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Digital simulation of delta modulation Matsushita, Jack Shigeo 1960

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D I G I T A L S I M U L A T I O N O F D E L T A M O D U L A T I O N J A C K S H I G E O M A T S U S H I T A B . A . Sc. , University of B r i t i s h Columbia, 1959 A THESIS S U B M I T T E D IN 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 T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in the Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia November I960 DIGITAL SIMULATION OF DELTA MODULATION by JACK SHIGEO MATSUSHITA B . A . S c . , 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 , 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE I n t h e Department of E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e s t a n d a r d s r e q u i r e d from c a n d i d a t e s f o r the degree of M a s t e r o f A p p l i e d S c i e n c e Members o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA November 1960 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of fc I t s i t H ( A c £ ^ ' ' ^ g ' The U n i v e r s i t y of B r i t i s h Columbia, Vancouver #, Canada. Date M cTQ - I S , ] f\UU i ABSTRACT T h i s t h e s i s d e s c r i b e s t h e s i m u l a t i o n of a d i g i t a l communication system on a d i g i t a l computer. D e l t a m o d u l a t i o n was chosen, as the s y s t e m , and i t s mode o f a c t i o n i s f i r s t d e s c r i b e d . S e v e r a l v a r i a t i o n s o f the b a s i c system a r e p o s s i b l e . I n o r d e r t o get t h e b e s t t r a n s m i s s i o n q u a l i t y , a c a r e f u l c h o i c e must be made o f t h e system and o f i t s d e s i g n p a r a m e t e r s . C o n v e n t i o n a l methods o f f i n d i n g t h e s e optimum p a r a m e t e r s have d i f f i c u l t i e s which d i g i t a l s i m u l a t i o n can c i r c u m v e n t . The programming of t h e ALWAC I I I - E computer f o r t h i s t a s k i s d e s c r i b e d . D i f f i c u l t i e s were e n c o u n t e r e d due t o t h e modest speed o f t h e computer. The s i m u l a t i o n experiments y i e l d e d many r e s u l t s o f i n t e r e s t c o n c e r n i n g the o p e r a t i o n o f b o t h s i m p l e and complex d e l t a m o d u l a t i o n systems w i t h d i f f e r e n t d e s i g n p a r a m e t e r s , and a l l o w e d a n optimum system to be d e s i g n e d . Where i t i s p o s s i b l e t o compare r e s u l t s w i t h p r e v i o u s e x p e r i m e n t a l work, the agreement i s good. i i TABLE OF CONTENTS Page j^l) S "trH/C o o o o o o o o o o o o o o o o o o o o o o o o o o i List of Illustrations . • • • » . « • « • « * > . . . . - . . . iv Acknowledg©n»ent . o o o , « o . « . . . . » « . . . o . « o v 1 o Int r Q&U^ C tXOn o o o o . o o o . o . o a o . o o a . . . . 1 2. The Delta Modulation System . . . . . . . . . . . . . . 1 2.1 Delta Modulation . . . . . . . . . . . . . . o a e 2 2.2 Variations -of Delta Modulation Systems . . . . . . 4 3. Signal to Noise Considerations . . . . . . . . . . . . 10 3.1 Overload and Quantizing Noise . . . . . . . . . . . 10 3.2 Standardizing Overload Characteristics . . . . . . 12 3.3 The Signal-to-Noise Ratio Determinations . . . . . 13 3.3.1 Calculation of Signal-to-Noise Ratios . . . 13 3.3.2 Experimental Measurements of Signal-to-Noise R a t i O S 0 . 0 0 0 0 0 0 0 0 0 O O O O O .16 • i 3.3.3 Listening Tests . . . . . . . 17 4. Digital Simulation of a Delta Modulator . 19 4.1 Program Requirements . . . . . . . . . . . . . . . 19 4.2 Choosing the Input Signal . . . . . . . . . . . . . 21 4.3 Basic Delta Modulati on Program . . . . . . . . . . . 22 4.4 Remarks on the Programming . . . . . . . . . . . . 24 4.4.1 The Kicksorter . . . . . . . . . . 24 4.4o2 Linear Filters . . . . . . . . . . . . . . . 26 4.4.3 Calculation of Noise Power . . 30 4.5 Actual Operation of Program . . . . . . . . . . . . 32 i i i 5. Variations of th.e Basic Program . . . . . 34 5.1 Adding D-C Component to the Input . . . . . . . . 34 5.2 Adding Higher Orders of Integration to the Feedback Loop . . . . . . . . . . . . . . . . . . 34 5.3 Varying the Integration Time Constant . . . . . . 35 6::© S U l t S e o « o o o « o o o o o o « « o o o o o o o o o 36 6.1 Basic Delta Modulator Program . . . . . . . . . . 36 6.2 Addition of D-C Component 39 6.3 Multiple Integration Systems . . . . . . . . . . . 41 6.3.1 Double Integration System . . . . . . . . . 41 6.3.2 Adding Third Order Integration . . . . . . 44 6.4 Changing .the F i l t e r Time Constant 44 T. Conclusions , O O O O O O O . . O O O O O O O O O O Q « . 48 7.1 Conclusions Regarding the Delta Modulatiom 7.2 Conclusions Concerning Digital Simulation Technic C£UeS . t t o . e . o o . o o e . o o o . o o o . . . 48 Appendix I . . . . . . . . . . . . . . . . . . . . . . . . 50 Appendix I I . . . . . . . . . . . . . . . . . o . . . . . . 53 Appendix III . . . . . . . . . . 0 . 0 . . . . o o o . . . 54 Reference • • . • • • • . * . . . « . • • . . « . • • • • .- 56 iv LIST OP ILLUSTRATIONS Figure Page 2 d Delta Modulation System . a o o . . . . . . . . . . . 3 2.2 Typical Waveforms of Simple Delta Modulation . . . . 5 2.3 Instability in Second Order Systems . . . . . . . . . 7 204 Double Integration With Prediction . . . 0 . . o . 0 7 205 Network for Double Integration with Prediction . . . 8 3ol Threshold Effect o . o . . . . . . . . . . . . . . . 11 3.2 Standardizing the Overload Characteristics of a Double Integration Delta Modulation System ...<.. 14 3o3 Slope Overload 0 0 . . 0 . . . . . . . . . . . . . a 15 4ol Digital Simulation Arrangement . . . . . . . . . . . 19 4»2 Sample Input Signal . o . . . 0 . . < , . . . . . . . 23 4.3 Basic Delta Modulation Program . . . . . . . . . . . 25 4o4 LOWpaSS Filter . 0 0 . . . . . . . o o o o . o o o o 27 405 ErrOr SignalS . o o o o o o o o o o o o o o o i o o o o 29 406 Kicksorter Contents . . o . . . . . . . . . . . . . 30 407 Sample Computer Output 0 . » . e . . . . . . o • « . 33 6ol Results of Basic Delta Modulation Program . . « . . . 37 6»2 Signal—to—Noise Ratios o o o o . o o o o o . a o o . 38 6.3 Addition of d-c Component to Input Signal . . . . . . 40 6.4 Opt 6.5 Signal-to-Noise Ratios for Optimum System . . . . . . 43 6.6 " Adding Third Order Integration . . . . . . . . . . . 45 6.7 Changing the F i l t e r Time Constant . . . . . . . . . . 46 V ACKNOWLEDGEMENT Acknowledgement i s g r a t e f u l l y g i v e n t o t h e Defence R e s e a r c h B o a r d , Department o f N a t i o n a l D e f e n c e , f o r a r e s e a r c h a s s i s t a n t s h i p and f o r t h e i r s p o n s o r s h i p o f t h i s p r o j e c t (Grant No. 2 8 0 3 - 1 9 ) , and a l s o t o t h e N a t i o n a l R e s e a r c h C o u n c i l f o r t h e i r a s s i s t a n c e under b l o c k t e r m g r a n t BT 6 8 . The a u t h o r a l s o wishes t o thank t h e B r i t i s h Columbia T e l e p h o n e Company f o r t h e s c h o l a r s h i p awarded him i n 1959. The a u t h o r i s i n d e b t e d t o P r o f e s s o r P . K. Bowers f o r h i s g u i d a n c e t h r o u g h p u t t h e p r o j e c t , and to D r . F . Noakes and o t h e r members of the E l e c t r i c a l E n g i n e e r i n g Department f o r t h e i r h e l p . S p e c i a l thanks go t o M r . H . Dempster pf t h e U . B . C . Computing C e n t r e f o r h i s a i d i n the programming. 