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UBC Theses and Dissertations

Effect of channel transmission errors on DPCM systems Palffy-Muhoray, Peter 1969

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EFFECT OF CHANNEL TRANSMISSION ERRORS ONDPCM SYSTEI1SbyPE9ER PALFFY-MUHORAYB.A.Sc., University of British Columbia, 1966A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIR1ENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEin the Department ofElectrical EngineeringWe accept this thesis as conforming to thestandards reciuired from candidates for thedegree of Master of Applied Science,Members of the Departmentof Electrical EngineeringTHE UNIVERSITY OF BRITISH COLUMBIAMarch, 1969In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s , i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada ABSTRACT D i f f e r e n t i a l pulse code modulation (DPCM) i s a p r a c t i c a l encoding scheme f o r speech, t e l e v i s i o n and telemetry signals. In t h i s thesis, the mean square error £*• i s minimized i n situations where the noise contribution which r e s u l t s from channel transmission errors i s s i g n i f i c a n t . Expressions f o r the quantizer and channel noise were developed for both uniform and non-uniform quantization. These expressions were then used to obtain the mean square error as a function of the system parameters f o r binary symmetric channels, natural coding and previous sample feedback. Sampling was at the Kyquist rate, and the non-uniform quantization was logarithmic. Mean square error was minimized, and the optimum system parameters were obtained using a computer search. Gaussian, t e l e v i s i o n and speech were considered as message signals. DPCM was found, f o r a given channel capacity, to have a lower mean square error than PCM i n a l l cases. Logarithmic quantization was found to y i e l d better performance than uniform quantization f o r speech signals f o r the PCM case, and for speech and t e l e v i s i o n signals f o r the DPCM case. In general, the r a t i o of the mean square errors of DPCM;.to.PCM increases while the difference i n performance decreases as channel noise increases. i i TABLE OF CONTENTS Page ABSTRACT. i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS . . i . v LIST OP TABLES .. v i i ACKNOWLEDGEMENT... v i i i I . INTRODUCTION 1 1.1 Transmission of Messages over D i g i t a l Communication Channels 1 1.2 Review of Previous Research.. 1 1.5' Scope of the Thesis 2 I I . GENERAL MEAN SQUARE ERROR EQUATIONS FOR DPCM SYSTEM 4 2.1 Mean Squared Error i n DPCM 4 2.2 Evaluation of Quantizer Rate Distortion Function ^ f o r Uniform Quantizer 6 2.5 Evaluation of Quantizer Rate Distortion Function f o r Non-Uniform Quantizer.................. 8 2.4 Evaluation of Channel Noise f o r Uniform Quantizer and Natural Binary Code..... 9 2.5 Evaluation of Channel Noise f o r Non-Uniform Quantizer and Natural Binary Code 12 I I I . MEAN SQUARED ERROR FOR GAUSSIAN, TELEVISION, AND SPEECH SIGNALS 14 5.1 Signal S t a t i s t i c s 14 3.2 Mean Squared Error f o r Gaussian Signals 16 5.3 Mean Squared Error f o r Television Signals 17 3.4 Mean Squared Error f o r Speech Signals 18 i i i Page IV. RESULTS AND DISCUSSION OF RESULTS 21 4.1 Computational Procedure 21 4.2 PCM Results.... 21 4.3 DPCM Results 23 4.4 Results f o r Companded Systems 24 4.5 Concluding Remarks • 25 APPENDIX I 50 i v LIST OF ILLUSTRATIONS Figure Page 1.1 General d i g i t a l communication system 1 1.2 DPCM source encoder •••• 3 2.1 Communication system considered 5 2.2 Uniform quantizer characteristics 6 2.3 Alternative representation of non-uniforra quantizer 8 2.4 General system with alternative representation of quantizer 9 2.5 Modified d i g i t a l channel 10 4.1 Gaussian signal - PCM - Coherent phase channel -Uniform quantization - B ) V and y vs. P/2Mo{c ..... 26 4.2 Gaussian signal - PCM - Incoherent phase channel -Uniform quantization - W and y vs. P/^Nofe ..... 27 4.3 Gaussian signal - PCM - f a d i n g channel -Uniform quantization - W and y vs. P/lHo^c 28 4.4 Gaussian signal - PCM - Incoherent phase channel -Uniform quantization - Be; Vie and Ye. vs. VllHdp, 29 4.5a Gaussian signal - DPCM - Incoherent phase channel -Uniform quantization - 6} £> and.W vs. P/lNo^o 30 4.5b Gaussian signal - DPCM - Incoherent phase channel -Uniform quantization - o< and Y vs. P/lKo^- < 31 4.6 Gaussian signal - Uniform quantization - vs. P/2Ncfc f o r PCM - Coherent and incoherent phase channels and vs. ?/a^0^. for DPCM - Incoherent phase channel 32 4.7 Television signal - PCM - Incoherent phase channel -Uniform quantization -£JB,W and y vs.P/lMo^- 33 4.8a Television signal - DPCM - Incoherent phase channel -Uniform quantization - €) E> and W vs. p/;i^0-^o 34 4.8b . Television signal - DPCM - Incoherent phase channel -Uniform quantization - c< and y vs. •• 35 v Figure P a g e 4.9 Television signal - DPCM - Incoherent phase channel -Uniform quantization - 6 vs. P/at40^c f o r fix e d V 36 4.10 Television signal - DPCM - Incoherent phase channel -Uniform quantization - 7 vs. P/lHofc f o r fixed V/ 37 4.11 Television signal - DPCM - Incoherent phase channel -Uniform quantization - *.vs. p/O-Ho^c f o r fix e d W 38 4.12a Television signal - DPCM - Incoherent phase channel -Logarithmic quantization . ^  6 and V/vs. p/2U^c. 39 4.12b Television signal - DPCM - Incoherent phase channel -Logarithmic quantization - c andy^-vs. ?/Q.HO-^ C. .... 40 4.13 Television signal - Incoherent phase channel -? vs. P/lMcfo f o r PCM - Uniform quantization and p vs. P/zNofc f o r DPCM - Uniform and logarithmic quantization 41 4.14 Speech signal - PCM - Incoherent phase channel -Uniform quantization - T) B, W and Y vs. P/lvU^W, 42 4.15a Speech signal - DPCM - Incoherent phase channel -Uniform quantization - E> and W vs. ?/2Vl0-^cs • 43 4.15b Speech signal - DPCM - Incoherent phase channel -Uniform quantization -«Xand Y vs. p/2Mo-£cs 44 4.l6a Speech signal - PCM - Incoherent phase channel -Logarithmic quantization - 6^  fc> and v/ vs.P/2N0^cs . . . . . 