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The design and performance evaluation of a pre-emphasis network for FM voice communication Beaulieu, Norman C. 1982

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a/. THE DESIGN AND PERFORMANCE EVALUATION OF A PRE-EMPHASIS NETWORK FOR FM VOICE COMMUNICATION by NORMAN C. BEAULIEU B.A.Sc, University Of British Columbia, 1980 A THESFS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of Electrical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1982 © Norman C. Beaulieu, 1982 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Department of Date 2<L 2_1 DE-6 (3/81) i i ABSTRACT The pre-emphasis problem for FM radiotelephony over an additive white Gaussian noise channel is investigated. A Gaussian model and an experimentally measured frequency spectrum density for speech are used to solve the optimum f i l t e r i n g problem analytically. The optimum pre-emphasis characteristic is expressed in terms of the message spectral power density, the noise power density and the LaGrange multiplier. An iterative algorithm for the determination of the multiplier is presented and solved numerically for typical values of the signal-to-noise ratio. The optimum f i l t e r is unrealizable. It is approximated by a whitening f i l t e r in cascade with a two pole network. The selection of the pole positions involves a trade-off between mean square error performance and spectral compression. The transmitted signal spectra are computed numerically for three choices of pole positions and compared to those for the currently employed p r e f i l t e r . The mean square error is computed and compared to the optimum mean square error for typical values of the signal-to-noise ratio. The new designs have less distortion than the present design. The r o l l - o f f rate of the spectrum in the adjacent channel region is determined to be greater than 4 for the proposed f i l t e r . This compares favourably with the rate of 2 for the current f i l t e r . Interference coefficients are defined to measure the interference between adjacent channel signals. The cases of voice-to-voice, voice-to-digital and digital-to-voice i i i interference are investigated. Four types of data signals are considered, PAM signals with a = 0 and a = 1, MSK and PSK signals. An improvement ranging from 4 to 19 dB isnoted for the new network. The improvement is greatest for the bandlimited PAM signals. The effects of a different speech model on the previous results are investigated. Spectra and interference coefficients are computed for a second speech spectral density function. A computer simulation verifies that the proposed new design offers improved performance. i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i LIST OF ILLUSTRATIONS v i i ACKNOWLEDGEMENT x i I. INTRODUCTION 1.1 Background 1 1.2 Summary of Relevant Work 4 1.3 Scope and Outline of the Thesis 5 II. PRE-EMPHASIS NETWORKS FOR ANALOG FM RADIOTELEPHONY 2.1 Problem Statement 7 2.2 Analysis 9 2.3 Determination of The LaGrange Multiplier 14 2.4 Approximation of the Non-Realizable F i l t e r ........ 17 2.5 Design of Fi l t e r s and Calculation of Signal Spectra 18 2.6 Optimum Mean Square Error and Realizable Mean Square Error 36 2.7 The De-emphasis Network 40 III. INTERFERENCE AND NOISE CONSIDERATIONS 3.1 Spectral Tails 41 3.2 Calculation of Interference Coefficients 49 3.3 De-emphasis and Signal-to-Noise Ratio 3.4 Heuristic Discussion 3.5 Interference from Adjacent Channel Data Signals 3.6 Interference to Adjacent Channel Data Signals IV. AN ALTERNATIVE MODEL FOR SPEECH SPECTRA 4.1 The Speech Spectrum Model 4.2 Transmitted Signal Spectra and Mean Square Error 4.3 Interference Coefficients V. IMPLEMENTATION AND TESTING 5.1 Realization 5.2 Measurement 5.3 Simulation VI. CONCLUSION 6.1 Summary of Results 6.2 Suggestions for Future Research APPENDIX A. Estimating the Error in the T a i l v i LIST OF TABLES Table Page 2-1 F i l t e r Pole Positions 18 2-2 Representative values of 2P/No> , and X ( B = 2900 Hz.) 37 2- 3 Representative values of mean square error ( B = 2900 Hz.) 39 3- 1 Spectral t a i l r o l l - o f f rate, n 49 3-2 Voice-to-voice signal interference coefficients, J , and J _ 58 v l v2 3-3 Digital-to-voice interference coefficients, and J._ 80 az 3- 4 Voice-to-digital interference coefficient, TC v .... 85 4- 1 Mean square error for speech model C^(f) 93 v i i LIST OF ILLUSTRATIONS Figure Page 2.1 Range of s t a r t i n g value of X 16 2.2 Normalized au t o c o r r e l a t i o n function, p(x), for integrated speech f i l t e r e d with f i l t e r s A, B and C 27 2.3 Autocorrelation function V(T) - v(°°) for FM speech modulated s i g n a l using f i l t e r A 28 2.4 Autocorrelation function V ( T ) - vi00) for FM speech modulated s i g n a l using f i l t e r B 29 2.5 Autocorrelation function V(T) - v(°°) f o r FM speech modulated s i g n a l using f i l t e r C 30 2.6 Power s p e c t r a l d e n s i t y ( f ) f o r FM speech modulated s i g n a l using f i l t e r A 32 2.7 Power sp e c t r a l density V (f) f o r FM speech modulated s i g n a l using f i l t e r B 33 2.8 Power s p e c t r a l density v a c ( 0 for FM speech modulated s i g n a l using f i l t e r C 34 2.9 V ( f ) vs rms bandwidth, B, f o r ac ' f = 20 kHz 35 3.1 Transmitted s i g n a l spectra S(f) i n the adjacent channel region using f i l t e r A 42 3.2 Transmitted s i g n a l spectra S(f) i n the adjacent channel region using f i l t e r B 43 v i i i Figure Page 3.3 Transmitted s i g n a l spectra S(f) i n the adjacent channel region using f i l t e r C 44 3.4 L o g 1 0 ( 1 0 U S ( f ) ) versus l o g 1 Q ( f x IO - 3) in the adjacent channel region f o r f i l t e r A 46 3.5 L o g 1 0 ( 1 0 U S ( f ) ) versus l o g 1 Q ( f x IO - 3) in the adjacent channel region f o r f i l t e r B 47 3.6 L o g 1 Q ( 1 0 1 1 S ( f ) ) versus l o g 1 Q ( f x 10~ 3) in the adjacent channel region for f i l t e r C 48 3.7 Block diagram of conventional FM receiver 50 3.8 Spectra V(f) of transmitted FM speech s i g n a l using p r e f i l t e r A 55 3.9 Spectra V(f) of transmitted FM speech s i g n a l using p r e f i l t e r B 56 3.10 Spectra V(f) of transmitted FM speech si g n a l using p r e f i l t e r C 57 3.11 Interference c o e f f i c i e n t . J ., as a ' v l function of channel width, W. The rms bandwidth B = 2900 Hertz 59 3.12 Interference c o e f f i c i e n t , J , as a function of channel width, W. The rms bandwidth B = 3155 Hertz 69 3.13 Break point p l o t s of f i l t e r A and the CCF f i l t e r 71 3.14 Interference c o e f f i c i e n t , J ^ j , as a function of channel width 81 ix Figure Page 3.15 D i g i t a l data receiver demodulator f o r coherent detection of PAM signals with zero intersymbol interference 83 3.16 Interference c o e f f i c i e n t , TC , as a v function of channel width 86 4.1 Normalized autocorrelation function, p ( x ) , fo r integrated lowpass speech f i l t e r e d with f i l t e r s A and CCF 88 4.2 Autocorrelation function V(T) - v(°°) f o r FM speech modulated s i g n a l using f i l t e r A 89 4.3 Autocorrelation function V(T) - v(<») f o r FM speech modulated s i g n a l using f i l t e r CCF 90 4.4 Power s p e c t r a l density V (f) of FM signal 3C modulated by lowpass speech p r e f i l t e r e d with f i l t e r A 91 4.5 Power s p e c t r a l density V (f) of FM s i g n a l modulated by lowpass speech p r e f i l t e r e d with f i l t e r CCF 92 4.6 Interference c o e f f i c i e n t , J v^» as a function of channel width 94 4.7 Interference c o e f f i c i e n t , J d l , as a function of channel width 95 4.8 Interference c o e f f i c i e n t , TC v, as a function of channel width 96 X Figure Page 5.1 Break point plots of H^f) and rL/f) 99 5.2 Circuit for realization of H 3(f) 100 5.3 Circuit for realization of H^(f) 101 5.4 Schematic diagram of the pre-emphasis f i l t e r 105 5.5 Simulation values for the interference coefficient J , (W) I l l vl x i ACKNOWLEDGEMENT I w i s h t o t h a n k my s u p e r v i s o r D r . R.W. D o n a l d s o n f o r s u g g e s t i n g t h e p r o j e c t a n d f o r t h e p a t i e n c e a n d h e l p h e h a s g i v e n t h r o u g h o u t t h e c o u r s e o f t h e w o r k . I am g r a t e f u l t o t h e N a t u r a l S c i e n c e s a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a f o r t h e f i n a n c i a l s u p p o r t r e c e i v e d u n d e r t h e i r s c h o l a r s h i p p r o g r a m a n d G r a n t A - 3 3 0 8 . F i n a l l y , I w i s h t o e x p r e s s my a p p r e c i a t i o n t o t h e o f f i c e s t a f f f o r t h e i r w i l l i n g a s s i s t a n c e a n d c h e e r f u l n e s s . 1 I INTRODUCTION 1.1 Background Pre-emphasis is the technique of accentuating certain frequency components of a communications signal relative to the other components prior to the transmission of the signal over a communications channel. The technique is used to improve the signal-to-noise ratio of the received signal. It is in FM broadcasting and FM land mobile services that the system finds its major use. The two cases differ in that a speech signal is communicated in the land mobile service whereas music as well as speech is transmitted in the broadcast service. This thesis deals with the pre-emphasis of speech in the context of frequency modulation radiotelephony systems. The history of the subject is closely linked to the history of FM and by way of introduction a short summary follows. In the communications f i e l d where major new developments are announced each decade the subject of frequency modulation is an old one. A good historical account of the development of FM radiotelephony until the early 1930's is given by Armstrong!1]. A brief description of the early history of FM drawn from this source follows. wnile the exact origin of this method of signalling is not clear i t appears to be shortly after the invention of the Poulsen arc in 1903. This date precedes the invention of the vacuum tube oscillator in 1912. Among the early proposals for FM systems in these days before signal amplification was possible 2 were the use of electrostatic microphones in the oscillator c i r c u i t cf the transmitter and the use of mistuned resonant circuits in the receiver. There were, however, no successful practical implementations of frequency modulation and amplitude modulation was unanimously considered superior. After the invention of the vacuum tube oscillator the subject was revived. It was proposed to reduce the bandwidth required for a communications signal by using FM rather than AM transmission. The deviation of the carrier frequency was to be held below a fraction of the highest frequency component of the modulating signal. In 1922 Carson[2] showed that the bandwidth required was at least double the value of the highest modulating frequency and concluded that, "this method of modulation inherently distorts without any compensating advantages whatsoever." During the next decade the subject was examined in a number of mathematical treatments by authors who confirmed Carson's results and concluded that frequency modulation was of no practical value. In 1936 Armstrong[l] showed that noise reduction can be obtained by using FM signalling at the expense of increased bandwidth of the transmitted signal. Both the theory of the process and the practical implementation of i t were described. Armstrong himself funded an experimental FM broadcast station in i539 t o prove i t s worth. Following this, FM slowly established it s e l f in the broadcast industry. The merits of narrowband FM as opposed to AM for mobile land services were investigated prior to World War II. The theory and preliminary tests favoured AM. The advent of the war 3 intensified the examination and testing of the two systems. In 1941 i t was reported[3] that FM proved better than AM in fi e l d tests conducted by the American military. The early AM mobile radio systems had serious flaws. They were very susceptible to static and noise interference and to received signal level variations. The noise and static is due to co-channel and adjacent channel interfering signals, ignition noise, receiver generated noise and industrial and meteorological electrical disturbances. The signal level variation is caused by changing multipath interference, changes in path transmission characteristics and variations in the transmitter-receiver distance. The FM systems provided better discrimination against noise and fading because the receiver detects changes in frequency and is nonresponsive to amplitude changes. Shortly after the introduction of FM into broadcasting and the land mobile service i t was found that significant gain in the output signal-to-noise ratio could be achieved through pre-emphasis of the modulating signal at the transmitter and de-emphasis of the demodulated signal at the receiver. This technique offers some improvement in AM systems but i t is most efficient for FM systems. This is because the noise power spectral density at the output of the demodulator has a squared frequency dependence in the case of FM but a linear dependence in the case of AM, both cases being considered for a white additive channel noise. Though there was some discussion as to the optimum amount of pre-emphasis in the years following its introduction the pre- and de-emphasis system has changed l i t t l e since i t s inception. 