1 DIGITAL SIMULATION OF DELTA MODULATION 1. INTRODUCTION The work r e p o r t e d i n t h i s t h e s i s s t a r t e d as a c o n t i n u a t i o n o f P o J o d e F a y e ' s project"*". He had c o n s t r u c t e d t r a n s i s t o r i z e d equipment f o r t r a n s m i t t i n g speech i n t h e form o f b i n a r y p u l s e s , u s i n g " D e l t a Modulation"« The o b j e c t i v e o f the p r e s e n t work was t o examine v a r i a t i o n s o f e n c o d i n g schemes t o f i n d one a b l e t o t r a n s m i t t h e s i g n a l w i t h the l e a s t p o s s i b l e amount o f d i s t o r t i o n . A f t e r c o n s i d e r i n g v a r i o u s methods of p e r f o r m i n g t h i s o p t i m i z a t i o n , i t was d e c i d e d t o t r y an i n d i r e c t a p p r o a c h . I n s t e a d o f b u i l d i n g a c t u a l equipment u s i n g v a r i o u s schemes, the ALWAC I I I - E computer was programmed t o compute what t h e equipment would do i f i t were b u i l t . Due t o the modest speed o f t h i s r computer, t h i s a p p r o a c h posed some c h a l l e n g i n g p r o b l e m s . However, the method was a f r u i t f u l one, and gave i n s i g h t b o t h i n t o t h e i c h a r a c t e r i s t i c s o f d e l t a m o d u l a t i o n systems and i n t o the c a p a b i l i t i e s o f t h i s type of computer f o r s i m u l a t i n g d i g i t a l communication systems g e n e r a l l y . The s i m u l a t i o n p r o c e s s was t h e r e f o r e p u r s u e d much f u r t h e r t h a n had been o r i g i n a l l y i n t e n d e d , and t h e c o m p i l a t i o n o f d i f f e r e n t computer programs forms a l a r g e p a r t of t h i s t h e s i s . No e f f o r t was made t o a p p l y the r e s u l t s o f t h e s e s i m u l a t i o n t r i a l s t o the a c t u a l equipment. 2 2. THE DELTA MODULATION SYSTEM 2.1 Delta Modulation 2 Delta Modulation is a method of communicating infor-mation, sudh as speech, by means of a series of binary pulses. It may be regarded as a simple form of pulse code modulation, requiring only a single digit code. In this system, the signal ' i . to be transmitted i s f i r s t sampled. The sampling rate used is much higher than the theoretically required minimum of 2f, where f is the upper cut-off frequency of the input signal. At each sampling instant, the amplitude of the input signal is compared with the amplitude of a waveform similar to the signal recon-structed at the receiver from previously transmitted information, and the polarity of the difference i s transmitted by a pulse. Thus a single digit binary code is sufficient. The simplicity of the required code suggests the possibility of inexpensive equipments simpler than that needed for other forms of pulse code modulation, but s t i l l retaining their desirable characteris-ticsa the pulse train to be transmitted i s a rugged signal and can, i n principle, be transmitted, switched, or stored with no deterioration whatever. 1 2 3 Various investigators ' * have reported on slightly differing delta modulation systems, but they a l l follow the basic circuit"illustrated in Figure 2.1. In the coder, the pulse generator i s triggered by the clock pulses, emitting either a positive or a negative pulse depending on the polarity at i t s input. These pulses form the output e Q ( t ) . The output is also introduced into a feedback loop containing a network Q equivalent 3 input e.(t) e(t) (t) CLOCK i P. Q output - ^ e -(t) CODER •0(t) Q e(t) Xi © 1? © p= f i l t e r e d output DECODER Figure 2»1 Delta Modulation System 4 t o t h e network i n the r e c e i v e r . The f e e d b a c k waveform e ( t ) , w h i c h w i l l be s i m i l a r t o the o u t p u t o f t h e r e c e i v e r j u s t b e f o r e t h e low pass f i l t e r , i s compared t o the i n p u t s i g n a l e ^ ( t ) , and the d i f f e r e n c e forms the e r r o r s i g n a l e ( t ) , w h i c h d e t e r m i n e s t h e p o l a r i t y a t the i n p u t to t h e p u l s e g e n e r a t o r . T h i s feedback a c t i o n t h u s t e n d s t o make the r e c e i v e r o u t p u t f o l l o w the coder i n p u t . 2.2 V a r i a t i o n s o f D e l t a M o d u l a t i o n Systems ' . • •' • • i The v a r i a t i o n s i n t h e s e v e r a l w o r k i n g systems l i e i n the network Q u s e d i n the feedback l o o p and i n the r e c e i v e r n e t w o r k . The most e l e m e n t a r y l o o p i s s i m p l y an i n t e g r a t o r , u s u a l l y a c a p a c i t a t i v e network. As i l l u s t r a t e d i n F i g u r e 2 . 2 , a t each s a m p l i n g i n s t a n t a p o s i t i v e or a n e g a t i v e p u l s e o f h e i g h t h w i l l i n c r e a s e o r d e c r e a s e the s t e p waveform e ( t ) p r o d u c e d a t the i n t e g r a t o r by one u n i t s t e p o f h e i g h t h . T h i s s t e p wave w i l l t e n d t o f o l l o w the i n p u t s i g n a l q u i t e c l o s e l y . The d i f f e r e n c e , or " q u a n t i z i n g " e r r o r w i l l be a f u n c t i o n o f the s a m p l i n g f r e q u e n c y and of the s t e p h e i g h t h . I n a p r a c t i c a l s y s t e m , a t r u e i n t e g r a t o r i s h a r d t o a c h i e v e because o f the f i n i t e t ime c o n s t a n t a s s o c i a t e d w i t h c a p a c i t o r s , b u t w i t h a s a m p l i n g f r e q u e n c y i n t h e o r d e r o f 100 k s / c , a good a p p r o x i m a t i o n t o a p e r f e c t i n t e g r a t i o n c o u l d be o b t a i n e d . However, r a f i n i t e t ime c o n s t a n t i s n o t n e c e s s a r i l y a h a n d i c a p , i t can a l s o be u s e d t o good a d v a n t a g e . The A m e r i c a n Army S i g n a l Corps has d e v e l o p e d a system w h i c h t h e y c a l l E x p o n e n t i a l D e l t a 3 M o d u l a t i o n . A f i n i t e t ime c o n s t a n t RC network i s u s e d i n s t e a d o f an i n t e g r a t o r , r e s u l t i n g i n a waveform w i t h an e f f e c t i v e l y 5 1 Figure 2<,2 Typical Waveforms of Simple Delta Modulation 6 n o n - u n i f o r m s t e p s i z e , as opposed t o t h e e q u a l i n c r e m e n t s due t o t r u e i n t e g r a t i o n . T h i s system has an advantage i n t h a t i t i s now p o s s i b l e t o t r a n s m i t d - c l e v e l s . They are t r a n s m i t t e d by p u l s e p a t t e r n s v a r y i n g from one o f a l l p u l s e s , t o one o f a l l i spaceso S e v e r a l more complex feedback l o o p s have been t r i e d i n the c o d e r , i n t h e hope t h a t t h i s w i l l produce a c l o s e r resemblance between t h e i n p u t s i g n a l and the r e p r o d u c e d s i g n a l . One of t h e most e f f e c t i v e ways t o improve on t h e s i m p l e c o d e r i s t o use m u l t i p l e i n t e g r a t i o n networks i n the f e e d b a c k l o o p . I n h i s 2 o r i g i n a l p a p e r on d e l t a m o d u l a t i o n , de J a g e r d e s c r i b e s a l s o a second o r d e r system u s i n g double i n t e g r a t i o n i n the feedback l o o p . A f t e r p a s s i n g t h r o u g h two s t a g e s o f i n t e g r a t i o n , a s i n g l e p u l s e r w i l l produce a change i n t h e s l o p e o f e ( t ) o f ±A. U n f o r t u n a t e l y , such a system has a t e n d e n c y t o o s c i l l a t e , as i s i l l u s t r a t e d i n F i g u r e 2 . 3 . T h i s shows t h e b e h a v i o u r of a second o r d e r system f o l l o w i n g a sudden change i n t h e i n p u t s i g n a l e ^ ( t ) . The o s c i l l a t i o n t h u s p r o d u c e d d i f f e r s from t h o s e n o r m a l l y found i n feedback; l o o p s i n t h a t i t s p e r i o d c a n have many d i f f e r e n t v a l u e s , depending on i n i t i a l c o n d i t i o n s . To overcome t h i s i n s t a b i l i t y , de J a g e r u s e d what he c a l l s " p r e d i c t i o n " . The a p p r o x i m a t i n g c u r v e e ( t ) , i s b u i l t up o f s t r a i g h t l i n e s so t h a t i f no change o f s l o p e o c c u r s , the v a l u e w h i c h w o u l d be r e a c h e d a f t e r a t i m e i n t e r v a l TJ c a n be p r e d i c t e d . T h i s e x t r a p o l a t e d v a l u e o f t h e a p p r o x i m a t i n g c u r v e i s compared t o t h e v a l u e o f the i n p u t s i g n a l a t t h a t t i m e , and the d e c i s i o n as t o t h e p o l a r i t y q f t h e n e x t output p u l s e i s made. ( F i g u r e 2 . 4 ) . 7 Figure 2»4 Double Integration with Prediction 8 The electrical system required for this prediction must make the output signal e 2 a ^ * n e second integrator at time t + X equal to the input signal e^ at time t, A network must be constructed which generates the extrapolated value of e 2 such that e(t,) = e~(t +T) This network, i f placed in the feedback loop, w i l l provide the i necessary predicting action. Figure 2.5 shows such a network. R, e Q(t) R2 - r e(t) • i ( t ) C _ X (t) Now, and Figure 2.5 {Network for Double Integration with Prediction. de-e,(t + X ) = e 0(t) + X 2 dt d t " c 2 X * The desired feedback voltage e(t) can be obtained by adding ^ ! 