45 4.l6b Speech signal - PCM - Incoherent phase channel -Logarithmic quantization - Y, c and^u_vs. P/2 0^^ s . . . . . . 46 4.17a Speech signal - DPCM - Incoherent phase channel -Logarithmic quantization - "i* £> and V/ vs. P/lHo^ s 47 4.17D Speech signal - DPCM - Incoherent phase channel -Logarithmic quantization - Yj c andy^vs. P/2NO{CS 48 4.18 Speech signal - Incoherent phase channel -vs. P/bHsfcs f o r PCM and DPCM and uniform and logarithmic quantization 49 A . l Power density spectrum of speech 51 v i L I S T OF TABLES Page Signal s t a t i s t i c s 16 f , vs. i 5 0 v i i ACKNOWLEDGEMENT Grateful acknowledgement i s given to the Defence Research Board of Canada and the National Research Council of Canada f o r f i n a n c i a l support received. I would l i k e to thank Dr. R.W. Donaldson, the supervisor of t h i s project, f o r h i s valuable suggestions and constant encouragement, I am greatly indebted to Dr. F. Noakes f o r his understanding and support. I would l i k e to thank Dr. M.P. Beddoes f o r reading the manuscript and f o r his helpful suggestions. I am grateful to Mr. J. Cavers and Mr. D. Chan for t h e i r many valuable discussions. I would also l i k e to thank my parents f o r t h e i r support and encouragement, Mrs. A. Andersen f o r typing the manuscript and Mrs. B. Lange f o r a s s i s t i n g with the i l l u s t r a t i o n s . v i i i I . INTRODUCTION 1.1 Transmission of Messages over D i g i t a l Communication Channels P r o c e s s i n g a message x(l) f o r tr a n s m i s s i o n over a no i s y d i g i t a l channel can be considered to c o n s i s t of two operations: source and channel encoding (see F i g . l . l ) . The source encoder transforms the message i n t o S o u r c e C\nov-vvnel pm F i g . 1.1 General d i g i t a l communication system. a sequence of d i s c r e t e symbols chosen from a f i n i t e s e t . The channel encoder then assigns some sequence of symbols, chosen from a f i n i t e set to each member of ^ A}- « I d e a l l y , the source encoding renders a l l members of jflj s t a t i s t i c a l l y independent and equiprobable, and the channel encoding adds redundancy to the message s t a t i s t i c s i n a way that i s optimum f o r the p a r t i c u l a r channel. A c t u a l source encoders do not remove a l l the redundancy, w h i l e a c t u a l channel encoders are a l s o suboptimum. The channel and source decoders perform the i n v e r s e mappings to y i e l d a delayed and d i s t o r t e d r e p l i c a of the o r i g i n a l message. 1.2 Review of Previous Research Many coders which improve the redundancy r e d u c t i o n e f f i c i e n c y over 2 t h a t of ordinary pulse code modulation (PCM) have been proposed. D i f f e r e n t i a l p u l s e code modulation (DPCM) employs such a source encoder, shown i n P i g . 1.2. PCM and d e l t a modulation (AM) are s p e c i a l cases of DPCM; PCM i s DPCM with no feedback i n the source encoder, while AM i s i d e n t i c a l to pre v i o u s sample feedback DPCM .-with one b i t of q u a n t i z a t i o n . Numerous o p t i m i z a t i o n s t u d i e s of PCM haye been c a r r i e d out. Op t i m i z a t i o n of the q u a n t i z a t i o n process have been considered by [l ] , I2-] , and t^ iH-J . The problem of o p t i m i z i n g the combined process of q u a n t i z i n g , sampling, and r e c o n s t r u c t i o n (source decoding) i n the absence of channel noise was considered by J5~] , [&] , and 1^3 . S e v e r a l i n v e s t i g a t o r s have analysed source encoders which use feed-back around the q u a n t i z e r i n order to reduce the redundancy i n the encoder output. Approximate formulas f o r the s i g n a l to noise r a t i o of DPCM systems w i t h no channel noise have been d e r i v e d by [8^], and M . S u b j e c t i v e o p t i m i z a t i o n and general mathematical a n a l y s i s of DPCM systems, again i n the absence of channel n o i s e , was done by Apparently, the e f f e c t of channel n o i s e on DPCM systems has r e c e i v e d l i t t l e a t t e n t i o n . 1.3 Scope of the Thesis In t h i s t h e s i s , the DPCM system shown i n P i g . 2.1 i s considered. The i n p u t to the system i.8 f i r s t low-pass f i l t e r e d , then compared with a l i n e a r estimate based on the past h i s t o r y of the f i l t e r e d i n p u t . For the sake of s i m p l i c i t y , only previous sample feedback i s considered. The d i f f e r e n c e 3 between the a c t u a l s i g n a l and i t s estimated value i s quantized and sampled before being presented to the channel encoder f o r t r a n s m i s s i o n over the d i g i t a l channel. Channel encoders u s u a l l y a s s i g n b i n a r y words to the output of the source encoder, that i s , ^ o j u s u a l l y c o n s i s t s of-the binary alphabet. Both the modulator and demodulator f o r b i n a r y a n t i p o d a l s i g n a l l i n g can e a s i l y be implemented. Among the b i n a r y codes, the n a t u r a l b i n a r y code i s the sim p l e s t and most commonly used; t h i s code has the added advantage t h a t i t i s e a s i l y analysed. The method of a n a l y s i s presented i n t h i s t h e s i s can, i n p r i n c i p l e ^ b e extended to i n v e s t i g a t e more complex b i n a r y coders, as o u t l i n e d i n Sec.4 . 4 . The purpose of t h i s work was to evaluate the mean squared e r r o r £ x and to minimize by proper s e l e c t i o n of the parameters of the system i n P i g . 2 . 1 . A mathematical model of the system was developed, and an IBM 7044 d i g i t a l computer was used t o o b t a i n the l e a s t mean squared e r r o r and the optimum system parameters. Pre-Rlter F . l t e r 4 r c c U o v w e l ev\co<Xe.v . + F i g . 1.2 DPCM Source encoder 4 I I . GENERAL MEAN SQUARE ERROR EQUATIONS FOR DPCM SYSTEM 2.1 Mean Squared E r r o r i n DPCM The communication system considered i s shown i n F i g . 