4 1.2 Summary of Relevant Work Preliminary work suggests that use of a pre-emphasis f i l t e r different from the one currently employed may be advantageous. Costas[4] derived the optimum design equations for linear pre-and post-filtering to minimize the mean square error of a message received after transmission over a noisy channel. The optimization assumed an average signal power constraint. A moderate improvement in system performance due to the pre- and de-emphasis networks was demonstrated. It was noted, however, that the model used for the message was not well chosen and nonlinear systems were not considered. Boardman and Van Trees[5] analysed the following formulation of the optimum pre-emphasis problem. A continuous stationary Gaussian random process angle modulates a sinusoidal carrier and is transmitted over an additive white Gaussian noise channel. The linear pre- and post-filters which minimize the mean square error of the received signal are to be found. The system operation is constrained to be linear (above threshold) and the rms bandwidth of the transmitted signal is also constrained. The optimum f i l t e r derived is dependent upon the message input, the noise input and the carrier power level. The synthesis of the optimum pre-emphasis network was not undertaken but the d i f f i c u l t y of this problem is noted. A mathematically t w i . v c i i i c u l message spectrum was assumed and spectra were presented for both the message and the pre-emphasized message. It was concluded that the typical FM pre-emphasis system performs almost as well as an optimum filtered angle modulation system. This conclusion, however, was not based upon an 5 experimentally determined model of the message spectra and the spectra of the the transmitted signals were not determined. It should be noted that determination of the signal spectra is important because two signals with the same rms bandwidth can be very different from the point of view of spectral occupancy. This has important consequences for the adjacent channel interference problem. Jakes[6] considered the signal-to-noise ratio improvement for a number of pre- and post-filters and a number of speech models. The analysis was based upon an earlier result derived by Schwartz[7 ]. This result was derived for an oversimplified statement of the pre-emphasis problem. It is only intended to be an illu s t r a t i o n of the problem rather than a rigorous solution of i t . 1.3 Scope and Outline of the Thesis This thesis considers the pre-emphasis problem for FM transmission of speech over a noisy channel. The following issues are important for assessing the usefulness of pre- and post-filtering schemes for FM radiotelephony and are examined in the sequel. 1. The spectral density, S(f), of the transmitted signal. In particular, bandwidth restrictions and adjacent channel interference can be determined when S(f) is known. 2. The mean square error performance in noise. This may be compared with the optimum to give a measure of performance. 3. The pre- and post-filter implementations. A realization 6 u t i l i z i n g current hardware is desired. 1. Ths testing of promising f i l t e r i n g schemes with actual speech. It is important to verify the validity of the models used in the analysis. In chapter 2 the problem is formulated analytically and the solution is derived using variational calculus. The synthesis of the mathematically described f i l t e r is undertaken. Spectra of the transmitted FM signals are presented and considered with reference to a spectral emission constraint. The mean square error of the f i l t e r i n g schemes is calculated and compared to the optimum (unrealizable) mean square error. The chapter concludes with a description of the de-emphasis network. Chapter 3 considers the adjacent channel interference problem. The t a i l s of the transmitted spectra are presented and characterized by a ro l l - o f f rate. An interference coefficient is defined and computed to measure the disturbance caused a voice transmission by an adjacent channel voice transmission. The interference problem for voice-to-digital and digital-to-voice situations are similarly treated. The effect of the de-emphasis network on the output signal-to-noise ratio is examined. In chapter 4 an alternative model for speech is presented and results based on this model for transmitted signal spectra and interference coefficients are presented. Chapter 5 offers a design implementation of the f i l t e r i n g scheme. The experimental measurement of the system is discussed. A computer simulation is used to verify previous results. 7 II PRE-EMPHASIS NETWORKS FOR ANALOG FM RADIOTELEPHONY 2.1 Problem Statement The trend towards di g i t a l transmission of radiotelephony is evident. There i s , however, continued widespread use of analog FM radiotelephony in the land mobile radio channels. The use of analog FM in these channels may well be expected to continue for many years. The owners and manufacturers of communications equipment are reluctant to accept new standards which will render their systems useless. In view of the ever increasing demands for spectrum space in the land mobile channels i t is appropriate to ask whether technical changes can be effected to provide more user channels in a given spectrum allocation. Ideally, the technical changes would be of such a nature that existing equipment can be easily modified to be compatible with the new standard. In the following a new audio pre-emphasis function is proposed. It is shown that the new function gives considerable spectral bandwidth savings over the pre-emphasis function currently employed. Since only audio processing is involved this approach to spectrum conservation is easily implemented on new as well as existing systems. The problem to be solved is the following. Design a pre-ar.d de emphasis characteristic for the processing of speech in FM radiotelephony transceivers which will maximize the root mean square (rms) bandwidth of the transmitted signal subject to a 8 distortion constraint. Equivalently, the problem can be formulated as that of finding a characteristic to minimize the mean square error given a bandwidth constraint on the transmitted signal. The latter problem statement is the one pursued in the following analysis. The following assumptions are made in the analysis. 1. The audio message, m(t), is a wide sense stationary random process. 2. The f i l t e r in the FM modulator is linear and time invariant. 3. The FM transmitted signal is s(t)=/2F ? cc>s{2iif t + e + /^mfOdt}, where P is the power of the transmitted signal, f is the carrier frequency and © is a random phase angle. 4. The system operation is in the linear region allowing the use of linearized FM system models. 9 2.2 Analysis Let S(f) denote the power spectral density of the transmitted FM signal. When the message is non-Gaussian the spectrum is d i f f i c u l t to calculate. Speech is not a Gaussian random process and analytical determination of the transmitted signal is not tractable. To fa c i l i t a t e computation we assume that speech is a Gaussian random process. We can then arrive at a solution which can be verified experimentally or by computer simulation. For a Gaussian message[8] S(f) = f- [ V(f + f c ) + V ( f - f c ) ] , where V ( f ) is the lowpass equivalent power spectral density of the transmitted signal, V(f )=2/Q v(i)cos{2nfx}dx (2.1) and M(f) is the Fourier transform of the audio message m(t). 1 0 The transmitted root mean square bandwidth, B, i s determined / " f 2S(f)df B 2= ; s ( f ) d f —00 The rms bandwidth expressed in terms of V(f) i s / " f 2 V ( f ) d f B2=-; v ( f ) d f or B2 = ! m f2v(f)df since / v(f)df=l. The mean square error (assuming use of an optimum demodulator) i s given by[5] C(f)(N /2P)df ( 2- 3) .°° g __ m . s . e . = / ^ 9 o o (N /2P) + (C(f)!H(f)|Z/(An O ) o where C(f) i s the power spec tra l density of the speech, 11 H(f) i s the f i l t e r c h a r a c t e r i s t i c and N q/2 i s the two-sided noise power spectral density. For Gaussian random processes the rms bandwidth of the modulated signal is[9] m 211 / M(f)df — 0 0 1/2 (2.4) where a i s the rms value of the message process. The bandwidth m constraint can be expressed as /°° C(f) |H(f)|2< 4 n 2 B 2 Let Y(f)= |n(f)| 2, then y(f) i s chosen to minimize the mean square error given a bandwidth constraint. The solution uses the calculus of variations with the LaGrange m u l t i p l i e r , ro C(f)(N /2P)df L , ( e ) = / - — 8  6 -"(N /2P) + (C(f)( Y(f) + e 6 ( f ) ) / ( 4 n 2 f 2 ) ) o + X/ C(f)( Y(f) + e6(f))df (-l)C(f)(N /2P)C(f)6(f)/(4n 2f 2)df oo r\ L » r ( e ) = / e  6 (N /2P) + (C(f)( Y(f) + e6(f))/(4Tt 2f 2)} 2 o + X/ C(f)6(f)df 1 2 L ' f i ( 0 ) - / (-l)C 2(f)(N n/2P)6(f)/(4n 2f 2)df {(N0/2P) + c ( f ) Y ( f ) / ( 4 n 2 f 2 ) } 2 + X/. C ( f ) 6 ( f ) d f L' ( 0 ) = 0 V 6 ( f ) -=> - 0 A ' - u ; - - — 9 9 9 " " {(N Q/2P) + C ( f M f ) / ( 4 n V ) } Z X{(N /2P) + C ( f ) Y ( f ) / ( 4 n 2 f 2 ) } 2 - C ( f ) ( N / 2 P ) ( l / ( 4 T I 2 f 2 ) ) = 0 o o Y ( f ) - « ( C ( f ) / X ) ( N / 2 P ) / ( 4 n 2 f 2 ) } 1 / 2 - (N / 2 P ) ) ( 4 n 2 f 2 ) / C ( f ) 0 0 Y ( f ) - ( { ( C ( f ) / X ) ( N / 2 P ) 4 n 2 f 2 } 1 / 2 - (N / 2 P ) 4 n 2 f 2 ) / C ( f ) o o f({(C(f)/X)(N / 2 P ) ( 2 n f ) 2 } 1 / 2 - (N / 2 P ) ( 2 n f ) 2 ) / C ( f ) , ( 2 . 5 ) 1 3 where W i s the range of f where c(f)N o(2nf)' 2PX rN\(2nf)' or (2.6) c(f)> x^N oy2nf)' An equation to determine the value of X can be obtained by combining (2.5) with (2.4) y i e l d i n g (2.7) Note that the l i m i t s of integration depend on x and are given by (2.5). The determination of X w i l l be discussed in more d e t a i l l a t e r . When X has been determined the optimum pre-emphasis c h a r a c t e r i s t i c i s H(f) - Y 1 / 2 ( f ) (U(C(f)/X)(N o/2P)(2IIf) 2) 1 / 2 - (N o/2P)(2nf) 2} 1 / 2/C 1 / 2(f) , 1 feW =) (2.8) \p, feW . 1 4 When c h o o s i n g t h e r o o t we p i c k t h e r o o t w i t h i t s p o l e s and z e r o s i n t"he l e f t hand s i d e ( n e g a t i v e r e a l c o m p o n e n t ) o f t h e c o m p l e x p l a n e f o r s t a b i l i t y . The P a l e y - W i e n e r c r i t e r i o n g u a r a n t e e s t h a t t h i s f i l t e r i s n o t r e a l i z a b l e . T h e mean s q u a r e e r r o r o f t h e op t imum u n r e a l i z a b l e f i l t e r ( o . m . s . e . ) o b t a i n e d by c o m b i n i n g ( 2 . 8 ) w i t h ( 2 . 3 ) i s o.m.s.e. = (N X / 2 P ) 1 / 2 / { ( 2 n f ) 2 C ( f ) } 1 / 2 d f + / n C ( f ) d f . (2.9) o W w 2.3 D e t e r m i n a t i o n o f The L a G r a n g e M u l t i p l i e r In o r d e r t o s o l v e ( 2 . 7 ) f o r X ( t h e L a G r a n g e m u l t i p l i e r ) t h e l i m i t s o f t h e i n t e g r a t i o n must be known. T h e s e l i m i t s a r e d e t e r m i n e d ( f r o m ( 2 . 6 ) ) by t h e r o o t s o f t h e e q u a t i o n C(f) - X(N /2P)(2TIf) 2 = 0 , (2.10) o w h i c h a g a i n r e q u i r e s k n o w l e d g e o f t h e v a l u e o f A . G i v e n t h e s e c o n s i d e r a t i o n s , i t i s n a t u r a l t o seek an i t e r a t i v e n u m e r i c a l a l g o r i t h m f o r t h e d e t e r m i n a t i o n o f X. L e t W1 and Wu d e n o t e t h e l o w e r a n d u p p e r l i m i t s , r e s p e c t i v e l y , o f t h e r e g i o n o f i n t e g r a t i o n W. S t a r t i n g w i t h ( 2 . 