2 to the voltage on Examination of Figure 2.5 shows that this network w i l l provide the necessary voltage, with X « r C 2 , TJ being small compared to R ^ l and R 2C 2« The network of Figure 2.5 may be also be regarded in a different light. Since e^(t) and egCt) a r e respectively the result of one and of two stages of integration, the output e(t) can be written as a linear combination of these two signals, where the proportion of second-order integration may be judiciously adjusted to obtain the closest resemblance of signals e Q(t) and e(t) consistent with adequate stabi l i t y . It may then be possible to improve the performance of the system s t i l l further by a small admixture of third-order integrations. There i s hardly any limit to the possible types of feedback loop that might be tried, some perhaps designed to f i t a particular type of input signal. Very l i t t l e study has been made of such complex delta modulation systems, but i t is clear there is a wide scope for t r i a l and error, and a great need exists for methods of evaluating the performance of such systems quantitatively, and speedily. The development of one such method is the subject of this thesis. 10 3. SIGNAL TO NOISE CONSIDERATIONS 3«1 Overload and Quantizing Noise In transmission, pulses can in theory be regenerated so that no error is introduced, but even then the reproduced signal w i l l not be identical with the input signal. If the input signal is 1 • i • too large, very great discrepancies appear. The overload amplitude is an important characteristic b:f any communications system. However, even for smaller inputs there will be differences or distortions due to the f i n i t e size of the step height h. Such distortions are called "quantizing noise" and every effort i s made to keep this type of noise to a minimum. In particular, i t i s desired to have the largest possible "signal-to-noise ratio", that i s , the ratio of the overload signal power to the average quantizing noise power. A particularily noticeable form of quantizing noise occurs in some delta modulation systems when the input is very small, with amplitude less than h, as illustrated in Figure 3.1. The system then transmits the zero level signal and a "threshold effect" i s observed. Both ordinary quantizing noise and the threshold effect are reduced i f the step height is reduced. This, however, also decreases the overload point, and does not alter the overall signal-to-noise ratio or the dynamic range of the system. Considerable improvement results i f the sampling frequency f is increased: the overload point increases, while quantizing noise is decreased. However, the greater sampling frequency requires a wider trans-mission bandwidth, and the real test of the performance of a 11 Figure 3 01 Threshold Effect 12 particular pulse communication system should be the quantizing ' i noise i t produces with a given overload point and a given pulse frequency* 3o2 Standardizing Overload Characteristics When the signal-to-noise ratios of several variants of delta modulation systems are to be compared, i t i s generally necessary to find for each system both the overload characteristics and the quantizing noise. However, there i s a technique for ensuring that a l l these systems have the same overload characteristics, and this w i l l now be described. If a linear network N were added to- the receiver, this can have a drastic effect on the frequency response of the whole system, but at any frequency, i t w i l l have the same effect on the maximum amplitude that can be received before overload occurs as i t b.as on the quantizing noise at that frequency. Hence N does not alter the signal-to-noise ratio of tjtie received signal. To undo the damage done to the frequency response of the entire system by this network, an inverse network N1 can be placed in front of the trans-mitter, such that the combination of N and N1 has a f l a t frequency i response. By a suitable choice of N-N' networks, i t i s possible to manipulate the overload characteristics of any variant system until i t i s the same as that of the standard system, say simple delta modulation, without altering the signal-to-noise ratjj.o of the variant system. To optimize the sjystem, then, i t is necessary to examine only the quantizing noise, and to look for a system which wil l make this a minimum. 1 3 Figure 3 o 2 illustrates the procedure for a double integration delta modulation system. One of the integrators in the receiver must be omitted to give i t the same overload spectrum as single integration, so that in this case N is a differentiator network, and N1 is an integrator. After some reorganization of the block-diagram (Figure 3 o 2 d ) , i t jis seen that this variant can be regarded as having the same feedback loop as simple delta modulation, but differing from i t in the manner in which the decision between positive and negative pulses is made. In simple delta modulation, the decision function d(t) is e(t), the error signal, whereas in the double integration system, .d(t) is Jc(t)ib* A l l other variations of the feedback loop, using linear ele-ments, can be similarly re-drawn as a circuit with a loop containing one integrator, but with more complicated decision functions. This is the viewpoint adopted for the simulation experiments, and atten-tion need then be directed only at the quantizing noise accompanying assorted input signals. 3 . 3 The Signal-to-Noise Ratio Determinations Signal-to-noise ratios are normally found by three possible methods: by calculation, by measurement, and by listening tests. Each method w i l l be discussed in. turn, and each w i l l be found to have i t s particular d i f f i c u l t i e s . 3 . 3 . 1 Calculation of Signal-to-Noise Ratio The overload point for simple delta modulation is easily calculated. As can be seen from Figure 3 . 3 , the maximum slope the step wave e(t) can follow is given by 14 e ^ t ) -Transmitter e(t)=d(t) Receiver PG ( t l •H-t ) — [ 0 — «(t> a) The "standard" system: Simple Delta Modulation. (I = Integrator, PG = Pulse Generator, with pulse p o l a r i t y depending on the p o l a r i t y of the decision function d ( t ) . ) - PG J I 1 I - Q — —o-o— b) Double Integration Delta Modulation. , (Overloads when second derivative of input signal i s too large.) I PG N' N c) Networks N and N' added to b) to get the same overload spectrum as systenr (a);: overloads when f i r s t derivative i s excessive. (D =• D i f f e r e n t i a t o r . Addition of N and N1 does not change the signal-to-noise r a t i o at any frequency.) ( t ) ^ T } £ i ± i | 7 ] ^ PG e(t) - ^ e o ( t ) [TJ >• e(t) d) Re-organisation of (c) to show s i m i l a r i t y to (a). (Note that the difference between two integrals i s the same as the integral of the difference signal.) Figure 3.2 Standardizing the Overload Characteristics of a Double Integration Delta Modulation System. 15 max I 7 F Thus t h e maximum a m p l i t u d e a s i n e wave of f r e q u e n c y f can h a v e , i s g i v e n by h f . •il h f S\ 2 •max ~ 2nf o r S ~ *2itf) max s i g n a l * - t F i g u r e 3.3 S l o p e O v e r l o a d A t t h e r e c e i v e r t h e decoded s i g n a l i s u s u a l l y p a s s e d t h r o u g h a l o w - p a s s f i l t e r which has an upper c u t - o f f f r e q u e n c y £ the « u t - o f f f r e q u e n c y o f t h e i n p u t s i g n a l . Hence o n l y the low f r e -quency components, t h o s e l e s s t h a n f Q , a r e o f i n t e r e s t i n s t u d y i n g the n o i s e . The q u a n t i z i n g n o i s e a r i s e s from the low f r e q u e n c y components i n t h e d i f f e r e n c e s i g n a l e ( t ) = e^(t) " e Q ( t ) . F o r most s i g n a l s which a r e n o t t o o l a r g e or t o o s m a l l , the a m p l i t u d e s o f s u c c e s s i v e segments of e are randomly d i s t r i b u t e d over the range i h . ( c f . F i g . 2 . 