2.1. The quantizer i s ' a non- l inear device that produces an output y^ i f the amplitude of the quantizer input f a l l s i n t o the input range (tf^ jL; *w*i] ^ o r k*!,2/"L-The input ranges ai*e ordered p a r t i t i o n s on the r e a l l i n e i n the i n t e r v a l •^-eo^coj such that any point belonging to the p a r t i t i o n i s s t r i c t l y less than any point belonging to the.(y**f* p a r t i t i o n . The y j t are monotonically increas ing with k . The coder maps each ^ i n t o a sequence of B binary d i g i t s , where BHo^L. The decoder attempts to reconstruct. v(fc). Let R X l (x) andR.^(f) denote the normalized autocorre lat ion functions of jq-U) and Rx(^ .Cc) the normalized c r o s s - c o r r e l a t i o n function of x ( Ct) and < (^t) r e s p e c t i v e l y . I f the amplitude variance of the dif ference between vGO and Ok) i s denoted Cn,1" and the pov/er density spectrum of xGO I s denoted XCf) then i t fol lows from [ l l ] that r °x fxcf) J f w i <• ^-1« R* 6/0 + oc* io) - > iu^ /f.) 1 r r -\NtL i (VO where i s the normalized mean squared e r r o r , i . e . £*= (^4)-x(0)x/x,'(-t) and |Q i s the r a t e - d i s t o r t i o n f u n c t i o n of the q u a n t i z e r . Note t h a t cC i s the amount by which the feedback s i g n a l i n F i g . 2.1 i s attenuated, and . Note that c< appears l a t e r as a v a r i a b l e of i n t e g r a t i o n and i s not to be confused with the present. 1*. -5a»wp\er& of ^recjMeno^ \ pulse wloHU A LP., 0 [Z-uwtj CotJer —f*- Mock.- j Decode la-tor Vector 1 / 'W\v\arn dibits tw\pu\$«- ripens* Delav. L.P.- M. ! F i g . 2.1 Communication system considered 6 i s the r e c i p r o c a l of the s i g n a l to noise r a t i o . w For both r^ <|-(cJ and ^ ^ . ( r ) a r e assumed to be much small than R^fc) f o r most t*. Some al g e b r a shows t h a t , i f er aw w .w i - of1 f o r a l l -l<e6Cl . Thus, The three terms on the r.h.s. of (l.l) represent d i s t o r t i o n due t o f i l t e r i n g , q u a n t i z a t i o n and channel n o i s e r e s p e c t i v e l y . 2.2 E v a l u a t i o n of Quantizer Rate D i s t o r t i o n F u n c t i o n . ^ l o r Uniform Quantizer The input-output c h a r a c t e r i s t i c s o f a symmetric uniform q u a n t i z e r w i t h m i d - r i s e r are shown i n F i g . 2.2, ) v(t) , 1 H 1.-S 3£ 3-X C-t) o u t p u t : 2. -3J-o i n p u t F i g . 2.2 Uniform q u a n t i z e r c h a r a c t e r i s t i c s 7 The expected noise power i s given by - ( x ( ^ - v ( ^ | ^ v ( j : ) ; thus ^d~c^/cj^x. I f U denotes the number of output l e v e l s , then cT= 2 Y « ~ 0 • I f the output of the l e v e l i s designated and i f ^  i s halfway betv;een the end-points of the i n p u t range, then al l k The i n t e g r a t i o n , which a p p l i e s even i f the q u a n t i z e r i s non-uniform, i s over the input range, (a.k.b,,{ . In p a r t i c u l a r , r where L(U)=7iw[LV "l J ; U(V)= [u^? "0 and i s the p r o b a b i l i t y d e n s i t y f u n c t i o n of v Ct) . S i m p l i f i c a t i o n g i v e s L e t t i n g X^lo^ 1 - , s u b s t i t u t i n g f o r ^ , u t i l i z i n g the symmetry of the q u a n t i z e r and n o r m a l i z i n g w.r.t. {o^- y i e l d s U s 1-3 OS or where U - i ¥ - 1 8 and F<[o<-) i s the u n i t variance p r o b a b i l i t y d e n s i t y f u n c t i o n of v(t) , i . e . -<<> 2.3 E v a l u a t i o n of Quantizer Rate D i s t o r t i o n Function -W* f o r Non-Uniform Quantizer • . The non-uniform q u a n t i z e r may be represented as the uniform q u a n t i z e r of Sec. 2.2 preceeded by a compressor and f o l l o w e d by an expander, as shown i n F i g . 2.3. Hon-Uniform f J GLuant lie r J Q u a i n t r Co wnpressor F i g . 2.3 A l t e r n a t i v e .-representation .of non-uniform q u a n t i z e r The compressor and expander c h a r a c t e r i s t i c s are s i n g l e - v a l u e d continuous d i f f e r e n t i a b l e monotonic f u n c t i o n s such that when the inp u t i s the outputs are -["K(x) and { i00 » r e s p e c t i v e l y . Since the d i f f e r e n c e between the i n p u t and the output of the non-uniform q u a n t i z e r vanishes, as the number of q u a n t i z a t i o n b i t s B> and the magnitude of the quan t i z e r output s i g n a l Yvn become l a r g e , -C(fv(V)) =y . cp i s then given from si n c e the U output of the non-unifoi srm -I q u a n t i z e r i s -(xC^ u) , and the end-points of the i n p u t range are -(i (at) and (t>0, or J^CQU") and -ji.C'bu) . I f the expander c h a r a c t e r i s t i c i s given by 9 e wherejm. i s the compression parameter, ^(y.) i s symmetric. Proceeding as i n Sec. 2.2 y i e l d s \- where q-^ i i ^ L a u l ( p,^)^^ + C U U M ) L c and Fv (°^ ) i s the u n i t v a r i a n c e p r o b a b i l i t y d e n s i t y f u n c t i o n of v(t) • 2.4 E v a l u a t i o n of Channel Noise f o r Uniform Quantizer and N a t u r a l B i n a r y . Code F i g . 2.4 i s a diagram of the general system u t i l i z i n g the a l t e r n a t i v e r e p r e s e n t a t i o n f o r the non-uniform q u a n t i z e r suggested i n Sec. 2.J. The modifie d channel, shown i n F i g . 2.5, i s discussed l a t e r i n t h i s s e c t i o n . r > r - M - j r r r - - j - - - i L.R, Quant. Co w>,p t-c.>S C . . ' E x p a n d e r - 4.* F i g . 2.4: General system w i t h a l t e r n a t i v e representationf'of q u a n t i z e r 10 The n o i s e amplitude variance 0^ i n (2.1) i s the varia n c e of n(t) where n(t) i s given by * ( l ) . s ( l ) = v(l)-te(t)*-q-ft)]-S(V) and . 6 ( t ) i s the impulse t r a i n used f o r sampling. From F i g . 2,4 vCt)a£ Jtfito)) «• c l^-s(t) W*)Js(t) where' r^ GO i s the noise amplitude c o n t r i b u t i o n of the modified d i g i t a l channel. Since ( t ( < 4 ) . <0) - (t)"). s ( l ) , v C ^ ^ E ^ e C ^ ^ ^ ^ ^ . ^ ^ S ^ ) and *(tVf[f.(eCO)+c^(*) - (e&)*<j60V ft-8) Furthermore, (i (e (0) v c^ iCOll " e(0 *-<^ (0 and •f (eCOl *<^Oej^e&) + ^ t&))cf (e&) +<j-C-0) S u b s t i t u t i o n i n t o fr*) gives nGt) = £ ' ^  [ e U ) v <^ )1 * A (-Oj -(e(4) vc .^(X)). A Ta y l o r s e r i e s expansion of the f i r s t term about H(lVo y i e l d s v i C t V ^ ^ ^ L e ^ V f ^ l ^ ( H A1. I n the case of uniform q u a n t i z a t i o n , the expander c h a r a c t e r i s t i c i s the i d e n t i t y f u n c t i o n , and (2.