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UBC Theses and Dissertations

Display system for a digital statistical analyser Venditti, Comenico Antonio 1967

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A DISPLAY SYSTEM FOR A DIGITAL STATISTICAL ANALYSER by DOMENICO ANTONIO VENDITTI B. Eng., M c G i l l U n i v e r s i t y , 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the 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 con f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a 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 , I a g r e e t h a t th.v; 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 by the 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 . I t i s u n d e r s t o o d t h a t c o p y i n g " o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d v/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 o f 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 D a t e A IL ; I ? 6 7 ABSTRACT A d i s p l a y system f o r a p o l a r i t y - c o i n c i d e n c e d i g i t a l c o r r e l a t o r i s d e s c r i b e d i n t h i s t h e s i s . The d i s p l a y of n o r -m a l i z e d s t a t i s t i c a l e s t i m a t e s of the computer o c c u r s whenever the sample s i z e e q u a l s 2 n ( n = 1, 2, 1 8 ) . T h i s a l l o w s one t o v i s u a l l y determine convergence of the e s t i m a t e s as they are b e i n g computed. A b r i e f d e s c r i p t i o n of the c o r r e l a t o r i s i n c l u d e d , p e r m i t t i n g the r e a d e r t o r e l a t e more e a s i l y i t s output i n f o r -m a t ion and c o n t r o l f u n c t i o n s w i t h the d i s p l a y system. The o r d e r of magnitude of e r r o r s i n t r o d u c e d by s a m p l i n g f l u c t u a t i o n s and by the d i g i t a l - t o - a n a l o g u e c o n v e r s i o n i s shown t o be l e s s than 2$ of, f u l l s c a l e a t the output of the 18 m o n i t o r i n g system (sample s i z e = 2 ), w i t h about 95$ c o n f i d e n c e l i m i t s a s s i g n e d t o the s t a n d a r d d e v i a t i o n of t h e c o r r e l o g r a m . D i s p l a y s of the e s t i m a t e s s t o r e d i n the memory of the computer- are i l l u s t r a t e d f o r v a r i o u s modes of o p e r a t i o n of the c o r r e l a t o r . I n p a r t i c u l a r , i t i s shown how the d i s p l a y system f a c i l i t a t e s o b s e r v a t i o n of the r a t e o f convergence of c o r -r e l a t i o n e s t i m a t e s towards t h e i r f i n a l v a l u e s . i i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT v i i 1. INTRODUCTION 1 1.1 Statement of the Problem 1 1.2 T urner's C o r r e l a t o r 1 2. FUNCTIONAL OPERATION OF THE DISPLAY SYSTEM 10 2.1 The D i v i s i o n P r o c e s s 10 2.2 F u n c t i o n a l O r g a n i z a t i o n of the D i s p l a y System .. 12 2.3 Main S u b s e c t i o n s of the D i s p l a y System 19 2.3.1 The I n t e g e r Exponent D e t e c t o r 19 2.3.2 The S h i f t R e g i s t e r 22 2.3.3 The D i g i t a l - t o - A n a l o g u e C o n v e r t e r 25 2.3.4 The C o n t r o l L o g i c 28 3. DESIGN AND IMPLEMENTATION 35 3.1 L e v e l C o n v e r t e r s 36 3.1.1 Input L e v e l C o n v e r t e r s 36 3.1.2 Output L e v e l C o n v e r t e r s and D r i v e r s 37 3.2 NOR-Logic Implementation 39 3.3 Waveform-Shaping C i r c u i t s 43 3.3.1 The A s t a b l e and Monostable M u l t i -v i b r a t o r s 43 3.3.2 D i f f e r e n t i a t i n g and I n t e g r a t i n g C i r -c u i t s 44 3.4 The D i g i t a l - t o - A n a l o g u e C o n v e r t e r . 46 3.5 The D i s p l a y System Layout 50 4. MEASUREMENT OF STATISTICAL ESTIMATES 52 4.1 Sampling F l u c t u a t i o n s and Measurement E r r o r s ... 52 4.2 Some Examples of Output D i s p l a y s 56 5. CONCLUSIONS 62 APPENDIX A 64 i i i Page APPENDIX B 71 APPENDIX C 74 REFERENCES 79 i v LIST OF ILLUSTRATIONS F i g u r e Page 1-1 Q u a n t i z a t i o n L e v e l s of A u x i l i a r y N o i s e , y^, and 1- 2 Turner's C o r r e l a t o r i n Symmetrical-Sampling Mode 5 2- 1 S i m p l i f i e d B l o c k Diagram of D i s p l a y System w i t h A s s o c i a t e d Timing S i g n a l s from Computer 13 2-2 Timing Waveforms from Computer 14 2-3 T i m i n g Waveforms f o r D i s p l a y System 14 2-4 The D i s p l a y System 16 2-5 I n t e g e r Exponent D e t e c t o r s f o r N = b n 20 2-6 The B i n a r y I n t e g e r Exponent D e t e c t o r 21 2-7 The D i g i t a l - t o - A n a l o g u e C o n v e r t e r 26 2-8 The C o n t r o l L o g i c 30 2-9(a) Tim i n g Waveforms a t B e g i n n i n g of Computation .... 33 2- 9(b) Timing Waveforms D u r i n g T r a n s i t i o n from N = t o N = 2 2 34 3- 1 Output L e v e l C o n v e r t e r s 38 3-2 Input Gates t o D i s p l a y - S y s t e m S h i f t R e g i s t e r .... 40 3-3 Input Gates t o D/A C o n v e r t e r 41 3-4 C o n t r o l L o g i c Gates 42 3-5 The uL914 Module Connected as a M u l t i v i b r a t o r ... 45 3-6 B i t D r i v e r 47 3-7(a) Model f o r Upper Reference V o l t a g e 47 3-7(b) T y p i c a l C h a r a c t e r i s t i c s of 1N4154 47 3- 8 The D i s p l a y System Layout 51 4- 1 Sources of E r r o r i n the Measurement of C o r r e l a -t i o n E s t i m a t e s . . 53 v F i g u r e Page 4-2 Output E r r o r Due t o In h e r e n t S t a t i s t i c a l Noise and D/A C o n v e r s i o n .... 55 4-3 A u t o - C o r r e l a t i o n E s t i m a t i o n of a S i n u s o i d a l Wave-form a f t e r 2 n (n = 1, 2, 12) S a m p l e s — Sy m m e t r i c a l Sampling Mode 58 4-4 A u t o - C o r r e l a t i o n E s t i m a t i o n of a S i n u s o i d a l S i g -n a l P l u s A d d i t i v e N o i s e a f t e r 2 n (n = 1, 2, ..., 12) S a m p l e s — A s y m m e t r i c a l Sampling Mode 59 4-5(a) Amplitude Cumulative P r o b a b i l i t y D i s t r i b u t i o n of Noise Used f o r F i g u r e 4-4 60 4-5(b) Amplitude P r o b a b i l i t y D e n s i t y E s t i m a t i o n of Noise Used f o r F i g u r e 4-4 60 A - l Input L e v e l C o n v e r t e r and A s s o c i a t e d S t a t e s 65 A-2 A l l o w a b l e V a l u e s of R^, R^, and j3 f o r O p e r a t i o n of C o n v e r t e r „ 70 B - l C o n v e r s i o n of an AND/OR-Logic Tree t o a NOR-Logic Tree 73 B-2 AIL Symbols f o r RTL Implementation of F i g u r e B - l 73 C - l A Ladder-Type D/A C o n v e r t e r 74 AV , out„ C-2 — ^ f o r Three Case S t u d i e s 78 ACKNOWLEDGEMENT I w i s h t o expr e s s my s i n c e r e g r a t i t u d e t o Dr. A.D. Moore f o r h i s i n v a l u a b l e guidance and encouragement w h i l e s u p e r v i s i n g t h i s p r o j e c t . A l s o , I would l i k e t o thank Dr. J.S. MacDonald f o r r e a d i n g the manu s c r i p t and f o r h i s h e l p f u l s u g g e s t i o n s . Ideas a re b e s t enhanced by d i s c u s s i n g them w i t h o t h e r p e o p l e . F o r t h i s r e a s o n , I am i n d e b t e d t o