2 ) . On t h i s a s s u m p t i o n , de J a g e r has o b t a i n e d an e x p r e s s i o n f o r the s i g n a l - t o - n o i s e r a t i o . F o r a sinewave of f r e q u e n c y f , w i t h s t e p h e i g h t h , and s a m p l i n g f r e q u e n c y f , t h e ratio is given by 3 f 2 S = c s » • 1 f o The constant c^ was found to be approximately 0.20. For a system with double integration, the corresponding expression turns out to be 5 f s o Here the constant was found by numerical methods to be 0.026. For a mixture of higher orders of integration, i t is not possible^to calculate their effect on the noise. Similarly, computation of noise for signals near the threshold is not possible. 3.3.2 Experimental Measurements of Signal-to-Noise Ratios There is in principle no d i f f i c u l t y in such measurements. A reliable delta coder must f i r s t be constructed, and i t must then be supplied with a variety of input signals. The quantizing noise i s most conveniently measured by examining the error wave-form e(t) present in the coder, and f i l t e r i n g out a l l the irrelevant high frequency components. Two d i f f i c u l t i e s arises 1) It is very hard to be sure that the noise is not influenced by small imperfections in the apparatus, and 2) The noise w i l l not be completely gaussian and uncorrelated with the signal, and i t s subjective effect on transmission quality may be very different from that of the same amount of gaussian noise. 17 3 o 3 o 3 L i s t e n i n g T e s t s The o n l y r e a l c r i t e r i o n o f t h e q u a l i t y of a t r a n s m i s s i o n system i s t o have s e v e r a l o b s e r v e r s , l i s t e n t o the t r a n s m i t t e d s i g n a l s , and compare them w i t h the sound of the o r i g i n a l s i g n a l p l u s v a r i o u s amounts o f added random n o i s e . A n o i s e f i g u r e can be a r r i v e d a t when a n amount o f added n o i s e i s f o u n d w h i c h i s j u s t as o b j e c t i o n a b l e as the q u a n t i z i n g n o i s e accompanying the t r a n s m i t t e d s i g n a l . Such t e s t s r e q u i r e a l o n g t i m e , many p a t i e n t o b s e r v e r s , and a l a r g e v a r i e t y o f i n p u t s i g n a l s . They a r e s u b j e c t t o human e r r o r s - o b s e r v e r s become accustomed to c e r t a i n t y p e s of d i s t o r t i o n - and a l s o t o e r r o r s due to i m p e r f e c t i o n i n t h e equipment. A l t h o u g h s u c h t e s t s a r e the o n l y f i n a l c r i t e r i o n by which a system i s to be j u d g e d , t h e y a r e not p r a c t i c a l f o r t h e d e t e r m i n a t i o n of the optimum v a l u e o f a l a r g e number o f parameters;. F o r t h i s , a f a s t e r method o f e v a l u a t i o n i s needed, p r e f e r a b l y one t h a t does n o t depend on p e r f e c t i o n o f the a p p a r a t u s . B o t h s i g n a l - t o - n o i s e r a t i o measurements and l i s t e n i n g t e s t s a r e slow and u n r e l i a b l e methods o f d e t e r m i n i n g the v a l u e of a s y s t e m . I n b o t h c a s e s , a w o r k i n g model must f i r s t be c o n s t r u c t e d , o f t e n a slow and c o s t l y p r o c e s s . E x p e r i m e n t a l i m p e r f e c t i o n s of t h e model can i n v a l i d a t e r e s u l t s u s i n g e i t h e r method. A s p e e d i e r and more r e l i a b l e method of e v a l u a t i o n i s r e q u i r e d . One such method, u s e d i n t h i s t h e s i s , i s s i m u l a t i o n o f t h e system on a d i g i t a l 4 computer . V i t h the i n c r e a s i n g a v a i l a b i l i t y o f g e n e r a l - p u r p o s e c o m p u t e r s , d i g i t a l s i m u l a t i o n becomes more and more p r a c t i c a l . D i g i t a l s i m u l a t i o n has an advantage i n i t s h i g h speed and low 1 8 costs, and more important, i n the case of sampled data systems, i n i t s r e l i a b i l i t y . For i n t h i s case, the simulation can be made exact. 19 4 . DIGITAL SIMULATION OF A DELTA MODULATOR 4»1 Program Requirements The arrangement r e q u i r e d f o r d i g i t a l s i m u l a t i o n i s i l l u s t r a t e d i n F i g u r e 4 . 1 . A n a l o g Input A n a l o g - D i g i t a l D i g i t a l I n p u t C o n v e r t e r D a t a Program o f O p e r a t i o n s P u l s e T r a i n S i g n a l Tape Recorder N o i s e Mean Power e t c . OUTPUT F i g u r e 4.1 D i g i t a l S i m u l a t i o n Arrangement The a n a l o g s i g n a l i s f i r s t p r o c e s s e d and c o n v e r t e d to d i g i t a l form,. The d i g i t a l d a t a a r e t h e n s t o r e d i n some c o n v e n i e n t f o r m , such as on t a p e , and f e d i n t o the computer, where t h e a c t u a l o p e r a t i o n s a r e c a r r i e d o u t . The r e s u l t s o f t h e s e o p e r a t i o n s form the o u t p u t , w h i c h w i l l be i n d i g i t a l f o r m . T h i s i s t h e n c o n v e r t e d back t o an a n a l o g s i g n a l . V a r i o u s o t h e r forms o f output may a l s o be d e s i r e d . I n the case of the d e l t a m o d u l a t o r , the d e s i r e d o u t -Are ; p u t s A t h e p u l s e t r a i n , t h e s i g n a l a t the r e c e i v e r , the a m p l i t u d e s o f t h e q u a n t i z i n g n o i s e , t h e mean power and o t h e r c h a r a c t e r i s t i c s o f t h e q u a n t i z i n g n o i s e , such as i t s low f r e q u e n c y components. The i d e a l , c o m p u t e r f o r t h i s purpose would h a v e : a) Input and output elements c a p a b l e of h a n d l i n g whole segments o f a c t u a l s p e e c h . b) L a r g e d a t a s t o r a g e f a c i l i t i e s t o a l l o w a s i g n i f i c a n t t e s t . 20 c) A program compiler to allow easy changes in the system. A computer to f i t these specifications was not available. However, this thesis w i l l show that even on a modest sized computer useful results may be obtained. The ALWAC III-E, a medium sized, medium A „ speed general purpose computer was programmed to optimize design parameters in a delta modulation system. In the AL¥AC III-E, inputs and outputs are handled either through a flexowriter with punched tape control, or through a high speed paper tape reader and punch. The speeds at which they operate are, for inputs, 10 and 150 characters per second respectively, and for outputs, 10 and 50 characters per second. These speeds severely limit the data handling capacity, especially at the input. Taking into account these limitations, a model delta modulator must be programmed which wi l l allow studies of the pertinent characteris-tics of the system. The main points of interest w i l l be the changes in the noise power at the output due to various changes in the feedback loop parameters. Also of interest w i l l be a closer look at the nature of the signal-to-noise ratio, a study of i t s dependence on such parameters as input level. The model must be flexible enough to allow changes in the feedback loop without extensive changes in the main programj i t must yield i t s output, especially noise power and filt e r e d noise power, in a form that allows rapid comparison and evaluation; and above a l l , i t must give significant, results in a short time - say ten minutes - of machine time. This i s most essential* since, once the basic system is established, various changes wi l l be made in the feedback loop, and the resulting changes in the output w i l l be evaluated, with the aim of optimizing the whole system. As 21 there are many parameters to he adjusted, the time for a single run through the program must be kept to a minimum. This last requirement rules out the use of ALCOM, the automatic program compiler available in the ALWAC computer, and instead, an "optimum" program must be written. 