7 ) we d e r i v e 1 5 'W c ( f ) / N o \ ( 2 n f ) ' X \2P/ 1 / 2 / N \ ( 2 n f ) 2 i - f e • 2 2 df - An B x" 1 / 2/ u C(f) / N \ ( 2 H f y ( I ) 1 / 2 W / N \(2nf) 2 0 9 df - / „ u { - i ] df = 2 n 2 B 2 W 1V2P/ 1 /9 W w l c(f) |N Q j(2nf)' 1/2 df -2P/ An' l w u 3 - w x 3 ) = 2n 2B 2 w 9 1/9 / T 7 U {C(f)(N /2P ) (2nf) /} 1 /^df W1 O x l / 2 = — 2TI 2{B 2 + (N / P ) ( ( W 3 - W 3)/3> 0 U -A-(N /2P)1/2ZU {C(f)f2}1/2df 0 "1 l / 2 =  J I{B 2 + (N / P ) ( ( W 3 - W 3)/3)} O U J. 16 (N /2P) {/„ U { C ( f ) f 2 } l / 2 d f > 2 0 VL (2.11) X n 2{B 2 + (N / p ) ( (w n 3 - w 3 ) / 3 ) } 2 o U J. One iteration consists of finding the roots w1 and ^ u of (2.10) using the estimate of x determined from the previous iteration and then using (2.11) to find a new estimate of X. To start the algorithm a value of X must be chosen such that the line y = X(N /2P) intersects the curve y = C(f)/(2nf) 2 at four o points. This is illustrated graphically in Fig. 2.1. F i g . 2.1 Range of s t a r t i n g value of X. 1 7 A s t a r t i n g value of * must thus be chosen which s a t i s f i e s the c o n d i t i o n (2M)' > * <- > o s t a r t _J maximum The convergence of t h i s a l g o r i t h m to the proper value of X can be j u s t i f i e d by i n s e r t i n g the estimate of X i n ( 2 . 1 0 ) and ( 2 . 1 1 ) and v e r i f y i n g that e q u a l i t y o b t a i n s . 2 . 4 Approximation of the Non-Realizable F i l t e r The t r a n s f e r f u n c t i o n of the optimum f i l t e r i s zero for few". Again, l e t Wu and W"1 denote the upper and lower band edges of W. Then Wu and W][ depend on the audio s p e c t r a l d e n s i t y C ( f ) , the s i g n a l to noise r a t i o ( 2 P / N q ) and the bandwidth c o n s t r a i n t B. A simple approximation to the u n r e a l i z a b l e f i l t e r uses a two pole network f o r the second term, G(f) = . — , (K i s a constant) c 1 / z ( f ) ( f x + J f ) ( f u + j f ) where f, and f ( f < f ) are the l o c a t i o n s of the two poles. The l u l U s e l e c t i o n of the pole p o s i t i o n s i n v o l v e s a t r a d e - o f f between mean square e r r o r performance and adjacent channel i n t e r f e r e n c e . This w i l l be demonstrated by three f i l t e r designs. We denote the designs as F i l t e r A, F i l t e r B and F i l t e r C. The pole p o s i t i o n s 18 of the three f i l t e r s are given in Table 2-1. Poles F i l t e r A F i l t e r B F i l t e r C f l 7 0 Hz. 260 Hz. 120 Hz. f u 7 00 Hz. 17 00 Hz. 4000 Hz . Table 2-1 F i l t e r Pole P o s i t i o n s 2.5 Design of F i l t e r s and Calculation of Signal Spectra The long term average spectral power density of speech has K a o n measured!10] rC(f) = c r 0  ( f 2 + 70 2)(f 2 + 180 2)(f 2 + 400 2)(f 2 + 7002) (2.12) The pre-emphasis function using (2.12) i s then H(f) = K(f + j70)(f + j!80)(f + j400)(f + J700) f ( f x + j f ) ( f u + j f ) c o (2.13) 19 The message power spec tra l density i s 2 2 9 K f (2 14) M(f) = ccniHcor = • 9 2 2 2 ( f x Z + f ) ( f u + f > The lowpass equivalent power spec tra l dens i ty , V ( f ) , of the transmitted s igna l can be ca lcu la ted using (2.1) and ( 2 . 2 ) . For the f i l t e r s under considerat ion here K 2 s i n 2 { n f i } = exp{-2/°° • ! df} ~~ 9 2 2 2 2 4n 2 ( r + f p ( f + f u z) Since 1 9 9 2 2 (f Z + f ^ ) ( f Z + f^) 2 2 f + 2 2 f + f u 2 0 i t f o l l o w s t h a t df < f l 2 + f 2 > < f u 2 + f 2 ) C ~ f l 2 df 2 2 f 2 + f , 2 - / df 2 2 f + f u 2 2 f " f i u 1 I L _ 2L f i f i . 1 u J <fu + W l A l s o , cos { 2 n f x } c o s { 2 n f x > n r df = 2 / " df = — e x p { - 2 n l a x | } f + a r 2 2 f + a Thu s , K 2sin 2{nfx} I = / "df 4 n 2 ( f 2 + f x 2 ) ( f 2 + f u 2 ) df - r 8JI2 ( f 2 + f x 2 ) ( f 2 + f u 2 ) cos {2T I fx } df} 9 o o ° cr + f^)(fz + £u~) 21 1 .2 1TI 8 n ' < f u + f l ) f u f l | f u 2 - f l 2 1 IL exp{-2TIf1|T|} exp{-2nfu|TlJ u i d 8TI (f + f„)f f, v u Y u 1 f uexp{-2nf 1lTlJ - f i e x P { - 2 n f u l T l J ( fu 2 - f l 2 ) f u f l 8lI<fu + fl> V l f exp{-2nf. |x|} - £ exp{-2Hf | T | } u x - i u (f - f-)f f. u l u l (2.15) and V ( T ) = exp(-2I). From (2.4) 2TIB = {/" M(f)df) a n d f r o m ( 2 . 1 4 ) 22 / M(f)df = / 2 2 K f 2 2 2 2 ( f ^ + f ) ( f u + f ) df 2 oo 9 KV f Z 2 2 f + f x ? 2 2 2 f 1 - f, f, - f u 1 1 u _ 2 2 f + f u df K . CO 2 2 ^ - c o f - f . u 1 2 2 f + f ] _ 2 2 f + f u df K f 2 - £ , 2 u 1 1 - 2 2 f + f l - 1 + _U_ 2 2 r + f u df K 2 2 f - f i u 1 u 2 2 2 2 f + f u f + f l _ i df —^(nf u -nf 1 )= T 4r f - f u 1 g i v i n g 2 2 4TTB Z = K2n f + f. u 1 or K 2 = 4 n B 2 ( f u + f ) Combining t h i s r e s u l t w i th (2.15) g i v e s B 2 1 f exp{-2nf.. | T J } - f .exp{-2nf I T | > _ r U 1 1 1 1 U ' ' ^ ~ v T ( f u - f i ) f u f i and hence V (T ) = exp{-21} exp{ }exp{ f f, f f. u 1 u 1 f expf^n^lx l } - f ^ e x p ^ n f j - r l u 1 24 T h i s c a n be p u t i n a more c o n v e n i e n t f o r m by u s i n g t h e i n t e g r a t i o n p r o p e r t y o f t h e F o u r i e r t r a n s f o r m t o d e r i v e a u s e f u l r e s u l t . I f M ( f ) and R ( O a r e r e l a t e d by t h e F o u r i e r t r a n s f o r m , m i t f o l l o w s t h a t M(£)/(4!T2f2) a n d R CO a r e r e l a t e d by t h e F o u r i e r t r a n s f o r m where p ( t ) = m f t J d t . 1 Then R CO « > M(f) 4TI f K 2 / ( 4 n 2 ) f 2 - f l 2 U -L f 2 + f 2 ' f 2 + f 2 1 u _ 2 2 P f 2 - f 2 u 1 n n — exp{-2nf1lT|} exp{-2TIfu|T|} f. f . and s i n c e R (0) P 1C/(4II) ^ 2 " f l 2 u 1 f l fu K 2/(4n) (f + f-)f f, u l u l (2.17) i t f o l l o w s t h a t v ( 0 = exp{Rp(x) - Rp(0)> . 1This i s true for message processes which have a zero at the origin in their spectral power density function. 25 L e t P ( T ) = R (T)/R p(0) , then V ( T ) = exp{R (0)(P(T) - 1)} . (2.18) Combining ( 2 . 1 6 ) and ( 2 . 1 7 ) gives R (0) = and ( 2 . 1 8 ) becomes V ( T ) = exp{-R (0)}exp{R (O )p (x ) } . There is a spectral impulse of magnitude v(») = exp{-Rp(0)} at f = 0. Removing this impulse will f a c i l i t a t e the numerical computation of V(f). Let v a c ( f ) denote the lowpass equivalent power spectral density function of the sidebands of the transmitted signal. Then V (f) = 2/n{v(x) - v(°°)}cos{2TIfT}dT ac . 26 V (f) = 2/o{exp{-R (0)}exp{R (0) P(T)} - exp{-R (0)}}cos{2nfT}dT ac P P P = 2/oexp{-R p(0)} {exp{R p(0)p(x)} - 1}cos{2nft}dx (2.19) where f exp{-2T!T!1 |T I > - f.exp{-2nf IT| } p ( x ) = . f ~ f l U 1 The normalized autocorrelation function, p ( x ) , of the integrated prefiltered speech is shown in Fig. 2.2 for the three f i l t e r designs. Figs. 2.3, 2.4, and 2.5 show the autocorrelation function V(T) - V(«) V S T for rms bandwidths of 1000, 2500, 5000 and 7500 Hertz for f i l t e r A, B and C respectively. The power spectral density v a c ( f ) can be computed numerically using (2.19). Note that the upper limit of integration extends to infinity and hence the application of numerical techniques will require truncation of the integral. It is thus desireable to bound the error in the truncated t a i l . This topic is treated in greater depth in Appendix A. The integrand of (2.19) is oscillatory and care must be exercised in the numerical integration. The integral is evaluated here by 27 0.0 2 0 4 0 6.0 8.0 10.0 TIME X (msec.) 0 . 0 B -i 1 1 1 1 0 2.0 3.0 4.0 5.0 TIME X (msec.) o.o -I 1 1 r~ 2 0 4 . 0 6 . 0 B . O TIME X (msec.) 10.0 Fig. 2.2. Normalized autocorrelation function, p(x), for integrated speech f i l t e r e d with f i l t e r s A, B and C. B=2500 0.24 . (msec.) 0.32 0.4 Fig. 2.3. Autocorrelation function V ( T ) - V ( » ) for FM speech modulated signal using f i l t e r A. 29 o.o B = 2500 T 1— 0.32 0.46 TIME X (msec) 0 . 6 4 0.B F i g . 2.4. Autocorrelation function V ( T ) - v(°°) for FM speech modulated signal using f i l t e r B. 30 Fig. 2.5. Autocorrelation function V ( T ) - v(») for FM speech modulated signal using f i l t e r C. 31 using an adaptive cautious Romberg extrapolat ion algori thm[11] . F i g s . 2.6, 2.7 and 2.8 show v (f) vs f for various values of ac rms bandwidth for f i l t e r s A, B and C r e s p e c t i v e l y . The emission l eve l s are r e l a t i v e to the unmodulated c a r r i e r l e v e l . The "top-hat" i s the spectra l emission constra int of Communications Canada Radio Standards S p e c i f i c a t i o n RSS-119[12]. The spectra l emission constra int at f = 20 kHz. determines the maximum allowable value of B. F i g . 2.9 shows V (f) vs B for f=20 kHz. for the three 3 ac f i l t e r s . The largest rms bandwidth for which the spec tra l emission constra int i s s a t i s f i e d i s B=7000, 6910 and 6740 Hz. for f i l t e r A, B and C re spec t ive ly . The Communications Canada proposed radio standards s p e c i f i c a t i o n previous ly mentioned requires f i l t e r i n g the speech by a network with transfer function H ( f ) = Kp j f + 3000 > ( K p i s a constant) (2.20) where f i s frequency in Hertz . It has been shown that the largest rms bandwidth for which the spectra l emission constra int i s s a t i s f i e d using the proposed standard is B=2900 Hz. [13] . In the absence of fading the output s ignal to interference r a t i o improves as the cube of the rms bandwidth of the modulated 2 FM s igna l (since the s ignal output i s proport ional to B and the interference output i s inverse ly proport ional to B) . In the presence of fading the s igna l to interference r a t i o i s approximately constant but by using d i v e r s i t y the dependence on 32 o F i g . 2 . 6 . Power s p e c t r a l density V^Cf) for FM speech modulated si g n a l using f i l t e r A. 33 Fig. 2.7. Power spectral density V (f) for FM speech modulated signal using f i l t e r B. Fig. 2.8. Power spectral density V (f) for FM speech modulated signal using f i l t e r C. 35 7 . 5 (kHz) Fig. 2.9. V (f) vs rms bandwidth, B, for f = 20 kHz 3.C the cube of B is regained!6]. It is thus advantageous to achieve a greater transmitted signal bandwidth by speech processing. The advantage of, for example, f i l t e r A over the proposed standard (2.20) in this regard is a 2.4-fold increase in the bandwidth which corresponds to a 11.5 dB increase in the signal-to-interference ratio in the nonfading and diversity cases. An attractive alternative to maximizing the signal-to-interference ratio by increasing B is to fix B and with the same signal-to-interference ratio reduce the channel spacing. This increases the number of channels available in a fixed spectrum space. This is discussed further in Chapter 3. 36 2.6 Optimum Mean Square Error and Realizable Mean Square Error The mean square error of the optimum unrealizable f i l t e r was given in (2.9) o.m.s.e. = N X o 2P 1/2 / w { ( 2 n f ) 2 c ( f ) } 1 / 2 d f + / g c ( f ) d f Combining this with (2.12) yields o.m.s.e. = N X o 2P 1/2 'w 2 n { C f o } 1 / 2 d f ( f 2 + 7 0 2 ) ( f 2 + 1 8 0 2 ) ( f 2 + 4 0 0 2 ) ( f 2 + 7 0 0 Z ) C f o ( f 2 + 7 0 2 ) ( f 2 + 1 8 0 2 ) ( f 2 + 4 0 0 2 ) ( f 2 + 7 0 0 2 ) d f ( 2 . 2 1 ) Calculation of the optimum mean square error requires knowledge of the corresponding values of X , the signal— t o — noise ratio (2P/N ). w, and w . An iterative algorithm for their 0 1 u determination has previously been presented. Some representative values are tabulated in Table 2-2. The mean square error of the realizable f i l t e r is given by (2.3), m.s.e. = / C(f)(NQ/2P) d f . (NQ/2P) + (C(f)|H(f)| 2/(4n 2f 2)} 37 2P/N 0 W l W u X 6.000000 X i o 5 | 0.0026415 11002.26 1.642723 X 10" •12 9.509359 X 10 5 0.0017519 12619.07 1.145202 X 10" -12 1.507132 X 10 6 0.0011595 14482.88 7.950010 x 10" •13 2.388 643 X i o 6 0 . 0007 65 9 | 16632.09 5.497423 X 10" -13 3.785744 X i o 6 0.0005050 19111.16 3.787737 X 10" -13 6.000000 X i o 6 0.0003324 21971.51 2.601007 X 10" -13 9.509359 X i o 6 | 0.0002184 | 25272.67 1.780511 X 10" -13 1.507132 X i o 7 | 0.0001434 j 29083.56 1.215292 X 10' -13 2 . 38 8 643 X i o 7 | 0.0000939 | 33483.94 | 8.272424 X 10' -14 3.785744 X i o 7 | 0.0000615 | 38566.23 5.616621 X 10' -14 6.000000 X i o 7 | 0.0000402 I 44437.38 ! 3.8043 1 9 X 10 -14 Table 2-2 Representative values of 2P/N Q, V^, Wu and X < B = 2900 Hz ) . The u n i t s of 2P/N o, W1 and Wu are Hertz. The un i t of X i s Watts/Hertz . Combining this with (2.14) and (2.16) gives C(f)(N /2P) 00 " m.s.e. = / o d f (N /2P) + ( { B 2 ( f + f . ) } / { n ( f 2 + f 2 ) ( f 2 + f 2 ) » o u J. j . « C ( f ) ( N /2P) = 2/°° - - (.2.22) ° (N /2P) + ({B 2 ( f + f , ) } / ( n ( f 2 + f 2 ) ( f 2 + f 2 ) » 0 11 X x 38 The mean square error of the Communications Canada proposed f i l t e r (CCF) can be obtained by combining (2.3) with (2.20) giving C(f)(No/2P) m.s.e. = / df (N /2P) + ({C(f)K 2 ( f 2 + 3002)}/{4n2f2(f2 + 30002)}) o p C(f)(N /2P) = - df . (2.23) (N /2P) + ({C(f)K 2 ( f 2 + 3002)}/{4n2f2(f2 + 30002)}) o p The gain constant, Kp , can be expressed in terms of the rms bandwidth, B, by using (2.4) with M(f) = C(f )| H(f )|2 . The derivation is similar to that which lead to equation (2.16). The result is K = (28.35)(B). Equation (2.22) can thus be written P to show the dependence on B as C(f)(NQ/2P) (N /2P) + ({C(f)(803.8)B2(f2 + 3002)}/{4TI2f2(f2 + 30002)}) C(f)(N /2P) . 