many persons who have h e l p e d d u r i n g the course o f t h i s work. I n p a r t i c u l a r , I extend my a p p r e c i a t i o n t o Mr. R.A. S t e i n f o r v e r i f y i n g the d e r i v a t i o n s i n Appendix C, t o Messrs. R.M. Turner and L.A. Warman f o r u s e f u l d i s c u s s i o n s of the c o r r e l a t o r , and t o Mr. E.A. Voth f o r h i s a s s i s t a n c e i n t r o u b l e - s h o o t i n g the composite system. Acknowledgement i s g r a t e f u l l y g i v e n t o the N a t i o n a l R esearch C o u n c i l f o r a B u r s a r y and a S t u d e n t s h i p awarded i n 1965 and 1966 r e s p e c t i v e l y , and t o the Quebec M i n i s t r y of E d u c a t i o n f o r t h e i r "Bourses Symboliques" awarded i n the same y e a r s . The i m p l e m e n t a t i o n of the d i s p l a y system was c a r r i e d out under the a u s p i c e s of NRC Grant A - 3 3 5 7 . S p e c i a l thanks go t o God and t o my p a r e n t s who have been the prime s o u r c e s of my s u c c e s s . v i i 1. INTRODUCTION 1.1 Statement of the Problem In computing s t a t i s t i c a l f u n c t i o n s f o r l o w - f r e q u e n c y s i g n a l s i t i s u s e f u l t o mo n i t o r the output of the computer d u r i n g t h e i r e v a l u a t i o n . One can the n cease computation when s a t i s f a c t o r y convergence t o a f i n a l v a l u e has been reached. I n the absence of such m o n i t o r i n g , u n n e c e s s a r i l y l o n g time i n t e r -v a l s may be needed i n c o m p i l i n g s t a t i s t i c a l i n f o r m a t i o n from such s o u r c e s as g e o p h y s i c a l phenomena, oceanographic or meteor-o l o g i c a l d a t a , market t r e n d s , e t c . Turner's d i g i t a l c o r r e l a t o r f o r l o w - f r e q u e n c y s i g -n a l s , de s i g n e d and b u i l t a t the U n i v e r s i t y of B r i t i s h Columbia, can compute v a r i o u s s t a t i s t i c a l f u n c t i o n s . I t i s the purpose of t h i s t h e s i s t o d e s c r i b e a d i g i t a l d e v i c e used to n o r m a l i z e i n f o r m a t i o n from Turner's computer f o r v i s u a l d i s -p l a y of these f u n c t i o n s d u r i n g t h e i r c o mputation, i n orde r t o observe a f i n a l - v a l u e convergence. 1.2 Turner's C o r r e l a t o r Turner's c o r r e l a t o r i s a ge n e r a l - p u r p o s e s t a t i s t i c a l a n a l y s e r c apable of computing c o r r e l a t i o n f u n c t i o n s , p r o b a b i l i t y d e n s i t i e s and d i s t r i b u t i o n f u n c t i o n s , or of o p e r a t i n g as a computer of average t r a n s i e n t s . V a r i o u s output and t i m i n g v o l t a g e s are o b t a i n e d from the computer t o f e e d and c o n t r o l the d i s p l a y system (and v i c e v e r s a ) . A b r i e f d e s c r i p t i o n of the computer i s t h e r e f o r e g i v e n i n t h i s s e c t i o n w i t h p a r t i c u l a r emphasis on the s y m m e t r i c a l -s a m p l i n g mode of o p e r a t i o n i n e s t i m a t i n g c o r r e l a t i o n f u n c -2. t i o n s of e r g o d i c random p r o c e s s e s . The o t h e r modes o n l y r e q u i r e s l i g h t l o g i c c o n t r o l m o d i f i c a t i o n s , which do not impose any c i r c u i t changes i n the d i s p l a y system i t s e l f . The computer operates on a p o l a r i t y - c o i n c i d e n c e b a s i s . T h i s p r i n c i p l e of p o l a r i t y comparison i s c o n t a i n e d i n a theorem (2) d e r i v e d by J e s p e r s , Chu, and F e t t w e i s ' f o r the case where the i n p u t s and n o i s e comparison l e v e l s are c o n t i n u o u s random v a r i a b l e s , A m o d i f i e d theorem a p p l i c a b l e t o the case where the i n p u t s i g n a l s and/or the a u x i l i a r y n o i s e comparison l e v e l s are q u a n t i z e d i s quoted from Turner's t h e s i s : "Let the i n t e r v a l - A^<y^<A^ be d i v i d e d i n t o 2n e q u a l i n t e r v a l s . L e t the y^ be d i s c r e t e and independent r a n -dom v a r i a b l e s t a k i n g on the v a l u e s l y i n g a t the m i d - p o i n t s of these i n t e r v a l s w i t h e q u a l p r o b a b i l i t y , f o r each l e v e l . L e t x. be a c o n t i n u o u s random v a r i a b l e i n the range - A ^ ( l - 2n) < x j _ < ~ 2n") ' w ^ e r e "^e hounds are l e v e l s of y^. L e t x^ be the m i d p o i n t of the i n t e r v a l between q u a n t i z e d l e v e l s of y^ w i t h i n which x^ f a l l s (see F i g u r e 1-1). Then, X-, x ' ' . . . x = A-, A 0 . . . A sgn Z U ^2 ' ' ' q - JL"2 q where Z = Z,Z 0 ... Z and Z. = x. - y.. 1 2 q l l J i i T h i s theorem shows t h a t , i n s o f a r as xV i s a good e s t i -mate of x^, sgn Z p r o v i d e s an e s t i m a t e of the moment. The e f f e c t of q u a n t i z a t i o n of the a u x i l i a r y f u n c t i o n s y^ i s an apparent q u a n t i z a t i o n of the random v a r i a b l e s x^ t o d i s c r e t e l e v e l s midway between those f o r y^. Thus the r e s u l t i s the same whether one or bo t h of x^ and y^ are q u a n t i z e d , p r o v i d e d QUANTIZATION LEVELS FOR Jt - A . T 1 1 ?1 1 1 1 I , I , 1 « 1 1 | 1 ( M 1 -2 1 0 1 + 1 1 1 1 1 1 1 +3 n-z -n-i Ar QUANTIZATION LEVELS FOR r-F i g u r e 1-1 Q u a n t i z a t i o n L e v e l s of A u x i l i a r y N o i s e , y., and S i g n a l , x^ the l e v e l s a r e i n t e r l a c e d i f bo t h are q u a n t i z e d . " I n an e r g o d i c random proc e s s ( n e c e s s a r i l y s t a t i o n a r y ) the c o r r e l a t i o n f u n c t i o n s ( o r ensemble averages) and time averages of the p r o d u c t s of s i g n a l s ' a r e e q u i v a l e n t . I f the f o r e -g o i n g theorem i s t h e r e f o r e used t o determine the c r o s s - c o r r e l a t i o n f u n c t i o n , R ^ 2 ( T ) , of two s i g n a l s , and x 2 , i n such a p r o c e s s , then R 1 2 ( r ) = <x 2(o) x2(v)y l i m . T |rjT ^ X 1 ( t ) X 2 ( t + T) -T dt = x x ( t ) x 2 ( t +r ) = x 1 c t - r ) x 2 ( t ) «x^(t - T ) x' 2(t) = A 1 A 2 u , where u = u-^u2 and u^ = s g n ( x ^ - y^) , -A-^<y-^<A-^ u 2 = s g n ( x 2 - y 2 ) , - A 2 < y 2 < A 2 > L e t t i n g C-^2(T) be an e s t i m a t e of R-^2(T) o b t a i n e d by 4. u s i n g a f i n i t e i n t e g r a t i o n t i m e , we have T c1 2(r) = ^ ^ X l ( t ) x 2 ( t +r)dt = A 1 A 2 [ u ] T 0 where [ u ] ^ i s u a t the f i n i t e t i m e , T. [u]]rp f o r one v a l u e of T can be determined e l e c t r o n i -c a l l y as f o l l o w s : x^ and x 2 are b o t h compared w i t h c o r r e s p o n d i n g n o i s e s o u r c e s , y^ and y 0 , i n l e v e l comparators whose outputs y i e l d s g n ( x ^ - y ^ ) , i = 1, 2. D e l a y i n g the output of the f i r s t com-p a r a t o r by a time i n t e r v a l , Tf, w i l l generate u ^ ( t - X). The s i g n s of u-^(t - T ) and u 2 ( t ) are th e n m u l t i p l i e d i n a p o l a r i t y -comparison c i r c u i t whose output i s u ( o r sgn Z ). F i n a l l y , i n t e -g r a t i o n f o r a time T produces an output p r o p o r t i o n a l t o [u ] r p . The e v a l u a t i o n of [u ] ^ , f o r t h i r t y - t w o e q u a l l y - s p a c e d v a l u e s of X i s a c c o m p l i s h e d s i m u l t a n e o u s l y by using, a s h i f t r e g i s t e r f o r d e l a y i n Turner's c o r r e l a t o r i n the s y m m e t r i c a l -s a m p l i n g mode, as o u t l i n e d i n F i g u r e 1-2. (The cascade of f u n c t i o n a l b l o c k s a t the top of the diagram r e p r e s e n t s the sequence of monostable command c o n t r o l s f o r the computer. The p u l s e outputs and/or the time d u r a t i o n of these monostables w i l l be r e f e r r e d t o by the s u b s c r i p t e d M's i n the b l o c k s . A r e s e t l i n e , B.