quency. (See Section 3.1.2 f o r d e f i n i t i o n of lowpass s i g n a l . ) For such a frequency set ft, the analog f i l t e r s used i n the d i g i t a l recording and playback systems are lowpass with passband l e s s then h a l f the sampling r a t e (see F i g . 13). Furthermore, the d i g i t a l f i l t e r s used i n the optimal p r e f i l t e r - c o n s t a n t amplitude p o s t f i l t e r scheme are lowpass d i g i t a l f i l t e r s . -In step c ) , loudness was c o n t r o l l e d by monitoring the record a m p l i f i e r output of the S c u l l y analog tape recorder, and adjusting the record l e v e l so the recorded speech samples sounded equally loud. I t was shown that close agreement e x i s t s between the l i s t e n e r s ' and the experimenter's judgment of equal loudness. Analog tapes f o r the l i s t e n i n g tests were made by s p l i c i n g the speech samples i n t o the desired format. A l l l i s t e n i n g tests were conducted i n a quiet room using Sharpe HA-10-MK II stereo headphones. External volume controls were provided with each headphone. P r i o r to each l i s t e n i n g session, a p a i r of speech samples was played i n order to allow the l i s t e n e r s to adjust t h e i r volume controls. During the course of the l i s t e n i n g session the l i s t e n e r s were not allowed to readjust t h e i r loudness l e v e l s . The l i s t e n e r s were 17 male graduate students and s t a f f members whose ages ranged from 21 to 41 years. The mean age was s l i g h t l y over 25 years, and a l l except one l i s t e n e r was under 28 years. Only 16 of the 17 l i s t e n e r s were used on any one l i s t e n i n g t e s t . A l l l i s t e n e r s showed no hearing abnormalities and a l l had l i t t l e or no previous experience i n l i s t e n i n g t e s t s . 6.2.2 Determination of Isopreference Contours Isopreference contours connecting points of equal subjective q u a l i t y on a sampling rate versus channel s i g n a l - t o - n o i s e r a t i o plane are presented :'in F i g . 17 f o r PAM systems. Contours shown i n F i g . 17a are f o r lowpass pre . and p o s t f i l t e r s (LPF) and F i g . 17b are f o r weak noise pre and p o s t f i l t e r s (WNF). S i m i l a r i s o p r e f e r e n c e contours on sampling r a t e versus number of q u a n t i z a t i o n b i t s planes are presented i n F i g s . 18 and 19 f o r PCM and DPCM systems, r e s p e c t i v e l y . A l s o shown are curves of constant channel c a p a c i t y . The i s o p r e f e r e n c e contours were obtained from pair-comparison t e s t s . For each of the 6 planes shown i n F i g s . 17, 18, and 19, a random sequence of approximately 96 p a i r s of speech samples were heard by each l i s t e n e r . Three l i s t e n i n g sessions w i t h 4 t e s t runs per sessions were conducted during the course of a week. A t e s t run c o n s i s t e d of 48 p a i r e d comparisons and l a s t e d about 15 minutes. A f t e r each t e s t run, a r e s t p e r i o d of approximately 5 minutes f o l l o w e d . No systematic v a r i a t i o n of t e s t r e s u l t s was detected due to l i s t e n e r f a t i g u e i n sessions of t h i s d u r a t i o n . P r i o r to each l i s t e n i n g s e s s i o n , the l i s t e n e r s . r e a d the f o l l o w i n g i n s t r u c t i o n s . I n t h i s l i s t e n i n g t e s t you w i l l hear p a i r s of speech s i g n a l s . Each p a i r i s separated by a 5 second s i l e n t i n t e r v a l . A f t e r l i s t e n i n g to a p a i r , i n d i c a t e i n the a p p r o p r i a t e column which speech s i g n a l of the p a i r you would p r e f e r to hear. I f both speech s i g n a l s sound e q u a l l y good, make an a r b i t r a r y choice. The f i r s t speech s i g n a l of each p a i r i s designated as "A", and the second, as "B". The speech m a t e r i a l used throughout the t e s t s c o n s i s t s of the two sentences, "Joe took f a t h e r ' s shoe bench out.' She was w a i t i n g at my lawn". Approximately 5 seconds were r e q u i r e d to hear each speech s i g n a l and a 1 second i n t e r v a l separated each speech s i g n a l i n the p a i r . As an example of how the p o i n t s i n F i g s . 17, 18, and 19 were ob-6 12 18 24 30 36 CHANNEL SIGNAL-TO-NOISE RATIO S/N0WC (dB) fa) 6 12 18 24 30 36 CHANNEL SIGNAL-TO-NOISE RATIO S/N0WC (dB) (b) Fig. 17 PAM isopref erence -contours. Plus-and minus standard deviations' of "each experimental point are denoted by the bar through the point. Reference--points "associated with each isopref erence contour are drawn solid. (S/N)^^..' v a l u e s are given in dB and Sc values are enclosed in brackets. The bandwidth of the. {Jre and postfilter equals f /2. (a) Lowpass f i l t e r i n g scheme (LPF). s (b) Weak noise f i l t e r i n g scheme (WNF). NO OF QUANTIZATION BITS d m 0 F QUANTIZATION BITS d (a) (b) F i g . 18 PCM i s o p r e f e r e n c e contours. See F i g . 17 ca p t i o n f o r f u r t h e r comments. (a) LPF. (b) WNF. 75 t a i n e d , consider p o i n t s A and B i n F i g . 17a. PAM LPF speech, samples having f and S/N W values of p o i n t A were p a i r e d w i t h f o u r d i f f e r e n t PAM LPF samples S O C r r r having a S/N W of 24 dB, but d i f f e r e n t values of f . The o r d i n a t e of F i g . o c s 20a shows the percentage of l i s t e n e r s who p r e f e r r e d A to the sample f o r which f was as defi n e d by the a b s c i s s a . The range i n f was chosen l a r g e enough s s so that the preference judgments would vary from 0 to 100 percent. From the psychometric curve drawn through the experimental p o i n t s , the 50 percent p o i n t (2.76) was obtained. The corresponding a b s c i s s a value i n F i g . 20a defines p o i n t B i n F i g . 17a. A l l other p o i n t s hear the i s o p r e f e r e n c e contour passing through r e f e r e n c e p o i n t A were obtained by comparing speech samples corresponding to p o i n t A w i t h other samples. The expected shape of the i s o p r e f e r e n c e contour determined whether f or S/N^ W^  was constant f o r the samples being compared w i t h those of p o i n t A. Other i s o p r e f e r e n c e contours i n F i g . 17a were obtained by an i d e n t i c a l procedure, although a d i f f e r e n t r eference p o i n t was used f o r each contour. The reference p o i n t s are drawn s o l i d i n F i g . 17a. When i t was expected that f g was the main determinant of speech q u a l i t y , S/NQWc was h e l d constant and f v a r i e d , as i n F i g . 20a. Thus, S/N W was h e l d constant s o c i n the lower r i g h t r e g i o n of each f -S/N W plane i n F i g . 17, w h i l e f was s o c s h e l d constant i n the upper l e f t r e g i o n . S i m i l a r l y , i s o p r e f e r e n c e contours on f g-rd planes were obtained i n F i g s . 18 and 19 f o r PCM and DPCM systems. I n p l o t t i n g a l l psychometric curves, i t was found t h a t normal d i s t r i b u t i o n curves f i t t e d a l l data p o i n t s . For t h i s reason, the p r o p o r t i o n p^ o f the l i s t e n e r s p r e f e r r i n g the speech sample corresponding to the refer e n c e p o i n t was converted to u n i t normal deviates y^, and a weighted l e a s t squares technique was used to f i t a s t r a i g h t l i n e to the data p o i n t s . A l l p^ values of 0.00 and 1.00 were changed to 0.01 and 0.99, r e s p e c t i v e l y , before conversion 76 to y^. The weight U)^  attached to each d e v i a t e y_^  was given by 2 0). = N . e ~ y i /2irp. (1-p . ) 1 1 1. .1 where N. i s the number of judgments on which y. i s based. In our case N.=16 1 x i f o r a l l i . The weight OK , which i s p r o p o r t i o n a l to the Muller-Urban weight, equals the r e c i p r o c a l o f the v a r i a n c e a s s o c i a t e d w i t h y.[73,74]. F i g . 20b shows the psychometric curve i n F i g . 20a p l o t t e d i n u n i t normal d e v i a t e s . An i n d i c a t i o n o f the goodness of f i t of the s t r a i g h t l i n e y=ax+b (y being the u n i t normal deviate and x being e i t h e r f , S/N W , or d) to the data p o i n t s s o c 2 2 2 2 was obtained by c a l c u l a t i n g y =<(a-a^) >/a ( a ) , where a (a) i s the s t a t i s t i c a l 2 d e v i a t i o n expected i n a, and <(a-a^) > i s the mean squared e r r o r between the f true c o e f f i c i e n t a^ and the f i t t e d c o e f f i c i e n t a[75] . A v a l u e of y s i g n i f i -c a n t l y l a r g e r than u n i t y i m p l i e s t h a t the f u n c t i o n used i n the l e a s t squares f i t i s i n c o r r e c t 1 ^ . For the psychometric curve i n F i g . 20b, y=0.12. For a l l psychometric curves used to o b t a i n the i s o p r e f e r e n c e p o i n t s shown i n F i g s . 17, 18, and 19, Y<1.96. The slope of the l i n e i n F i g . 