2r _ . °— d f 0 (N /2P) + ({C(f)(803.8)B2(f2 + 3002)}/{4n2f2(f2 + 30002)}) (2.24) 39 The integrals of (2.21), (2.22) and (2.24) can be easily evaluated numerically using Simpson's technique. Representative values are given in Table 2-3 for a high and a low value of signal-to-noise ratio. 2P/N = 6.0 x 107 Hz. o . , F i l t e r Mean Square Error m.s.e./o.m.s.e. Optimum 0.0000202 1.000 CCF 0.0001591 7.876 A 0.0000626 3.099 B 0.0000357 1.7 67 C 0.0000298 1.475 2P/N = 6.0 x 10 5 0 Hz. F i l t e r Mean Square Error m.s.e./o.m.s.e. Opt imum 0.0009352 1.000 CCF 0.0020993 2.245 A 0.0018345 1.962 B 0.0013368 1.429 C 0.0016254 1.738 Table 2-3 Representative values of mean square error (B = 2900 Hz.) It is seen that a l l three of the two pole f i l t e r s (A, B, and C) have smaller mean'square error than the proposed f i l t e r (CCF). F i l t e r A which offers the greatest constrained bandwidth has the greatest error of the three f i l t e r s . A trade-off between bandwidth (or spectral compression) and mean square error is 40 thus i nd i ca ted . 2.7 The De-emphasis Network For a given s ignal power density spectrum, C ( f ) , a given pre-emphasis network, H ( f ) , and a given noise function with power density spectrum, S n ( f ) , the optimum de-emphasis network, D ( f ) , i s given by[14] D(f) H(-f)C(f) H(f)| 2C(f) + Sn(f) (2.25) Since the impulse response of a r e a l i z a b l e network must be rea l |H(-f)| = |H(f) | . Then, under the condi t ion of a large s i g n a l -to-noise r a t i o (2.25) becomes |D(f)| = |H(f) -1 ( 2 . 2 6 ) This resu l t w i l l be used in the sequel in the determination of the SNR improvement of the pre- and de-emphasis system. 41 III INTERFERENCE AND NOISE CONSIDERATIONS 3.1 Spectral Tails The t a i l s of the transmitted signal spectra are of considerable importance since they determine the adjacent channel interference. The behaviour of the spectra in the region 15 to 45 kHz. off the carrier frequency is shown in Figs. 3.1, 3.2 and 3.3 for f i l t e r s A, B and C respectively. This region corresponds to the adjacent channel when the channel spacing is 30 kHz. which is currently the standard in the 138-174 MHz. land mobile services band. The rate at which the spectral t a i l s f a l l off is of particular interest for the adjacent channel interference problem. This is especially true in a mixed analog-digital environment since i t has been shown that for many dig i t a l signalling schemes of interest FM radiotelephony causes more adjacent channel interference than does another digital data signal of similar type[l3]. This interference can be reduced by decreasing the transmitted signal bandwidth, B, by increasing the channel separation, or by pre-processing the voice signal prior to FM modulation (apart from employing digital transmission of the voice signals). Since the output signal-to-noise ratio for the demodulated voice signal increases with B i t is not desirable to reduce B. Increasing the channel separation implies a reduction in the number of available channels in a fixed spectrum space. Hence, this is not a viable solution either. One hopes that speech pre-processing can result in 42 F i g . 3 . 1 . Transmitted signal spectra S(f) i n the adjacent channel region using f i l t e r A. 43 F i g . 3.2. Transmitted signal spectra S(f) i n the adjacent channel region using f i l t e r B. 44 F i g . 3 . 3 . Transmitted signal spectra S(f) i n the adjacent channel region using f i l t e r C. 45 transmitted spectra with good r o l l - o f f . The rol l - o f f of the "trahsmitted signals for the three f i l t e r designs will be considered below. Assume that there is a simple inverse power relation between the transmitted signal spectra in the region of the adjacent channel and the frequency, that is S ( f ) = C f ~ n where C is an undetermined constant. Then, i o n s ( f ) = c f ~ n = c i o _ 3 n f - n = c ( f x i o " 3 ) n and l o g 1 0 ( 1 0 U S ( f ) ) = l o g 1 Q C - n l o g 1 0 ( f x 10~3) where scaling constants have been introduced to fa c i l i t a t e graphing. Figs. 3.4, 3.5 and 3.6 show log 1 Q(10 1 1S(f)) plotted versus l o g 1 Q ( f x 10 ) for the f i l t e r s A, B and C respectively for different values of the rms bandwidth, B. The average values of n in the adjacent channel region are tabulated in Table 3-1. It has been shown that for B = 2900 Hz. the transmitted signal using the Communications Canada proposed 2 f i l t e r f a l l s off as 1/f or n = 2[13]. It is because of this relatively slow spectral r o l l - o f f of FM voice signals that the use of different types of receiver f i l t e r s for data demodulation wi l l not significantly decrease interference from adjacent channel FM radiotelephony. F i l t e r A offers a ro l l - o f f rate of n = 4.4 for B = 2900 Hz. One would expect significantly better performance as regards interference to adjacent channel data 46 Fig. 3.4. L o g 1 0 ( 1 0 U S ( f ) ) versus l o g 1 Q ( f x 10"3) in the adjacent channel region for f i l t e r A. 47 F i g . 3.5. L o g 1 0 ( 1 0 U S ( f ) ) versus l o g 1 Q ( f x 10 _ 3) i n the adjacent channel region for f i l t e r B. 48 49 signals as well as adjacent voice signals by using Fi l t e r A rather than the proposed f i l t e r . R.M.S. Bandwidth F i l t e r A Fi l t e r B Fi l t e r C 1000 4.0 4.0 4.0 2500 4.3 4.3 4.1 2900 4.4 4.3 4.2 5000 5.8 5.1 4.5 7500 6.6 5.6 4.8 Table 3-1 Spectral t a i l r o l l - o f f rate, n. The rms bandwidth is in Hertz. 3.2 Calculat i on of Interference Coef f ic ients The conventional receiver for demodulation of FM radiotelephony is shown in Fig. 3.7. The mixer wi l l contain a local oscillator and probably a radio frequency amplifier. The mixer wi l l translate the received signal r(t) to the intermediate frequency (IF). It will be assumed in the following analysis that the only effect of the mixer stage is to translate the received signal to the intermediate frequency. That i s , the carrier frequency, f is equal to the channel frequency before tne mixer and the intermediate frequency after the mixer. To ease the notational burden f £ w i l l be used in both cases to denote the carrier frequency. The input to the IF f i l t e r is then r ( t ) . The IF f i l t e r is chosen to maximize the f i l t e r ' s output i R E C E I V E D S I G N A L r ( t ) I F F I L T E R F M »• B A S E B A N D G ( f ) D E M O D U L A T O R F I L T E R O U T P U T S I G N A L ai o Fig. 3.7. Block diagram of conventional FM r e c e i v e r . 51 signal-to-noise ratio. This requires!14,15] - S(f) | G ( f ) r = « : , (3.1) S(f) + S ( f ) + S (f) n I where S ( f ) , S R ( f ) and S i ( f ) denote the power spectral densities of the received FM signal, the received noise and the received interference respectively. The noise and interference on the channel wi l l in general vary and are not known a p r i o r i . It is thus necessary to approximate (3.1). When the signal— to - noise plus interference ratio (SNIR) is large | G ( f ) | 2 « 1 and when the SNIR is small , S(f) | G ( f ) | 2 S (f) + S.(f) n i In words, the optimum IF f i l t e r w i l l pass without attenuation those frequencies for which the SNIR is high and strongly attenuate those frequencies for which the SNIR is low. This suggests two models to approximate (3.1). The f i r s t model uses the transmitted signal spectrum and is denoted G 1 ( f ) , where |GxCf>|2 = S(f) - exp{-R p(0)}6(f - f c ) and R (0) = B 2 f u f l as in section 2.5. The spectral impulse at the carrier frequency has been removed. This impulse contributes a d.c. term to the demodulated signal which is removed by the baseband f i l t e r and 52 d o e s n o t e n h a n c e t h e o u t p u t s i g n a l . The s e c o n d m o d e l i s a b a n d p a s s f i l t e r a n d i s d e n o t e d by G 2(f) where - f + f f - f |G (f)T - n( -) + n( £-) . W W The f u n c t i o n n(f) i s t h e u n i t r e c t a n g u l a r p u l s e d e f i n e d a s fl. |f| < 1/2 n(f) - J ( 0, otherwise and W i s t h e c h a n n e l w i d t h . The i n t e r f e r e n c e t o a s i g n a l c a u s e d by an a d j a c e n t c h a n n e l s i g n a l w i l l now be c a l c u l a t e d . L e t t h e i n t e r f e r e n c e c o e f f i c i e n t J v be d e f i n e d a s t h e r a t i o o f t h e power i n t h e i n t e r f e r i n g s i g n a l t o t h e power i n t h e d e s i r e d s i g n a l a t t h e o u t p u t o f t h e I F f i l t e r when b o t h t h e r e c e i v e d s i g n a l s a r e o f u n i t p o w e r . N o t e t h a t t h e c o e f f i c i e n t i s a f u n c t i o n o f t h e c h a n n e l s p a c i n g , W. The p o w e r , S, f r o m t h e I F f i l t e r r e s u l t i n g f r o m a s i g n a l w i t h c a r r i e r f r e q u e n c y f i s P CO 9 S =-/J0O |G(f)r ( V ( f + f c ) + V(f - f c ) } d f 2 ~°° where P i s t h e power i n t h e r e c e i v e d s i g n a l a n d V ( f ) i s t h e l o w -p a s s e q u i v a l e n t power s p e c t r a l d e n s i t y o f t h e t r a n s m i t t e d s i g n a l a s i n s e c t i o n 2.2. F o r t h e f i r s t m o d e l P s < v , ( f + f J + v (f - f )}{V(f + f ) + v ( f - f )}df (3.2) —°° ac c ac c c c where v a c ( f ) i s t h e l o w p a s s e q u i v a l e n t s p e c t r a l power d e n s i t y o f t h e s i d e b a n d s o f t h e t r a n s m i t t e d s i g n a l a s i n s e c t i o n 2.5. The p o w e r , Sv, f r o m an i n t e r f e r i n g a d j a c e n t c h a n n e l s i g n a l c a n 53 be obtained from (3.2) by replac ing f c by f £ + W in the second -ractoj" of the integrand g iv ing P S - — / " {V (f + f ) + V (f - f )}{V(f + f + W) + V(f - f - W)}df v , - 0 0 ac c ac c c c 4 where P v i s the power in the received i n t e r f e r i n g s i g n a l . Let P g denote the power in the des ired received s i g n a l , then P S v i P S v / " {V (f + f ) + v (f - f )}{V(f + f + W) + V(f - f - W)}df -» ac c ac c c c . (3.3) / " {V (f + f ) + V (f - f )}{V(f + f ) + V(f - f )}df -» ac c ac c c c where J v j / W ) denotes the interference c o e f f i c i e n t for the f i r s t model. For the second model P « f + f c f ~ f -/ JU - ) + TK 2 ~ W W and the power from the output of the IF f i l t e r r e s u l t i n g from the i n t e r f e r i n g s ignal i s p „ „ f + f f - f - 0 " C * ) + n( ' 2 W W which i s again obtained by replac ing f £ by f c + w in the second Idctor of the integrand in ( 3 . 4 ) . Let J v 2 ( w ) denote the s =-/°° {it( -) + n( )}{v(f + f ) + v(f - f )}df (3.4) —00 c <-s = {n( ^ )  ( )}{v(f + f + w) + v(f - f - w)}df 54 interference coefficient for the second model. Then, P s v S f + f f - f -) + n(-w -)}{V(f + f + W) + V(f - f c - W)}df w (3.5) f + f f - f W -) + n(-w -)}{V(f + f ) + V(f - f )}df c c Equations (3.3) and (3.5) can be numerically integrated once the transmitted signal spectra are known. The spectra shown in Figs. 3.8, 3.9 and 3.10 for f i l t e r A, B and C respectively yield the interference coefficients of Table 3-2 for W = 30 kHz. It is seen that the interference coefficient depends on the choice of model for the IF f i l t e r . The signal spectrum model yields, in a l l cases, a coefficient of greater magnitude than the rectangular bandpass model. The difference is small, about 4 dB, when the rms bandwidth equals 7500 Hertz and increases to about 15 dB for 1000 Hertz. The interference coefficient as defined here offers a measure of the gain in spectrum space to be had by using f i l t e r A rather than the Communications Canada f i l t e r . One may choose as the new channel spacing that width which results in the same adjacent channel interference as the CCF f i l t e r . This channel width w i l l , of course, depend upon the IF f i l t e r model chosen and the desired maximum signal bandwidth. A value for the interference coefficient, J j , using the CCF f i l t e r has been determinedf13]. The value of -18.0 dB is specified for a rms bandwidth of 2900 Hertz. Fig. 3.11 shows the interference 55 Fig. 3.8. Spectra V(f) of transmitted FM speech signal using prefilter A. The rms bandwidth, B, is in Hertz. The spectral impulses at the carrier frequency are not shown. T 0 0 10 0 20.0 30.0 40.C F R E Q U E N C Y FROM UNMODULATED C A R R I E R F R E Q U E N C Y B=7500 B=5000 B=2900 B=2500 B=1000 50.3 (kHz] Fig. 3.9. Spectra V(f) of transmitted FM speech signal using prefilter B. The rms bandwidth, B, is in Hertz. The spectral impulses at the carrier frequency are not shown. 