4.2 Choosing the Input Signal Ideally, the input should be a "typical" sample pf speech or of a random signal with the same spectral characteristics as speech. Since in the computer t r i a l s i t would be possible to process only about 50 milliseconds worth of input signal, i t would be d i f f i c u l t to make sure that representative samples of such input signals are selected. Furthermore, even with so short a sample, the process of supplying tQ the computer values of the input at 5000 instants of time i s quite tedious. It was therefore decided to use an a r t i f i c i a l input signal which can be generated by the computer i t s e l f , given some i n i t i a l conditions. In speech, several frequency components are usually present simultaneously; the average power spectrum has a maximum below 1000 c/s, with power decreasing rapidly at higher frequen-cies. The final choice for an a r t i f i c i a l input signal, determined partly by convenience in programming, was a sine wave of 500 c/s, accompanied by a fixed proportion of fourth harmonic, 20$ in amplitude. In later t r i a l s , an adjustable amount of d-c was also r added. The amplitude of the composite signal is specified at the beginning of each t r i a l , and many different amplitudes were used. Due to the regularities of this a r t i f i c i a l signal, i t was 22 only necessary to process about 4 milliseconds worth of input. This gave computer times of about 2 minutes per t r i a l . Some checks were performed to show that the performance of the system tested does not depend c r i t i c a l l y on the exact ratio of the components of the input signal, and there is good reason to hope that the performance would also be very similar for samples of actual speech. The subroutines for sines and cosines available in the computer library were too slow, and the values of the sine waves were therefore generated as part of the program. The method used is the simultaneous solution of two differential equations, as —3 -2 shown in Appendix I. The values chosen for k were. 2 and 2 These values yielded sine waves with frequencies of 500 c/s and 2000 c/s respectively, with the values being generated at 10 ^ s e c . intervals, corresponding to a~ sampling frequency of 100 kc/s. ¥hen the constants can be expressed as powers of 2 the computation can be carried out by shifting, rather than by the lengthy multiply or divide operation. The amplitudes of the two waveforms are set by printing in the i n i t i a l values for the cosine waves at the start of the program. The i n i t i a l values of the sine waves are zero. A sample of the input signal is shown in Figure 4.2 4.3 Basic Delta Modulator Program The basic delta modulator program starts by generating the f i r s t value of the input signal as described in the previous section. From the input is subtracted the i n i t i a l value of the integrated waveform e, which is usually zero. The error signal e thus computed is used to make the decision whether to add or Figure 4.2 Sample Input Signal 24 subtract a step height h from e: i f the particular value of e is positive or zero, h i s added; i f negative, h i s subtracted. This forms the new value of e which i s ready to be subtracted from the next input. The program processes the input in exactly the same manner as an ideal physical delta modulator. The program, along with the output routine, i s reproduced in Appendix II. It is written in machine language, and the instruc-tions are arranged in a rather complicated order. A rearranged set of commands, together with explanatory remarks, i s provided in Figure 4.3 to help cl a r i f y the programming. 4.4 Remarks on the Programming In the rearranged set of commands shown in^Figure 4.3, the instructions are grouped in blocks; each of these blocks will be dealt with in turn. The f i r s t block of instructions generates'the two signals, and adds them. The next block performs the comparison, and forms i e and e. e i s stored in i t s assigned space,1 but e cannot be handled as easily. Obviously i t would be impractical to store a l l the individual errors, since the number of storage spaces required would be equal to the number of inputs used, which in this case was usually four hundred. Instead, a "kicksorter" was programmed to perform this counting task. 4.4.1 The Kicksorter The "kicksorter" receives1 the errors and sorts them according to magnitude. It records the number of errors less than one unit in magnitude, between one and two units, and so on. The units Enter -Clear Ch. II Print i n z z_l 1 h ' h Store z,z* Carriage Return i Pick up y Start loop Generate 7 Generate y 8 ' Store e Pick up h Add d-c Add y&y8 Form e Store c in kick-sorter Form new e Fir s t F i l t e r 00 04 08 Oc 10 14 18 lc 01 05 09 Od Oa 74 7d 62 4b 4f 53 lb If 6b 6f 73 77 7b 41 6a 6e 72 54 64 61 65 69 71 5511 2800 4840 1704 4 I 8544 551d 1160^  e740 c460 c54e f782 794c 571 e f — 795c f — a505 r ~ 615e a505 f 655c 3000 a 503 c55c 3132 c747 4140 3a00 6178 615c f c742 871e 5b0f< 3000 allO 178oJ 7915 8745 3a00 110a 121 117d 3132 114b — i c55e 7942 111b 6l4e Ile7i a503* c54e 6542 3800 0000 1141 t 510b 6934 6934 683f 683f 3600 3000 3000 a502 lle6 6747 rlfdO 2600 l l e l i 3600 rlleO W2600 1161 J OcOO* 6147 674c llc 5 80 84 88 8c 90 94 98 9c 81 85 89 8d 8a f4 fd e2 cb cf d3 9b 9f e7 eb ef f3 f7 fb cl e6 ea ee f2 dO d4 eO e4 el e5 e9 f l Second Filt e r Store e" in kick-sorter Pick up y Copy C h e 46&43 Print out e Print out e" Form E e 2 Print out E e 2 Form Print out 2 Stop 49 4d 51 55 5b 63 68 6c 70 56 02 06 Oa Oe 12 00 04 08 Oc 10 14 18 leH 01 05 09 Od 11 15 Oa Oe 12 Ob Of 13 17 lb If 02 OOOO 3aOO OOOO a502 — * c547 614c 3800 6744 115b c74c 1163 6144 r~ 5159 6926 6926 6831 r~ 2600 3600 795c 1168 i •lfd2 2600 11861 0600 1102 6831 3a0O*| 1774 1 c?44 8143 ,871 f 7840 1160 5bld 790c 1704 OeOO 1901 3000 c516 643f eF54 1798 5b06 5503 7832 111b 3000 c519 6431 e654 170e 5bla 1102 8546 1100 5503 5b07 7832 1160 f701 5503 7830 1111 e260 7841 3000 bdl6 3000-1 1160 OeOO 190b e260 7833 3000 bdl9 3000 1160 c5 c9 cd dl d5 db e3 e8 ec fO d2 d6 82 86 8a 8e 92 80 84 88 8c 90 94 98 9c 81 85 89 8d 91 95 8a 8e 8b 8f 93 97 9b 9f Figure 4„3 Basic Delta Modulation Program 26 u s e d h e r e are d e c i m a l numbers* Twelve s t o r a g e spaces were a v a i l a b l e f o r t h i s p u r p o s e , so the f i r s t e l e v e n " b i n s " were u s e d t o s t o r e the number of e r r o r s l e s s t h a n e l e v e n u n i t s , w h i l e t h e l a s t b i n was r e s e r v e d f o r the " o v e r f l o w " , that i s , f o r the number o f e r r o r s over e l e v e n u n i t s i n m a g n i t u d e . S i n c e t h e e r r o r s seldom exceeded one s t e p h e i g h t h i n magnitude, s e t t i n g h a t e i g h t u n i t s took c a r e o f most of the e r r o r s , n e i t h e r o v e r l o a d i n g the o v e r f l o w b i n , nor l e a v i n g too many of the h i g h e r v a l u e d b i n s empty. I f e i t h e r of t h e s e c o n d i t i o n s o c c u r s , the s e n s i t i v i t y of the k i c k s o r t e r can be changed s i m p l y by c h a n g i n g the v a l u e of h , i n c r e a s i n g i t to i n c r e a s e s e n s i t i v i t y ( too many empty b i n s ) , or d e c r e a s i n g i t to d e c r e a s e s e n s i t i v i t y ( o v e r l o a d i n g ) . The k i c k s o r t e r makes up the n e x t b l o c k of i n s t r u c t i o n s . A l o n g w i t h t h e e r r o r s i g n a l , i t i s d e s i r a b l e t o have a f i l t e r e d e r r o r s i g n a l , s i n c e a t the r e c e i v e r , the o u t p u t s i g n a l i s u s u a l l y p a s s e d . t h r o u g h low pass f i l t e r , and o n l y the low f r e q u e n c y components o f the e r r o r s i g n a l r e m a i n as the n o i s e . To* a c c o m p l i s h t h i s f i l t e r i n g a d i g i t a l e q u i v a l e n t o f a low-pa&s. f i l t e r was programmed, as w i l l now be d e s c r i b e d . 4 . 4 . 2 L i n e a r F i l t e r s To s t u d y a l i n e a r f i l t e r f o r sampled d a t a systems, a n a l y s i s must be c a r r i e d out i n the t i m e , r a t h e r t h a n f r e q u e n c y d o m a i n . The b e h a v i o u r of a l i n e a r f i l t e r can be e x p r e s s e d i n terms o f 5 i t s o u t p u t c o r r e s p o n d i n g to a u n i t - s t e p f u n c t i o n i n p u t . I f , when the i n p u t i s ^ Q * ' * n e 0 U "kP u '* ' * s A ( t ) , t h e n c o r r e s p o n d i n g to any i n p u t f ( t ) the output i s g i v e n by 27 P(t) oo A' ("C)f ( t-X ) d r + A(0)f(t) where A(0) = lim A(t). t*0+ In the discrete system this output is given by a sum F(ks) = s f l A'(ns)f(ks-ns) + A(0)f(ks) n=l where s is the sampling interval, and P(t) and f(t) are adequately determined by their values at t = ks. In the case of a simple RC lowpass f i l t e r (Figure 4.4), R AAAAAAA f(t) F(t) Figure 4.4 Lowpass F i l t e r A(t) i s given by A(t) = 1 - e _ t / T where Also, and T = RC A(0) = 0 s o t ha t eo F(ks)= s f ^ | e " n s / T f [(k-n)s] n=l The output can be regarded as being made up of a weighted sum of the present input and past values of input. For programming 28 purposes, the sum can be expressed as a recurrence relationship. In the case of the low pass f i l t e r , the past values are weighted exponentially. If: the error signal e i s f i l t e r e d to y i e l d the f i l -tered error signal e', the recurrence relationship for e' can be written down as e^' = ae^ + (1-a) , a £ l where i s the l a t e s t value of the error signal, e^' the l a t e s t value of the f i l t e r e d error signal, and the l a t e s t previous value of the f i l t e r e d error s i g n a l . The constant f r a c t i o n a i s determined by the required cut-off frequency of the f i l t e r . For th i s work, the cut-off frequency was set at about 4QGj0 c/s. This gives a value for a of 2.n (4000) TC  = 100,000 = 1275 for a sampling frequency of 100 kc/s. For convenience i n pro-gramming, a » ^ was used, giving a cut-off frequency of 3980 c/s. A single.