20b equals 1/a, where a i s the standard d e v i a t i o n of the normal d i s t r i b u t i o n curve f i t t e d to the p o i n t s . The standard d e v i a t i o n of the i s o p r e f e r e n c e p o i n t i s given by C u l l e r ' s formula [73] N 1/2 a = cr/( Z u).p.) I .,11 1=1 where N i s the number of p o i n t s f i t t e d . The plus and minus standard d e v i a t i o n a s s o c i a t e d w i t h each i s o p r e f e r e n c e p o i n t i s i n d i c a t e d i n F i g . 17, 18, and 19. t 2 2 2 A l t e r n a t i v e l y , y =<(b-b ) >/a ( b ) , where b i s the t r u e c o e f f i c i e n t . t't 2 I f a (b) i s normally d i s t r i b u t e d w i t h mean a^ (b^) and v a r i a n c e a (a) ( a 2 ( b ) ) , then has a chi-squared d i s t r i b u t i o n w i t h one degree of freedom [76]. For a 95 percent confidence l i m i t , the c r i t i c a l v a l u e of x 2 i s Xc =3-84, and the c r i t i c a l value of y . i s y = Y =1.96. c A c LL 78 The i s o p r e f e r e n c e curves, which are based on v i s u a l f i t s to the data p o i n t s , were drawn c l o s e to p o i n t s of s m a l l v a r i a n c e and were con s t r a i n e d to have the same general shape as the neighboring curves. 6.2.3 S c a l i n g Isopreference Contours I t has been i m p l i e d i n the previous sub-Section that t r a n s i t i v i t y e x i s t s along an i s o p r e f e r e n c e contour. That i s , a l l speech communication systems defined by p o i n t s on an i s o p r e f e r e n c e contour w i l l be equal i n pre-ference when compared d i r e c t l y w i t h each other. Furthermore, i f t r a n s i t i v i t y i s assumed among planes d e f i n e d by F i g s . 17, 18, and 19, then s c a l e values could be assigned to the i s o p r e f e r e n c e contours that would i d e n t i f y and rank order them [66]. One method of s e l e c t i n g a s e t of numbers f o r a preference s c a l e i s to use an e a s i l y measurable p h y s i c a l c h a r a c t e r i s t i c of a f a m i l y of continuously degradable standard reference s i g n a l s . In t h i s study, a f a m i l y of refe r e n c e s i g n a l s which i s e a s i l y generated by d i g i t a l means and has a uniquely d e f i n e d s i g n a l - t o - n o i s e (S/N) r a t i o i s u t i l i z e d . Such a fa m i l y i s defined by [69] r Q ( t k ) = ( 1 + a V 1 7 2 [ s ( t k ) + c*.n(t k)] where n ( t k ) = e ( t ^ ) • s ( t k ) i s a noise process d e r i v e d by m u l t i p l i n g s i g n a l samples sCt^) by a zero mean d i s c r e t e s t o c h a s t i c process e ( t k ) = +1, which i s u n c o r r e l a t e d w i t h s i g n a l sCt^) and whose samples form a sequence of u n c o r r e l a t e d random _2 v a r i a b l e s . Parameter a determines the S/N r a t i o of r (tj ) , that i s , S/N=ct Ot K. , The s c a l i n g task i s to determine i n pair-comparison t e s t s which of a one-parameter f a m i l y of reference s i g n a l s r (t, ) w i t h d i f f e r e n t S/N r a t i o s i s eq u i v a l e n t i n preference to the t e s t speech s i g n a l . The S/N r a t i o of the is o p r e f e r e n c e reference s i g n a l i s then a t t r i b u t e d to the t e s t s i g n a l as i t s 79 s u b j e c t i v e s i g n a l - t o - n o i s e r a t i o 0>/N) s u I n the s u b j e c t i v e e v a l u a t i o n , s i g n a l samples s ( t ^ ) are obtained by d i g i t a l l y r e c o r d i n g "Joe.... lawn". Lowpass f i l t e r i n g w i t h an e f f e c t i v e band-width of 4 kHz and sampling at an e f f e c t i v e r a t e .of 8 kHz were used. This i s approximately the r a t e used i n the B e l l System v o i c e frequency PCM system, TI C a r r i e r System [77]. S i m u l a t i o n of reference s i g n a l s r ( t ^ ) w i t h a so v a r i e d that S/N= -9, -6, -3, 0, 3,...33 dB i s performed i n the IBM 360/67 computer usi n g a pseudo-random number generator [78] to generate d i s c r e t e samples e ( t ] c ) * Speech samples f o r pair-comparison t e s t s are obtained using the d i g i t a l playback system de s c r i b e d p r e v i o u s l y . I n the l i s t e n i n g t e s t s , speech samples r e p r e s e n t i n g the simulated communication systems to be s c a l e d are p a i r e d w i t h the reference s i g n a l s to form a random sequence. The l i s t e n e r s , t e s t i n s t r u c t i o n s , and psychometric method are i d e n t i c a l to those used p r e v i o u s l y i n S e c t i o n 6.2.2. Two weeks separated these t e s t s from the l i s t e n i n g t e s t s f o r determining i s o p r e f e r e n c e contours. Two l i s t e n i n g s e s sions were conducted during the course of a week. In the f i r s t s e s s i o n , 3 t e s t runs c o n s i s t i n g of 60 p a i r e d comparisons each were presented. In the second s e s s i o n , another 3 t e s t s runs c o n s i s t i n g of 60 p a i r e d comparisons were presented, along w i t h an e x t r a t e s t run. This f o u r t h run c o n s i s t e d of 48 p a i r s and was conducted to o b t a i n an a l t e r n a t i v e q u a l i t y s c a l e . F u r t h e r d e t a i l s are presented i n the l a t t e r p a r t of the S e c t i o n . The (S/N) . . and standard d e v i a t i o n of v a r i o u s p o i n t s are i n d i -subj cated i n F i g s . 17, 18, and 19 i n u n i t s of dB. Since t r a n s i t i v i t y i s assumed, each i s o p r e f e r e n c e contour i s assigned the (S/N) , . v a l u e equal to the SUD J v a l u e of i t s reference p o i n t (the p o i n t drawn s o l i d ) . P o i n t s designated by an "X" were used as a simple check on t r a n s i t i v i t y . The d e v i a t i o n i n (S/N) , . subj values between the "X" p o i n t and the adjacent i s o p r e f e r e n c e contours can be 80 used as a measure to which the t r a n s i t i v i t y c r i t e r i o n i s met. Taking i n t o account the standard d e v i a t i o n a s s o c i a t e d w i t h the (S/N) ,. v a l u e s , the subj r e s u l t s show approximately the c o r r e c t rank o r d e r i n g . An a l t e r n a t i v e procedure, which i s based on the s u b j e c t i v e estimate method [70] and s i m i l a r i n p r i n c i p l e to the category-judgment method [68,70] commonly used i n the o v e r - a l l r a t i n g of telephone systems, i s presented f o r o b t a i n i n g a q u a l i t y s c a l e . The procedure i s d i v i d e d i n t o two phases, f a m i l i a r -i z a t i o n and e v a l u a t i o n . In the f a m i l i a r i z a t i o n phase, a p o i n t of re f e r e n c e f o r l i s t e n e r s ' responses i s e s t a b l i s h e d by p r e s e n t i n g a p a i r of reference s i g n a l s which are r e p r e s e n t a t i v e of the extreme p o i n t s on the q u a l i t y s c a l e . This i s commonly known as "anchoring". A s u b j e c t i v e s c a l e value of 0 i s assigned to speech samples which are j u s t u n i n t e l l i g i b l e and a v a l u e of 10 i s assigned to 6 kHz lowpass speech samples. Since the e v a l u a t i o n t e s t was the f o u r t h t e s t run i n the f i n a l l i s t e n i n g s e s s i o n f o r determining s u b j e c t i v e S/N r a t i o s , i t was not deemed necessary to o r i e n t the l i s t e n e r s to the range of q u a l i t i e s to be encountered i n the e v a l u a t i o n phase. P r i o r to the e v a l u a t i o n t e s t the l i s t e n e r s heard a sample r e p r e s e n t a t i v e of the upper anchor p o i n t , 6 kHz lowpassed speech, and was t o l d the samples heard i n the f i r s t 3 t e s t runs were r e p r e s e n t a t i v e of the speech samples to be evaluated. I n an e f f o r t to p r o v i d e a lower anchor, i t was suggested to the l i s t e n e r s that some of the samples heard i n the previous t e s t runs could q u i t e p o s s i b l y be assigned 0 s c a l e v a l u e s . The experimenter was r e f e r r i n g s p e c i f i c a l l y to the standard refe r e n c e s i g n a l s having S/N r a t i o s of -9 and -6 dB. In the e v a l u a t i o n phase, the t e s t samples to be s c a l e d were p a i r e d w i t h the 6 kHz lowpass reference samples and presented to the l i s t e n e r s i n a random sequence. Each t e s t sample was evaluated twice by the 16 l i s t e n e r s ; once when the p a i r c o n s i s t s of t e s t sample f o l l o w e d by re f e r e n c e sample, and once when the p a i r order i s reversed. Reference samples were i n t e r s p e r s e d so as to r e f r e s h the l i s t e n e r s of the standard of r e f e r e n c e . P r i o r to the e v a l u a t i o n t e s t , the l i s t e n e r s read the f o l l o w i n g i n s t r u c t i o n s . In t h i s l i s t e n i n g t e s t you w i l l hear p a i r s of speech s i g n a l s . Each, p a i r i s separated by a 5 second s i l e n t i n t e r v a l . I f a speech s i g n a l which i s JUST UNINTELLIGIBLE has a s c a l e value of 0, and one of the speech s i g n a l of the p a i r . ( " A " or "B") has a s c a l e v a l u e of 10, how would you r a t e the other speech s i g n a l of the p a i r on an equal h a l f -i n t e r v a l s c a l e , t h a t i s , 0,0,5,1.0,...,9.5,10? The speech s i g n a l which has a s c a l e value of 10 i s i n d i c a t e d by the number "10" i n column "A" or "B", depending on whether the f i r s t o r the second speech s i g n a l of the p a i r has the s c a l e value of 10. I n d i c a t e your s c a l e value i n the a p p r o p r i a t e blank space provided. The speech m a t e r i a l used throughout the t e s t c o n s i s t s of the two sentences, "Joe took f a t h e r ' s shoe bench out. She was w a i t i n g at my lawn". The s c a l e values Sc of each t e s t sample was obtained by averaging the 32 r a t i n g s of the 16 l i s t e n e r s . The standard d e v i a t i o n was a l s o computed. The s c a l e value and standard d e v i a t i o n of the reference p o i n t s a s s o c i a t e d w i t h each i s o p r e f e r e n c e contour are shown i n F i g s . 