^ , f o r the whole system i s o m i t t e d t o a v o i d c l u t -t e r i n g the diagram.) Each of the s i g n a l s , x^ and x 2 , i s compared w i t h i t s companion d i s c r e t e n o i s e s o u r c e , y^ and y 2 , r e s p e c t i v e l y . The comparator outputs have two p o s s i b l e s t a t e s r e p r e s e n t i n g the p o s i t i v e or n e g a t i v e s i g n of ( x ^ - y ^ ) . Each output i s sampled CLOCK h£H^l^^ COMPAR-ATOR M9 ENABLE MOD-2 SUM COUNT RESET I MONO-STABLE SHIFT_ • PULSE 5. 32-BIT SHIFT REGISTER I FLIP SHIFT REGISTER FLOP SCANNING CIRCUIT POLARITY COINCIDENCE CIRCUIT REVERSIBLE COUNTER CONTROL CIRCUITS SIGN S INDICATOR I » REVERSIBLE COUNTER CC) • » • IB 2-INPUT Mb QAJES • , • T t READ RESET SAMPLE ^ G i T < 3 / a ^ I WRITE - f t , 9 COUNTER (A/) ^I^EMRVARRAY tr r I * I S-8/T COUNTER 32 5-IN PUT AND GATES T F i g u r e 1-2 Turner's C o r r e l a t o r i n S y m m e t r i c a l - S a m p l i n g Mode 6. s i m u l t a n e o u s l y by M q through two AND gates . A F l i p - F l o p s t o r e s the l a s t sampled output of the ch a n n e l - 2 comparator; the f i r s t f l i p - f l o p of a 3 2 - b i t s h i f t r e g i s t e r s t o r e s t h a t of the c h a n n e l - 1 comparator. The co n t e n t of the l a t t e r i s s h i f t e d i n t o the second f l i p - f l o p of the s h i f t r e g i s t e r and the f i r s t i s r e s e t and made ready t o s t o r e the next sampled output of the c h a n n e l - 1 comparator. T h i s p r o c e s s i s r e p e a t e d f o r each sample t a k e n as the s h i f t r e g i s t e r becomes f i l l e d w i t h samples d e l a y e d i n time r e l a t i v e t o the l a s t sample s t o r e d i n the c h a n n e l - 2 f l i p - f l o p . U n t i l the t h i r t y - t w o f l i p -f l o p s i n the r e g i s t e r have the f i r s t t h i r t y - t w o samples from channel 1 i n s t o r a g e , the c o n t r o l v o l t a g e F^ p r e v e n t s any i n f o r m a t i o n from e n t e r i n g the memory a r r a y . A c o u n t e r keeps count of the number of samples t a k e n by means of M^. T h i s c o u n t e r i s e f f e c t i v e l y d i v i d e d i n t o two p a r t s i n cascade. The f i r s t p a r t has f i v e f l i p - f l o p s ( 5 - B i t Counter) whose outputs f e e d t h i r t y - t w o 5-input AND g a t e s . The out p u t s of the AND gates a re d e f i n e d as S ( r = 0 , 1, 3 1 ) . Each o c c u r s once f o r eve r y sample t a k e n (note t h a t P g = t°-S ). These t h i r t y - t w o o u tputs are used as w o r d - s e l e c t commands f o r 32 x 19 c o r e s (32 words) of 'a 32 x 32 memory a r r a y and f o r s c a n n i n g the c o n t e n t s of the s h i f t r e g i s t e r . The second p a r t c o n s i s t s of n i n e t e e n f l i p - f l o p s and i s c a l l e d the Sample Counter. When the t h i r t y - s e c o n d sample i s t a k e n , F^ permanently enables the i n p u t of the memory a r r a y (which i s c l e a r e d by then * The AND-gate symbol w i l l be r e p r e s e n t e d as shown i n F i g u r e 1-2 i n the c o n t e x t o f t h i s t h e s i s ; the OR-gate i s t o be s i m i l a r w i t h a ,"+" s i g n r e p l a c i n g the "•" i n s i d e the h a l f - c i r c l e . 7. because of i t s d e s t r u c t i v e readout f e a t u r e ) and r e s e t s the sample c o u n t e r through a monostable. The c a l c u l a t i o n of [u]rp e f f e c t i v e l y s t a r t s when t h i r t y - t w o samples from c h a n n e l 1 are s t o r e d i n the s h i f t r e g i s t e r , each d e l a y e d r A t u n i t s of time ( A f = r a t e ^ " o f M ^ r e l a t i v e t o the t h i r t y - s e c o n d , sample o s t o r e d i n the f l i p - f l o p of c h a n n e l 2. F o r each sample t a k e n , the.computer proceeds a c c o r d i n g t o the f o l l o w i n g sequence of e v e n t s . By s c a n n i n g the s h i f t r e g i s t e r w i t h S r, the two p o l a r i t y s i g n a l s u ^ ( t - r AT) and u 0 ( t ) a r e compared i n a p o l a r i t y comparison c i r c u i t . M^ com-mands readout of a c o r r e s p o n d i n g word from the memory a r r a y i n t o a r e v e r s i b l e c o u n t e r . The output u ( r A f ) i s enabled by the monostable M^ which produces an output of one s i g n i f u-^  and u 2 are of l i k e p o l a r i t y and of the o p p o s i t e s i g n i f they are of u n l i k e p o l a r i t y (mod-2 sum). Depending on the s i g n of u ( r Af)» c o n t r o l c i r c u i t s a l l o w the r e v e r s i b l e c o u n t e r t o count up or down ( i n t e g r a t e ) a t the occurence of Mg. P r o v i s i o n i s made i n the r e v e r s i b l e c o u n t e r c o n t r o l c i r c u i t s f o r the case where a word r e a d from the memory a r r a y has a z e r o count. A d e l a y , M-^ , occurs b e f o r e the new word i s w r i t t e n back i n t o the memory upon command of M^. T h i s d e l a y a l l o w s p r o p a g a t i o n t h r o u g h the r e v e r s i b l e c o u n t e r . The r e v e r s i b l e c o u n t e r i s r e s e t by M^ f o l l o w i n g M^. The above sequence of o p e r a t i o n s occurs d u r i n g the time i n t e r v a l I t i s r e p e a t e d f o r each word s e l e c t command S ( r = 0, 1, 31)• I n f o r m a t i o n i s s h i f t e d from one f l i p -f l o p t o the nex t i n the s h i f t r e g i s t e r on the occurrence of the 8. s h i f t p u l s e , P . The t h i r t y - f o u r t h samples are t a k e n from b o t h s channels on the next M q p u l s e and the pr o c e s s i s r e p e a t e d . Because i n t e g r a t i o n i s a "smoothing" p r o c e s s , the s t a t i s t i c a l e s t i m a t e r e p r e s e n t e d by each n o r m a l i z e d word of the memory i s improved as the number of samples i s . i n c r e a s e d . A f t e r F^ has enabled the i n p u t t o the memory a r r a y , N, the count c o n t a i n e d i n the sample c o u n t e r i s r e l a t e d t o a p a r t i c u l a r time t = T. At time T each word