°^ reduces to tn(V) =vu (V) . Noise ^ (0 i s caused by the modifi e d d i g i t a l channel of F i g . 2,4. This m o d i f i e d channel i s shown i n more d e t a i l i n F i g . 2 .5 . fx pcmder f Mod,. F i g . 2.5 M o d i f i e d d i g i t a l channel I Modified Decoder I V View, I t i s d e s i r e d to c a l c u l a t e = (x(V)-^G-))* • A d d i t i o n of b i a s Yw 1.1 and s c a l i n g by O- a f f e c t s n* as f o l l o w s : cx being chosen such th a t the in p u t to the coder i s a p o s i t i v e i n t e g e r ; thus o_= where B i s the number of b i t s of q u a n t i z a t i o n . The coder then maps i n t e g e r tf^ i n t o a b i n a r y word, as discussed i n Sec. 2.1. The channel i s assumed to be a memoryles3 b i n a r y symmetric channel (BSC) w i t h t r a n s i t i o n p r o b a b i l i t y p . Since V x and y x are i n t e g e r s , they can be expressed a t any p a r t i c u l a r time i n s t a n t as where U U j N l are O o r \ . V-v, and Yj are the vec t o r s <u0 a^-i) and <V0 ^Q-C) r e s p e c t i v e l y . tx-%•* 21 2 L(ui~yl[)'^2.lei note: £^(-^0, l \ . 9-1 B-i where €<. and C; are independent, and Now Ti = -\-P(uo-o) ? + l?(u;=O P^lP(u,= 0 - V C u ^ c O ! .where P(uL«o) andp(u;= | )^ i s the p r o b a b i l i t y that U ; ^ o and , r e s p e c t i v e l y . Since P(ul=0^ = P(ul=LV h~ i n t h i s woi'k, as eCt") has a symmetric p r o b a b i l i t y d i s t r i b u t i o n GL = 0 . (^rSj'' 2 - ^ ° C-Ol-PCui-o)? • (M) 1 . PcW0p = p fex-Vx)l=p Oj^ii . Since O^W = <~A_ ( Y > - Y j * 3 Q X ^ ^ - h L p V - M (a.io) 2) r Cu-0 I f the t r a n s m i t t e r sends one of two orthogonal s i g n a l s i n the 12 presence of a d d i t i v e white Gaussian noise p"Q(^ &->) f o r a coherent phase channel, ff s being the energy of the t r a n s m i t t e d s i g n a l and H o i s the mean noi s e power l ' 4 ^ . I f there are u n i f o r m l y d i s t r i b u t e d phase p e r t u r b a t i o n s . present between the t r a n s m i t t e r and the r e c e i v e r , p=-^e a W o f o r a n incoherent R 2 phase channel jj5~] . I f both u n i f o r m l y d i s t r i b u t e d phase p e r t u r b a t i o n s and Ral e i g h f a d i n g are present between the t r a n s m i t t e r and r e c e i v e r , ^ = \ /(2tt &/N 0) [l£>l • I f R i s the number of b i t s sent i n one second,. £s=rVR where P i s the t r a n s m i t t e r power. Since R*2\/E> f o r Nyquist r a t e sampling, Es = ?/zv/fe 2.5 E v a l u a t i o n of Channel Noise f o r Non-Uniform Quantizer and N a t u r a l Binary Code For the case where the q u a n t i z e r i s non-uniform, i t f o l l o w s from fr.<0 t h a t but [eCt^  ^ C^VfiCeX-VJl^CO = "2&) where denotes the output of the uniform q u a n t i z e r . Since the channel noise i s assumed to be independent of the message sent, For the l o g a r i t h m i c compressor c h a r a c t e r i s t i c of (l.^O j L. The higher moments of w(l) are obtained as f o l l o w s from Sec. 2.4. . rt}-t t %-i e.-' §j±, S-i ... i=o J >(=o ">a=o i j . o V. i Note that €i; i s independent of G:^  f o r Wk and Q<=o f ° r odd, s i n c e J B-* A . -P(a:= o) = P(u:; = i ) . In p a r t i c u l a r , for. ^ 2 , K/Q^-IZI &i as before. 15 I f VT> r l ( c-r ) l then f o r \c-4 and k*-6 , r e s p e c t i v e l y , . &-l 6-1 B - I B-J . B - i 6-1. e - 1 J/*0 V l a o » ^ where p i s the s i n g l e d i g i t e r r o r p r o b a b i l i t y . Since ^ « \ ) pV"' P"™" ° ' i t c a n t e s h o w n t h a t t o a g° 0 (* ap p r o x i -mation Y\\la* = pj|3 P ^ ^ ^ B ~ t l . S u b s t i t u t i o n f o r a* g i v e s of the double sum y i e l d s To o b t a i n e"^~ consider the f o l l o w i n g , g -> t Pfey)e^"where ?«. i s the f o r o ^ 3 > . From ( l . u ) i t f o l l o w s that where i s even. E v a l u a t i o n C m ) output l e v e l of the uniform q u a n t i z e r i n Fig. 2.4 and P & O i s given by PeC°0<W where h^(?0 and ^ C«) are the p r o b a b i l i t y d e n s i t y f u n c t i o n s of.<z(V) and.eG.)of F i g . 2.4 r e s p e c t i v e l y , and._2^-= l i l k l L L , •LM- avCk and U(U)^YW,(UM) . Then -£0-C0> cY^U^iJ tj and £(u(0) cVv^  W^Ce^-1) * N o O T' a l i z i nS ^ ) y i e l d s - - 5 where feCjs") i s the u n i t variance p r o b a b i l i t y d e n s i t y f u n c t i o n of e£4r) . 14 I I I . MEAN SQUARED ERROR FOR GAUSSIAN ..TELEVISION, AND SPEECH SIGNALS 3.1 S i g n a l S t a t i s t i c s Input s i g n a l x(L) i n Fig.'2.1. i s assumed to be a s t a t i o n a r y random process.. Such a process may be described i n terms of a sample space ^ c o n t a i n i n g an i n f i n i t y of points.u> , a set of r e a l time f u n c t i o n s and a p r o b a b i l i t y measure a s s o c i a t e d w i t h each point.0 i n Q . The random process x(0 at any given time i s one of the set of sample f u n c t i o n s ^y^t^ o being s e l e c t e d a t random but i n accordance w i t h the p r o b a b i l i t y measure assigned to Q . I f the p r o b a b i l i t y of the event x ^ O - K ^ x t a ^ K ^ - j X G ^ U ' v i s ( ? ) where x(l)=-< x h\ X (U/-j X (k) > and oT = < o a , .-^  c<c^ ? then the j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n of x(t) i s defined as ^c7j)6*)= Bc(5 Furthermore, i f P-^(o?) i s independent of the time o r i g i n , then. xCt) i s s a i d to be s t a t i o n a r y . A s t a t i o n a r y random process i s s p e c i f i e d i f a r u l e i s given or i m p l i e d f o r determining i t s j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n f o r any f i n i t e s et of times {ti^ -U^ -• • • For the a n a l y s i s presented i n t h i s t h e s i s , only f i r s t and second order s t a t i s t i c a l c h a r a c t e r i z a t i o n i s r e q u i r e d . In p a r t i c u l a r , knowledge of the f i r s t order p r o b a b i l i t y d e n s i t y f u n c t i o n H f^eO and the a u t o c o r r e l a t i o n f u n c t i o n Qy.(tr)(or e q u i v a l e n t l y the power density spectrum X {^)) °f ^he message. x(t) were s u f f i c i e n t to determine the mean squared e r r o r £ v between the tr a n s m i t t e d and re c e i v e d message. Three types of messages were considered. 