17, 18, and 19. These values are enclosed i n brackets to d i s t i n g u i s h them from t h e i r corresponding (S/N) , sub v a l u e s . 6.3 F u r t h e r Results and D i s c u s s i o n of S u b j e c t i v e E v a l u a t i o n I n s p e c t i o n of each of the planes defined i n F i g s . 17, 18, and 19 shows that p o i n t s on the i s o p r e f e r e n c e contours which are r e p r e s e n t a t i v e of the communication systems r e q u i r i n g the l e a s t channel c a p a c i t y f o r a given speech q u a l i t y l i e approximately along a s t r a i g h t l i n e . L i n e s of minimum 8 2 c h a n n e l c a p a c i t y a r e f i t t e d t o t h e s e p o i n t s by v i s u a l e x a m i n a t i o n . I n F i g . 2 1 , s u b j e c t i v e r a t i n g s f o r t h e PAM LPF and WNF, PCM LPF and WNF, and DPCM LPF and WNF i s o p r e f e r e n c e c o n t o u r s shown i n F i g s . 17, 18, and 19, r e s p e c t i v e l y , a r e p l o t t e d v e r s u s t h e i r a s s o c i a t e d minimum c h a n n e l c a p a c i t y , d e f i n e d as t h e p o i n t o f i n t e r s e c t i o n o f t h e l o c u s o f minimum c h a n n e l c a p a c i t y w i t h t h e i s o p r e f e r e n c e c o n t o u r . S u b j e c t i v e s c a l e (S/N) , . i s us e d i n F i g . subj 21a, and Sc i s us e d i n F i g . 21b. Some o f t h e d a t a p o i n t s have been s l i g h t l y d i s p l a c e d h o r i z o n t a l l y i n o r d e r t o r e p r e s e n t t h e p l u s and minus s t a n d a r d d e v i a t i o n o f the p o i n t . T h e i r a c t u a l a b s c i s s a v a l u e s a r e d e s i g n a t e d by "X"'s on t h e c u r v e s f i t t e d t o t h e p o i n t s . I f t h e r e i s o n l y one "X" i n a c l u s t e r o f d a t a p o i n t s , t h e n a l l p o i n t s have a common a b s c i s s a v a l u e . One c u r v e i s drawn t h r o u g h t h e PAM LPF and PAM WNF p o i n t s s i n c e t h e p o i n t s a r e n o t s u f f i c i e n t l y s e p a r a b l e i n v i e w o f t h e i r o v e r l a p p i n g s t a n d a r d d e v i a t i o n s . S i m i l a r l y , o n l y one c u r v e i s f i t t e d t o t h e PCM L P F , PCM WNF, DPCM L P F , and DPCM WNF p o i n t s . I n F i g . 21a, t h e p o i n t s were f i t t e d by s t r a i g h t l i n e s , whereas i n F i g . 21b, t h e p o i n t s were a p p r o x i m a t e d by n o r m a l d i s t r i b u t i o n c u r v e s . I n r e l a t i o n to t h e range o f s u b j e c t i v e s c a l e v a l u e s spanned by t h e p o i n t s i n F i g . 21, n o t e t h e a p p r e c i a b l y l a r g e r s t a n d a r d d e v i a t i o n s a s s o c i a t e d w i t h t he p o i n t s i n F i g . 21b when compared w i t h t h o s e o f F i g . 21a. T h i s i s t o be e x p e c t e d s i n c e t h e s u b j e c t i v e e s t i m a t e method i s v e r y s t r o n g l y i n f l u e n c e d by the v a r i e t y and v a r i a b i l i t y o f t h e l i s t e n e r s . F u r t h e r m o r e , c o m p a r i s o n o f the PAM c u r v e and t h e PCM-DPCM c u r v e shows t h a t o v e r the q u a l i t y r a n g e t e s t e d , PAM c o m m u n i c a t i o n systems o p e r a t i n g w i t h f g a n d S/N v a l u e s d e f i n e d b y t h e l o c i o f minimum c h a n n e l c a p a c i t y i n F i g . 17 a c h i e v e a g i v e n s p e e c h q u a l i t y a t a c a p a c i t y w h i c h i s l e s s t h a n t h a t a c h i e v e d by PCM and DPCM systems o p e r a t - . i n g w i t h f and d v a l u e s d e f i n e d by t h e l o c i o f minimum c h a n n e l c a p a c i t y i n F i g s . 18 and 19. • L P F )PAM '}PCM A WNFJ 4 6 B 10 20 MINIMUM CHANNEL CAPACITY (kbits/sec) REQUIRED BY ISOPREFERENCE CONTOUR (a) 40 SO * L P F ) PAM A LPF &WNF m LPF a WNF JPCM } DPCM 10 20 MINIMUM CHANNEL CAPACITY (kbits/sec) REQUIRED BY ISOPREFERENCE CONTOUR (b) 40 60 21 Q u a l i t y r a t i n g os is o p r e f e r e n c e contours shown i n F i g . 17, 18, and 19 versus t h e i r corresponding minimum channel c a p a c i t y . (a) S u b j e c t i v e s c a l e i n (S/N) . (b) S u b j e c t i v e s c a l e i n Sc. s u h J 84 The r e l a t i o n between (S/N) , . and Sc i s given i n F i g . 22. This subj was obtained by p l o t t i n g (S/"*0 g u versus Sc fo r each, reference point (points drawn s o l i d ) shown i n F i g s . 17, 18, and 19. As expected by eliminating the common ax i s , channel capacity, i n Fi g s . 21a and 21b, the normal d i s t r i b u t i o n curve also f i t s the points. For purposes of comparison, the isopreference contours shown i n F i g s . 17a and 17b are redrawn i n F i g . 23a, and those i n F i g s . 18a, 18b, 19a, and 19b are redrawn i n F i g . 23b. The (S/N) , . and Sc values attached to the i s o -subj preference contours are obtained from F i g . 21 and represent values which were derived by smoothing the o r i g i n a l ratings of the isopreference contours of Figs . 17, 18, and 19. These values are given by the ordinates of the "X" points i n F i g . 21. Also redrawn are the curves of constant channel capacity and the l i n e s of minimum channel capacity. • A l l isopreference contours i n F i g . 23 display the same general behaviour. As S/N W (d) increases along a l i n e of constant f , a region i s o c s reached i n which continued increase i n S/NQWC (d) does not y i e l d any s i g n i f i c a n t improvement i n q u a l i t y . In th i s region, q u a l i t y i s l i m i t e d by speech band-width. S i m i l a r l y , as f g i s increased along a l i n e of constant S/NQWc (d), a region i s reached i n which f u r t h e r increase i n f re s u l t s i n l i t t l e im-s provement i n speech q u a l i t y . In t h i s region, transmission (quantization) noise l i m i t s q u a l i t y . Comparison of LPF and WNF isopreference contours shows that only i n the n o i s e - l i m i t e d region does weak noise f i l t e r s have any e f f e c t on subjec-t i v e performance. An explanation follows. For a given f g weak noise f i l t e r s and lowpass f i l t e r s have i d e n t i c a l passbands; so when speech bandwidth i s the main f a c t o r l i m i t i n g q u a l i t y , LPF and WNF isopreference contours converge. In the f i l t e r passband, however, weak noise f i l t e r s have frequency character-0 4 8 12 16 20 24 SUBJECTIVE SIGNAL-TO-NOISE RATIO (S/N)subj (dB) 28 Fig. 22 Relation between scale value Sc and subjective signal-to-noise ratio (S/N) , .. subj 00 Fig. 23 Isopreference contours with quality ratings derived from the curves fit t e d to the raw data in Fig.-21. (S/N) , . values are in dB and Sc values are enclosed i n brackets. (a) PAM. S U b j (b) PCM and DPCM. oo 87 i s t i c s which are "best" suited to minimize noise. Hence, when inband noise i s the main factor limiting speech quality, weak noise f i l t e r s may improve the subjective preformance of the communication system. The improvement, i f any, is shown in the upward r o l l - o f f sections of the isopreference contours. The r o l l - o f f i n WNF contours i s less steep in comparison to LPF contours. The subjective performance of PAM communication systems i s shown in Fig. 23a. Over the range of f and S/N W considered, the improvement in s o c speech quality afforded by the use of weak noise pre and postfilters is negligible i n comparison to the performance achieved by lowpass f i l t e r s . This can be seen by comparing the isopreference contours of the two f i l t e r i n g schemes. Note that the high quality PAM LPF and PAM WNF contours are identical in shape and rating, whereas, only a slight discrepancy exists between low quality PAM LPF and PAM WNF contours. In Fig. 23b, the subjective performance of PCM and DPCM systems i s shown. In the region below and to the right of the locus of minimum channel capacity, PCM LPF, PCM WNF, DPCM LPF, and DPCM WNF isopreference contours are identical both i n form and subjective rating. In this region where speech bandwidth limits quality, speech quality is essentially independent of the type of communication system (PCM or DPCM) and of the choice of f i l t e r i n g scheme (LPF or WNF), but only essentially