of the memory a r r a y c o n t a i n s a count, C ( r A f ) = {N» u ( r A 7J )~j ^ , a l o n g w i t h a c o r r e s p o n d i n g s i g n b i t . Hence, C 1 2 ( r A f ) = A 1 A 2 [u(r A t ) ] ^ = A ^ so t h a t C^2(r) can be o b t a i n e d by a s c a l e f a c t o r , J~£~ , s i m p l y by s c a n n i n g the c o n t e n t s of the memory. The readout s e c t i o n and a s s o c i a t e d c o n t r o l c i r c u i t s of Turner's c o r r e l a t o r are not shown i n F i g u r e 1-2, but the o p e r a t i o n i s as f o l l o w s . A sample count N = 2 n (n = 8, 9, i s p r e s e l e c t e d f o r a p a r t i c u l a r c o mputation, and when t h i s count i s reached on the sample c o u n t e r , the computation i s stopped a u t o m a t i c a l l y . The computer i s then s w i t c h e d manually to the readout mode ( r e p e t i t i v e or n o n - r e p e t i t i v e ) which may operate a t a d i f f e r e n t c l o c k r a t e . The words are r e a d from the memory i n t o the r e v e r s i b l e c o u n t e r , and r e - w r i t t e n u n a l t e r e d i n t o the memory, e s s e n t i a l l y as i n the compute mode. E i g h t a d j a c e n t b i t s are s e l e c t e d from the outputs of the r e v e r s i b l e c o u n t e r and are f e d t o one of two d i g i t a l - t o - a n a l o g u e (D/A) c o n v e r t e r s f o r an analogue d i s p l a y of the t h i r t y - t w o words s t o r e d i n the memory Note t h a t no p r o v i s i o n i s made f o r m o n i t o r i n g the output of the C ( r A t ) N 9. r e v e r s i b l e c o u n t e r d u r i n g computation, s i n c e b o t h D/A c o n v e r t e r s are t h e n used as d i s c r e t e n o i s e g e n e r a t o r s t o produce and y 2 , the d i s c r e t e n o i s e comparison l e v e l s f o r channels 1 and 2. F o r f u r t h e r d e t a i l s of the t h e o r y of o p e r a t i o n and c o n s t r u c t i o n of Turner's c o r r e l a t o r , the r e a d e r i s r e f e r r e d t o the o r i g i n a l t h e s i s . 10. 2. FUNCTIONAL OPERATION OF THE DISPLAY SYSTEM In o r d e r t o determine s t a t i s t i c a l e s t i m a t e s from Turner's c o r r e l a t o r , i t i s n e c e s s a r y t o d i v i d e the words s t o r e d i n the memory by the sample s i z e , N, f o r n o r m a l i z a t i o n , and t o c o n v e r t the b i n a r y q u o t i e n t s i n t o analogue v o l t a g e s . I n the c o r r e l a t i o n p r o c e s s , each such q u o t i e n t r e p r e s e n t s an e s t i m a t e of the c o r r e l a t i o n f u n c t i o n f o r a s p e c i f i c c o r r e l a t i o n t i m e , when m u l t i p l i e d by the s c a l e f a c t o r , A-^L,, where A^ and are the peak a m p l i t u d e s of the n o i s e comparison l e v e l s f o r the s i g -n a l i n p u t s , x^ and x 2 r e s p e c t i v e l y . 2.1 The D i v i s i o n P r o c e s s A l t h o u g h computer a d d i t i o n , s u b t r a c t i o n and m u l t i p l i -c a t i o n can be performed e x a c t l y (assuming no o v e r f l o w ) , the q u o t i e n t of a d i v i s i o n o p e r a t i o n cannot always be so determined. D i v i s i o n u s u a l l y c o n s i s t s of an i t e r a t i v e p r o c e s s c o n t i n u i n g u n t i l the remainder i s s u f f i c i e n t l y s m a l l . Such c o n s t r a i n t s as speed and a c c u r a c y f o r a p a r t i c u l a r system might govern the c h o i c e of one method of d i v i d i n g over o t h e r s . ( 3 ) At f i r s t , a method of d i v i s i o n proposed by M i t c h e l l appeared p r o m i s i n g and was i n v e s t i g a t e d i n some d e t a i l . I t uses an a p p r o x i m a t i o n t o the b i n a r y l o g a r i t h m of a number determined from the number i t s e l f by s i m p l e s h i f t i n g and c o u n t i n g . Adding or s u b t r a c t i n g such a p p r o x i m a t i o n s f o r two numbers then y i e l d s the approximate l o g a r i t h m of t h e i r p r o d uct or q u o t i e n t r e s p e c t i v e l y . However, i t i s shown i n h i s paper t h a t the q u o t i e n t thus o b t a i n e d can be i n e r r o r by as h i g h as +12. 5f° depending on the r e l a t i v e magnitudes of the m a n t i s s a s of the d i v i s o r and 11. d i v i d e n d . I f the m a n t i s s a of the d i v i s o r i s z e r o , the d i v i s i o n i s e x a c t . Because of the h i g h maximum e r r o r i n h e r e n t i n t h i s method, i t was not used f o r the d i s p l a y system. D i v i s i o n t a k e s on i t s s i m p l e s t form when d i v i d i n g a g i v e n d i v i d e n d by i t s r a d i x r a i s e d t o an i n t e g e r exponent. T h i s s i m p l y r e q u i r e s a s h i f t of the r a d i x p o i n t of the d i v i d e n d , and the d i v i s i o n i s e x a c t . T h e r e f o r e , i t was d e c i d e d t o d i s p l a y the n o r m a l i z e d c o n t e n t s of the t h i r t y - t w o words of the memory f o r each o c c u r r e n c e of N = 2 , where n i s an i n t e g e r exponent.. The consequent l o g a r i t h m i c r a t e of d i s p l a y i n g the output of the computer i s a u s e f u l f e a t u r e i n t h a t the "growth" of the e s t i -mate t o the t r u e v a l u e i s f a s t e r f o r a s m a l l e r number of samples than f o r l a r g e r sample s i z e s . Synchronous d i v i s i o n i s used i n the d i s p l a y system because of a p p r o p r i a t e e x i s t i n g t i m i n g waveforms a v a i l a b l e from the c o r r e l a t o r . A l s o , s u f f i c i e n t d i s p l a y time i s a v a i l a b l e d e s p i t e t h a t t a k e n by d i v i s i o n . A l t h o u g h Turner's c o r r e l a t o r c o u l d be m o d i f i e d f o r t h i s o p e r a t i o n , i t used m e c h a n i c a l s t e p p i n g s w i t c h e s . To e l i m i n a t e the r i s k of l o s s of i n f o r m a t i o n due t o n o i s e produced by these s w i t c h e s , and a l s o t o reduce the volume of the computer, the d i s p l a y system d e s c r i b e d i n the f o l l o w i n g s e c t i o n i s d e s i g n e d t o p e r f o r m the d i v i s i o n p r o c e s s and t o produce a d i s p l a y com-p l e t e l y e l e c t r o n i c a l l y . F o r b e t t e r comprehension of the o p e r a t i o n of the system, s e p a r a t e d i s c u s s i o n s of i t s f u n c t i o n a l r e p r e s e n t a t i o n and i t s i m p l e m e n t a t i o n are g i v e n i n t h i s c h a p t e r and the next r e s p e c t i v e l y . 12. 