1. Gaussian 2-0^1 too-1 15 2. T e l e v i s i o n Rc<*y 3. Speech . • _ The equation f o r the pov;er d e n s i t y spectrum of speech was obtained by a weighted l e a s t mean square c u r v e - f i t , shown i n F i g . A . l , to data presented by jVQ . . I n order to evaluate the q u a n t i z a t i o n n o i s e , ' i t i s necessary to know the p r o b a b i l i t y d e n s i t y f u n c t i o n of the compressor input eCt). Consider the PCM (°(=o)case f i r s t . Here e ( i ) w i l l have approximately the same p r o b a b i l i t y d e n s i t y f u n c t i o n as the random process ~f(i) . When Y(4.) i s Gaussian, e(X) i s Gaussian When x(,W) i s t e l e v i s i o n or speech, i f the bandwidth of L . P . \ i s s u f f i c i e n t l y g r e a t , i . e . 1< = I where V.* Jn )((^)df - rV;wr then one expects the p r o b a b i l i t y d e n s i t y f u n c t i o n of eftH° he approximately equal to the p r o b a b i l i t y d e n s i t y f u n c t i o n of . In the DPCM (o/^o) case, experimental r e s u l t s i n d i c a t e t h a t the i n p u t to the q u a n t i z e r e&) has L a p l a c i a n d i s t r i b u t i o n i f i s t e l e v i s i o n [_& 1 or speech [}°\. When x(C) i s Gaussian, e GO w i l l be Gaussian, s i n c e 16 the fed-back q u a n t i z e r noise t/^.(v-^) i s n e g l i g i b l e and the sum of the two Gaussian processes and —c<y.(t-1/^ ) i s Gaussian The r e l e v a n t p r o p e r t i e s of the s i g n a l s considered are summarized below i n Table 3.1. Xjpe e f F u n c V i o w O^ tfG=) PrdoaWil *\ty _De«5^ F ^ ^ c V i o v i oy Pctf P P C M Gaussian A t - o r 1 ' e 1 e _ or 1 1 e i , , J \ e [ O - c ^ i c ^ o i 1 Table 3«1 S i g n a l s t a t i s t i c s 3.2 Mean Squared E r r o r f o r Gaussian S i g n a l s The t o t a l mean squared e r r o r f o r the system can now be evaluated from (l.l) . S u b s t i t u t i o n f o r the normalized power d e n s i t y spectrum of from Table 3.1 y i e l d s lo v/ (3.X) (3.3) For a Gaussian s i g n a l and Q. 17 where Q(*)»l _Leri/.X_ \ and R^ ) is the unit variance probability density function of . For uniform quantizer IQ becomes, from (l. u) and (2-5~) F i n a l l y , substitution f o r the channel noise term from, (i.to) y i e l d s In (2>.4) V has been replaced by W . The mean squared error for the PCM case i s obtained by setting.o/=o i n . ( 3 . 4 ) . For the Gaussian process, only uniform quantization was considered, since optimal compressor ch a r a c t e r i s t i c s were not readily available. 3.3 Mean Squared Error f o r Television Signals For t e l e v i s i o n signals, the int e g r a l of the power density spectrum and the autocorrelation function are given by (3.l) and (3."b) , respectively. The rate d i s t o r t i o n function must be considered separately f o r the PCM and DPCM cases. For uniform quantization and PCM UPefcW. J A l - ax) and 4 J^_(d-c ) where fe) is the unit 18 variance p r o b a b i l i t y d e n s i t y f u n c t i o n of Q.QC) . The r a t e d i s t o r t i o n f u n c t i o n £3 becomes, from (l.L^and (2.5\) 1Q =_J—I - * — U ^ - \ 3 \ 3 l f o r a l l O^c, ^ 4s . S u b s t i t u t i o n f o r the channel noise term from (2.\o) 0 y i e l d s Bandwidth \J has been replaced b y W / ^ c . ' F o r the DPCM case Jo r and^Cs^) i s the u n i t variance p r o b a b i l i t y d e n s i t y f u n c t i o n of eCi) * For uniform q u a n t i z a t i o n S u b s t i t u t i o n f o r the channel n o i s e from (2.\o) y i e l d s - e (3.8) For l o g a r i t h m i c q u a n t i z a t i o n , which i s optimum f o r t h i s case i f the channel n o i s e i s small fe] , i s obtained from (l.k) and (2.7) , 19 where VJ i s normalized w . r . t . f , and J^Q and are given by and ( 3 .\o) r e s p e c t i v e l y . The PCM case w i t h compressor was not considered, s i n c e i t has been shown that f o r uniform 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 , as i n t e l e v i s i o n w i t h PCM, uniform q u a n t i z a t i o n i s optimum i f the channel noiee i s s m a l l . 3.4 Mean Squared E r r o r f o r Speech S i g n a l s The i n t e g r a l of the normalized speech power d e n s i t y spectrum of y(A) i n Table 3*1 i s given by Since f o r a speech s i g n a l f o r both PCM and DPCM 20 e ku-L avid. \?e(°0ck " - t i e -fxW e - e. 1 where P£(y) i s the u n i t variance p r o b a b i l i t y d e n s i t y f u n c t i o n of eC-t) -To f o r uniform q u a n t i z a t i o n i s given by (3.7) . Let lyC J^if = Ol^i , 1 lX(f) d f= I- 6iJ". S u b s t i t u t i o n f o r the channel noise from (2. lo) y i e l d s 3 • The mean squared e r r o r f o r the PCM case i s obtained by s e t t i n g (3. l l ) 0(-O i n (3-\x) . For l o g a r i t h m i c q u a n t i z a t i o n , which i s optimum f o r speech i f the channel noise i s small [l8] , £3 i s given by fa])» S i m i l a r l y , e ^ i s given by (2>.U>) and s u b s t i t u t i o n f o r the channel noise from (2.12-)yields The mean squared e r r o r f o r the PCM case i s again obtained by settingc/=o i n . (3>.l3>^ . 21 IV. RESULTS AND DISCUSSION OF RESULTS 4.1 Computational Procedure M i n i m i z a t i o n of. £%. was c a r r i e d out by a simple search procedure on an IBM 7044 d i g i t a l computer. The parameter space formed by Yj &yt«-and c was quantized i n t o a f i n i t e number of p o i n t s by r e s t r i c t i n g the para-meters to assume d i s c r e t e . v a l u e s i n some reasonable range, and.^was then evaluated f o r a gi v e n . P/N\0 from the equations i n Sec. 3.2 - 3»4 f o r various values of the parameters. The minimum £ lwas then recorded, along w i t h the optimum system parameters, and the process was repeated f o r v a r i o u s values of P/Mo . Use of arguments s i m i l a r to those i n . [ i j ] provides s t r o n g evidence t h a t the minimum.£% f o r a PCM or DPCM system operating over a given coramuuni-c a t i o n channel i s uniqueJ t h a t i s , there i s but one s e t of optimum system parameters. Rigorous proof of t h i s statement appears d i f f i c u l t , however, and i s not presented here. 4.2 PCM R e s u l t s F i g s . . 