dependent on the system bandwidth. This follows from the fact that the differential aspect of DPCM and the spectral-shaping of WNF only affect the inband distortion and not the out-of-band distortion which arises from bandlimiting. Furthermore, i f PCM and DPCM systems are operated along the locus of minimum channel capacity, then the channel capacity required by each system to achieve a given speech quality i s identical for PCM LPF, PCM WNF, DPCM LPF, and DPCM WNF systems (also see Fig. 21). The DPCM LPF and DPCM WNF contours are also identical in form and subjec-88 t i v e r a t i n g . Weak noise f i l t e r s , on the otherhand, can improve the s u b j e c t i v e performance of PCM systems i n the r e g i o n which i s above and to the l e f t of the l i n e of minimum channel c a p a c i t y . Comparison of PCM LPF and PCM WNF contours shows th a t i n t h i s r e g i o n weak n o i s e f i l t e r s may reduce by almost one b i t the number of q u a n t i z a t i o n b i t s r e q u i r e d to achieve a given speech q u a l i t y . This b i t r e d u c t i o n w i t h respect to PCM LPF systems, however, i s l e s s than that achieved by DPCM systems. I n r e t r o s p e c t , a h e u r i s t i c e x p l a n a t i o n f o r the r e l a t i v e s u b j e c t i v e behaviour of WNF systems w i t h respect to LPF systems can be p o s t u l a t e d from the o b j e c t i v e r e s u l t s of Chapter 4. L e t the speech s i g n a l , d e f i n e d by the s t a t i s t i c s given i n F i g . 16, be approximated by a s t a t i o n a r y Gaussian s i g n a l w i t h f i r s t - o r d e r Butterworth spectrum * ( f ) = a a / [ T r ( f 2 + a 2 ) ] (74) where a = 600 Hz. I f the input speech s i g n a l to the LPF and WNF systems simulated i n Chapter 5 i s assumed to have these s t a t i s t i c a l p r o p e r t i e s , then the o b j e c t i v e r e s u l t s f o r the PR (optimal p r e f i l t e r - c o n s t a n t amplitude p o s t -f i l t e r ) and the 0 (optimal pre and p o s t f i l t e r s ) systems discussed i n Chapter 4 are i n d i c a t i v e of the o b j e c t i v e performance t h a t could be achieved by LPF and WNF speech communication systems. F i g . 5 shows the inband s i g n a l - t o - n o i s e r a t i o (SNR) defined by equation (53) f o r PAM PR and PAM 0 systems. The o b j e c t i v e advantage gained by using o p t i m a l pre and p o s t f i l t e r s can be expressed i n terms of the s i g n a l -to-noise improvement f a c t o r SNRIF, which i s simply the r a t i o of SNR achieved using 0 f i l t e r s over the SNR achieved using PR f i l t e r s . Note from F i g . 5 that SNRIF i s an i n c r e a s i n g f u n c t i o n of W /a. The maximum W /a considered i n the c c s u b j e c t i v e i n v e s t i g a t i o n of PAM i s W /a] = f ] /2a=12000/(2 • 600) = 10. c max s max 89 F i g . 5 shows that f o r W^/aflO, the SNRIF < 1.6. Since such a r e l a t i v e l y s m a l l improvement i n SNR can be"achieved by WNF systems, only s l i g h t s u b j e c t i v e -d i f f e r e n c e s are expected between PAM LPF and PAM WNF systems. Examination of the i s o p r e f e r e n c e contours shown i n F i g . 23a suggests that such i s the case. I n F i g . 9, the SNR f o r PCM systems are presented. Although d i g i t a l channel e r r o r s are taken i n t o c o n s i d e r a t i o n i n the a n a l y s i s , system performance approaches the e r r o r - f r e e case when S/N W becomes l a r g e and exceeds the o c th r e s h o l d p o i n t on the SNR curve. F i g . 9 shows that f o r e r r o r - f r e e t r a n s m i s s i o n , the d i f f e r e n c e i n inband s i g n a l - t o - n o i s e r a t i o SNR between the PR and 0 curves i s s m a l l when W /a=10. However, i f the W /a of the PCM system i s i n c r e a s e d to c c 100, the SNRIF < 3. The exact value i s dependent on the number of q u a n t i z a -t i o n b i t s d used i n the PCM system. For the systems simulated here, values of W /a=f d/2a ranging up to 70 i s used. I n t e r p o l a t i n g the r e s u l t s shown i n c s F i g . 9, values of SNRIF up to approximately 2 would apply i n the s i m u l a t i o n . Based on these o b j e c t i v e r e s u l t s , s u b j e c t i v e performances of PCM LPF and PCM WNF are expected to be moderately d i f f e r e n t . F i g . 23b shows the PCM LPF and PCM WNF i s o p r e f e r e n c e contours. Note the moderately l a r g e d e v i a t i o n between the LPF and WNF contours i n the upper l e f t r e g i o n of the f ~d plane. I n F i g . 10b, e/o i s shown f o r DPCM. The o v e r - a l l d i s t o r t i o n e/cr-, due to inband noise and f i l t e r i n g e r r o r , f o r both PR and 0 systems are the same which i n d i c a t e s that whatever redundancy r e d u c t i o n i s not done by the p r e f i l t e r w i l l be done by the d i f f e r e n t i a l aspect of the DPCM system. Hence, the s u b j e c t i v e performance f o r DPCM LPF system i s expected to be the same as f o r the DPCM WNF system. F i g . 23b shows t h a t such i s the case. Since d i g i t a l computer s i m u l a t i o n s of PAM and s u p p r e s s e d - c a r r i e r , amplitude modulation (AM/SC) systems are i d e n t i c a l , the s u b j e c t i v e r e s u l t s 90 f o r PAM can a l s o apply to AM/SC. Sampling r a t e f must be i n t e r p r e t e d as twice the bandwidth W, of the baseband speech s i g n a l and S, as the average b t r a n s m i t t e d power. For s i n g l e sideband (SSB) AM/SC, w c = w b » whereas f o r double sideband (DSB) AM/SC, W = 2W, . Hence, i n F i g s . 17 and 23, f must c b s be r e p l a c e d by 2W^ , and the values f o r the constant channel c a p a c i t y curves must be l e f t unchanged f o r SSB AM/SC, but doubled f o r DSB AM/SC. A l s o i n F i g . 21, the a b s c i s s a must be s c a l e d by a f a c t o r of 2 when the PAM curves are used to represent DSB AM/SC s u b j e c t i v e performance. 6.4 Concluding Remarks 6.4.1 Summary and Comparison w i t h Previous Works Re s u l t s of the present e v a l u a t i o n show the e f f e c t s of two f i l t e r i n g schemes, namely, lowpass f i l t e r i n g (LPF) and weak noise f i l t e r i n g (WNF), on speech q u a l i t y i n PAM, PCM, and DPCM communication systems. S u b j e c t i v e e f f e c t s o f sampling r a t e f and channel s i g n a l - t o - n o i s e r a t i o S/N W i n PAM s o c systems, and s u b j e c t i v e e f f e c t s of sampling r a t e and number of q u a n t i z a t i o n b i t s d i n PCM and DPCM systems are als o considered i n a d d i t i o n to the e f f e c t s of f i l t e r i n g . I t was observed that no s i g n i f i c a n t s u b j e c t i v e d i f f e r e n c e s e x i s t between PAM LPF and PAM WNF systems f o r a l l f and S/N W considered . s o c i n the s i m u l a t i o n . A l s o , no s i g n i f i c a n t d i f f e r e n c e s e x i s t between DPCM LPF and DPCM WNF systems. For PCM LPF and PCM WNF, however, s i g n i f i c a n t d i f f e r e n c e s i n s u b j e c t i v e performance can e x i s t . I t was shown that such s u b j e c t i v e behaviours can be h e u r i s t i c a l l y e x p l a i n e d using the o b j e c t i v e r e s u l t s of Chapter 4. A measure of confidence i n the s u b j e c t i v e r e s u l t s can be obtained by p l o t t i n g the s u b j e c t i v e s i g n a l - t o - n o i s e r a t i o (S/N)^^.. or the s c a l e value Sc of a l l i s o p r e f e r e n c e contours shown i n F i g . 23 versus t h e i r corresponding 91 s a t u r a t i n g f v a l u e s . I f a l l p o i n t s l i e approximately on the f i t t e d curve i r r e g a r d l e s s of whether the p o i n t s are r e p r e s e n t a t i v e of PAM LPF, PAM WNF, PCM LPF, PCM WNF, DPCM LPF, or DPCM WNF systems, then the c r e d i b i l i t y of the s u b j e c t i v e r e s u l t s i s suggested. This f o l l o w s from the f a c t t h a t when the is o p r e f e r e n c e contours i n F i g . 23 s a t u r a t e h o r i z o n t a l l y , the type of communi-c a t i o n system (PAM, PCM or DPCM) and the choice of f i l t e r i n g scheme (LPF or WNF) are i r r e l e v a n t f a c t o r s i n determining speech q u a l i t y . In t h i s s a t u r a t i o n r e g i o n , speech q u a l i t y i s , i n e f f e c t , s o l e l y dependent on the bandwidth of the system (equal to -^f ). F i g . 24 shows (S/N) , . and Sc versus the z s subj s a t u r a t i n g f . Examination r e v e a l s that a l l p o i n t s l i e c l o s e to the f i t t e d curve, thereby i m p l y i n g the c r e d i b i l i t y of the s u b j e c t i v e r e s u l t s . The s a t u r a t i n g f values used i n F i g . 24 correspond to the o r d i n a t e values of the i s o p r e f e r e n c e contours i n F i g . 