2.2 F u n c t i o n a l O r g a n i z a t i o n of the D i s p l a y System A s i m p l i f i e d b l o c k diagram of the d i s p l a y system w i t h a s s o c i a t e d t i m i n g s i g n a l s from the computer i s shown i n F i g u r e 2-1. The t i m i n g waveforms from the computer and f o r the d i s -p l a y system are shown i n F i g u r e s 2-2 and 2-3 r e s p e c t i v e l y . The d i s p l a y system e s s e n t i a l l y c o n s i s t s of an i n t e g e r exponent d e t e c t o r , a s h i f t r e g i s t e r , a D/A c o n v e r t e r and the i n t e r c o n n e c t i n g c o n t r o l and g a t i n g l o g i c . A f t e r e n a b l i n g the i n t e g e r - e x p o n e n t d e t e c t o r w i t h Fy = 1, d e t e c t i o n of an i n t e g e r n v a l u e of n i n the sample s i z e , N = 2 , i n i t i a t e s the d i s p l a y of the n o r m a l i z e d c o n t e n t s of the memory, d u r i n g which compu-t a t i o n i s t e m p o r a r i l y stopped. The t i m i n g of the monostable p u l s e s Mp, M^, and M^, shown r e l a t i v e t o S Q i n F i g u r e 2-2, determines the o r d e r w i t h w h i c h the s h i f t r e g i s t e r r e c e i v e s i t s r e s e t ( V ^ ) , f e e d (Vp), and s h i f t (Vg) commands r e s p e c t i v e l y . Each word from the memory i s dumped i n t o the r e g i s t e r and i s s h i f t e d , by an amount dependent on the v a l u e of n, towards a f l a g b i t . The f l a g b i t generates the c o n t r o l waveform, P g , whose 1-s t a t e s t o p s the s h i f t i n g p r o c e s s y i e l d i n g the d i g i -t a l q u o t i e n t a t a f i x e d s e c t i o n of the r e g i s t e r . An e i g h t - b i t D/A c o n v e r t e r i s gated w i t h F t o d i s p l a y the q u o t i e n t i n analogue form ( V ^ ) . The s i g n , S, of the word i n q u e s t i o n determines whether V-^  i s p o s i t i v e or n e g a t i v e . T h i s i s r e p e a t e d f o r a l l t h i r t y - t w o words c o n t a i n e d i n the memory, a f t e r w h i c h computation i s resumed u n l e s s the computer i s stopped manually. A more d e t a i l e d r e p r e s e n t a t i o n of the d i s p l a y system (C) (A/) REVERSIBLE COUNTER 1 13. SAMPLE COUNTER ENABLE DETECTOR M. Mi RESET FEED SHIFT FIRST WRb SELECT SIGN INblCATOR COMPUTER INTEGER EXPONENT DETECTOR \ ft e o T R 0 L ALIGNMENT A A/D STROBE GATES 5 5*^ SHIFT REGISTER Z)/S/r/»/- QUOTIENT SELECT GATES 3-g/T fl/fo COH VERIER ANALOGUE QUOTIENT F i g u r e 2-1 S i m p l i f i e d B l o c k Diagram of D i s p l a y System w i t h A s s o c i a t e d T i m i n g S i g n a l s from Computer F i g u r e 2-2 Timing Waveforms from Computer (n o r . s c a l e : ~ 2 0 usee./cm.; v e r t , s c a l e : ~ 2 0 7./cm.) F i g u r e 2-5 Timing Waveforms f o r D i s p l a y System (nor. s c a l e : ~ 10 usee./cm.; v e r t , s c a l e : ~ 5 V./cm.) 15. i s shown i n F i g u r e 2-4. T h i s w i l l s e r v e as the b a s i s f o r w r i t i n g f u n c t i o n a l Boolean e x p r e s s i o n s i n t e r c o n n e c t i n g the main s e c t i o n s of the d i s p l a y system. The sample c o u n t e r of the c o r r e l a t o r f e eds the i n t e g e r - e x p o n e n t d e t e c t o r the output of which i s gated by F^. The d e t e c t o r produces a 1-output a t C (n =..1, 2, 18) between the times when N = 2 n and N = 2 n + 1 - 1; a l l o t h e r C .'s ( j / n ) remain i n the 0 - s t a t e d u r i n g t h i s time i n t e r v a l . The i n t e g e r n i s thus d e t e c t e d as the sample s i z e N i n c r e a s e s . D e t e c t i o n of the b e g i n n i n g of the 1 - s t a t e of C i s a v a i l a b l e as the o u t p u t , V-g, and i s used i n the c o n t r o l s e c t i o n . E i t h e r the r e v e r s i b l e c o u n t e r outputs or the d e t e c -t o r o u t p u t s p a r a l l e l - f e e d the s h i f t r e g i s t e r upon s i m u l t a n e o u s g a t i n g by a v o l t a g e V-p. The i n f o r m a t i o n c o p i e d i n t o the i n p u t s o f the s h i f t r e g i s t e r can be r e p r e s e n t e d by the f o l l o w i n g B o o lean e x p r e s s i o n : F. = V ( C j - l + D j + 1 } ' j = 0, 1, 19 where C_± = C Q = D i g = D O Q k 0. We note from F i g u r e 2-4 t h a t corresponds t o the presence of 2^ 1 i n the sample c o u n t e r , D. , t o 2^ i n the r e v e r s i b l e c o u n t e r . T h e r e f o r e , because the count, C, i n the r e v e r s i b l e c o u n t e r i s never g r e a t e r than the sample count, N, and ^-j +2. a r e n e v e r ^ e 1 - s t a t e s i m u l t a n e o u s l y . The i n p u t Boolean e x p r e s s i o n s f o r the 8-input D/A c o n v e r t e r are the f o l l o w i n g : * LSB = l e a s t s i g n i f i c a n t b i t ; MSB = most s i g n i f i c a n t b i t 16. SAMPLE\ I CNTR DIGITAL-TO-ANALOGUE CON VER TER (3 BITS) MD M3 M4 V 7 S^7 S c RESET 0 N FEED T R SHIFT 0 L F, CLOCK F i g u r e 2 - 4 The D i s p l a y System 17, K o = H o + J l + ^O' 1!! + J2 and K. = H.. + J ± + + J 2 » j = 1» 2 7 o r K o = V10' ( I2 + I10' I11 ) + ^O' 1!! + J2 i and Z. = V 1 Q . ( I . + 2 + I 1 0 . I 1 : L ) + + J 2 , 0=1, 2, ..., 7 The voltage V^Q is in the 0-state un t i l a sample count of N = 2 1 0 is reached. For N>2 1 0, V 1 Q is switched and remains in the 1-state. ^ s ^ e c o m P l e m e n " ' : ; °^ ^ 10*^ For ease of comprehension consider the case where N>2"^ (or V^Q =1). The foregoing equations reduce to K.. = I.. + J 0 , j = 0 , 1, ...,7. Upon o c c u r r e n c e of ¥ = 2 n(n>10), V E sends a p u l s e t o the con-t r o l s e c t i o n which, i n t u r n , g i v e s r e s e t ( v ^ ) , f e e d (V-p), and s h i f t (Vg) commands t o t h e s h i f t r e g i s t e r i n s y n c h r o n i z a t i o n w i t h the monostable p u l s e s M^, y[ f and r e s p e c t i v e l y . The c l o c k f r e q u e n c y i s s e t a u t o m a t i c a l l y f o r the compute mode o r the d i s p l a y mode by the c o n t r o l s e c t i o n b e f o r e i t a r r i v e s a t the computer. Because of the r e l a t i v e d i s p l a c e m e n t of N- and C - i n f o r m a t i o n f e d i n t o t he s h i f t r e g i s t e r , C n l e a d s the word o b t a i n e d from the r e v e r s i b l e c o u n t e r i n the s h i f t i n g d i r e c t i o n . T h i s f e a t u r e forms the essence of the d i v i s i o n p r o c e s s . S i n c e C i s t o be d i v i d e d by 2 n, i t s b i n a r y p o i n t must be s h i f t e d n p l a c e s towards the most s i g n i f i c a n t d i g i t . However * h a v i n g a " f l a g b i t " t o d e t e c t the a r r i v a l of C a t one end o f th e s h i f t n r e g i s t e r a c c o m p l i s h e s the same o b j e c t i v e i f a proper c h o i c e o f 1 8 . I - j + n ' s f e e d s the D/A c o n v e r t e r . F i g u r e 2 - 4 shows how d i v i s i o n i s a c h i e v e d when the f l a g b i t d e t e c t s the 1 - b i t l e a d i n g the s h i f t e d i n f o r m a t i o n i n the r e g i s t e r . When I^Q = 1 ( " f l a g goes on") the v o l t a g e F^ enables the output of the D/A c o n v e r t e r , ( p o s i t i v e or n e g a t i v e v o l t a g e depending on M^*S—an "S"-i n d i c a t o r f l i p - f l o p i s s e t a t and r e s e t a t M^), and a l s o sends i n f o r m a t i o n t o the c o n t r o l s e c t i o n . As i s shown i n S e c t i o n 2 . 