4. | , k 1. and H.^ show the minimum and the optimum values ofWjfc and y when a Gaussian s i g n a l w i t h spectrum XCf) = P\^- A " C f a * ^ c 1 ") i s used i n con j u n c t i o n w i t h a uniform q u a n t i z e r . Note t h a t i s p l o t t e d vs. P/zHo^c- • The optimum s i g n a l bandwidth V , number of q u a n t i z a t i o n b i t s & and amplitude of normalized q u a n t i z e r end-point a l l i n c r e a s e w i t h the channel s i g n a l to noise ratio;;.. The f a d i n g channel i s n o t i c e a b l y i n f e r i o r 22 t o e i t h e r the incoherent or coherent one, as expected from Sec. 2,4. The r e s u l t s f o r the incoherent and coherent channels are i n very c l o s e agreement, as expected. To obtain some measure of the s e n s i t i v i t y of the minimum. <?* t o v a r i a t i o n s i n the optimum parameters,^ ( ^ s* — / £ mi cjo was c a l c u l a t e d f o r values of P/O-Ho^o where £1 i s the average of the two mean squared e r r o r s evaluated when one parameter i s 1.1 times as l a r g e and 0.9 times as l a r g e as i t s optimum value, (except i n the case of fc ., where the d e v i a t i o n i s . ^ l ) , the other parameters being optimum. S e n s i t i v i t y curves. B e ,Vie-j v s . . .appear i n F i g . . S e n s i t i v i t y curves were p l o t t e d f o r a l l three types of channels, but f o r the sake of b r e v i t y , only the ones f o r the incoherent phase channel are i n c l u d e d . Mean square e r r o r £ l f o r t e l e v i s i o n (see F i g . 4,7 ) i s s l i g h t l y l e s s than the corresponding C\for Gaussian s i g n a l s , s i n c e the uniform q u a n t i z e r i s optimum i n t h i s case, as discussed i n Sec. 3«3« Note th a t the r e s u l t s f o r speech are p l o t t e d v s . , where . -^cs - U5"oV\i i s a p p r o x i -mately the mean frequency of speech. Although r e s u l t s were obtained f o r t e l e v i s i o n and speech s i g n a l s on coherent channels and f a d i n g channels, these r e s u l t s are not i n c l u d e d i n the t h e s i s . In a l l cases, the minimum f o r coherent channels was s l i g h t l y l e s s than t h a t of incoherent channels, while the minimum. £ a f o r f a d i n g channels was much g r e a t e r than t h a t of coherent and incoherent channels f o r a given value of P/lsIo . 25 4.3 DPCM Results Some i n i t i a l d i f f i c u l t i e s were encountered i n o b t a i n i n g the minimum £ 4 v s . P/NO f o r DPCM. Combining (2-2), (2-to) and s u b s t i t u t i n g 7*,*- <nfX i n t o ( l . l o ) (a3 i n Sec. 2 ;2), where o-e!1" i s the va r i a n c e of the qu a n t i z e r i n p u t , y i e l d s Equation (k.l) suggests that £l-?c> asV/-) o^ > , V s i n c e Co) =A . This anomaly r e s u l t s from the approximation, made i n Sec. 2.1, that the fed-back q u a n t i z e r noise i s n e g l i g i b l e compared t o the p r e d i c t e d s i g n a l . I n c l u s i o n of the e f f e c t of the feedback q u a n t i z a t i o n noise shows that. £ x i s never z e r o . This d i f f i c u l t y was overcome by f i x i n g W a t some reasonable value, thereby c o n s t r a i n i n g the q u a n t i z e r i n p u t variance to be g r e a t e r than some nonzero p o s i t i v e constant. Assuming, as i n Sec. 4«1 that a unique minimum £ x e x i s t s , and that the optimumW v a r i e s continuously w i t h the values of V/chosen are i n f a c t optimum f c r some value of ?Mo , as can be seen i n the minimum _£*..,.curves of F i g s . hSa.} k$o- and tu5"«. . . This argument i s f u r t h e r s u b s t a n t i a t e d by examining optimum W values obtained by another method, discussed l a t e r i n t h i s s e c t i o n . In g e n e r a l , then, the optimum parameters were obtained f o r DPCM as f o l l o w s . A f a m i l y of minimum curves f o r f i x e d values of.V were p l o t t e d , t ogether w i t h the corresponding system parameters. The approximate range of o p t i m a l i t y of each w . c h o s e n was determined by examination of the £* curves. The optimal system parameter curves were then obtained by s e l e c t i n g the appropriate curve i n the region where W i s optimum. This process i s 24 i l l u s t r a t e d i n F i g s . l\.<\ H.lo andH.\\ , Some other methods of approach to o b t a i n i n g the optimum system parameters and the r e s u l t i n g £ l are o u t l i n e d below* S i m u l a t i o n of the p r e d i c t o r - q u a n t i z e r p a r t of the system on the computer was c a r r i e d out i n order to o b t a i n an expression f o r the q u a n t i z e r inp'ut variance as a f u n c t i o n of WjB and << . Gaussian samples w i t h Butterworth s p e c t r a were generated, but the convergence r a t e of the fed-back q u a n t i z e r noise v a r i a n c e was too low f o r p r a c t i c a l a p p l i c a t i o n . T i m e - t r a j e c t o r y c a l c u l a t i o n s of the p r o b a b i l i t y d e n s i t y f u n c t i o n and moment-generating f u n c t i o n (Fokker-Planck equations) of e(t) were attempted, but the mathematics proved unwieldy. Approximating the q u a n t i z e r by a white Gaussian n o i s e source f o l l o w e d by a c l i p p i n g - a m p l i f i e r gave r i s e to an approximate c l o s e d form expression (omitted here f o r the sake of b r e v i t y ) f o r the q u a n t i z e r n o i s e which gave good asymptotic behaviour, t h a t i s , U ? = | . This expression was used to determine a l l the optimum V/ - > o f -» I system parameters f o r the Gaussian DPCM u n i f o r m q u a n t i z a t i o n case. The optimum w values thus obtained were p l o t t e d vs. p / x H o ^ i n F i g . k.5h. , together w i t h the op t i mur. •. V values obtained by the method discussed e a r l i e r i n t h i s s e c t i o n . The r e s u l t s obtained by the two methods are i n very good agreement. Examination of the minimum e 1 curves f o r DPCM shows an improvement over £*• f o r PCM, as w e l l as an average s a v i n g of approximately one q u a n t i z a t i o n b i t . 