23 when a S/N W a b s c i s s a value of 36 dB o c i s used f o r PAM, and when a d a b s c i s s a value of 7 b i t s i s used f o r PCM and DPCM. At these a b s c i s s a values the is o p r e f e r e n c e contours are approximately f l a t . PAM ( a c t u a l l y AM/SC) LPF, PCM LPF, and DPCM LPF speech systems have been considered i n previous s u b j e c t i v e i n v e s t i g a t i o n s [41,42]. In the previous works, i t was suggested that DPCM LPF systems were u n c o n d i t i o n a l l y s u p e r i o r to PCM LPF systems. The present work, shows that DPCM LPF systems, p r o p e r l y designed, w i l l always be at l e a s t as good as PCM LPF systems, and that under c e r t a i n c o n d i t i o n s DPCM LPF systems may be s u p e r i o r to PCM LPF systems. When PCM LPF and DPCM LPF systems are compared w i t h regards to minimum channel c a p a c i t y r e q u i r e d to achieve a c e r t a i n speech q u a l i t y , PCM LPF and DPCM LPF appear to be s u b j e c t i v e l y e q u i v a l e n t (see F i g s . 21 and 23b). On the otherhand, when the systems are operated "nonoptimally", that i s , o f f the l i n e of minimum channel c a p a c i t y , DPCM LPF systems may y i e l d b e t t e r ( 03 20 .18 -Q -«> \ to ^- 14 o ^ 12 Q: Uj 10 to c5 i s i § to UJ o Uj io 8 4 -2 . 0. -2 I I 2 3 4 5 6 7 8 SATURATING SAMPLING FREQUENCY (kHz) (a) x PCM LPF, PCM WNF, DPCM LPF, DPCM WNF 0 L P F ] P A M O oWNFJr 10 o to Uj : D - J UJ —i o io Ul o Ul § to / a> x 2 3 4 5 6 7 8 SATURATING SAMPLING FREQUENCY (kHz) (b) F i g . 24 Quality r a t i n g of isopreference contours versus the isopreference contour s a t u r a t i n g values of sampling rate f o . (a) Subjective scale i n (S/N)^ (b) Subjective scale i n Sc. subj 93 performance. I n p a r t i c u l a r , f o r systems o p e r a t i n g i n the upper l e f t r e g i o n of the f -d plane shown i n F i g . 23b, DPCM LPF systems r e q u i r e a lower channel ca p a c i t y to achieve a c e r t a i n q u a l i t y than do PCM LPF systems. There i s agreement w i t h previous work [42] that SSB AM/SC LPF systems can be s u p e r i o r to both PCM LPF and DPCM LPF systems when the channel c a p a c i t y r e q u i r e d by the systems i s low, say l e s s than 40 k b i t s / s e c . This can be seen i n F i g . 21. Since the speech m a t e r i a l and s u b j e c t i v e t e s t i n g methods used i n the i n v e s t i g a t i o n s are s i m i l a r , the discrepancy may be a t t r i b u t e d p a r t l y to the v a r i e t y and v a r i a b i l i t y of the l i s t e n e r s , and p a r t l y to the d i f f e r e n c e s in-equipment and methods used i n generating the speech samples. I n the present i n v e s t i g a t i o n , a b e t t e r microphone was used to record the o r i g i n a l speech sample; a b e t t e r analog tape, recorder was used to record the play the speech samples; and f i n a l l y , b e t t e r headphones were used i n the l i s t e n i n g t e s t s . Furthermore, a d i g i t a l computer was used to simulate the v a r i o u s communication systems considered i n t h i s study, whereas previous i n v e s t i g a -t i o n s were performed i n r e a l - t i m e using hardware models of the AM, PCM, and DPCM systems. The problems of d r i f t , s e n s i t i v i t y , and component t o l e r a n c e s inherent i n hardware models, but e l i m i n a t e d i n d i g i t a l computer s i m u l a t i o n , i n t r o d u c e a d d i t i o n a l inband n o i s e to the e x i s t i n g inband q u a n t i z a t i o n n o i s e which i s designed i n t o PCM LPF and DPCM LPF systems. This a d d i t i o n a l n o i s e prolongs the s a t u r a t i n g e f f e c t of inband n o i s e on speech q u a l i t y and p a r t i a l l y n u l l i f i e s the predominant e f f e c t of bandwidth on speech q u a l i t y i n the band-w i d t h - l i m i t e d s a t u r a t i o n r e g i o n of the is o p r e f e r e n c e contours. Furthermore, the added presence of inband n o i s e on speech q u a l i t y has the tendency to make DPCM LPF systems appear u n c o n d i t i o n a l l y s u p e r i o r to PCM LPF systems s i n c e the inband noise r e d u c t i o n c a p a b i l i t y of DPCM LPF systems has grea t e r i n f l u e n c e over a l a r g e r p o r t i o n of the is o p r e f e r e n c e contour. Comparison of the PCM LPF and DPCM LPF isopreference contours i n F i g . 23 with previous work [41] shows that the isopreference contours obtained previously saturate from inadequate speech bandwidth much les s abruptly than those of the present i n v e s t i g a t i o n . This suggests the presence of unwanted inband noise i n the previous work, and suggests uncertainty i n the previously reached conclusion that PCM LPF systems are unconditionally superior to PCM LPF sys terns. 6.4.2 Subjective Weighting Function f o r Speech Extensive subjective t e s t i n g was mandatory i n t h i s i n v e s t i g a t i o n because an objec t i v e measure of o v e r - a l l speech q u a l i t y from p h y s i c a l parameters of the communication systems was not a v a i l a b l e . I t i s tempting, • therefore, to try to f i n d a mathematically t r a c t a b l e evaluation c r i t e r i o n f o r speech which i s based on p h y s i c a l l y measurable q u a n t i t i e s of the speech communication system alone, and which nevertheless agrees with subjective measurements. One such o b j e c t i v e measure i s the frequency weighted mean-integral-squared e r r o r c r i t e r i o n (see F i g . l ) CO WMSE = / $ ( f ) | w ( f ) | 2 df ( 7 5 ) —CO where $ £ ( f ) i s the power density spectrum of the e r r o r s i g n a l , and weighting function W(f) represents the r e l a t i v e s e n s i t i v i t y of human auditors to erro r at that frequency. The underlying basis behind (75) i s the t a c i t assumption that the subjective e f f e c t of the e r r o r s i g n a l i s a d d i t i v e i n power. The f e a s i b i l i t y of the WMSE c r i t e r i o n has been demonstrated i n the design of PCM noise-feedback systems f o r s t i l l monochrome t e l e v i s i o n p i c t u r e s [23]. Since a l l communication systems which are represented by points on an isopreference contour are s u b j e c t i v e l y equivalent, an obj e c t i v e measure of 95 speech quality must yield the same figure of merit for a l l systems on the contour. Furthermore, the figure of merit should increase as isopreference contours of increasing quality are considered. When the objective measure is the WMSE criterion, weighting function W(f) must be chosen such that the WMSE is constant along an isopreference contour, and the WMSE must increase as isopreference contours of higher quality are considered. The physical parameters that were varied to obtain the isopreference contours i n the PAM systems were sampling rate f and channel signal-to-noise ratio S/N^ W^ . In PCM and DPCM systems, the parameters were f and number of quantization bits d.- Since the passband of the pre and postfilters i s a function of f and s since transmission noise and quantization noise are functions of S/N W and o c d, respectively, the planes defining the isopreference contours may be inter-preted as planes of out-of-band f i l t e r i n g error versus inband noise. There-fore, an indication of the relative importance of out-of-band f i l t e r i n g error and inband noise on speech quality can be obtained from the isopreference contours shown i n Fig. 23. Over part of the contour (horizontal), f i l t e r i n g error is the main determinant on speech quality, and over the other part (vertical), inband noise is the main determinant. In an attempt to satisfy the constant WMSE condition over the entire isopreference contour, l e t $ e(f) include out-of-band, as well as inband error. Therefore, W(f) must not only take into consideration subjective effects of inband noise, but also speech bandwidth, However, because of the saturating effect of f i l t e r i n g error and of inband noise on speech quality, the only solut ion for W(f) i s the t r i v i a l one, W(f) =0 for a l l frequency f. The t r i v i a l solution suggests that the subjective effect of fi l t e r i n g error and inband noise i s not additive, and indicates the inadequacy of the WMSE criterion as an objective measure of speech quality. 