-stag.e,v.low-pass f i l t e r attenuated the high frequencies sta r t i n g at the cut-off frequency, and1 t h i s attenuation increases at a rate of 6 ,db/octave. This i s not a very sharp cut-off, and i t was f e l t that a steeper " r o l l o f f " was required. Hence the error signal.was passed through two stages of f i l t e r i n g , providing a 12db/octave r a l l a f f . The second stage of f i l t e r i n g i s similar to the f i r s t , the signal e' being used to produce the f i n a l f i l t e r e d error signal e". Figure 4.5 shows a t y p i c a l error signal, and the effects of the f i l t e r i n g . The values of e" were stored i n another kicksorter. The f i l t e r s and the second kicksorter constitute the next two blocks i of i n structions. The output routine completes the program. As a measure of h 4 e(t) - h + 1.4 0 . 5 h + - 0 . 5 h f Figure 4.5 E r r o r Signals to MO 30 the average quantizing noise power, i t is desirable to compute and print out the average of the squares of the errors, both for the original error signal e and for the filtered error signal e" < Along with noise power, i t would be convenient to have some sort of a count or measure of the individual errors themselves0 Hence the contents of the bins in the kicksorter were printed out. The noise power is calculated from the number of errors in each bin. 4o4«»3 Calculation of Noise Power There is no exact method of calculating the noise power of the contents of the kicksorter bins. There are several ways in which this noise power can be approximated, and one of these is described below. The contents of the bins can be plotted as in Figure 4.6. Number of Errors in bins n n Bin Values Figure 4.6 Kicksorter Contents 31 The simplest method would be to form the sum 1*1 = E T n X n n that i s , the sum of the products of the number in each bin and the average value of the bin squared. This sum, divided by the total number of errors w i l l yield a value of the average noise power. Another way of obtaining noise power would be to integrate the product _n x dx over the interval (X - -x , X + -z) , and sum. These methods, however, do not take into account any rapid changes in the number of errors in adjacent bins. The method fi n a l l y adopted does take into account this change. 2 As before, gives the average noise power. Let where Then (x - Xn)J x 2dx 32 The —g values can be calculated beforehand and stored, as can 2 <T 2 the X Q + Y2 values. The program now has to take the number xn each bin and multiply i t by the appropriate constant, add another constant, sum, and print out this sum. This output routine constitutes the final block of instructions. 4.5 Actual Operation of Program Several constants must be printed in before the actual program is started. The step height h must be set. In this work, h = 8 was found to be satisfactory for a l l the t r i a l s . The starting values of the two sine waves are printed in as part of the program, e is set at zero. After the program is placed in the computer, and is started, i t w i l l c a l l for two inputs. These will be the starting values of the two cosine waves, which wi l l determine the maximum, amplitude of input into the coder. The amplitudes must be printed in through the flexowriter at the start of each run. The most useful amplitude range was found to be between O.lh and 20h. A few runs were tried beyond this range, but did not result in much additional information. The amplitudes are printed in as multiples of h. Thus a starting input (10, 2) w i l l produce an input to the coder with maximum amplitude about twelve.^ step heights, and with the two frequencies mixed in an amplitude ratio of 5si. As i t was programmed, the coder can handle four hundred input values in just over a minute. This includes the time required to process the error signal and calculate noise power, but not the printout time. At the end of each run, the computer prints out the number of errors in each bin, for both the fi l t e r e d and u n f i l -33 tered error, in two vertical columns, with the sum of the squares of the errors at the foot of each column. A sample is shown in Figure 4.7. 10 2 19.00 . .00 15.00 .00 23.00 .00 30.00 .00 36.00 .00 28.00 .00 24.00 .00 37.00 .00 35.00 4.00 40.00 17.00 44.00 73.00 69.00 306.00 13876.92 376.56 Figure 4.7 Sample Computer Output In addition to the regular program, another program was written to print out individual errors. Since this program was to be used for just one run, the time requirements were not so stringent, and the program was written using ALCOM, the automatic compiler program. This program takes a l i s t of algebraic commands and writes a program, allotting a l l the necessary storage spaces automatically. It is quite inefficient as far as speed is concerned, but for this purpose, and also for printing out values of the input in one case, the compiled program worked very well. The data comes out on punched tape, and this tape was fed into a Mosely X-Y plotter. The results are seen in Figures 4.2 and 4.5. 34 5. VARIATIONS OF THE BASIC PROGRAM Once the basic program was tried successfully with various input amplitudes, changes were introduced with several objectives in mind. 5.1 Adding D-C Components to the Input The purpose of thes6 tests was to demonstrate that the quantizing noise did not depend c r i t i c a l l y on the exact value of the input signal, as long as i t s amplitude was well above the threshold. The modification was easy to carry out. After generating in the computer the values of the sine waves representing the fundamental and fourth harmonic, a constant was added to their sum. This constant was readily accessible and could be easily changed. Several values were tried. 5.2 Adding Higher Orders of Integration to the Feedback Loop The object of these t r i a l s was to produce a system with less quantizing noise than simple delta modulation. These program changes were more d i f f i c u l t to carry out. As the original program had been "optimized" to save computer time, a change in the program involved re-writing a whole section of i t i f optimi-zation was to be maintained. As explained in section 3.2, systems involving higher order integration can be created by basing the decision of whether to add or subtract h from e(t), not on the polarity of the error e, but on the polarity of a decision function d(t), which can be a combination of e and integrals of e. Now for the purpose of com-35 p u t i n g q u a n t i z i n g n o i s e , the computer i s a l r e a d y s e t up to f i l t e r e and produce e ' and e". (See s e c t i o n 4 . 4 . 2 ) S i n c e t h i s f i l t e r i n g p r o c e s s i s an i n t e g r a t i o n w i t h a f i n i t e t i m e - c o n s t a n t , i t was d e c i d e d t o use e , e ' , and e" t o form the d e c i s i o n f u n c t i o n d ( t ) . The amended program, w i t h p r o v i s i o n f o r a d d i n g d - c t o t h e i n p u t , and w i t h a d e c i s i o n f u n c t i o n d = e + me' + ne" i s shown i n Appendix I I I . Many v a l u e s of m and n were t r i e d . 