4.4 R e s u l t s f o r Companded Systems The advantage of non-uniform q u a n t i z a t i o n i s t h a t the compressor ( F i g . 2.4) tends to f l a t t e n out the p r o b a b i l i t y d e n s i t y f u n c t i o n of e l i ) , 25 making the d e n s i t y f u n c t i o n of the input of the uniform q u a n t i z e r , more uniform. The q u a n t i z a t i o n noise i s minimum when the d i s t r i b u t i o n f u n c t i o n of the uniform q u a n t i z e r i n p u t s i g n a l i s uniform pw] . However, f o r the n a t u r a l b i n a r y code, the channel noise i n c r e a s e s very r a p i d l y w i t h the compression p a r a m e t e r ^ . This i n c r e a s e i n channel n o i s e i s caused by the term e c V~ i n \2M) , or e q u i v a l e n t l y , the e f f e c t of the expander before the coder i n F i g . 2.5. Thus, the optimum values of y j - obtained are f a r below the values g e n e r a l l y assumed f o r systems w i t h no channel n o i s e . I t i s i n t e r e s t i n g to note t h a t i n s e r t i o n of another compressor-expander p a i r before and a f t e r the modified coder and decoder of F i g . 2.5 would be eq u i v a l e n t to u s i n g a more s o p h i s t i c a t e d code, which would s u b s t a n t i a l l y reduce the channel n o i s e . I n theory, t h i s compressor-expander p a i r could be analyzed by the method used i n Sec. 2.4, but the t r a c t a b i l i t y of such an approach has not been i n v e s t i g a t e d i n d e t a i l . 4.5 Concluding Remarks DPCM was found, f o r a given channel c a p a c i t y , to have a lower mean square e r r o r than PCM i n a l l cases. Logarithmic q u a n t i z a t i o n was four.d to y i e l d b e t t e r performance than uniform q u a n t i z a t i o n f o r speech s i g n a l s , f o r the PCM case, and f o r speech and t e l e v i s i o n s i g n a l s f o r the DPCM case. In gene r a l , the r a t i o of the mean square e r r o r of DPCM to t h a t of PCM inc r e a s e s w h i l e the d i f f e r e n c e i n performance decreases as channel noise i n c r e a s e s . 26 E\10 5 Y w/f 3.06 r120 1000 2000 3000 4000 5000 6000 7000 8000 Fig. 4.1 Gaussian signal - PCM ^Coherent phase channel -Uniform quantization -E'E^ W and Y vs. P 2Nofc 2N 0f c 27 1000 2000 3000 4000 5000 6000 7000 8000 P 2Nof c Fig. 4.2 Gaussian signal - PCM ~_Incoherent phase channel -Uniform quantization - £' B,W and Y vs.P2Nofc 29 Be Ye 4 8 i 3 4 n 1000 2000 3000 4000 5000 6000 7000 8000 - TUolc F i g . 4.4 Gaussian s i g n a l - PCM - Incoherent phase channel -Uniform q u a n t i z a t i o n - Be, V/e and Ye vs. F/2 Nofc Be/Ye and We show percentage i n c r e a s e of £ lwhen B,Y and W are sub-optimum (Sec. 4«l)» 30 Fig. 4«5& Gaussian signal - DPCM - Incoherent phase channel - Uniform quantization -£)B and Wvs.P/2Nofc 2N 0 f c 32 EVlO5 50 45 40-35 30 25 20 15 10 5-1000 2000 3000 4000 5000 6 000 7000 8000 9030 pF i g . 4«6 Gaussian s i g n a l - Uniform q u a n t i z a t i o n -E J v s . r/2Nofc f o r PCM - Coherent and i n -coherent phase channels and E vs. P/2Nofc f o r DPCM - Incoherent phase channel 2N0fc 33. i 1 , , , , , , , 1000 2000 3000 4000 5000 6000.7000 9000 P 2Nofc P i g . 4.7 Television signal - PCM_- Incoherent phase channel -Uniform quantization - E^W and Y vs. P/2Nofc 1000 2000 3000 4000 5000 6000 • 7000 8000 p 2 Note Fig. 4.8a Television signal - DPCM - Incoherent phase channel - Uniform quantization - E) B andW vs. P/2Nofc 35 B 81 7-4-3-w= 50f c w=100fc w=150fc w=200fc loOO 2000 3000 4 000 5000 6000 7000 8000 9000 36 F i g . 4.9 T e l e v i s i o n s i g n a l - DPCM - Incoherent phase 2 Nofc channel - Uniform q u a n t i z a t i o n - B v s . P 2 Nofc f o r f i x e d W S o l i d l i n e i n d i c a t e s range where a given curve i s optimum. 37 2-_w= 50_fc_ i 1-w-100fc j 1000 2000 3000 4000 5000 6000 7000 8000 9000 2 Nofc F i g . 4.10 T e l e v i s i o n s i g n a l - DPCM ~ Incoherent phase channel - Uniform q u a n t i z a t i o n - Y vs. f/2 Nofc f o r f i x e d W S o l i d l i n e i n d i c a t e s range where a gi v e n curve i s optimum. 38 1.0 .9 8 .9 6 .9 4 .9 2 .90 -.8 8 -.8 6 .84 .8 2 w=150fc leafed w=1 OOfc w=50fc i I I • 1000 2000 3000 4000 5000 6000 7000 8000 9000 2Nofc F i g . 4.11 Tele-vision s i g n a l - DPCM - Incoherent phase channel - Uniform q u a n t i z a t i o n - </~ vs. P2Nofc f o r f i x e d W S o l i d l i n e i n d i c a t e s range where a given curve i s optimum. 39 45 i 40 35 i 30 25 2 0 15 10 1000 20 0 0 3000 4 0 0 0 5 0 00 6000 . 7000 80 00 p 2NDf, O'C F i g . 4.12a T e l e v i s i o n s i g n a l - DPCM - i n c o h e r e n t phase channel Logarithmic q u a n t i z a t i o n - E*B and W vs.r/2Nofc 41 50 4 5 40 3 5 -3 0 25 20 1 5 -PCM no c o m p r e s s i o n DPCM no compress ion DPCM logaritnmMc c o m-pre s s i o n 10 1000 2000 3000 4000 5000 6000 7000 8000 9000 p 2Nof c P i g . 4.13 Te l e v i s i on s igna l - Incoherent phase channel -£* vs . P/2Nofc\ f o r PCM - Uniform quant iza t ion and E* v s . P/2Nofc f o r DPCM - Uniform and logar i thmic quant iza t ion 42 1 m3 E*10 125 250 375 500 625 750 875 1000 2 Notes F i g . 4'14 Speech s i g n a l - PCM - Incoherent phase channel -Uniform q u a n t i z a t i o n - E* B,W and Y vs. P/2Nofcs • 125 250 375 500 625 750 875 1000 p 2Nofe F i g . 4.15a Speech s i g n a l - JDBCM - i n c o h e r e n t phase channel -Uniform q u a n t i z a t i o n - £*B and W vs.P/2Nofcs 45 • 125 250 375 500 625 750 875 1000 p 2NoTcs F i g . 4 . l 6 b Speech s i g n a l - PCM - Incoherent phase channel -Logarithmic q u a n t i z a t i o n - y,c and A vs .P/Nofcs 47 1 . , , , , , - r 125 250 375 500 625 750 875 1000 p 2NoTcs F i g . 4.17a Speech s i g n a l - DPCM - Incoherent phase channel -Logarithmic q u a n t i z a t i o n -£,*B andW vs.P/2Nofcs 48 c 4 i 125 250 375 500 625 750 875 1000 _P 2Nofc5 F i g . 