96 The t r i v i a l solution can be seen by '.considering thePAM LPF isopre-ference contours shown in Fig. 23a. Note that as S/N W is increased beyond o c approximately 18 dB, the lower quality isopreference contour remains constant at fg=2.8 kHz. If A and B are any two different points in this saturating portion of the contour, then W(f) must satisfy CO 00-/ $g A(f)|w(f)| 2 df= / $> B(f)|w(f)I df ( 7 6 ) — 0 0 — 0 0 where $ (f) and $ D ( f ) are the error spectra associated with PAM LPF systems eA ec represented by points A and B, respectively. Since the cutoff frequencies of the lowpass f i l t e r s are W, =^f =1.4 kHz, and since additive white Gaussian b 2 s . noise comprises the inband noise (f) , |f|>W f i l t e r i n g error y^ , |f |<w inband noise $ (f) , |f|^ W, f i l t e r i n g error *eB ( f ) = i " , , <77b> y^ , |f|<W^  inband noise Substituting (77) into (76) yields y / b| W(f)r df = y / b| W(f)r df (78) -W, -W. b b However, since points A and B represents PAM systems with different S/N W >y^ vg' Hence, the only solution to (78) is W(f)=0,IfI<W, . Solution W(f)=0 for a l l b f follows by application of the same arguments to isopreference contours of higher quality. Note that for these contours, f saturates at higher values, thereby yielding higher values of W, . b 6.4.3 Application to Television Signals Computer simulations and subjective evaluation of pre and postfilters 97 i n PAM, PCM, and DPCM speech communication systems have demonstrated the usefulness of the a n a l y t i c a l r e s u l t s i n Chapter 4 i n i n t e r p r e t i n g the r e l a t i v e subjective behaviour of communication systems. The s i g n a l - t o - n o i s e r a t i o improvement f a c t o r SNRIF, which i s simply the r a t i o of SNR's f o r systems using optimal pre and p o s t f i l t e r s (0) and systems using optimal p r e f i l t e r -constant amplitude p o s t f i l t e r (PR), was shown to be a reasonable i n d i c a t o r of the r e l a t i v e subjective advantage which may be gained by using optimal pre and p o s t f i l t e r s . For a SNRIF value close to unity, small differences i n speech q u a l i t y was observed between communication systems using WNF and systems using LPF. Values of SNRIF appreciably greater than unity implies that the subjective advantage to be gained i n using weak noise f i l t e r s may be s i g n i f i c a n t . P i c t u r e q u a l i t y , l i k e speech q u a l i t y , cannot be judged by an objec-t i v e measure alone. However, by analogy with the speech problem, an i n d i c a t i o n of the r e l a t i v e subjective performance between video systems using WNF and systems using LPF can be obtained from the a n a l y t i c a l r e s u l t s of Chapter 4. The envelopes f o r the power s p e c t r a l density of Picturephone (PP) signals and standard broadcast t e l e v i s i o n (BCTV) s i g n a l s have approximately f i r s t -order Butterworth spectra (73) with corner frequencies at 15 kHz and 49 kHz, r e s p e c t i v e l y [79]. In a d d i t i o n , PP s i g n a l s are normally bandlimited to 375 kHz, and BCTV signals are approximately bandlimited to 4.5 mHz. Hence, f o r PP signals transmitted by PAM systems, Wc/a]pp=375/15=26 and from i n t e r p o l a t i o n of the r e s u l t s shown i n F i g . 5, SNRIF=2. For such a value of SNRIF, only a moderate improvement i n p i c t u r e q u a l i t y i s expected from PAM systems which use weak noise f i l t e r s to pre and p o s t f i l t e r Picturephone s i g n a l s . For BCTV the parameter W / a l =4500/49=90, and from F i g . 5, SNRIF=5. Such a C DwXV t ® J Picturephone i s a low bandwidth t e l e v i s i o n system of the B e l l System. 98 large value of SNRIF implies that a large subjective advantage can be gained by using WNF to pre and po s t f i l t e r BCTV signals for PAM transmission. In PCM systems, W /a]„ =375d/15=26d and W /a]_.-_T:=4500d/49=90d.where d is the number c PP c B^rv of quantization bits used i n the PCM video communication system. For typical values of d, approximately 4 to 8, PP SNRIF=4, and BCTV SNRIF=10. These values suggest that a large subjective advantage i s possible by f i l t e r i n g PCM video signals with WNF. On the other hand, because of the efficient redundancy reduction aspect of DPCM systems, no improvement in picture quality i s expected from DPCM video systems using WNF instead of LPF. In general, SNRIF values for systems transmitting video signals are significantly larger than the corresponding values for systems transmitting speech signals. This suggests greater advantages can be achieved using weak noise f i l t e r s for video signals than for speech signals. However, due to the obvious differences in subjective nature of the two signals, f i n a l judgement about the use of weak noise f i l t e r s in PAM, PCM, and DPCM video f communication systems must be based on subjective viewing tests . The author is aware that some subjective results for PCM and DPCM television systems exist in the literature [20,21,23]. However, i n a l l these investi-gations the sampling rate, and hence, the bandwidth of the system was fixed. Since this is an important physical parameter, the subjective effects of sampling rate, as well as noise (transmission and/or quantization) i n optimal pre and postfiltering of television systems should be considered. 99 7. CONCLUSIONS Pre and postfilters which minimize the frequency weighted mean-integral-squared error i n noisy sampled systems have.been considered. In the analysis, no restrictions are imposed on the input signal spectrum and the noise spectrum, and negligible cross-correlation between the signal and noise i s not assumed. An algorithm for determining the jo i n t l y optimal pre and postfilters i s presented, and the v a l i d i t y of the algorithm proved. The principal conclusion to be drawn from the algorithm i s that the joi n t l y optimal pre and postfilters are bandlimited to a frequency set of total measure less than or equal to the sampling, frequency, of which no two points coincide under integer multiple translations of the sampling frequency. An important practical consequence of this conclusion i s that the optimal pre and postfilters can be synthesized by combinations of analog bandpass and di g i t a l spectral-shaping f i l t e r s . Several suboptimal pre and postfiltering schemes have been inves-tigated. One scheme that was studied results when the magnitude of the pre-f i l t e r transfer function is constrained to have a constant amplitude i n the passband and the po s t f i l t e r chosen to minimize the mean-squared error of the system. Another suboptimal scheme occurs when the p o s t f i l t e r transfer function i s constrained to have a constant magnitude i n the passband and the pr e f i l t e r optimized. A third suboptimal scheme, designated as weak noise f i l t e r i n g , was also investigated. Weak noise f i l t e r s yield v i r t u a l l y the same performance as jointly optimal pre and postfilters i n many cases of interest and have the practical advantage that their f i l t e r transfer characteristics are dependent only on the input signal spectrum and the relative spectrum of the noise. Applications of the optimization algorithm to PAM, PCM, and DPCM 100 communication systems have been considered. A M-channel time-multiplexed PAM system with no intersymbol or interchannel distortion i n the channel is analyzed. In the PCM analysis, d i g i t a l channel errors are included, and correlation functions for quantized signals transmitted over discrete memoryless channels are derived and shown to be a necessary part of the f i l t e r optimization. In the DPCM analysis, error-free d i g i t a l transmission i s assumed and the cross-correlation between signal and quantization noise is taken into consideration. Performance characteristics, showing mean-squared error and inband signal-to-noise ratio versus channel signal-to-noise ratio, are determined ex p l i c i t l y for optimal pre and postfiltered PAM and PCM systems with f i r s t -order Butterworth input spectrum. These characteristics are compared with the performance characteristics achieved by PAM and PCM systems which use suboptimal f i l t e r i n g schemes and with the optimal performance theoretically attainable. Performance characteristics, showing mean-squared error versus channel capacity, are also determined for PAM, PCM, and DPCM systems when the system parameters are optimized to yield the least mean-squared error for a given channel capacity. Examination, of the performance characteristics show that significant reduction i n mean-squared error and significant improvement in inband signal-to-noise ratio are possible for PAM and PCM systems which use optimal pre and postfilters in place of the more conven-tional lowpass pre and po s t f i l t e r s . For DPCM systems which use jointly optimal f i l t e r s , however, negligible reduction in mean-squared error was observed. This suggests that whatever redundancy reduction is not done by the p r e f i l t e r w i l l be done by the differential aspect of the DPCM system. Because of the subjective nature of speech, the effect of pre and postfiltering in PAM, PCM, and DPCM communication systems for speech transmission is studied by simulation methods and evaluated with subjective tests. Weak noise pre and postfilters (WNF), which yield v i r t u a l l y the same performance as optimal pre and pos t f i l t e r s , are considered i n the subjective evaluation as well as lowpass pre and postfilters (LPF). The di g i t a l simulation f a s c i l i t i e s and the subjective testing methods are des-cribed, and the subjective results interpreted. As a direct result of the isopreference test method u t i l i z e d in the subjective investigation, isopreference contours showing the subjective