5.3 V a r y i n g the I n t e g r a t i o n Time C o n s t a n t I t has b e e n assumed, b o t h i n t h i s work and i n o t h e r e x p e r i m e n t s , t h a t the time c o n s t a n t of t h e i n t e g r a t o r or i n t e g r a t o r s does n o t a f f e c t t h e amount of q u a n t i z i n g n o i s e , as l o n g as i t i s l o n g e r t h a n l/t t , where o> = 2rcf , and f i s the c u t - o f f f r e q u e n c y of o o o o the i n p u t s i g n a l . A c h e c k on t h i s a s s u m p t i o n was d e s i r a b l e . T h i s change r e q u i r e d t h a t a s e p a r a t e f i l t e r be programmed to operate on e , s i n c e the output n o i s e s i g n a l had t o be p r o c e s s e d as b e f o r e , b u t t h e d e c i s i o n f u n c t i o n d ( t ) had t o be formed from a c o m b i n a t i o n o f e and the s i g n a l s , r e s u l t i n g from p a s s i n g c t h r o u g h f i l t e r s w i t h t i m e c o n s t a n t s o t h e r t h a n the ones a l r e a d y i n u s e . F o r t u n a t e l y , t h e r e was s u f f i c i e n t room i n the program to add two more s t a g e s of f i l t e r i n g , and t h i s was done, l e a v i n g t h e c o n s t a n t a (See s e c t i o n 4 . 4 . 2 ) as a v a r i a b l e p a r a m e t e r . S e v e r a l v a l u e s o f t h i s parameter were t r i e d . J ) 36 6. RESULTS 6.1 Basic Delta Modulation Program For the basic program, t r i a l s vere run using maximum amplitudes ranging from O.lh to 25h for the fundamental frequency. The results are shown i n Figure 6.1. The noise is expressed as a root-mean-square amplitude and plotted versus the input signal amplitude. Both input and noise amplitudes are expressed in i decibels with respect to the step-height h. Some characteristics of simple delta modulation can be seen from the plot. The threshold effect can be seen, starting just below -5 db (maximum amplitude of 0.56h). The overload begins at about 22 db. In between these limits, the noise i s irregular, but reasonably f l a t . In this "usable" band, the noise amplitude is on the average just below -14 db. There are several common ways of specifying the signal-to-noise ratio of a communication system. One way i s to use the maximum signal-to-noise ratio in the system; another is to use the ratio of the signal that w i l l just overload the system to the minimum noise in the usable band; and the third method i s to i quote the "dynamic range" ofi the system, which for our purposes is taken to mean the ratio of the largest signal to the smallest signal which can be transmitted with a signal-to-noise exceeding some specified value, say 20 db. Figure 4.2 shows the signal-to-noise ratio of the received signal for various values of signal amplitude. From this, the three signal-to-noise ratios for the system can be determined. These values were obtained% 0 Figure 6.1 Results of Basic Delta Modulation Figure 6„2 Signal-to-Noise Eatios 39 a) For the maximum signal-to-noise ratio in the system, (l>max = 3 3 ' 6 d b ± 2 d b b) For the ratio of the overload signal to the minimum noise in the usable band, S o v = 39.8 db ±2 db N . mxn c) For the ratio of the largest signal to the smallest with signal-to-noise ratio greater than 20, ( j ^ S / N ^ O = 2 8 ' 8 d b i 2 d b The value obtained in (b) can be compared with a theoretical 2 value as calculated by de Jager . He assumes a signal ,Asin2itft, f g h and for this signal, the overload amplitude i f • (See Section 3.3.1). For the composite signal used in the present work, which can be written as A (sin 2nft + i-sin 4(2nftl, the overload occurs V 5 f h(1+1/5) 2 f h when the amplitude i s approximately = ?(> T p ) ° Using 2it (144/5 )f this expression, the signal-to-noise ratio as derived by de Jager becomes ^\ | = (|) 2 x 0.20 x - 8 — f . f ^ 2 O Substituting f = 100 kc/s, f = 3980 c/s, f = 500 c/s s o I = 39.5 db This i s in excellent agreement with the value found in (b) above, 6.2 Addition of D-C Component In general, adding a d-c component to the input signal affects only the low level signal. In the usable band, there is negligible effect. The results of adding O.lh, 0.3h, and 0.5h d-c to the input signal are shown in Figure 6.3. As might be expected, the threshold effect noise is altered appreciably, but 41 there is very l i t t l e effect i n the usable band, or on the overload. 6.3 Multiple Integration Systems 6.3.1 Double Integration System At several fixed input signal amplitudes, the constant n was set at zero, and m was varied until the noise power was at a minimum. (See Section 5.2). There was a slight variation in the optimum value of the constant m at different input amplitudes, but a l l the optimum values of m obtained were in the range 6 to 8, and within this range, the system was quite insensitive to changes in the value of m. A value m = 7.5 was f i n a l l y decided upon as being most suitable for a l l input signal amplitudes which would be used. Results of t r i a l s using m = 7.5 are plotted in Figure 6.4. Here, as before, r.m.s. noise amplitude i s plotted versus input signal amplitude, both in db with respect to h. Figure 6.1 is superimposed on this figure for comparison. Figure 6.5 shows signal-to-noise ratio plotted versus signal amplitude. As can be seen from Figures 6.4 and 6.5, this optimized double integration system has an improvement of noise level in the usable band of about 7 db. The threshold effect is virtually eliminated, extending the dynamic range of the system by about 7 db. This result can be compared to the optimum double integration 2 system cited by de Jager . The f i r s t point of comparison is between the optimum m and his optimum TT . (See Section 12.2). Now T can be expressed in terms of m as T m ' where T 2 i s the time constant of the f i l t e r which integrates C. Figure 6.4 Optimum System Signal-to-Noise ratio r of received signal i n db 44 F o r t h i s f i l t e r , m 1 2 ~ 3980(211) so the optimum T i s g i v e n by T ^ o p t . - 5 ^ " " 5 ' 3 5 I* s e c - » u s i n g n»0p .^ = 7 . 5 . Now a c c o r d i n g t o de J a g e r , t h e o p t i m u m ^ i s about o n e - h a l f the s a m p l i n g t i m e , o r i n t h i s c a s e , 5 \x s e c . The v a l u e found i n - these t r i a l s i s i n v e r y good agreement w i t h de J a g e r ' s v a l u e . The second p o i n t i s the improvement i n t h e system due t o i n t r o d u c i n g . s e c o n d o r d e r i n t e g r a t i o n . De J a g e r quotes a f i g u r e o f 10 db as h i s improvement, and t h i s f i g u r e i s s l i g h t l y h i g h e r t h a n the improvement shown i n F i g u r e 6 . 5 . 6 . 3 . 2 Adding T h i r d Order I n t e g r a t i o n The next s t e p was t o form the d e c i s i o n f u n c t i o n d ( t ) u s i n g p o r t i o n s o f f i r s t and second i n t e g r a t i o n s o f e. (See s e c t i o n 5 . 2 ) . A c c o r d i n g l y , f o r f i x e d v a l u e s of i n p u t s i g n a l a m p l i t u d e s , t h e c o n s t a n t s m and n were v a r i e d . At most lower v a l u e s o f i n p u t s i g n a l , t h e r e was l i t t l e o r no improvement due t o a d d i n g the second i n t e g r a t i o n of c . S l i g h t improvement a t h i g h e r i n p u t a m p l i t u d e s were o b t a i n e d , b u t a t the expense o f h i g h e r n o i s e a m p l i t u d e s a t the m i d d l e v a l u e d i n p u t s . The b e s t r e s u l t s were o b t a i n e d w i t h m = 5 and n = 1. These r e s u l t s a r e shown i n F i g u r e 6 . 6 , t o g e t h e r w i t h the optimum second o r d e r system f o r c o m p a r i s o n . I t was f e l t t h a t t h e s e r e s u l t s d i d n o t m e r i t f u r t h e r i n v e s t i g a t i o n o f t h i s l i n e o f a t t a c k . 6.4 Changing the F i l t e r Time C o n s t a n t I n t h e o r i g i n a l f i l t e r s u s e d t o f i l t e r e, t h e c o n s t a n t a Figure 6„6 Adding Third Order Integration -4--6 • -8" 10-12--14 16-18-20--Optimum Second Order System Different F i l t e r Time Constant -18.4 -10 0 10 Signal Ampl. ^ h * —6-20 Figure 6.7 Changing the Fi l t e r Time Constant 47 was set at |, giving a cut-off frequency o f 3980 c/s, (See section 4,4,2). For these t r i a l s , the values a = -g- and a = j ^ -were tried in the f i l t e r s which generated the integrated errors for the decision function. These values represent cut-off frequencies of 1990 c/s and 995 c/s respectively. For each value, a new optimum m had to be found. The f i r s t change, with m = 5, produced no significant change'. There was slight, improvement for some amplitudes, and a slight worsening of the noise for others. These results are illustrated in Figure 6,7. The second change seemed to increase the noise power at a l l values of input, and so these t r i a l s were abandoned. 