3 * 4 , f o r a g i v e n N = 2 N, t h e r e i s c o i n c i d e n c e w i t h the t h i r t y - s e c o n d f l a g and S q w h i c h , a f t e r the l a s t n o r m a l i z e d word of the core memory has been d i s p l a y e d , t r i g g e r s c i r c u i t r y t o c o n t i n u e computation u n t i l N = 2 where a b e t t e r e s t i m a t e of the t h i r t y - t w o words i s a g a i n d i s p l a y e d . To reduce the number of s h i f t p u l s e s r e q u i r e d f o r the f l a g t o go on, a second f l a g is l o c a t e d near the c e n t r e of the s h i f t r e g i s t e r and o p e r a t e s in c o n j u n c t i o n with the gates pro-d u c i n g H. ( j = 0, 1, 7 ) . This part of the d i s p l a y system j is enabled by the v o l t a g e V ^ Q at the beginning of computation for N < 2 1 0 . We thus have the f o l l o w i n g e q u a t i o n s for 2 1 < N < 2 1 8 : F g = V 1 0 * 1 1 1 + X 2 0 P g l = V 1 0 * X l l + V 1 0 ' X20> Both th e s e v o l t a g e s a re used i n the c o n t r o l s e c t i o n of the d i s -p l a y system. We n o t e , of c o u r s e , t h a t F ^ = F ^ f o r V-^Q = 1 or 0 . i The outputs = V]_Q. * "*"10 * ^ 1 1 A N C ^ ^2 = "'"19*"I"20 S I M P-'-y Q f e e d the number 2 - 1 t o the D/A c o n v e r t e r whenever C = N. 19. 2 . 3 Main S u b s e c t i o n s of the D i s p l a y System 2.3.1 The I n t e g e r Exponent D e t e c t o r T h i s s u b s e c t i o n d e s c r i b e s a method f o r d e t e c t i n g the exponent, n, of an. i n c r e a s i n g modulo-b number, N ( = b n ) , whenever i t i s an i n t e g e r . I n p a r t i c u l a r , i f N i s the accumulated count o f a c o u n t e r , d e t e c t i o n of an i n t e g e r v a l u e of n i s c h a r a c t e r i z e d by the s t a t e of a Boolean v a r i a b l e , C (n = 1, 2 , . . . ) . C i s ' n ' ' ' n Yl T") _|_") i n the 1 - s t a t e between the tim e s when N = b and N = b -1 (n, i n t e g e r ) ; i n the 0 - s t a t e , f o r a l l o t h e r v a l u e s of N. C o n s i d e r the cascade of f i v e modulo-b c o u n t e r s i n F i g u r e 2 - 5 ( a ) . A " b r u t e - f o r c e " manner of o b t a i n i n g C^ (n = 0 , 1, 4) s i m p l y r e q u i r e s the memory of f i v e f l i p - f l o p s each of w h i c h i s s e t by the 1-output of a modulo-b c o u n t e r . B e f o r e the count b e g i n s , a l l f l i p - f l o p s are r e s e t t o the 0 - s t a t e . Once any f l i p - f l o p i s s e t d u r i n g the count, i t remains s e t f o r h i g h e r c o u n t s . Hence, the o u t p u t s , C , behave as r e q u i r e d f o r N<b 5-1. However, i t i s an i n h e r e n t p r o p e r t y of an e l e c t r o n i c c o u n t e r t o have a d j a c e n t s e c t i o n s of i t s e l f r e p e a t i n g s i m i l a r output p a t t e r n s d u r i n g the c o u n t i n g p r o c e s s . T h e r e f o r e , one can use t h i s f a c t a l o n g w i t h the i n h e r e n t memory of the c o u n t e r t o e l i m i n a t e some f l i p - f l o p s as shown i n F i g u r e 2 - 5(b) and y e t y i e l d the same o u t p u t s , C . Removal of a l l f l i p - f l o p s i s demonstrated i n F i g u r e 2 - 5 ( c ) . However, f o r a l a r g e c o u n t e r , one must r e a c h a compromise between g a t e s w i t h many i n p u t s and f l i p - f l o p s f o r a d d i t i o n a l memory. The c o n f i g u r a t i o n shown i n F i g u r e 2-6 i s a b i n a r y (a) 20. CLOCK PULSE: RESET Cb) 4* < *3 1 0 < s c J 0 CLOCK PULSE <- RESET (C) V 4 4 3 < t>z — 1 A ' <A 1 o I o | l o 1 » e> 1 o CLOCK PULSE F i g u r e 2-5 I n t e g e r Exponent D e t e c t o r s f o r N = b' 22. i n t e g e r - e x p o n e n t d e t e c t o r f o r the sample c o u n t e r . T h i s arrangement d i s t r i b u t e s the types of AND gates used e v e n l y a l o n g the c o u n t e r and r e q u i r e s t h r e e f l i p - f l o p s f o r a d d i t i o n a l memory. The c e n t r e f l i p - f l o p a l s o s e r v e s t o generate the v o l t a g e V-^Q d i s c u s s e d i n S e c t i o n 2.2. The outputs C. ( j = 1, 2, . 1 8 ) are used t o f e e d D the s h i f t r e g i s t e r of the d i s p l a y system and are a l s o d i f f e r e n -t i a t e d w i t h s i m p l e CR c i r c u i t s , whose o u t p u t s , . i n t u r n , f e e d an e i g h t e e n - i n p u t OR-gate y i e l d i n g the output V-g which s e t s a c o n t r o l f l i p - f l o p . Because the sample c o u n t e r counts the f i r s t t h i r t y -two samples t a k e n b e f o r e a c t u a l computation b e g i n s i n the s y m m e t r i c a l - s a m p l i n g mode of o p e r a t i o n , ( j = 1, 2, 3, 4) are g ated w i t h the v o l t a g e F^. F^ a l s o r e s e t s the sample c o u n t e r ; a 10 K-ohm r e s i s t o r i s n e c e s s a r y from the output of G^ t o the complementing i n p u t of G-^  t o prev e n t C^ from r e c e i v i n g a s m a l l - d u r a t i o n s p i k e . Without the r e s i s t o r , t h i s s p i k e i s of s u f f i c i e n t output a m p l i t u d e t o t r i g g e r the c o n t r o l f l i p - f l o p ( V ). (WYLE f l i p - f l o p s , type 4XC-RP, are used f o r the sample co u n t e r . ) 2.3.2 The S h i f t R e g i s t e r The s h i f t r e g i s t e r c o n s i s t s of twenty-one JK f l i p -f l o p s as shown i n F i g u r e 2-4. The JK f l i p - f l o p s used are con-v e n t i o n a l and have a "S" ( s e t ) i n p u t and a "C" ( c l e a r ) i n p u t , b o th of which determine the output s t a t e s upon s i m u l t a n e o u s a p p l i c a t i o n o f a n e g a t i v e p u l s e t r a n s i t i o n a t the t o g g l e i n p u t , "T". Each f l i p - f l o p a l s o has asynchronous p r e s e t and p r e c l e a r f e a t u r e s . 23. C o m b i n a t i o n a l l o g i c g a t i n g i s an a l t e r n a t i v e t o u s i n g a 2 1 - b i t s h i f t r e g i s t e r i n f e e d i n g the e i g h t e e n b i t s of the r e v e r s i b l e c o u n t e r i n t o the e i g h t i n p u t s of the D/A c o n v e r t e r . However, the l a t t e r i s the s i m p l e r s o l u t i o n , s i n c e n o r m a l i z a t i o n of the c o n t e n t s i n the core memory i s done f o r IT = 2 n (n = .1, 2, ..., 1 8 ) , and the i n p u t s of the D/A c o n v e r t e r have t h e i r p h y s i c a l l o c a t i o n f i x e d r e l a t i v e t o the e i g h t e e n o u t p u t s of the r e v e r s i b l e c o u n t e r . At l e a s t twenty f l i p - f l o p s a re t h e r e f o r e needed f o r the f l a g - b i t method of d i v i s i o n employed. An a d d i t i o n a l f l i p - f l o p a t the l e f t i s n e c e s s a r y t o a l l o w a l l i n f o r m a t i o n e n t e r i n g the r e g i s t e r t o be s h i f t e d a t l e a s t one p l a c e t o the l e f t b e f o r e the f l a g goes on. T h i s i s due t o the t i m i n g sequence w i t h which the c o n t r o l s e c t i o n o p e r a t e s ; t