effects of LPF and WNF on a sampling frequency f versus channel signal-to-noise ratio S/N W diagram are obtained for PAM speech communication systems, o c For PCM and DPCM systems, the isopreference contours for the two f i l t e r i n g schemes are obtained on a f versus number of bits of quantization d diagram. It was observed that a l l isopreference contours could be separated into two distinct saturation regions. In one saturation region, quality is limited by speech bandwidth and i n the other, inband noise limits quality. Examination of the LPF and WNF isopreference contours for PAM, PCM, and DPCM systems shows that only i n the noise-limited region do weak noise f i l t e r s have any effect on subjective performance. It was observed that no significant subjective differences exist between PAM LPF and PAM WNF systems and between DPCM LPF and DPCM WNF systems. For PCM systems, however, sig-nificant improvement i n subjective performance could be achieved by using WNF i n place of LPF. In the noise-limited saturation region of the iso-preference contour, PCM systems which use WNF reduce by almost one b i t the number of quantization bits required to achieve a given speech quality. This b i t reduction with respect to PCM LPF systems i s s t i l l less than that achieved by DPCM systems. Along each PAM isopreference contour, one combination of f and 102 S/N^ W^  yields the least channel capacity that is required by a PAM system in order to achieve the speech quality associated with the contour. Similarly, along each PCM or DPCM isopreference contour, one combination of f and d yields the least channel capacity required by the PCM or DPCM system in order to achieve the speech quality associated with the contour. Comparison of the minimum channel capacities required by PAM, PCM and DPCM systems i n order to achieve a given speech quality, shows that the minimum channel capacities required by.both PAM LPF and PAM WNF systems are equivalent, and that the minimum channel capacities required by a l l PCM LPF, PCM WNF, DPCM LPF, and. DPCM WNF systems are equivalent. Therefore, when system parameters i n PAM, PCM, and DPCM systems are chosen to minimize the channel capacity required to achieve a specific speech, quality, there i s no subjective advantage i n using weak noise f i l t e r s in place of lowpass f i l t e r s , and no subjective advantage in using DPCM systems i n place of PCM systems. Although particular application to speech communication systems is considered i n this investigation, application of pre and postfiltering in video communication systems i s also worthy of consideration. The useful-ness of the objective measure, signal-to-noise ratio improvement factor SNRIF, has been demonstrated i n the interpretation of the relative subjective behaviour of PAM, PCM, and DPCM systems which use LPF and WNF. Since, i n general, the SNRIF for video communication systems is significantly larger than the SNRIF for speech communication systems, greater improvement i n performance is expected in using weak noise f i l t e r s in place of lowpass f i l t e r s for video signals. However, ultimate judgement on the use of weak noise f i l t e r s in PAM, PCM, and DPCM video communication systems must be based on subjective viewing tests. 103 APPENDIX CORRELATION FUNCTIONS AND RECONSTRUCTION. ERROR FOR QUANTIZED GAUSSIAN SIGNALS TRANSMITTED OVER DISCRETE MEMORYLESS CHANNELS The purpose of t h i s Appendix i s to present a l t e r n a t i v e exact expressions, and some u s e f u l approximations and bounds f o r the c o r r e l a t i o n f u n c t i o n s discussed i n S e c t i o n 4.2.1, and to show how 0 (T) and 0 (x) a f f e c t n yn the mean-squared e r r o r which r e s u l t s when the r e c e i v e d samples v i n F i g . 6 are operated on by any l i n e a r t i m e - i n v a r i a n t f i l t e r to y i e l d an output x ( t ) which approximates any l i n e a r t i m e - i n v a r i a n t o p e r a t i o n on x ( t ) . For s u f f i -c i e n t l y good channels, the c r o s s - c o r r e l a t i o n between the q u a n t i z e r i n p u t s i g n a l and the q u a n t i z a t i o n p l u s channel noise i s shown to have l i t t l e e f f e c t on both the opt i m a l r e c o n s t r u c t i o n f i l t e r and the r e c o n s t r u c t i o n e r r o r when the number of l e v e l s N i n Max nonuniform and uniform q u a n t i z e r s i s s u f f i c i e n t l y l a r g e . This same c o n c l u s i o n does not apply, i n g e n e r a l , f o r s m a l l N o r f o r poor channels. A. 1 Exact Expressions f o r C o r r e l a t i o n Functions I n t h i s s e c t i o n , a l t e r n a t i v e exact expressions f o r 0"(x) and 0 ~(x) v yv are presented. S u b s t i t u t i n g (58) i n t o (57) y i e l d s the f o l l o w i n g equations, i n which 6=5 y(x), ..N N y.y. N N x. , , ' 2 (x -y)-5 (ct-y) 0,;(x) = 1 I - i - L - Z Z P..P . r i - exp[- - ^ - ] > ( e r f [ — ^ 3 (x 1 - y ) - 6 ( a - y ) - e r f [ — — ]}da (A.la) *4r2 (1-6 2) where x 2 e r f ( x ) = (2/vV) / e" r dr (A.lb) o 104 Equation ( A . l ) can be s i m p l i f i e d as f o l l o w s / w 2 N N y. P. . (x, -y) /a , , * , n . , r ( x ) = z E l i J E i *C6,n) exp(- —)dn ( A > 2 a ) v i = l k=l /2TT (x k_ 1-y)/a where N N y.P . ( x -y)/a . -6n (x -y)/a - 5 n ¥(6,n) = I- Z - ^ y 2 1 ( e r f [ m ' ] - e r f I m " 1 • — ] } j = 1 m = 1 Aa-S2) / 2 ( l - 6 2 ) (A.2b) E q u a t i o n . ( A . 2 ) f o r 0^(T) reduces to the r e s u l t obtained by K e l l o g g [22] when 2 y=0, a =1, and the d i g i t a l channel i n F i g . 6 i s n o i s e l e s s . S i m i l a r l y , from (62) we o b t a i n the f o l l o w i n g a l t e r n a t i v e expression f o r 0 ^ ( T ) . 2 0 * ( T ) = (a//2w) / n^Cfi.n) exp(-n /2)dn ( A . 3 ) ; —CO Functions 0 " ( T ) and 0 (x) are e a s i l y computed using the s e r i e s yv yn expansion s i n c e a Q and a^ are the only c o e f f i c i e n t s i n v o l v e d . Whether the s e r i e s r e p r e s e n t a t i o n s (60,63) o r the d i r e c t expressions ( A . l , A . 2 ) should be used to evaluate 0 ^ ( t ) and 0 n ( T ) depends on the number of terms i n the s e r i e s r e q u i r e d to y i e l d s u f f i c i e n t l y good approximations to the d e s i r e d f u n c t i o n s . When a l a r g e number of terms are needed, d i r e c t c a l c u l a t i o n may be more e f f i c i e n t . •A. 2 Approximation and Bounds Some u s e f u l approximations and bounds are now d e r i v e d . I t f o l l o w s from (65) t h a t M CO | R ; ( x ) / a 2 - E a 2 6 N ( T ) | < | 6 ^ F L ( T ) | E a 2 ( A . 4 a ) n=l 7 7 n=M+l M 00 |R ( T ) / a 2 - ( l - a i ) 2 6 ( T ) - E a 2 6 N ( T ) | < | 6 M F L ( T ) | E a 2 (A.4b) n 1 y n=2 n y ~ y n=M+l n 105 2 2 D i v i s i o n of (A.4a) by R^(0)/o and (A.4b) by R n(0)/a y i e l d s bounds r e l a t i v e to R*(0) and R (0) on the e r r o r which r e s u l t s when the f i r s t M terms of the s e r i e s v n are used to approximate R~(x) and R ( T ) . v n In many p r a c t i c a l s i t u a t i o n s the following equations apply f o r a l l 0<k<N, l£i£N and l<j<N. (y k-y) = ~ ( y N _ k - y ) (A.5a) P j i = PN+l-j > N+i-i (A.5b) V c = - ( v N + i - r c ) ( A - 5 c ) From (A.5c) i t follows that constant c i s given by the fo l l o w i n g equation, which holds f o r a l l l_£i-fN. C = ( v i + V N + l - i ) / 2 < A- 5 d) Equations (A.5a) and (A.5c) hold f o r v i r t u a l l y a l l p r a c t i c a l quantization schemes. Equation (A.5b) applies when the quantizer output l e v e l s are trans-mitted over a d i s c r e t e memoryless channel using a na t u r a l binary code. When (A.5a) a p p l i e s , i t follows from (61c) and (61d) that r n ( k ) = ( - i ) n r n(N+i-k) i<k<N . ( A > 6 ) S u b s t i t u t i o n of (A.5) and (A.6) i n t o (61b) shows that n even, p o s i t i v e 1 = 1 k=l • . (A.7a) where N even ( A < 7 b ) N odd S u b s t i t u t i o n of (A.5) into (61a) y i e l d s , a f t e r some algebra, the following equation. f o r a , 0. a Q = c/a (A.7c) 106 I t follows that when a =0 f o r n even and p o s i t i v e , i n e q u a l i t y M n 2. i„M , s i „ 2 R"(T)/CT |>_|6 (T) IZ a holds. D i v i d i n g (A.4a) by the i n e q u a l i t y y i e l d s • v n = i n f o r M>1, ' M .|*Jtt)/0 - Z a 2 6 N ( T ) | / | R ; ( T ) / a 2 | < | 6 ( T ) | B ; ( M ) ( A . 8 A ) n=l Y y v 00 B ; ( M ) = Z a 2 / Z a 2 ( A.8b) n= M+l n n=l n where from (65a) 0 0 M Z a 2 = R ; ( 0 ) / a 2 - Z a 2 (A.8c) n=M+l n=l n S i m i l a r l y , f o r M >_ 2 ' M | R n ( x ) / a 2 - ( l - a 1 ) 2 6 y ( x ) - Z^ a 2 6 n ( x ) | / | R r (X ) / a 2 | < | S y ( x ) | B n ( M ) (A.9a) 00 J { B (M) = Z a 2 / [ ( l - a i ) 2 + Z a 2] , ' . n=M+l n 1 n=2 n (A.9b) Note B^(i) = B^(i+1) and B n ( i ) = B n(i+1) f o r a l l p o s i t i v e even integers i . Th bounds i n (A.8) and (A.9) are a t t r a c t i v e because they give approximation errors r e l a t i v e to R"(x) and R (x). v n A.3 Optimal P o s t f i l t e r i n g of Quantized Signals Transmitted over Discrete  Memoryless Channels Although much work has been done on the reconstruction of sampled sign a l s [8,9], channel errors and c o r r e l a t i o n between s i g n a l and noise were assumed n e g l i g i b l e . In this Section of the Appendix, the e f f e c t of 0 n ( T ) and 0^ n(x) on the mean-squared e r r o r which r e s u l t s when the received samples v are operated on by a l i n e a r time-invariant p o s t f i l t e r i s considered. Let the samples v i n F i g . 