57 o o t—a . J B = 7 5 0 0 B = 5 0 0 0 B=2900 B=2500 B=1000 n n in n 20 0 30.0 40.0 50.0 FREQUENCY FROM UNMODULATED CARRIER FREQUENCY (kHz) Fig. 3.10. Spectra V(f) of transmitted FM speech signal using prefilter C. The rms bandwidth, B, is in Hertz. The spectral impulses at the carrier frequency are not shown. 58 F i l t e r RMS Bandwidth (Hertz) 1000 2500 2900 5000 7500 A -60.320 -46.209 -47.370 -37 .514 -45.141 -35.852 -34.550 -26.144 -20.394 -16.309 B -57.340 ^42.185 -43.819 -33.548 -41.520 -31.942 -31.551 -24.216 -20.385 -16.371 C -56.458 -39.112 ^1.669 -30.638 -39.239 -29.122 -29.564 -22.603 -20.409 -16.403 Table 3-2 Voice-to-voice signal interference coefficients, 3^ and J v 2 . values of J , are tabulated directly above the values of J .. The units vl ' , v^ both coefficients are decibels. The channel spacing i s 30 kHz. coefficient J j plotted as a function of the channel width for the rms signal bandwidth, B = 2900 Hertz. A channel spacing of 10.6 kHz. results in a value of J . = -18.0 dB. That i s , a FM vl radiotelephony system using pre-emphasis characteristic A has the same amount of adjacent channel interference when the channels are spaced 10.6 kHz. apart as does a system using the CCF f i l t e r when the channels are spaced 30 kHz. apart. The interference levels are the same in the sense of the measurement defined by the interference coefficient J j . This value i s , however, arrived at by exclusively considering the signal-to-interference problem. A more thorough estimate of a feasible c h a n n e l width must also consider the signal-to-noise ratio of the demodulated signal and, hence, the effect of the de-emphasis network on the SNR. 59 S i 1 1 1 1 I 10.0 15.0 20.0 25.0 30.0 CHANNEL WJDTH (kHz) F i g . 3.11. Interference c o e f f i c i e n t , J v l , as a function of channel width, W. The rms bandwidth B = 2900 Hertz. 3.3 De-emphasis and Signal-to-Noi se Rat io The SNR of the output signal i s dependent upon the de-emphasis network employed. Indeed, the motivation for using pre-emphasis i s to improve the SNR of the output s i g n a l . The optimum de-emphasis c h a r a c t e r i s t i c for a large signal-to-noise r a t i o was given in section 2-7. Combining (2.25) with (2.13) and 60 (2 .20) gives D (f) = C A ( j f + 180) ( j f + 400) ( j f + 3000) L L ( j f + 300) where D(f) i s the de-emphasis c h a r a c t e r i s t i c and the subscripts A and C denote f i l t e r A and the CCF f i l t e r respectively. The pole positions for f i l t e r A from Table 1-1 have been inserted and C and C are gain constants. The SNR improvement of these a. L» two f i l t e r s w i l l now be calculated. Assume that the received signal in F i g . 3 .7 i s r(t) = /2? cos{2JIf t + 0 + k/' m(T)dT) + n(t) where P i s the received signal power, n(t) i s the channel noise and k i s a constant that controls the rms bandwidth of the s i g n a l . Under the assumptions that the SNR at IF i s large, that the output noise from the FM demodulator i s independent of the modulating message and that the baseband f i l t e r cutoff frequency (a rectangular baseband f i l t e r i s assumed), f M , i s small compared to the noise bandwidth of the IF f i l t e r , B J T, i . e . f„ << B„, the output noise signal n_,(t) from the demodulator M N D 61 i s [ 6 ] where n (t) i s the quadrature component of the noise n(t) defined by n (t) - -n(t) sin{2IIf t ) + n ( t ) cos {2Hf t} S c c The f u n c t i o n n(t) i s the H i l b e r t transform of n ( t ) and i s given by The channel noise i s assumed to be a d d i t i v e White Gaussian and of power s p e c t r a l d e n s i t y s n ( f ) where N S ( f ) = - 2 . . n 2 The power s p e c t r a l d e n s i t y of »s(t) i s denoted S <f) and n s 0 The power s p e c t r a l d e n s i t y of n D ( 0 i s thus . N 2(Hf) 2 N S (f) = ( 2 n f ) ^ - 2 - -D 2P P s i n c e d i f f e r e n t i a t i o n i n the time domain corresponds to m u l t i p l i c a t i o n by ju> i n the frequency domain. The noise input to the de-emphasis network i s then S r (f) while the s i g n a l input i s 62 M(f). The output noise signal, S n (f), j s thus 0 2 2 2 2 2 ( I I f > No CA SAn ( f ) = S n ( F ) V F ) 2 2 ~ 2 T o nD A P(fZ + 180 )(f + 400Z) 9 2(nf)2N C2(f2 + 30002) SCn<f> " V f ) ° C ( f ) = -.J -„2 o lD u P(f" + 300 ) The output signal, M o(f),is MA (f) = M A(f) D 2(f) = k 2C 2C(f) 0 Mc (f) = M c(f) D 2(f) = k 2C 2C(f) o where M(f) is the power spectral density of the pre-emphasized speech and C(f) is the spectral density of the speech as in section 1.5. The baseband f i l t e r passes only frequencies up to f M . The output signal-to-noise ratio, SNR q, is thus 2 p k i r n c(f)df -f SNR = — -0 2 fM f 4 21 N / " df 0 -f 2 2 2 2 rM ( r + 180 ) ( f Z + 400 ) 2 f M ° - 6 ) Pkl/ " C(f)df -f M SNR, C ° 2 A / « f 2 » 2 + 3 0 0 ° 2 ) d f 0 -f 2 2 *M (f + 300 ) The integrals of (3.6) can be evaluated analytically by using a 63 partial fraction expansion. Let i , . / " ± « 1 _ f 9 9 2 2 *M ( r + 180 ) ( f + 400 ) then I 1 = / M f 2 { 1 } d f "fM ( f 2 + I80 2)(f 2 + 4002) = / M f 2 { 4 0 ° 2 - 1 8 ° 2 > df "fM 127,600 f 2 + 4002 f 2 + 1802 = _400 2_ / M { 1 . AOO2 } d f 127,600 ~fM f 2 + 400 2 - 1 8 ° 2 / M {1 - 1 8 ° 2 } df 127,600 ~fM f 2 + 1802 = 2{f + 45.7 arctan(f /180) - 501 arctan(f M/400)} (3.7) 64 Let = fM f 2 ( f 2 + 30002) , f -f 2 2 M (f + 300 ) then l 2 = / M f 2 {1 + 3 0 0 ° 2 - 3 0 ° 2 } df f 2 2 M f + 300 / M ( f 2 + 891 x 10A(1 ! 2 ° - _ ) } d f f 2 2 _ tM f + 300 = 2{f 3/3 + 891 x 10* f M - 2673 x 10 6 arctan(f M/300)} . (3.8) Using (2.12) and letting I = / M C(f)df one derives i , - / M c : — £ dt f 2 2 2 2 2 2 2 2 - tM (f + 70 Z)(f + 180^)(f + 400 Z)(f Z + 700^) 65 = C /M { 1-16 * 1 0 - 8 _ 6-54 x IO"7 + 3.92 x IO"6 " fM f 2 + 70 2 9 9 2 2 f Z + 180 f + 400 -6 3.28 x 10 2 2 f + 700 } df 1 n _Q - 2{1.66 x 10 arctan(f M/70) - 3.63 x 10 arctan(f /180) (3.9) + 9.80 x 1 0 - 9 arctan(f M/400) - 4.68 x IO - 9 arctan(f M/700)} . Equation (3.6) can now be rewritten as p k i T 3 SNR. A 0 2n2N i . o 1 SNRC o 2n2N I. o 2 A similar derivation gives the output signal-to-noise ratio without pre-emphasis, SNRW . The result is P k 2 / M C(f)df _ fM 3 P k 2 I 3 S N RW = 2^7^ 3 ' ° 2 fM 2 0 M 2n2N / M f 2 d f 0 - f M 66 Let 5 denote the improvement in the signal-to-noise ratio resulting from the use of pre-emphasis. Then S N \ k A 2 2 fM 5 - ° . (.A )2 _JL A SNI^ k 3Ij S N R c o kC 2 2 fM C r - ° - ( — ) 2 — SNI^ k 3I 2 o (3.10) In order to compare the SNR improvement of the two f i l t e r s the transmitted signal bandwidth must be the same. For a Gaussian random process the mean square frequency deviation of the transmitted signal is proportional to the rms value of the modulating message process. Hence, i t is required that the rms value of the baseband speech signal is the same as the rms value of the pre-emphasized signal. This puts a constraint on the variables kA, k c and k, ( f 2 + I80 2)(f 2 + 4002) f k 2C(f)df - /I k 2C(f) Z df f - r k2c f- df -*° A 0 o 9 2 2 .(£ + 70 )(f + 700*) (3.11) , co 9 (f2 + 3002) k zc(f)df - kjc(f) — 2 Tdf C (fZ + 3000*) 67 C ,- ,2 . 2-C ( f 2 + 70 2 ) ( f 2 + I80 2)(f 2 + A00Z) f A ( f 2 + 3002) df ( f 2 + 700 2)(f 2 + 30002) where ( 2 . 1 3 ) and ( 2 . 2 0 ) have been used. Let 1 4 = f ^ ( f 2 + 7 0 2 ) ( f 2 ^ O O 2 ) ^ then 2 2 1 m 700 70 H = n 2 " ~2" 2 } d f * 485,100 r + 700 f + 70 2 {700 arctan(»/700) - 70 arctan(<=°/70)> 485,100 63 on 485,100 Continuing, let * S = ^ - c o o 9 9 9 2 2 X ( f 2 + 70 2 ) ( f 2 + l S O ^ X r + 400^) 2 2 ( r + 3oo ) ( f 2 + 700 2)(f 2 + 30002) 68 t h e n 1.10 x 1 0 ~ 1 0 4.20 x 10 3.10 x 10  1 5 = ^ { f 2 + 70 2 " f 2 + 180 2 " f 2 + 400 2 1.54 x 10" 7 1.19 x IO" 7 2 2 f + 700 2 2 r + 3000 } df = 2{1.57 x I O " 1 2 arctan(»/70) - 2.33 x I O - 1 1 arctan(«/l80) - 7.76 x IO" 1 1 arctan(°°/400) + 2.20 x I O - 1 0 arctan(»/700) - 3.96 x 1 0 ~ U arctan(»/3000)} - (1.20 x 10~ 1 0 ) n . (3.12) C o m b i n i n g t h e s e r e s u l t s w i t h (3.11) a n d u s i n g (3.9) g i v e s k A 2 X3 * T4 ( V - = ^ k T 5 (3.13) F o r f„ = 3000 H e r t z ( 3 . 1 0 ) , ( 3 . 1 3 ) , ( 3 . 7 ) r (3.12) and (3.8) M y i e l d r = 6.86 dB A KQ = 7.59 dB The s i g n a l - t o - n o i s e r a t i o improvement o f f i l t e r A i s 0.73 6 9 dB l e s s t h a n t h a t o f t h e CCF f i l t e r . T h i s i s a v e r y m i n o r d e t e r i o r a t i o n i n p e r f o r m a n c e . I t c a n be c o r r e c t e d , however, by i n c r e a s i n g t h e b a n d w i d t h o f t h e t r a n s m i t t e d s i g n a l . The s p e c t r a l e m i s s i o n c o n s t r a i n t o f s e c t i o n 1.5 a l l o w s a maximum rms b a n d w i d t h o f 2900 H e r t z f o r t h e CCF f i l t e r . S i n c e t h e o u t p u t 2 s i g n a l - t o - n o i s e r a t i o i s p r o p o r t i o n a l t o B t h e 0.73 dB d e t e r i o r a t i o n c a n be c o m p e n s a t e d by u s i n g B = 3155 H e r t z f o r f i l t e r A. I n F i g . 3.12 t h e i n t e r f e r e n c e c o e f f i c i e n t , J v l , i s p l o t t e d a s a f u n c t i o n o f t h e c h a n n e l w i d t h f o r B = 3155 H e r t z . A c h a n n e l s p a c i n g of 11.6 kHz. r e s u l t s i n a v a l u e of J v l = -18.0 dB. T h a t i s , f o r t h e same o u t p u t SNR and t h e same a d j a c e n t 10.0 T ~ r 15.0 20.0 25-Q CHANNEL WIDTH (kHz) 30.0 F i g . 3.12. Interference c o e f f i c i e n t , J y l , as a function of channel width, W. The rms bandwidth B - 3155 Hertz. 70 channel interference f i l t e r A permits using a channel spacing of 11.6 kHz. where 30.0 kHz. is required by the CCF f i l t e r . This is a very substanial saving of spectrum resources. 3.4 Heuristic Discussion The pre-emphasis functions for f i l t e r A and the CCF f i l t e r are represented by break point plots in Fig. 3.13. The dominant feature of the CCF characteristic is that i t accentuates the high frequency components of the speech signal. This is done for the following reason. The noise power spectral density at the output of the FM demodulator is parabolic. The high frequency components contribute, therefore, strongly to the output noise power. The de-emphasis network w i l l have an inverse square depende nee on frequency for the high frequency components. This w i l l cancel the squared frequency dependence of the noise from the demodulator and consequently improve the output SNR. This improvement was shown to be 7.59 dB for the CCF f i l t e r in section 3.3. Consider now that i t is desired to design a network which w i l l increase the rms bandwidth of the transmitted signal given a spectral emission constraint such as that of section 1.5. This is equivalent to compressing the transmitted signal spectrum. It is intuitively clear that accentuating the high frequency components wi l l expand rather than compress the spectrum. However, one does not want to lose the gain in SNR achieved by so doing. The squared frequency dependence of the pre-emphasis network for the high frequencies is therefore retained. Spectral compression can s t i l l be achieved, however, 71 Fig. 3.13. Break point plots of f i l t e r A and the CCF f i l t e r . 72 by also accentuating the low frequency components. If a larger proportion of the energy of the modulating signal is put into the low frequencies i t is to be expected that the transmitted signal spectrum w i l l be narrower or, equivalently, that a greater transmitted signal bandwidth can be achieved for the spectral emission constraint. Since the output SNR varies as the square of the rms bandwidth, emphasizing the low frequency components will increase the ouput SNR. The trade-off involved is the following. Increasing the energy in the low frequency components will increase the output SNR by increasing the maximum rms bandwidth consistent with the spectral emission constraint but w i l l at the same time decrease the output SNR by increasing the noise power output from the demodulator. It has been shown, then, that the gain in SNR from the increased bandwidth attained by emphasizing the low frequencies far exceeds the loss due to the increase in the demodulator output noise power. The bandwidth can be increased by a factor of 2.4 corresponding to a 7.60 dB increase in the output SNR at the expense of a loss in the de-emphasis SNR gain of only 0.73 dB. This result is not surprising. The two designs are similar for f > 400 Hertz. Since the de-emphasis SNR improvement is pre-dominantly determined; by the high frequencies, the performance of the two networks will be nearly the same in this regard. The f i l t e r characteristics differ for the frequencies below 400 73 o Hertz. The new design achieves spectral compression by putting a greater proportion of the modulating signal energy in the low frequency components. From (3.9) i t is computed that without pre-emphasis the frequencies below 400 Hertz contain 51% of the speech signal energy. For speech pre-emphasized by f i l t e r A and the CCF f i l t e r the corresponding figures are 27% and 18% respectively. The optimum design, then, concentrates about 27% of the modulating speech signal energy in the frequencies below 400 Hertz which is 1.5 times the corresponding figure for the current design. Unprocessed speech has 1.9 times the optimum energy in these low frequency components. The design of f i l t e r A was done by mathematical analysis, using variational calculus. There were no a priori notions of what the f i l t e r characteristic should look like. It is very satisfying that an intuitive redesign of the current f i l t e r points to the new design. 