h a t i s , one cannot have the f l a g on a t the same i n s t a n t as i n f o r m a t i o n i s c o p i e d i n t o the r e g i s t e r . The monostable p u l s e , M , i s c o i n c i d e n t w i t h S . T h e r e f o r e , i f each F. ( j = 0 , 1, 19) c o n n e c t i o n were moved t o the a d j a c e n t f l i p - f l o p on i t s l e f t and the r i g h t - m o s t f l i p - f l o p were removed, t h e r e would be q i s c o i n c i d e n c e between F and S a t IT = 2 and H = 2 f o r each g o word b e i n g n o r m a l i z e d . However, f o r the analogue d i s p l a y t o c o n t i n u e f o r the t h i r t y - t w o words, the c o n t r o l s e c t i o n r e q u i r e s t h a t t h e r e i s c o i n c i d e n c e o n l y w i t h the t h i r t y - s e c o n d F p u l s e and S as mentioned e a r l i e r , o As s h i f t i n g p r o ceeds, a s e r i e s of O's are i n s e r t e d a t the l e a s t s i g n i f i c a n t end of the r e g i s t e r . T h i s p r e v e n t s any m o d i f i c a t i o n of a p a r t i c u l a r word as i t i s s h i f t e d t o the l e f t . 24. I f jam t r a n s f e r had "been used i n s t e a d of p a r a l l e l f e e d , t h e r e would have been no need f o r a r e s e t l i n e (V-^), However, a t l e a s t twenty-one a d d i t i o n a l AND gates and twenty-one i n v e r t e r s would have been r e q u i r e d . T h i s would a l s o have n e c e s s i t a t e d t w i c e the l o a d on the f e e d l i n e . Jam t r a n s f e r would remove any c o i n c i d e n c e problems which might e x i s t between the r e s e t -and f e e d - l i n e p u l s e s . However, such problems were e l i m i n a t e d by d i f f e r e n t i a t i n g the p u l s e and i n t e g r a t i n g the p u l s e M^. I n t h i s f a s h i o n , the r e s e t t i n g and s h i f t i n g o p e r a t i o n s cannot occur a t the same i n s t a n t as i n f o r m a t i o n i s f e d i n t o the r e g i s t e r . Because the ou t p u t s of the r e v e r s i b l e c o u n t e r a re a l r e a d y gated w i t h i n Turner's c o r r e l a t o r , an a l t e r n a t i v e method t o f e e d the i n f o r m a t i o n i n t o the r e g i s t e r i s t o gate the ( j = 1, 2, 18) ou t p u t s w i t h V-p and have e i t h e r o r D j + ^ f e e d the r e g i s t e r t h rough a s i m i l a r s e t of OR gates used i n F i g u r e 2-4. However, t h i s r e q u i r e s an e x t r a i n p u t t o a l l AND gates i n the b i n a r y i n t e g e r exponent d e t e c t o r . A l s o , d i r e c t NOR-logic i m p l e m e n t a t i o n i s not f e a s i b l e w i t h o u t e x t r a i n v e r t e r s . I f o n l y one f l a g b i t i s used a t one end of the s h i f t r e g i s t e r , a maximum of e i g h t e e n s h i f t p u l s e s a re r e q u i r e d f o r d i v i s i o n . P a r t i t i o n i n g the r e g i s t e r w i t h two f l a g b i t s , as i s done i n F i g u r e 2-4, reduces t h i s t o a maximum of n i n e s h i f t p u l s e s . l e t t 0 be the time t a k e n i n s h i f t i n g f o r d i v i s i o n , t - p . , i n d i s p l a y i n g a q u o t i e n t , and t ^ , i n + M^. I f t ^ i s the p e r i o d of t° (one r e a d - i n t e g r a t e - w r i t e c y c l e f o r the r e v e r s i b l e 25. c o u n t e r ) , t h e n t c = t M + t g + t D . Now, t ^ and t ^ have l o w e r l i m i t s f o r p r o p e r o p e r a t i o n of the c o r r e l a t o r . These l o w e r l i m i t s determine the speed w i t h which d i v i s i o n must be c a r r i e d out i n o r d e r t o ensure a s u f f i c i e n t d i s p l a y d u t y c y c l e , 7 — , f o r each q u o t i e n t . W i t h a s i n g l e c l o c k f o r the system, t ^ and t g are ^C r e l a t e d as t g = m > where the f a c t o r 12 i s i n h e r e n t a t the c l o c k i n p u t of Turner's c o r r e l a t o r (see F i g u r e 1-2), and m i s r e a l i z e d by an a p p r o p r i a t e cascade of k f l i p - f l o p s y i e l d i n g m = 2 k . A p r o p e r c h o i c e of m w i l l thus a l l o w s u f f i c i e n t time f o r d i s p l a y i n the method u s i n g o n e ( f l a g , p r o v i d i n g the upper f r e q u e n c y l i m i t of the s h i f t r e g i s t e r i s not exceeded. However, i n a n t i c i p a t i o n of f u t u r e i n c r e a s e of the number of p o i n t s on the c o r r e l o g r a m f o r a g i v e n s a m p l i n g r a t e , and the consequent need f o r h i g h e r f r e q u e n c y modules i n the c o r r e l a t o r i t s e l f , the method u s i n g two f l a g b i t s w i l l a l l o w t w i c e the d i s p l a y r a t e f o r the same v a l u e of m. I n the p r e s e n t system, the v a l u e s of t ^ and t ^ are r o u g h l y 15 and 50 microseconds r e s p e c t i v e l y . The two f l a g - b i t method i s employed w i t h m = 2, making the d i s p l a y d u t y c y c l e over t h i r t y p e r c e n t and the maximum c l o c k r a t e about 500 Kc/s. 2.3.3 The D i g i t a l - t o - A n a l o g u e C o n v e r t e r The e i g h t - b i t D/A c o n v e r t e r shown i n F i g u r e 2-7 c o n s i s t s of s i g n - p r o c e s s i n g g a t e s , b i t d r i v e r s , and a d e c o d i n g network. I n o r d e r t o decode b o t h p o s i t i v e (S = l ) and nega-F i g u r e 2-7 The D i g i t a l - t o - A n a l o g u e C o n v e r t e r 27. t i v e (S = 0) . numbers r e p r e s e n t e d by K ^ . ( j = 0 , 1, . .., 7 ) , one r e q u i r e s s i g n - p r o c e s s i n g gates whose outputs are r e p r e s e n t e d by the f o l l o w i n g B o o l e a n e x p r e s s i o n : L. = F g . ( S g + K - ) * ( S g + K.) , j = 0, 1, 7. T h i s e x p r e s s i o n can f u r t h e r be s i m p l i f i e d t o L . = F . S • K . + P - S ' - K ' . . 3 g g 3 g g 3 T h e r e f o r e , L_ can a l t e r n a t i v e l y be f u n c t i o n a l l y c o n s t r u c t e d by f e e d i n g the outputs of two t h r e e - i n p u t AND gates i n t o a t w o - i n p u t OR g a t e . However, the former method i s used because of more d i r e c t NOR-logic hardware i m p l e m e n t a t i o n . The b i t d r i v e r s have the p r o p e r t y of c o n n e c t i n g a v o l t a g e +E v o l t s t o the j t h i n p u t of the d e c o d i n g network i f 1. = 1 : -E v o l t s , i f L. = 0 . The d e c o d i n g network i s the 3 3 & R-2R l a d d e r t y p e . A l t e r n a t i v e l y , one c o u l d use the w e i g h t e d -r e s i s t o r t y p e . A d i g i t a l number p = Ky Kg ... K q ( o r p = K^ Kg ... K , f o r a n e g a t i v e number) i s decoded t o an analogue v o l t a g e a t the p o i n t d e s i g n a t e d by "MSB". I t s s i g n i s p r o c e s s e d w i t h an a d d i t i o n a l b i t c a l l e d the " s i g n b i t " which depends on the s t a t e of L Q = S + E ' . 