6 be m u l t i p l i e d by the impulse t r a i n 107 A(t) = T. E 6(t-kT+9) where 5(t) i s the un i t impulse, and l e t the r e s u l t i n g k=-°° s i g n a l be f i l t e r e d by a l i n e a r time-invariant f i l t e r having impulse response g(t) (see F i g . A . l ) . Let z(t) = a( t ) 8 y ( t ) be the desired output s i g n a l , where a(t) i s any r e a l time function with Fourier transform A ( f ) , and l e t n ( t ) = v ( t ) - y ( t ) . nft) A(t)= r-z s(t-kT+ e) RECONSTRUCTION FILTER IMPULSE RESPONSE g(t) x f t ) F i g . A . l Reconstruction system. Signal v ( t ) i s sampled at._.t=kT-0. If $ ( f ) , $ (f) and $ (f) are the Fourier transforms of 0 ( T ) , 0 (x) and y n yn y n 0y n ( T )» r e s p e c t i v e l y , then mean-integral-squared er r o r T • e = E {|- / [ z ( t ) - x ( t ) ] 2 d t } o = / [ IA(f) I $ (f)+|G(f) I E $ (f+kf )-2A(f)G*(f)$ ( f ) ] d f y i y s y —oo J k=—°° + •/' | G ( f ) | Z $ n ( f ) d f + 2 f { - [ A ( f ) G * ( f ) $ _ ( f ) - | G ( f ) | 2 Z $ _ ( f + k f = )]}df k = - c o where $ (f) = E $ (f+kf ) n n s s k=-°° (A.10) (A.11) 108 and * denotes complex conjugate. The f i r s t term i n (A.10) represents er r o r which r e s u l t s from the p o s t f i l t e r ' s i n a b i l i t y to make e=0 even when n(t)=0. The second term r e s u l t s from quantization plus channel noise, while the t h i r d term r e s u l t s from the i n t e r a c t i o n of y ( t ) with n ( t ) . When y ( t ) i s a Gaussian process and u=0 or a Q=y/a, then, from (64) $ (f) = (a-j-1)$ y(f) and the c o n t r i b u t i o n of the l a s t term r e l a t i v e to that of the second term i s / G * ( f ) [ A ( f ) - G ( f ) ] $ ( f ) d f - / | G ( f ) | 2 Z $ (f+kf )df y , y s -co J -co K=-oo J p = 2 [ i _ a i ] — _ _J<£0 (A.12) v A | G ( f ) | 2 $ ( f ) d f „ n _oo s In many cases of i n t e r e s t y ( t ) i s bandlimited i n the sense that i f $ y ( f ) ^ 0 then $ y(f+kf g)=0 f o r any non-zero integer k. I f G(f) i s bandlimited i n the same way then the second i n t e g r a l i n the numerator of (A.12) equals zero. I f , i n a d d i t i o n , A(f)=G(f) f o r a l l f for which $^(f)^0 then p=0 and CO e = / | G ( f ) | 2 $ n ( f ) d f -co S If y ( t ) i s Gaussian with u=0 or aQ=\i/o and i f G(f) i s chosen to minimize e then [ 25,26] CO G(f) = a x A ( f ) $ ( f ) / [ ( 2 a r l ) E • . • ( f f k f ) + • (£) ] ( A > 1 3 a ) k=-co y s I f G(f) i s chosen to minimize the f i r s t two i n t e g r a l s i n (A.10) and G(f) i s given by (A.13a) with a^=l, that i s CO G(f) = A(f)» ( f ) / [ E « ( f + k f ) + • • ( £ ) ] ( A . 1 3 b ) J k=-°° • 3 s 109 S u b s t i t u t i n g (A.13b) i n t o (A.12) y i e l d s P = 2 ( l - a ; L ) (A.14) From (A.12), (A.13), and (A.14) i t follows that i f 2 ( 1 ^ ) i s s u f f i c i e n t l y small then the c r o s s c o r r e l a t i o n between y ( t ) and n(t) can be neglected i n c a l c u l a t i n g both e and the optimal G ( f ) . ' Although $ (f) can be obtained from (A.11) and $ (f) from (63), s c a l c u l a t i o n i s tedious, p a r t i c u l a r l y when i t has to be performed f o r se v e r a l quantizers and s e v e r a l values of T. Ruchkin [34] and Robertson [35] have shown that with P.. given by (67), $ (f) i s approximately constant f o r a large 1 3 • n s class of $ (f) with a l l but very coarse quantizing and very small T. One t could expect t h i s conclusion to hold reasonably w e l l f o r any , i n which case $ (f)=T ' 0 (0) where n n s 0^(0) = E [ n 2 ] N N y. = E E P. . J - 1 (v.-a) p (a)d f k _ t-1 j - i y ± _ ± 3 ^ (A. 16) and Py(ct) i s the amplitude p r o b a b i l i t y density of y ( t ) . A.4 An Example L e t d d-d.. = p 1 J ( l - p ) 1 J (A.17) where N i s constrained to make d = lo g 2 N an integer , and where d^. i s the Hamming distance between quantizer output l e v e l s i and j . Equation (A.17) r e s u l t s when the quantizer output l e v e l s are n a t u r a l binary coded and trans-mitted over a d i s c r e t e memoryless channel having b i t e r r o r p r o b a b i l i t y p. ^The following example supports t h i s statement, since the a^'s are not strongly dependent on P^. when N >_ 4. 110 Tables A . l and A.2 and F i g . A.2 describe the behaviour of a (1 < n < 9), n — — 2 ( 1 - 3 , ) , A (0), B „(1) and B^(3) when Max [37] uniform and nonuniform quantizers 1 r n v v 2 are used. Max quantizers minimize <j>n(0) = E(n ) f o r any given N when p = 0, and make E(n) = 0 and a = 0 f o r n even, since (A.5) i s s a t i s f i e d . From Tables n 2 2 A . l and A.2 one see that a n >> a f o r N £ 4, and that f o r any N and p most I n same o f the magnitudes of (1-a^), a^, a^, a^, and a^ are of approximately the order, as one would expect f o r noise whose bandwidth i s considerably l a r g e r than that of s i g n a l y ( t ) . F i g . A.2a suggests that 2(l-a^) decreases as p decreases and N increases. As E,. approaches the values i n (67) each c o e f f i c i e n t a continu-l j n ously approaches the value which r e s u l t s when P.. i s given by (67), and both 2 the nonuniform and uniform quantizers which minimize E(n ) converge, respect-i v e l y , to uniform and nonuniform Max quantizers [38]. I t follows from (A.12-A.14) and F i g . A.2a that f o r any redundant or non-redundant coding scheme with optimum nonuniform or optimum uniform quantization, the cross-c o r r e l a t i o n of y with n has l i t t l e e f f e c t on both e and the optimal G(f) i f N i s s u f f i c i e n t l y large and i f the channel i s s u f f i c i e n t l y good. The previous statement i s not true i n general. For example, l e t G(f) minimize e on the assumption that ^ ^ ( T ) = 0. Then with a Max quantizer f o r which N=2 and a n a t u r a l binary code with p = 0.1, (A.14) shows that the l a s t i n t e g r a l i n (A.10) i s almost as large as the second i n t e g r a l , s i nce a^ = .5094 and p = .98. Tables A . l and A.2 show that <t>n(0) increases with p f o r any f i x e d value of N, and that a f i n i t e non-zero value of N minimized 4>n(0) f o r each p > 0, an e f f e c t noted and explained elsewhere [38]. F i g s . A.2b and A.2c show B^(l) and B*(3) vs. N and p. Not shown i s B*(M) f o r M >_ 5, whose behaviour was found to be v i r t u a l l y i d e n t i c a l to B^(3) f o r a l l values of N > 1 and p > 0.01. p 0.0 0.001 log2N 1 2 3 4 5 1 2 3 4 5 a i .6367 .8823 .9654 .9905 .9977 .6354 .8806 .9636 .9887 .9959 a 3 -.2599 -.1552 -.0611 -.0196 -.0056 -.2594 -.1549 -.0611 -.0198 -.0058 a5 .1744 .0108 -.0311 -.0197 -.0079 .1740 .0108 -.0310 -.0197 -.0079 a7 -.1345 .0362 .0243 .0002 -.0041 -.1343 .0361 .0242 .0003 -.0040 a9 .1110 -.0485 -.0044 .0083 .0027 .1108 -.0484 -.0044 .0082 .0026 0n(O) .3634 .1175 .0345 .0095 .0025 .3659 .1225 .0415 .0178 .0119 2 ( 1 - 3 ^ .7266 .2354 .0691 .0190 .0047 .7291 .2389 .0728 .0227 .0081 P 0.01 0.1 log2N 1 2 • 3 . V 5 1 2 3 4 5 a l .6240 .8647 .9469 .9724 .9802 .5094 .7058 .7780 .8051 .8167 3 3 -.2547 -.1521 -.0611 -.0210 -.0074 -.2079 -.1242 -.0574 -.0291 -.0188 a5 .1709 .0106 -.0299 -.0193 -.0082 .1395 .0087 -.0212 -.0154 -.0097 a7 -.1318 -.0354 .0238 .0009 -.0030 -.1076 .0289 .0199 .0058 .0044 a 9 .1088 -.0476 -.0047 .0072 .0017 .0888 -.0388 -.0067 -.0003 -.0047 0n(O) .3888 .1670 .1034 .0926 .0956 .6181 .6002 .6981 .8071 .8949 2 ( 1 - 3 ^ .7520 .2707 .1062 .0552 .0396 .9813 ,5883 .4440 .3898 ' .3667 Table A . l Max Nonuniform Quantizer p 0.0 0. 001 lo g 2 N 1 2 3 4 5 1 2 3 4 5 a l .6367 . 8812 .9626 .9885 .9965 .6354 .8794 .9606 .9865 .9945 a3 -.2599 -.1639 -.0800 -.0342 -.0134 -.2594 -.1635 -.0799 -.0341 -.0134 a5 .1744 .0220 -.0245 -.0261 -.0166 .1740 .0219 -.0245 -.0260 -.0166 a7 -.1345 .0241 .0280 .0097 -.0011 -.1343 .0240 .0279 .0097 -.0011 a 9 .1110 -.0362 -.0131 .0043 .0064 .1108 -.0362 -.0131 .0043 .0064 0 n(O) .3634 .1188 .0374 .0115 .0035 .3659 .1238 .0446 .0211 .0156 2 ( l - a i ) .7266 ,2377 .0749 .0231 .0070 .7291 .2412 .0787 .0270 .0110 P 0.01 0.1 lo g 2 N 1 2 3 4 5 1 2 3 4 5 a l .6240 .8635 .9433 .9687 .9766 .5094 .7049 .7700 .7908 .7972 a 3 -.2547 -.1606 -.0784 -.0335 -.0132 -.2079 -.1311 -.0640 -.0274 -.0107 a5 .1709 .0215 -.0241 -.0256 -.0163 .1395 .0176 -.0196 -.0209 -.0133 a7 -.1318 .0236 .0274 .0095 -.0011 -.1076 .0193 .0224 .0078 .0069 a9 .1088 -.0355 -.0129 .0042 .0063 .0888 -.0290 -.0105 .0035 . 0051 0 n(O) .3888 .1683 .1092 .1065 .1233 .6181 .6002 .7250 .9106 1.129 2 ( 1 - 3 ^ .7520 .2729 .1134 .0626 .0469 .9813 .5901 .4599 . .4185 .4056 Table A.2 Max Uniform Quantizer F i g . 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