3.5 Interference from Adjacent Channel Data Signals In this section the interference to a speech modulated FM signal caused by an adjacent channel digital data signal will be analyzed. In the analysis we will consider binary pulse amplitude modulation (PAM) signalling (with zero intersymbol interference) with rectangular pulses as well as raised cosine spectra pulses, and binary minimum-shift-keying (MSK) signals. Let the adjacent channel digital-to-noise interference coefficient, J d > be defined as the ratio of the power in the interfering d i g i t a l data signal to the power in the desired 74 voice signal at the output of the IF f i l t e r when both the received signals are of unit power. The subscripts 1 and 2 will refer to the signal model and the rectangular bandpass model of the receiver IF f i l t e r as in section 3 . 5 . Using the notation of the preceding sections and letting p d denote the power in the received interfering data signal and Sd the power in the output of the IF f i l t e r resulting from the interfering signal, and j ( w ) = _JL _ i ' P d S O l V f + f c + W ) ' 2 + | G A ( f " f c - W ) ! 2 }  x { V a c ( f + f c > + V ( f " f c > ) d f J d l ( W ) / " {V(f + f ) + V(f - f ) ) x (V (f + f ) + V f f - f ) )d f J d 2(w) ( 3 . 1 4 ) 0 | G A ( f + f c +W)| 2 + |G A(f - f c -W)| 2 } x {n( f + f c ) + n ( f ' f c )}df w w / " {V(f + f ) + V(f - f )} x OK f + f c ) + n( f " fc )}df w w where c.(f) is the normalized power spectral density of the di g i t a l data modulator pulse shape, g(t), with normalized energy 7 5 spectral density G p( f)» G (f) G.(f) - , A sp where T is the baud rate. The family of modulator pulse shapes g(t)= 5P~1{»^Fp(t)T) for sin{TIt/T} cos{ant/T} p(t) = k( )( * ) TIt/T l - ( 2 a t/T)^ where 0 < o < 1 and jf{«> denotes the Fourier transform has a raised cosine spectra. The parameter a influences the bandwidth and k the pulse energy!16). For PAM signalling with raised cosine pulse shape and a = 0, |G.( f )| 2 = T II(fT) . The signal-to-interference coefficient, obtained by using (3.14), is then, T / ^ { n ( ( f + f c + w)T) + n « f - f c - w)T)> x <Vac(f + V + V ( f " fc ) ) d f / " {V(f + f ) + v ( f - f ) ) —oo c x <V(f + fc> + V a c ( f " f c ) } d f 76 2WT J d 2 ( W ) /" {V(f + f ) + V(f - f )) x {n( f + f c ) + n( w , |WT| < 1/3 f - f t )>df w 1 - WT Ov(f + fc) + v(f - fc)> 1/3 < |WT| < 1 x TTTf f + f c ) + H ( f - f c ) ) d f W w , |WT| > 1 (3.15) In the-case of raised cosine PAM signalling with a - 1, 1 + cos{HfT} | G A ( f ) | 2 = T ( ) n(fT/2) and 77 oo 1 + cos{n(f + f + W)T} •) n((f + f + W)T/2) + ( j d l ( w ) -1 + cos {n(f - f - W)T} ) n((f - f c - W)T/2)} x (V <f + f c ) + V f l c(f - f c ) ) d f / {V(f + f ) + V(f - f )} x ( V a c ( f + f c ) + V a c ( f - f c ) } d f J d 2(w) -WT sin{3IIWT/2} - sin{nWT/2} 1 + 2 n x {n( f + f c ) + n( f ~ f c )}df (3.16) , IWT] < 2/3 W W WT sin{nWT/2} 1 -0V< f + f c ) + V ( f " f c ) } x WLIISL ) + n( f " f c ))df , 2/3 < |WT] < 2 W W , |WT| > 2 78 For rectangular pulses (conventional phase-shift-keying) sin2{nfT} 2 |cA(f)r • ;— = T S a ( n f T ) -A (nf)zT where Sa(x) denotes the sampling function and sin{x} Sa(x) = Then from (3.14) T/°° {Sa2(n(f + f + W)T) + Sa2(n(f - f - W)T)} x (V (f + f ) + V (f - f )>df ac c ac c Jdl(w) - — /" (V(f + f ) + V(f - f )) —oo C C x { V a c ( f + f c ) + V ( f " f c ) } d f T/"„{Sa2(n(f + f„ + W)T) + Sa 2OI(f - f c - W)T)} w w J d 2(w) -c c x {n( f * f c ) + n( f " f c )>df / > + y + v ( f - f c ) } x (n( H i e . ) + n( f ~ f c ))df w w (3.17) For binary MSK the normalized baseband spectrum is[17 ,18 ] 8T(1 + cos{4nfT» and 79 O W f + fc + w ) + W f - f c ' w ) } x {V(f + f ) + V(f - £ )}df x {V a c(f + f c ) + V a c ( f - f c ) ) d f O W f + f c + W ) + W f - f c - W ) ) + fc ) + n( ll w w 'MSK U + R C ^ "MSK x {n( f + f c ) + n( f ~ fc )}d£ Jd2(w) = /"• {V(f + f ) + V(f - f )} —DO C ^ x {TI (H!C ) +n( f * fc ))df w w (3.18) The interference coefficients can be computed from (3.15), (3.16), (3.17) and (3.18). Their value will depend upon the rms bandwidth of the voice signal and the baud rate of the data signal.. It has been shown[l3] that the maximum values of B (using the CCF f i l t e r ) and T which conform to the spectral emission constraint of section 1.5 are B = 2900 Hertz and T _ 1 = 31 Kb/s, 20 Kb/s, 3.95 Kb/s and 36Kb/s for raised cosine signalling with a = 0, a = 1, rectangular signalling, and MSK signalling respectively. These values are chosen for the computation of the interference coefficients since they maximize the output signal-to-noise ratio for the voice communication and the data rate for the d i g i t a l signal. The coefficients for f i l t e r A for a channel width of 30 kHz. are presented in Table 3-3. The coefficients for the CCF f i l t e r have been determined 8 0 f o r t h e f i r s t ( s i g n a l ) model o f t h e r e c e i v e r I F f i l t e r ! 1 3 ] , They a r e a l s o p r e s e n t e d i n T a b l e 3-3. • Signalling Bit Digital-to-Voice Interference Coefficient Scheme Rate F i l t e r A CCF PAM (a = 0) 31.0 -40.31 -21.2 -17.92 PAM (a » 1) 20.0 -41.53 -20.5 -19.04 PSK 3.95 -26.45 -20.3 -10.80 MSK 36.0 -28.69 -19.2 -14.28 dl a n d Jd2 The The units Table 3-3 Digital-to-voice interference coefficients, J values of J d l are tabulated directly above the values of J d 2 . of the coefficients are decibels. The bit rate is in kilobits per sec-ond and the rms bandwidth of the speech signal is 2900 Hertz in a l l cases, I t i s s e e n t h a t t h e r e c t a n g u l a r b a n d p a s s I F f i l t e r model y i e l d s a g r e a t e r d i g i t a l i n t e r f e r e n c e c o e f f i c i e n t t h a n t h e s i g n a l I F f i l t e r model i n a l l c a s e s . T h i s was a l s o t r u e f o r t h e v o i c e i n t e r f e r e n c e c o e f f i c i e n t . The s i g n a l model c h a r a c t e r i s t i c i s q u i t e s i m i l a r t o t h e shape o f a p r a c t i c a l I F f i l t e r c h a r a c t e r i s t i c a nd one e x p e c t s i t t o be a b e t t e r a p p r o x i m a t i o n t h a n t h e u n r e a l i z a b l e b a n d p a s s m o d e l . The t a b l e i n d i c a t e s t h a t f i l t e r A o u t p e r f o r m s t h e CCF f i l t e r f o r a l l t h e s i g n a l l i n g schemes c o n s i d e r e d . The i n t e r f e r e n c e r e d u c t i o n o v e r t h e CCF f i l t e r r a n g e s f r o m 6.2 dB f o r PSK s i g n a l l i n g t o 21.0 dB f o r PAM 8 1 signalling with o = 1. The interference coefficient, Jdl» is shown as a function of the channel width for the four signalling schemes previously considered in Fig. 3.14. PSK MSK 10.0 i r 15.D 20.0 25.0 CHANNEL WIDTH (kHz) 1 30.0 F i g . 3.14. Interference c o e f f i c i e n t , J ^ , as a function of channel width. The rms bandwidth of the voice s i g n a l i s 2900 Hertz. 3.6 Interference to Adjacent Channel Data Signals In the preceding section the interference to a voice 82 t r a n s m i s s i o n from an a d j a c e n t c h a n n e l d i g i t a l da ta s i g n a l was i n v e s t i g a t e d q u a n t i t a t i v e l y . The p r e s e n t s e c t i o n w i l l examine the i n t e r f e r e n c e to a d i g i t a l da ta s i g n a l caused by an adjacent channe l v o i c e communica t ion . As p r e v i o u s l y , both l i n e a r (PAM) and n o n - l i n e a r (PSK and MSK) m o d u l a t i o n w i l l be c o n s i d e r e d . The c o n v e n t i o n a l d i g i t a l da ta r e c e i v e r demodulator for coherent d e t e c t i o n of PAM or b i n a r y PSK s i g n a l s i s shown in F i g . 3 . 1 5 . The r e c e i v e d s i g n a l i s r ( t ) = / 2 P v p ( t ) cos{2TIf t + 0} + n ( t ) + i ( t ) s c where p s i s the power i n the d e s i r e d d a t a s i g n a l , p ( t ) i s the baseband da ta s i g n a l , 0 i s a random phase ang le w i t h u n i f o r m p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n on [0 , 2n ] , n ( t ) i s the c h a n n e l n o i s e f u n c t i o n and i ( t ) i s the i n t e r f e r i n g s i g n a l which i n the f o l l o w i n g a n a l y s i s i s a speech modulated FM s i g n a l . The c a r r i e r f requency f i s the i n t e r m e d i a t e frequency of the data r e c e i v e r . The input x ( t ) to the baseband f i l t e r i s x(t) = £s p( t ) ( l + cos{4TIf t + 0}) + (n(t) + i(t))cos{2IIf t + 0} . TJ2 c c The baseband f i l t e r h ( t ) i s u s u a l l y matched to the modulator p u l s e shape g ( t ) , tha t i s H(f) = G*(f) where H ( f ) denotes the F o u r i e r t r a n s f o r m of h ( t ) , G (f) the F o u r i e r t r a n s f o r m of g ( t ) and * denotes the complex conjugate o p e r a t i o n . The matched f i l t e r demodulator maximizes the output s i g n a l - t o - n o i s e r a t i o i n the absence of i n t e r s y m b o l i n t e r f e r e n c e and a l s o m i n i m i z e s the b i t e r r o r p r o b a b i l i t y when the cos(2TTfct*0) r(t) BASEBAND OPTIMUM FILTER X DECISION h(t) y(t) y(kT) DEVICE digital output Fig. 3.15. Digital data receiver demodulator for coherent detection of PAM signals with zero intersymbol interference. oo CO 84 disturbance (noise plus interference) is white and Gaussian. The baseband f i l t e r removes signals which are outside the data signal baseband. The response y(kT) resulting from the desired data signal at k=0 is y(0) = f G p ( f ) ! 2 d f (3.19) for the case of binary PAM signalling with matched f i l t e r i n g and zero intersymbol interference. The modulator pulse energy spectral density is assumed to be constrained so that ; l ! G A ( f ) | 2 d f = I / l | G p ( f ) | 2 d f = 1 O.20) as in section 3.6. Let the voice-to-data interference coefficient, C v, be defined as the ratio of the interfering voice signal power, S v, to the desired data signal energy, S E, at the output of the baseband f i l t e r when both received signals are of unit power. The coefficient will be a function of the channel spacing and using (3.19) and (3.20) one obtains •Co l G ( f ) ' { V ( f + W ) + V ( f " W ) > d f p S P c v(w) - = P v SE 2 { / - | G (f)| 2df> 2 = _ i /" |G A(f)| 2{V(f + W) + V(f - W)}df 2T -00 A 85 or TC (W) v 2 - |G (f)r(v(f + W) + V(f - W)}df (3.21) where P v and V ( f ) a r e the power i n the t r a n s m i t t e d v o i c e s i g n a l and i t s low-pass e q u i v a l e n t power s p e c t r a l d e n s i t y r e s p e c t i v e l y . The d i g i t a l data r e c e i v e r demodulator f o r coherent d e t e c t i o n of b i n a r y MSK s i g n a l s w i l l be s i m i l a r to the demodulator of F i g . 3 .15 though two matched f i l t e r s w i l l be r e q u i r e d . The v o i c e - t o - d i g i t a l i n t e r f e r e n c e c o e f f i c i e n t i s a g a i n g i v e n by ( 3 . 2 1 ) . V a l u e s of T C V a re p r e s e n t e d i n T a b l e 3-4 f o r f i l t e r A and the CCF f i l t e r [ l 3 ] . The c h a n n e l w id th i s 30 kHz. w h i l e the data baud r a t e and the v o i c e s i g n a l bandwidth are chosen as p r e v i o u s l y f o r T a b l e 3 - 3 . S i g n a l l i n g Scheme B i t Rate V o i c e - t o - D i g i t a l Interference C o e f f i c i e n t F i l t e r A CCF PAM (a = 0) 31.0 -80.21 -63.0 PAM (a = 1) 20.0 -81.42 -62.3 PSK 3.95 -66.35 -62.1 MSK 36.0 -68.58 -61.0 Table 3-4 V o i c e - t o - d i g i t a l interference c o e f f i c i e n t , TC^. The u n i t s of c o e f f i c i e n t s are d e c i b e l s . The b i t rate i s i n k i l o b i t s " p e r second and t rms bandwidth of the speech s i g n a l i s 2900 Hertz i n a l l cases. 86 It is seen that that the interference to a d i g i t a l signal caused by an adjacent channel voice signal is lower for a FM radiotelephony transmitter using f i l t e r A than the CCF f i l t e r . The improvement ranges from 4.3 dB for PSK signalling to 19.1 dB for PAM signalling with a = 1. The interference coefficient, T C V» is shown as a function of the channel width for the four signalling schemes in Fig. 3.16. 