48 7. CONCLUSIONS 7,1 Conclusions Regarding the Delta Modulation System The computed behaviour of simple delta modulation agrees well with that, previously observed, for a l l input levels from threshold to overload. An appreciable improvement in quantizing noise, (both at normal and at very low signal levels) results from the addition of a proportion of double integration to the feedback loop. The optimum proportion has been determined and i s recommended for a l l delta modulation systems used for speech signals. The addition of some third-order integration gives an Improve-ment which i s barely detectable, and probably not worth while. The time-constants of the integrators are not c r i t i c a l , and may conveniently be as short as 40 (isecond. .7.2 Conclusions Concerning Digital Simulation Tecfagrcpres Simulation rf a digital communication system on a digital computer ;can: be done with great accuracy, with freedom to alter parameters of the system easily, and with no fear that results are. influenced by imperfections in the apparatus. Furthermore crit e r i a for transmission quality (such as the quantizing noise power.) can be computed as part of the program with greater accuracy and r e l i a b i l i t y than can be expected of experimental measurements. For. a computer such as the ALWAC III-E, with modest speed and storage f a c i l i t i e s , the small sample of the input signal to be processed must be carefully chosen, and the program painfully optimized for reasonable computing times. Results of such t r i a l s are more questionable, but can s t i l l give much valuable 49 information in a short time. The work described here shows that where comparison with experiment is possible, the results are quite accurate. Such computer simulation trials are recommended as a method of selecting the best of several possible communica-tion systems, though the final choice should be verified with a "listening test". 50 APPENDIX I GENERATING A SINUSOIDAL FUNCTION The method u s e d t o g e n e r a t e v a l u e s o f a s i n e wave at i n t e r v a l s o f A t i s the s i m u l t a n e o u s s o l u t i o n of a p a i r o f d i f f e r e n t i a l e q u a t i o n s . S t a r t i n g w i t h y = A costot and z = A sintot, . . o o o o ( l ) we d i f f e r e n t i a t e t o get y = -toA s i n wt » -toz and z B S toA cos tot = coy (2) To g e n e r a t e v a l u e s a t i n t e r v a l s o f At, we c a n u s e Ay = - k z and Az s ky where k = to A t . „ . . . . (3) F o r t h e i t e r a t i o n p r o c e s s , t h e r e a r e s e v e r a l p o s s i b i l i t i e s : a) y n + l ~ y n = - k z n z n + l " z n = k y n b) y n + l " y n = ~ k z n + l Zn+1 ~ z n k y n + l c) y n + l ~ y n = - k z n z n + l - z n s k y n + l y n " k z n - i l y J n z n - k y n - mz„ n z n + l = z  ~ K y n ~ m z n o . » . (4) 2 where 1 and m are c o n s t a n t s w i t h v a l u e s e i t h e r 0 or k a Assume y and z t o of t h e form " n n y n = B X n and z n = C X n where A i s a complex number 0 . (5) 51 Then from (4) and (5) B A n + 1 = BA n - kCA n - lBXn , and C A n + 1 = CA n + kBA* - mCXn or A= 1 - k(|) - 1 , and A= 1 + k(§) - m So that and B ~ m - 1 2 Bring a l l the terms to the left-hand side, we have A2 - A(2-i-m) + 1 + k 2 + im - i - m = 0 .. .. (6) The interesting case is when = 0, and m = k 2(Case c) then A 2 - A(2 - k 2) + I = o ,2 r ~ — ~ 2 ~ i A= 1 - §- ±ik/ 1 - (7) 1 + |~ _ k 2 + k 2 - k 4 = 1 * 4 id So that one solution i s A = e . . . . . . (8) Then Y = e l n e , letting A = 1 Y Q = (cos © + i sin © ) n = cos n © + i sin n© ...... (9) So that using case (c), a pure stable sinusoidal signal may be generated. The relationship between n© and cot may be investigated. Equating the real and imaginary parts of (7) and (8) we have 2 C O S © = 1 - ~~ 2 and sin © = k / 1 - -r- .. .. •. (10) 52 53 APPENDIX II BASIC DELTA MODULATION PROGRAM The program i s stored i n channels 42 to 46 inclusive» 4204 55112800 48401704 8544871e 551d5bOf 11600000 00003000 e740all0 c460178c 4304 871f5503 78405b07 11607832 5bldll60 790cf701 17045503 Oe007840 19011111 4404 00080000 OOOOOOOO OOOOOOOO OOOOOOOO 00003600 683flle0 OOOOOOOO OOOOOOOO 4504 00002600 68 3 f l l 6 l 51591fd2 69262600 69261186 a505117d OOOOOOOO OOOOOOOO 4604 OOOOOOOO OOOOOOOO 0tD0000f5 OOOOOOaO 0000004b 00008455 00003855 00000c55 c54e7915 f7828745 794c3a00 571ell0a 00200000 5b680000 81431100 00020000 3000e260 C5167841 643f3000 e654bdl6 17983000 5b061160 OOOOOOOO 06000210 c74211e6 0000c547 0000614c 3a003800 00006744 a502115b OOObOOOO OOOOOOOO 36000cOO 30006147 3000674c a50211c5 OOOOOOOO OOOOOOOO OOOOOOOO 615e3132 OOOOOOOO OOOOOOOO OOOOOOeO 0000008b 00000035 00006e55 00002a55 00000655 26006831 36003a00 795cl774 C7448546 81431100 OOOOOOOO OOOOOOOO 01910000 11020000 0608020a 55030eOO 7832190b lllbOOOO OOOOOOOO 06080202 OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO 00003600 68311102 OOOOOOOO OOOOOOOO a505114b 00006747 51ObIfdO 69342600 693411el OOOOOOOO OOOOOOOO OOObOOOO OOOOOOOO OOOOOOOO OOOOOOcb 00000075 00000020 00005a55 00001e55 00000255 OOOOOOOO OOOOOOOO OOObOOOO 00100094 OOOOOOOO OOOOOOOO c55c614e 313211e7 OOOcOOOO 06000288 3000e260 C5197833 64313000 e654bdl9 170e3000 5blall60 OOOOOOOO OOOOOOOO 655cc55e 30007942 a503111b OOOOOOOO c74cll63 OOOOOOOO 61441168 OOOOa503 c744c54e 41406542 3a003800 61780000 615cll41 OOOOOOOO OOOOOOOO OOOOOOOO 000000b5 00000060 0000000b 00004855 00001455 00000055 APPENDIX III DELTA MODULATION PROGRAM with d(t) = e + m e 8 + n e " 42 83470000 00000000 8544871e 551d5bOf 11600000 00003000 e746all0 c460178c 43 871f5503 78405b07 11607832 5bldll60 790cf701 17045503 0e007840 19011111 44 000cOOOO 36006831 26001152 oooooooo 00000000 OOOOOOOO oooooooo oooooooo 45 OOOOOOOO C7650000 615bll4f c75dll74 OOOOOOOO 6l653aOO 3800675b a5021164 46 oooooooo OOOOOOOO OOOOOOf5 OOOOOOaO 0000004b 00008455 00003855 OOOOOc55 c55a7913 f7828745. 28000000 571elllb OOOOOOOO OOOOOOOO OOOOOOOO 00020000 3000e260 c5167841 643f3000 e654bdl6 17983000 5b061160 OOOOOOOO 06000210 OOOOOOOO 3a003800 6l4a6l5c 674e5140 c7421f66 69342600 693411ee OOOOOOOO 00000040 OOOOOOOO 6l5ac55c 30003200 a50311f7 a505117d 00000040 615e3132 OOOOOOOO OOOOOOOO OOOOOOeO 0000008b 00000035 00006e55 00002a'55 00000655 OOOOOOOO OOOOOOOO 0000615d Oc003000 lllbOOOO OOOOOOOO OOOOOOOO 01910000 11020000 0608020a 55030eOO 7832190b lllbOOOO OOOOOOOO 06080202 OOOOOOOO OOOOOOOO 00080000 OOOOOOOO oooooooo 68313600 495bll7e OOOOOOOO OOOOOOOO a505H6b 3600683f 2600683f 36003aOO 38006765 a502116c OOOOOOOO 3000e761 OOOOOOOO OOOOOOOO OOOOOOcb 00000075 00000020 00005a55 00001e55 00000255 30001775 c74e8546 81431100 00100094 5b680000 OOOOOOOO 415c614e 3a001103 OOOcOOOO 06000288 3000e260 c5197833 64313000 e654bdl9 170e3000 5blal l60 OOOOOOOO OOOOOOOO OOOOOOOO 51401f44 69262600 69261156 OOOOOOOO OOOOOOOO C5244165 67791121 655cc55e 30007942 a5031169 0000c55a 0000c74e £5421145 OOOOOOOO OOOOOOOO 000000b5 00000060 0000000b 00004855 00001455 00000055 47 OOOOOOOO 30006124 OOOOOOOO 4146118a OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO OOOOOOOO 56 REFERENCES 1» De Faye9 P.J., "Aspects of Delta Modulation,85 M0Ao Sc. Thesis, University of British Columbia,, 1959. 2» De Jager, F.9 "Deltamodulation, A Method of P. C. M. Transmission Using the l~Unit Code," Phjl\ips  Research Reports8 Vol. 79 pp. 442-466, 1952. 3o Holzer, J., "Exponential Delta Modulation for Military Communications," Signal Corps Engineering Laborato~  rieso Fort Monmouth, N 0Jo g Technical Memorandum No. M-1777, 1 June 1956. 4= David, E 0E 0 Jr O J, Mathews, M0V0 and MeDonald9 HoS.9 "Experiments With Speech Using Digital Computer Simulation," Bell  Telephone System Technical Publications, Monograph 3405. 5. Levinson, N., "The Wiener RMS Error Criterion in F i l t e r Design and Prediction 9" Journal of Math and Physics 0 Vol. XXV, No. 4, pp. 261-278, January 1947= 

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