4.17b Speech s i g n a l - DPCM - Incoherent phase channel -Logarithmic q u a n t i z a t i o n - o6.,y,c and /x vs.P/2Nofcs 49 E*103 50 i 125 250 375 500 625 750 875 1000 1125 p 2 NofcS Fig. 4.18 Speech signal - Incoherent phase channel -E 2 vs. r/2Nofcs for PCM and DPCM and uniform and logarithmic quantization 50 APPENDIX I P o w e r D e n s i t y S p e c t r u m o f S p e e c h C l o s e d - f o r m e x p r e s s i o n (3.1) f o r t h e p o w e r d e n s i t y s p e c t r u m o f s p e e c h w a s o b t a i n e d i n t h e f o l l o w i n g m a n n e r . T h e d a t a f o r t h e s p e e c h s p e c t r u m - S ^ ) , s h o w n i n F i g . A . l , o b t a i n e d f r o m IH| w a s l i n e a r l y I n t e r -p o l a t e d t o i n c l u d e f r e q u e n c i e s g r e a t e r t h a n J.5KHz, t h e n w a s p l o t t e d a s a f u n c t i o n o f f r e q u e n c y . V i s u a l e x a m i n a t i o n w a s u s e d t o m a k e a n i n i t i a l f u n c t i o n a l r e p r e s e n t a t i o n 5 , ( £ ) o f S ^ ) . S iO j ) w a s t h e n f i t t e d t o t h e d a t a p o i n t s 6 j ^ ) b y a w e i g h t e d l e a s t s q u a r e s m e t h o d . T h a t i s , t h e c o n s t a n t s o f SX(§) w e r e c h o s e n s u c h t h a t C w a s m i n i m i z e d , w h e r e all I T a b l e A . l s h o w s v s . I . 1* 1 2. zoo 3 6 Oo f eoo 5" \ooa 6 1 XOn y (Too S lono T a b l e A . l vs. I, ^ K ^ 0 v a s * n e n s u b t r a c t e d f r o m S<j(^), a n d a g a i n v i s u a l e x a m i n a t i o n w a s u s e d t o a r r i v e a t a f u n c t i o n a l r e p r e s e n t a t i o n . 61.Cf) o f S^Cf ) " SiC^ } • S.(.^ ) w a s t h e n r e p l a c e d b y S . ( ^ ) + S a ( C ) a n d t h e p r o c e s s w a s r e p e a t e d u n t i l S j j C f ) - ( ^ ) c o u l d b e c o n s i d e r e d n e g l i g i b l e . T h i s p r o c e s s l e a d t o ( 3 . 0 . T h e c o n s t a n t s g i v e n i n F i g . A . l w e r e n o r m a l i z e d s u c h t h a t I^SC^) c i ^ ~ i T h e t e r m L1 i n (ft,l) w a s i n s e r t e d i n o r d e r t o p r o d u c e a m o r e S(f) 1.0 .4 -500 - Normal ized Power Spectrum of Speech - S ( f ) a p p r o x i m a t i o n S(f )= A e b ( f - c ^ + F -9 ( f - h ) l + _ i L _ v+ f 4 % Curve ampl i f ied by \\a f ac to r of 5 •,.. f = f x 5 0 0 A= 0.927 6848 b= 2.06 c=08 F=0.034610 g=2.2S h=2.2 U=4.056 v=56.0 Vs(f ) d f= 2£9338077x500 A / l = 0.3.4 443136 F/1 = 0.01285015 U/1=1.5 05914.0 6 2 5 0 0 3000 3500 4 0 0 0 4 5 0 0 1000 1500 2000 2 5 0 0 3000 3 5 0 0 4000 4 5 0 0 5000 5000 VJ"! F i g . A-1 Power density spectrum of speech oo accurate curve f i t at higher f r e q u e n c i e s , s i n c e the e r r o r 1 which r e s u l t s from iowpass f i l t e r i n g r e q u i r e d accurate r e p r e s e n t a t i o n . 53 REFERENCES 1. J . Max, "Quantizing f o r Minimum D i s t o r t i o n " , IRE Trans, on Information Theory, v o l . I T - 6 , pp. 7-12, March, i960. 2. G. V i l l i a m s , " Q uantizing f o r Minimum E r r o r w i t h P a r t i c u l a r Reference t o Speech", E l e c t r o n i c s L e t t e r s , v o l . 3, PP» 134=135, A p r i l 1967. 3. J.D. Bruce, "Optimum Q u a n t i z a t i o n " , Research Laboratory of E l e c t r o n i c s , Massachusetts I n s t i t u t e of Technology, Cambridge, Tech. Rept. 429, . March, 1965. 4. J.D. Bruce, "On the Optimum Qu a n t i z a t i o n of S t a t i o n a r y S i g n a l s " , IEEE I n t e r n a t i o n a l Convention Record (USA), v o l . 12, pp. 118-124, 1964. 5. L.M. Goodman, "Optimum Sampling and Quantizing Rates", Proc. IEEE (Correspondence), v o l . 54, pp. 90-92, January, 1966. 6. J . Katzenelson, "On E r r o r s Introduced by Combined Sampling and Quanti-* z a t i o n " , IRE Trans, on Automatic C o n t r o l , v o l . AC-7, PP« 58-68, A p r i l , 1962 7. K. S t e i g l i t z , "Transmission of an Analog S i g n a l over, a F i x e d B i t - R a t e Channel", IEEE Trans, on Information Theory, v o l . IT-12, pp. 469-474, October, 1966. 8. J.B. O'Neal, " P r e d i c t i v e Quantizing Systems ( D i f f e r e n t i a l P u l s e Code Modulation) f o r the Transmission of T e l e v i s i o n S i g n a l s " , B e l l Sys. Tech. J . , v o l . 45* PP. 689-721, May-June, 1966. . 9 . J.B. O'Neal, "A Bound on S i g n a l - t o - Q u a n t i z i n g Noise R a t i o s f o r D i g i t a l Encoding Systems", Proc. IEEE, v o l . 55, PP« 287-292, March, 1967. 10. R.A. McDonald, " S i g n a l - t o - N o i s e and I d l e Channel Performance of D i f f e r e n t i a l P u l s e Code Modulation Sustems-Particular A p p l i c a t i o n s to Voice S i g n a l s " , B e l l Sys; Tech. J . , v o l . 45, PP» 1123=1151, September, 1966. 11. D. Chan, " A n a l y t i c a l and S u b j e c t i v e E v a l u a t i o n of D i f f e r e n t i a l P u l s e Code Modulation Voice Communication Systems", M.A. Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, B r i t i s h Columbia, Canada, 1967. 12. R.W. Donaldson and D. Chan, " A n a l y s i s and S u b j e c t i v e E v a l u a t i o n of D i f f e r e n t i a l Pulse Code Modulation Voice Communication Systems", IEEE Trans.'on Communication Technology, v o l . COM-17, no. 1, February, 1969. 13. I.T. Young and J.C. Mott-Smith, "On Weighted PCM", IEEE Trans, on Information Theory, v o l . IT-11, pp. 596-597, October, 1965. 54 14. J'.M. Wozencraft and I.M. Jacobs, P r i n c i p l e s of Communication Engineering, Hew York: J . Wiley and Sons, 1965, pp. 508-511-•15. i b i d . pp.511-527. 16. i b i d . pp.527-533. 17. R.W. Donaldson, " O p t i m i z a t i o n of PCM Systems Which Use N a t u r a l Binary Codes", Proc. IEEE (Correspondence), v o l . 56, pp. 1252-1253, J u l y , 1968. 18. B. Smith, "Instantaneous Companding of Quantized S i g n a l s " , B e l l Sy.s. Tech. J . , v o l . 36, pp. 653-709, March, 1957. 19. N.R. French and J.C, Steinberg, "Factors Governing the I n t e l l i g i b i l i t y of Speech Sounds", J . Acoust. Soc. Am., v o l . 19, pp. 9°-119> January, 1947. . 20. J.M. Wozencraft and J.M. Jacobs, P r i n c i p l e s of Communication Engineering. New York: J . Wiley and Sons, 1965, pp. 177-179. 21. i b i d pp. I64-I68. 22. P.F. Panter and V, D i t e , "Quantization D i s t o r t i o n i n Pulse-Count Modulation w i t h Nonuniform Spacing of L e v e l s " , Proc. IRE, v o l . 39» pp. 44-48, January, 1951. 

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