10.0 15 D 20.0 25.0 CHANNEL WIDTH (kHz) PSK MSK 30.0 Fig. 3.16. Interference coefficient, TC v, as a function of channel width. The rms bandwidth of the voice signal i s 2900 Hertz. 87 IV AN ALTERNATIVE MODEL FOR SPEECH SPECTRA 4.1 The Speech Spectrum Model The results obtained previously are a l l based on the speech spectrum model of section 2.5. In some cases the pre-emphasized speech might be lowpass fi l t e r e d by a relatively sharp cutoff f i l t e r at FH * 3 kHz.[13]. One expects such f i l t e r i n g to reduce the high frequency content of the transmitted FM spectrum S(f) since the modulating signal frequencies above FH wi l l be strongly attenuated. In addition, the response of audio transmitter c i r c u i t s usually f a l l s off below FL - 50 Hz. In order to assess these effects we employ the following power spectral density for a speech spectrum model, (f 2 + 50 2)(f 2 + 70 2)(f 2 + 1802)(f2 + 4002)(f2 + 7002)(f2 + 3000)5 (4.1) 4.2 Transmitted Signal Spectra and Mean Square Error An analysis similar to that of chapter 2 is used to obtain the following results. The autocorrelation function P(T) of the integrated prefiltered speech with spectrum CjCf) is shown in Fig. 4.1 for f i l t e r A and the CCF f i l t e r . The autocorrelation function V ( T) - v(») of the lowpass equivalent of the power spectral density of the transmitted signal, v a c( f)» is shown in 88 o' o o LP I 0.0 I 1 1— 2.0 4.0 6.0 TIME T (msec;) 8.0 10.0 0.0 2.0 4.0 TIME 6.0 (msec.) s . o 10.0 Fig. 4.1. Normalized autocorrelation function, p(x), for integrated lowpass speech fil t e r e d with f i l t e r s A and CCF. 89 Fig. 4.2. Autocorrelation function v(x) modulated signal using f i l t e r A. - v(°°) for FM speech 90 F i g . 4.3. Autocorrelation function v(x) - v(») for FM speech modulated signal using f i l t e r CCF. 9 1 a n d 4.3 f o r v a r i o u s v a l u e s o f t h e r m s b a n d w i d t h f o r t h e t w o f i l t e r s . F i g s . 4.4 a n d 4.5 s h o w v a c ( f ) f o r v a r i o u s v a l u e s o f B f o r f i l t e r s A a n d C C F r e s p e c t i v e l y . C o m p a r i s o n w i t h F i g . 3.8 s h o w s t h a t , a s e x p e c t e d , t h e t a i l s o f t h e t r a n s m i t t e d s p e c t r a f a l l o f f m u c h m o r e r a p i d l y f o r t h e s p e e c h m o d e l o f t h i s s e c t i o n t h a n t h a t o f s e c t i o n 2 . 5 . B=7500 T 0 0 10 0 20.0 30.0 40.0 50.0 FREQUENCY FROM UNMODULATED CARRIER FREQUENCY (kHz) V i e . 4.4. Power spectral density V (f) of FM signal modulated by lowpass speech prefiltered with f i l t e r A. 92 B=7500 FREQUENCY F R O M ' U N M O D U L A T E D CARRIER FREQUENCY (kHz) 10.D 20.0 Fig. A.5. Power spectral density V f l c(f) of FM signal modulated by lowpass speech prefiltered with f i l t e r CCF. The mean square e r r o r i s t a b u l a t e d and compared t o the optimum mean square e r r o r i n T a b l e 4-1 f o r v a r i o u s v a l u e s of the rms bandwidth f o r both p r e f i l t e r s . I t i s seen t h a t , f o r the speech model o f t h i s s e c t i o n , the d i s t o r t i o n of the two systems i s a p p r o x i m a t e l y the same. The d i f f e r e n c e i s l e s s than 20% i n a l l c a s e s . 93 2P/N = 6.0 x 10 H z . B F i l t e r A F i l t e r CCF Her tz m . s . e . / o . m . s . e . m.s.e./o.m.s.e. 1000 2.638 2.620 2500 3.007 3.281 2900 3.096 3.A32 5000 3 . A l l A . 033 7500 3.671 A.571 2P/N - 6 .0 x 1 0 5 H z . 0 B F i l t e r A F i l t e r CCF Her t z m . s . e . / o . m . s . e . m . s . e . / o . m . s . e. 1000 5.255 A.519 2500 2.856 2.555 2900 2.73A 2 . A 68 5000 2.5A7 2.386 7500 2.570 2.A88 Table A - l Mean square e r r o r f o r speech model C ^ ( f ) . 4.3 Interference Coefficients The interference coefficients defined in chapter 3 are calculated for the speech model of this section for the new pre-f i l t e r design. The voice-to-voice interference coefficient J d l ( W ) is presented in Fig. 4.6. Also shown is the coefficient for the CCF f i l t e r . The new design is seen to offer better adjacent channel interference performance than the current 4.6. Interference c o e f f i c i e n t , J y l , as a funct channel width. 95 d e s i g n . The improvement, however, i s n o t a s g r e a t f o r t h e s p e e c h model o f t h i s s e c t i o n as f o r t h e model o f s e c t i o n 2.5. (w) i s p r e s e The d i g i t a l - t o - v o i c e i n t e r f e r e n c e c o e f f i c i e n t J d l n t e d i n F i g . 4.7 and t h e v o i c e - t o - d i g i t a l i n t e r f e r e n c e PSK MSK 13.3 i r 15.3 20.0 25.3 CHANNEL WIDTH (kHz) 33.3 Fig. 4.7. Interference coefficient, J d l , as a function of channel width. The rms bandwidth of the voice signal i s 2900 Hertz. 96 coefficient TCv(w) is presented in Fig. 4.7. As previously, four type-s of dig i t a l signalling are !0.0 "I T 15.0 20.0 25.3 CHANNEL WIDTH (kHz) 33.3 Fig. 4.8. Interference coefficient, TC v, as a function of channel width. The rms bandwidth of the voice signal i s 2900 Hertz. 97 considered and the bit rates are those given in Table 3-3. The rms bandwidth of the voice signal is 2900 Hertz. It is seen by comparison with Figs. 3.14 and 3.16 that the interference between a voice and a data signal is more dependent upon the speech model employed in the case of PAM signalling than PSK and MSK signalling. For the range of values of channel width considered the interference coefficient is essentially the same for both speech models for PSK and MSK data signals. This is because the ro l l - o f f rate of these data signals is less than the ro l l - o f f rate of either speech model. The PAM signals are bandlimited and thus the r o l l - o f f rate of the voice signal determines the character of the interference coefficient for the case of interfering voice and PAM data signals. 98 V IMPLEMENTATION AND TESTING 5.1 R e a l i z a t i o n The p r e - e m p h a s i s n e t w o r k h a s t h e t r a n s f e r f u n c t i o n of (2.13) K(f + j70)(f + jl80)(f + j400)(f + J700) H(f) = — 1/2 f(f x + jf)(f u + jf)(c o ) I f *i - 7 0 Hz and f u = 7 0 0 Hz t h e n two p o l e - z e r o p a i r s w i l l c a n c e l a n d H ( f ) w i l l have f o u r f e w e r b r e a k p o i n t s t h a n o t h e r w i s e . S i n c e t h e p e r f o r m a n c e (mean s q u a r e e r r o r ) i s n o t a d v e r s e l y a f f e c t e d by t h i s c o n v e n i e n c e , i t c a n be u s e d w i t h a s a v i n g s i n h a r d w a r e i n an i m p l e m e n t a t i o n . The c h a r a c t e r i s t i c t o be i m p l e m e n t e d i s t h e n 7. (f + jl80)(f + jAOO) v f ) = -7172 o The p h y s i c a l i m p l e m e n t a t i o n c a n be f u r t h e r f a c i l i t a t e d by u s i n g t h e c h a r a c t e r i s t i c « 2 ( f ) r a t h e r t h a n ^ ( f ) , where H 2 ( f ) - K (f + jl80)(f + jAOO) (f + jf x ) (f + jf 2) By c h o o s i n g ^ t o be much l o w e r t h a n t h e l o w e s t f r e q u e n c y 99 components o f t h e s p e e c h w h i c h have s i g n i f i c a n t e n e r g y and c o r r e s p o n d i n g l y c h o o s i n g f 2 t o be h i g h e r t h a n a l l f r e q u e n c y components o f s i g n i f i c a n t e n e r g y t h e s e two a d d i t i o n a l b r e a k p o i n t s w i l l have n e g l i g i b l e e f f e c t on t h e p r o c e s s i n g o f t h e s p e e c h s i g n a l . T h i s i s i l l u s t r a t e d i n F i g . 5.1. LJ O =) Z o < f F R E Q U E N C Y ( H z ) Fig. 5.1. Break point plots of l l ^ f ) and H 2(f) The c h a r a c t e r i s t i c H 2(f) i s decomposed i n t o a p r o d u c t H 2(f) = K. (f + J180) (f + j f x ) K • (f + jAOO) (f + j f 2 ) where and K 2 a r e g a i n c o n s t a n t s . I m p l e m e n t a t i o n o f t h e f i r s t 100 factor requires a circuit which breaks at a pole before breaking at a zero whereas realization of the second factor requires a ci r c u i t which breaks at a zero before breaking at a pole. Let H 3(f) = Kx (f + jl80) (f + Jf x) fj < 180 and H 4(f) = K 2 (f + j400) (f + j f 2 ) f 2 > 400 then the c i r c u i t of Fig. 5.2 w i l l realize H 3(f) and that of Fig. 5.3 w i l l realize H^Cf) (within a gain factor). The analysis of these circuits follows. Fig. 5.2. Circuit for realization of H ( f ) . 101 R 3 Fig. 5.3. Circuit for realization of H (f) The v o l t a g e t r a n s f e r f u n c t i o n of the c i r c u i t of F i g . 5.2 i s R2 + l/(sC) sCR2 + 1 H 3 ( s ) = v. R + R_ + l/(sC) (R. + R 7)sC + 1 1 1 2 1 s + 1/(R2C) \ + \ s + 1/((R1 + R2)C) or e x p r e s s e d i n terms of the f requency f , l/CRjC) + j2Hf H 3(f) = Rx + R2 1/((R1 + R2)C) + j2nf 1 02 R2 1/(2TIR2C) + j f + ^ l/(2n(R1 + R2)C) + j f It is required that 1/(2IIR2C) = 180 Hertz, and i/(2n(R1 + R2)C) = f x . It is also required that f j < 180 . since R l + R2 " R l this condition is satisfied. The voltage transfer function of the circuit of Fig. 5.3 is Rj/sc R3 + 1/sC R 4(R 3 + 1/sC) R 4(sCR 3 + 1) R 4(R 3 + 1/sC) + Rj/sC sCR3R4 + R3 + R^  103 R^CCs + l/CP^C)) R 3R 4C(s + (Rg + R 4)/(R 3R AC)) (s + 1/CRgC)) (s + (R 3 + R 4)/(R 3R 40) or expressed in terms of f, (j2iif + l/CRgC)) H A(f) = ( j 2 n f + (R3 + R 4 ) / ( R 3 R 4 C ) ) ( i / ( 2 n R 3 c ) + j f ) + R 4)/(2nR 3R 4C) + j f ) It is required that 1/(2IIR3C) = 400 Hz and *3 + R4 2ITR3R4C It is also required that * 2 > 4 0 0 , s i n c e 1 04 R3 R4 ~ K R3 R3 + R4 this condition is satisfied. The realization of H 2 ( f ) is achieved by cascading the two circuits having transfer characteristics H 3 ( f ) and H 4 ( f ) • * n order to preserve the characteristics H 3 ( f ) and H^Cf) each ci r c u i t must see an infinite load. Some isolation must be provided after each circuit in order to keep the loading as light as possible. High input impedance operational amplifiers in a voltage follower configuration can be used to provide this buffering. The pre-emphasis f i l t e r is illustrated in Fig. 5 . 4 . The capacitor C £ isolates the d.c. output of the operational amplifier from the succeeding stage. It is chosen to have an reactance which is negligibly small compared to the input impedance of the circuit i t feeds. The input impedance, Z, of the voltage follower is Z = z.(l + Av) where z and A are the input impedance and open-loop gain of i v the operational amplifier respectively. Good buffering requires that {R2 + (l/(2nf 1c) 2} 1 / 2 « z and F i g . 5.4. Schematic diagram of the pre-emphasis f i l t e r . 106 R 4 << Z The d e - e m p h a s i s f i l t e r h a s t r a n s f e r f u n c t i o n H^Cf) w h i c h i s t h e i n v e r s e o f H 2 ( f ) , 1 (f + j f j ) ( f + j f 2 ) H 2 _ 1 ( f ) = . K j K 9 (f + j l 8 0 ) ( f + J400) The d e - e m p h a s i s f i l t e r c a n be r e a l i z e d w i t h t h e same c i r c u i t a s t h e p r e - e m p h a s i s f i l t e r i f t h e component v a l u e s a r e p r o p e r l y c h a n g e d . The d e - e m p h a s i s f i l t e r r e q u i r e s t h e f o l l o w i n g c o n d i t i o n s t o be met i n t h e s e l e c t i o n o f component v a l u e s , R3 + R4 = 180 Hertz 2nR„R,C 3 h l / ( 2 n ( R 1 + R 2)C) = 400 Hertz i / ( 2 n R 3 0 = f j 1/(21IR2C) = f 2 • A g a i n , g o o d i s o l a t i o n demands t h a t 1 07 {R 2 + ( i / ( 2 n f 1c) 2} 1 / 2 « z and R. « z From a practical standpoint using the same circ u i t for both pre-emphasis and de-emphasis f i l t e r s f a c i l i t a t e s construction since the same ci r c u i t board layout may be used for both. 5.2 Measurement Ideally, one wishes to obtain a measurement of the transmitted spectrum as a function of the frequency to verify the analytical results. This is needed for two reasons. F i r s t l y , speech has been assumed Gaussian in order to make the analysis possible. Secondly, the analysis is.based on a model for the average spectral density function of speech. Solution of the problem requires the assumption of some model and the best one available for the task at hand must be selected. A model, however, is only an approximation to the real phenomenon. In some cases accurate models are available for the elements of a problem. An accurate model for the spectral density of speech is not available and any model used is only approximate. One approach to experimental verification is to view the spectrum of a carrier frequency modulated by speech samples using a spectrum analyzer. This requires averaging the spectrum and the easiest way to do this is probably by using a storage 108 trace in the video display. A more stringent requirement to be ®et is that of the resolution bandwidth of the IF amplifier of the spectrum analyzer. For large scan widths the FM signal carrier w i l l appear on the video monitor as an impulse. As the scan width is decreased, however, the impulse representing the carrier w i l l broaden and flatten out so that i t resembles a smooth pulse. If the width of the pulse is too great i t will obscure the sidebands that we desire to measure. In the case of the FM speech modulated signal considered here the sidebands to be viewed are 5 to 10 kHz. off the carrier. This requires a spectrum analyzer of very high bandwidth resolution. Decreasing the scanning rate will help to narrow the carrier pulse but f u l l compensation for an inadequate bandwidth resolution cannot be made by this approach. An alternative approach is to measure the power in segments of the spectrum near the carrier frequency. The average power in, say, the regions 0-2.5, 2.5-5.0, 5.0-7.5kHz., etc., off the carrier is an indication of the average emitted spectral levels. As the width of the regions decreases to zero this approach becomes equivalent to the spectrum measurement described above. This method requires very sharp bandpass f i l t e r s . The selectivity required is not readily attainable at typical carrier frequencies. Since high Q circuits are more easily constructed for lower frequencies i t is indicated that the bandpass f i l t e r i n g must be done at low frequencies. 1 0 9 5.3 Simulation A computer simulation can be used to verify the results of the analysis. In particular, the modulating signal was assumed Gaussian while speech is known not to be Gaussian. Though this assumption is often encountered in the literature, the author is unaware of any verification of the validity of analytical results based on i t . The goal of the simulation is then, to verify that the analysis yields results reasonably close to the true values. The voice-to-voice interference curves for B = 5000 for f i l t e r A and the CCF f i l t e r w i l l be simulated for the speech model of section 4.2. Details of the simulation follow. The speech to be digitized was recorded monaurally on a Sony Elcaset Deck EL-5 using a Bruel and Kjaer type 4145 1-inch condenser microphone with type 2801 microphone power supply. The recording was done in an Industrial Acoustics Company sound chamber. The recording consisted of readings of the f i r s t page of the sports section of the local newspaper. Two readings were recorded, one by a 30 year old male and one by a 25 year old female. Both readers were western Canadians. The taped speech was digitized using a Kinetic Systems CAMAC 3553 12-bit A/D converter. The samples consisted of 240 blocks of 1024 sample points for both the male and the female reader. The sampling frequency was 10 kHz. The binary data was read into the MTS system of the UBC Computing Center and converted to floating point decimal format with a program written in FORTRAN. 1 1 0 The simulation consists of sampling the function s(t)»/2P*cos{2Hf t +6 • / ^ m ( T ) d t } , where f = 75kHz. at a sampling rate of 320 kHz. This means that m(t) must also be sampled at 320 kHz. The digitizing equipment, however, cannot run this fast. The system software limits the sampling rate to 10 kHz. To increase the time resolution the following technique was employed. A 1024 point FFT of the speech samples was taken. This was extended to a 32768 point FFT by f i l l i n g the middle 31744 points with zeros. The inverse FFT was then taken of the 32768 point frequency series. This approach is equivalent to lowpass f i l t e r i n g the sampled speech. A 32768 point FFT was taken of the modulated carrier time series. Prior to taking the FFTs the data was windowed to reduce leakage. A Kaiser window was selected for i t s high stopband attentuation. The interference coefficient was computed numerically from the discrete power spectra. The averaged interference coefficients are shown in Fig. 5.5. It is seen that the simulation results indicate that the proposed f i l t e r i n g scheme is better than the current one. The agreement with the previous analytical results is f a i r . For example, at a channel spacing of 30 kHz. the analysis predicts a 7 dB gain for prefilter A while the simulation indicates 4 dB. 1 1 1 Fig. 5.5. Simulation values for the interference coefficient J v l(W). The solid lines are the previous analytical results. The rms bandwidth of the transmitted signal is 5000 Hertz. 1 1 2 VI CONCLUSION 6 . 1 Summary of Results The pre-emphasis problem for FM radiotelephony has been analyzed using the calculus of variations and an experimentally determined model of the spectral density of speech. In this regard particular attention has been paid to the effects of prefiltering on the spectral characteristics of the transmitted signal. A simple approximation to the optimum unrealizable prefilte r was proposed and compared to the Communications Canada designated p r e f i l t e r . It was seen to result in more compact signal spectra. A significant improvement in the SNR of the received signal for a given bandwidth was demonstrated. The mean square error of the new f i l t e r was shown to be less than that of the current design. The effects of prefiltering upon the adjacent channel interference problem were investigated. To this end interference coefficients were defined and computed for the cases of voice-to-voice, voice-to-digital and digital-to-voice situations. Four types of data signal were considered, PAM (a = 0, a = 1 ) , PSK and MSK. In a l l cases improvement was noted using the new p r e f i l t e r . There was greatest improvement in the cases of the PAM bandlimited d i g i t a l pulse shapes. An experimental model for a lowpass speech signal was used as a second speech model and the performance of the prefilter for this model was analyzed. The transmitted signal spectra were computed as well as the d i g i t a l and voice interference coefficients. An improvement is noted for the lowpass model but 1 13 was found to be less than for the f i r s t model. The pre- and de-emphasis networks are easily implemented and ci r c u i t designs were presented. The testing of the scheme was described and a computer simulation verified that the new design offers a gain in performance. 6.2 Suggestions for Future Research The results of this work confirm that the currently used pre- and post-filtering scheme can be improved. A more complicated approximation to the optimum unrealizable f i l t e r may yield greater improvement. It is recognized that the quality of speech is highly subjective and is d i f f i c u l t to define or measure mathematically. Subjective evaluations of the two f i l t e r s are indicated. 1 1 4 Appendix A Est imating the Error in the T a i l Consider the i n t e g r a l of (2.18), I = V (f) = 2/-{V(T) - v(°°)}cos{2lIfT}dT ac 0 - 2/S*{v(T) - v(=°)}cos{2TIfT}dT + 2/~{v(x) - v(») }cos{2nf t}dr . 0 p Let e . 2/"{V(T) - v(»)}cos{2nfT>dT , p then one wishes to choose B to make e n e g l i g i b l e with respect to I = V ( f ) . w e have ac V ( T ) _ v ( < x > ) = ( exp{-Rp(0)} ) ( exp{R p ( 0 )p(x)> - 1 ) . The Taylor ser ies expansion gives exp^R (O)P(T)} - 1 = R (O)P(T) + {R t,(0)p(T)}2/2l + P P P + { R p ( 0 ) p ( T ) > n / n ! , where n runs to i n f i n i t y . If we truncate the ser ies af ter two terms we obtain exp{R (0)P(T)> - 1 = Rp(0)p(T) + ( exp{c} )(R p(0)p(O)~/2! where c i s some rea l number between 0 and R p(°)p( T)» that i s , 0 < c < R (O)P(T). Since the exponential function i s s t r i c t l y P 1 1 5 i n c r e a s i n g , exp{c} < exp{R (O )P (T)} , P a n d s i n c e 2! P i t f o l l o w s t h a t exp{c} 9 {R ( 0 ) p ( T ) } Z > 0 exp{R ( O ) p ( x ) } - 1 > R ( O ) p ( x ) P P a n d (exp{c}/2!){R p (0)p(T)r (exp{c}/2 !) (Rp(0) P (T) > exp{R p ( 0 ) p ( T ) } - 1 R p ( 0 ) p ( T ) 2 = (exp{c } /2)(R p (0)p (T)) < (exp{R p ( 0)p (T)»(R p ( 0)p (T ) ) / 2 t h a t i s , t h e r e l a t i v e e r r o r , i n e s t i m a t i n g exp{R p (0)p(x) } - 1 by R p (0)p(r) i s u p p e r b o u n d e d , e < (exp{R (0)p(x)})(R (0 ) P (T ) ) /2 . (A.l) r p p We h a v e = 2/°°{V(T) - v(»)}cos{2TIfT}dT 2/"(exp{-R (0)})(exp{R ( 0 ) p ( T ) } - l ) c o s { 2 n f T > d x p P P 2(exp{-R (0)})/ o(R (0 ) p (T ) ) c o s{2nfT }dx P p P 2(exp{-R p (0)})(R p (0)) x. f exp{-2nf.T>-f.exp{-2nf T > r°° _ _ cos{2nfT>dx j g  u 1 2(exp{-R (0)})(R (0)) P P x., f - f , t i 1 ^ f u e x p { - 2 n f 1 T } c o s { 2 H f T } d T - f^f1exp{-2TIfuT}cos{2TIfT>dx 2(exp{-R (0)})(R (0)) " f - f, u 1 X. f e x p { - 2 n f . T ] u 1 2 2 2 2 4n + 4irr (-2nf1cos{2nfT> + 2nfsin{2nfT}) f,exp{-2TIf T } 1 e u — (-2IIf {cos2nft> + 2nfsin{2nfx>) 11 2 2 2 ? An f + A n V u 2exp{-R (0)}(R (0)) P P 4n'<fu- f x) f,exp{-2IIf 6} 1 r u 2 2 f Z + f u -2M cos{2nfB) + 2nfsin{2nfB}) f exp{-2nflP> u 1 2 2 f l + f (-2nf1cos{2nfB} + 2nfsin{2nf3)) j 2exp{-R (0)}(R (0)) f f.,exp{-2nfuB> AH 2 ( f u - f x) 2 2 u (2nf + 2nfu) f exp{-2nf.. B) u 1 2 2 £1 + f (2nf1 + 2iif)| exp{-R (0)}(R (0)) n(f u - f,) f l < f + V exP{-2JIf B) + X"V2 ' T exP{-2Hf B) f 2 + f 2 U f * + f 2 u x f (f + M 1 18 where i t has been assumed (without loss of generality) that f > 0. The standard techniques of one-dimensional calculus may be used to mazimize the expression f, (f + f ) 1 u 2 T ' f + f u The f i r s t d e rivative with respect to f i s 2 2 f - f - 2ff u u (f 1 + f ) u This has one zero on f e [0 , 0 0 ) at f = (^2- 1 ) £ U - The f i r s t d erivative evaluated at f = 0 i s po s i t i v e and at f = 0 0 i s negative. Continuity implies that f = ( ^ - 1) i s the global maximum on [ 0 ,<° ). The maximum value of the expression i s /Iff, 1 + f. u 1 J-f 2 ( 1 + (^ 2 - 1)^) 2 f u Combining t h i s result with (A.2) gives Iiexp{-2IIf B) + — exp{-2nf..e} (A.3) f U f. 1 u 1 1 19 where B 2 R (0) * u 1 N u m e r i c a l l y , P may be chosen to make the r e l a t i v e e r r o r i n e s t i m a t i n g exp{ R p(0)p(i) } - 1 by Rp(0)p(x) s m a l l ( l e s s than the p r e c i s i o n of the computer , f o r example) u s i n g ( A . 1 ) and then ( A . 3 ) g i v e s a bound f o r the t a i l . T h i s bound i s compared to the n u m e r i c a l v a l u e of 1 = v a c ( f ) to determine whether & has been chosen l a r g e enough. 120 REFERENCES [1] Armstrong, E.H. "A method of reducing disturbances in radio signalling by a system of frequency modulation," IRE Proc, vol. 24, no. 5, pp 689-739, May 1936. [2] Carson, J.R. "Notes on theory of modulation," IRE Proc, vol. 10, no. 2, pp 57-83, February 1922. [3] Fagen, M.D. ed. A History of Engineering and Science in the Bell System: National Service in War and Peace (1925-1975), Bell Telephone Laboratories, USA, 1978. [4] Costas, J.P. "Coding with linear systems," Proceedings of the IRE, vol. 40, no. 9, pp. 1101-1103, September 1952. [5] Boardman, C. and Van Trees, H.C. "Optimum angle modulation," IEEE Trans, on Commun., vol. COM-13, no. 4, pp.452-469, December 1965. [6] Jakes, W.C. Jr. Microwave Mobile Communications, Wiley, New York, 1974. [7] Schwartz, M. Information Transmission, Modulation, and Noise, McGraw-Hill, New York, 1970. [8] Rowe, H.E. Signals and Noise in Communication Systems, Van Nostrand, Princeton, N.J., 1965. [9] Abramson, N. "Bandwidth and spectra of phase-and-frequency-modulated waves," IEEE Trans. Commun. Sys., vol. CS-11, no. 4, pp. 407-419, December 1963. [10] Fletcher, H. Speech and Hearing in Communication, Van Nostrand, New York, 1953. [11] Rice. J.R. Mathematical Software, Academic Press, New York, 1971. [12] Garrison, G.J. "A power spectral density analysis for di g i t a l FM,"IEEE Trans, on Commun., vol. COM-23, no. 11, pp.1228-1243, November 1975. [13] Donaldson, R.W. "Frequency assignment for land mobile radio system: di g i t a l transmission over land mobile channels-interference considerations," Report to Communications Canada, January 1980. 121 [14] Thomas, J.B. An Introduction to Statistical Communication  Theory. Wiley, New York, 1968. [15] Van Trees, H.L. Detection, Estimation and Modulation  Theory: Parti, Wiley, New York, 1968. [16] Lucky, R.W., Salz, J. and Weldon, E.J. Principles of Data  Communication, McGraw-Hill, New York, 1968. [17] Gronemeyer, S.A. and McBride, A.L. "MSK and offset QPSK modulation," IEEE Trans, on Commun., vol. COM-24, no. 8, pp.809-820, August 1976. [18] Simon, M.K. "A generalization of minimum-shift-keying (MSK)-type signalling based on input data symbol pulse shaping," IEEE Trans, on Commun., vol. COM-24, no. 8, pp. 845-856, August 1976. 

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