8 g g I f F g = S g = 1, L 8 = 1, L. = K. ( j = 0, 1, 7 ) , V D i s p o s i t i v e . I f P = 1, S = 0, 1 0 = 0, L. = K* "V-r, i s n e g a t i v e , g ' g ' 8 3 3 d I f E = 0, L Q = 1, L . = 0, V,, « 0 v o l t s , g ' 8 3 D The t e r m i n a t i o n r e s i s t o r r , connected t o the LSB, i s grounded so as t o have an e q u a l number of l e v e l s f o r b o t h p o s i -28. t i v e and n e g a t i v e v a l u e s of p. T h i s makes i t t h e o r e t i c a l l y i m p o s s i b l e t o have e x a c t l y 0 v o l t s output f o r Y - ^ . However, Y-Q i s of i n t e r e s t o n l y when F = 1. The r e a d e r w i l l note from F i g u r e 2-3 t h a t the duty c y c l e of Fg(and hence, of V-^ ) i n c r e a s e s as N i n c r e a s e s 1 9 from N = 2 t o N = 2 . Because of the p a r t i t i o n i n g of the s h i f t r e g i s t e r w i t h two f l a g b i t s , t h i s i s r e p e a t e d from N - 2^~® t o N = 2 1 8 . At If = 2 9 and N = 2 1 8 the duty c y c l e i s n e a r l y e q u a l t o 100$ of t°, e s p e c i a l l y i f a low d i s p l a y r a t e i s employed. These are u s e f u l f e a t u r e s , as they p e r m i t a t r a i n e d o p e r a t o r t o keep t r a c k of the sample count by o b s e r v i n g s u p e r -imposed d i s p l a y s h a v i n g the l e a d i n g edges of each word s l i g h t l y d i s p l a c e d f o r d i f f e r e n t v a l u e s of IT. 2.3,. 4 The C o n t r o l l o g i c A c o n t r o l - l o g i c system p r o v i d e s v a r i o u s f e a t u r e s i n d i s p l a y i n g n o r m a l i z e d s t a t i s t i c a l e s t i m a t e s computed by Turner's d i g i t a l c o r r e l a t o r . I t e s s e n t i a l l y f u n c t i o n s by d e t e c t i n g an i n t e g e r n i n N = 2 n , d i s p l a y i n g the n o r m a l i z e d c o n t e n t s of the core memory f o r t h a t p a r t i c u l a r v a l u e of n, and r e t u r n i n g a u t o m a t i c a l l y t o the compute mode a f t e r the t h i r t y - s e c o n d word has been d i s p l a y e d . W i th t h i s system one can have r e p e t i t i v e or non-r e p e t i t i v e r e adout a t t h e compute-clock r a t e o r a t a d i f f e r e n t d i s p l a y - c l o c k r a t e from a l o c a l a u x i l i a r y g e n e r a t o r . The d i s p l a y a u t o m a t i c a l l y appears on an a p p r o p r i a t e r e c o r d i n g d e v i c e whenever the sample s i z e e q u a l s 2 n (n = 1, 2, 1 8 ) . One would n o r -m a l l y have n o n - r e p e t i t i v e readout u n t i l the e s t i m a t e of the 29. s t a t i s t i c a l f u n c t i o n i n q u e s t i o n has s a t i s f a c t o r i l y converged t o a f i n a l v a l u e . T h i s convergence i s d e t e c t e d v i s u a l l y by comparing the l a s t two or t h r e e d i s p l a y s . A l s o , i f , f o r example, an e s t i m a t e of the a u t o - c o r r e l a t i o n f u n c t i o n of a n o i s y s i g n a l w i t h a p e r i o d i c component i s determined, one can r e p e a t i t s . c omputation u n t i l a pro p e r s e t t i n g of the compute c l o c k y i e l d s a s a t i s f a c t o r y number of c y c l e s i n the c o r r e l o g r a m . The computer can be i n h i b i t e d a t any time d u r i n g the compute or d i s p l a y modes. I f i n h i b i t e d a f t e r N = 2 n (n an i n t e -ger) d u r i n g the compute mode, i t c o n t i n u e s computation u n t i l N - 2 , d i s p l a y s the t h i r t y - t w o words f o r N = 2 , and s t o p s a u t o m a t i c a l l y a f t e r the t h i r t y - s e c o n d word has been d i s p l a y e d . I f i n h i b i t e d d u r i n g the d i s p l a y mode, i t c o n t i n u e s the d i s p l a y u n t i l a l l t h i r t y - t w o words have been p r o c e s s e d , and the n s t o p s . The computer s t o p s a u t o m a t i c a l l y i n a s i m i l a r f a s h i o n when 18 N = 2 . Once a p a r t i c u l a r d i s p l a y i s completed and the com-p u t e r has stopped, the c o n t e n t s of the memory can be d i s p l a y e d a g a i n i n a r e p e t i t i v e or n o n - r e p e t i t i v e sequence. F i g u r e 2-8 i l l u s t r a t e s the c o n t r o l l o g i c used. The major f u n c t i o n i n the c o n t r o l u n i t i s produced by a f l i p - f l o p y i e l d i n g the t i m i n g waveform V ( i n 1 - s t a t e d u r i n g d i s p l a y ). V enables M„ t o generate = M-»V and g i v e s command t o the n 3 F 3 n r e c o r d e r t o d i s p l a y the t h i r t y - t w o words as l o n g as V = 1. V enables the i n p u t s o f the sample and the r e v e r s i b l e c o u n t e r s . T h i s f l i p - f l o p a l s o a l l o w s e i t h e r the d i s p l a y c l o c k or the com-pute c l o c k s i g n a l t o be d i r e c t e d t o the p o l e of the s w i t c h T marked "GATED". 30... * 3 T i r T T \INTX ) DISPLAY CLOCK] •K2+x32) ^ 1 , U i k r32 s / C O 0 J l i 5 I C 0 RECORDER V&1 DELAY . FOR 32*4 WORD V DELAY FOR RECOR-DER V 'car '^8 t i S I c o NON-REP. 1 ff£SET r/2 C2 T32 3> V' SAMPLE COUNTER _ x : — » " REPETITIVE F i g u r e 2-8 The C o n t r o l L o g i c REVERSIBLE / * COUNTER 31.. Two g e n e r a l l y d i f f e r e n t c l o c k f r e q u e n c i e s are used throughout the system f o r the compute and d i s p l a y modes. However, the d e l a y l i n e (32-bit s h i f t r e g i s t e r ) and the c h a n n e l -2 f l i p - f l o p must c o n t i n u e r e c e i v i n g samples a t the compute c l o c k r a t e d i v i d e d by (24 x 32), or P , even d u r i n g the d i s -p l a y mode. I n t h i s manner, the c o n t e n t s of a d j a c e n t memory u n i t s i n the d e l a y l i n e w i l l have samples t a k e n a t a c o n s t a n t time i n t e r v a l . ( T h i s i s u n l i k e the method o r i g i n a l l y used i n Turner's c o r r e l a t o r where P^ = t°«S .) When g o i n g from the s o d i s p l a y mode t o the compute mode ( o r when r e s e t t i n g the computer w i t h R,p) , the system must f i r s t be s y n c h r o n i z e d w i t h P g . The l e a d i n g edge of P^ c o i n c i d e s w i t h t h a t of the f i r s t c y c l e of t°,corresponding t o S . T h i s i s done as shown i n F i g u r e 2-8. We r e c a l l from p r e v i o u s d i s c u s s i o n t h a t t h e r e i s no c o i n c i d e n c e between S and F u n t i l the t h i r t y - s e c o n d word i s o g J t o be d i s p l a y e d . At t h i s p o i n t , the c o i n c i d e n c e of S q and the l e a d i n g edge of F g t r i g g e r s two monostable m u l t i v i b r a t o r s (Rrp can a l s o t r i g g e r the two m o n o s t a b l e s ) . One monostable p r o v i d e s a d e l a y to a l l o w the t h i r t y - s e c o n d word t o be d i s p l a y e d . At the end of t h i s d i s p l a y the V - f l i p - f l o p i s r e s e t t o s i g n a l the end of the d i s p l a y . The o t h e r monostable has a s l i g h t l y l o n g e r d e l a y d u r i n g w h i c h n e i t h e r c l o c k f r e q u e n c i e s can r e a c h T^. T h i s a l l o w s a d d i t i o n a l t i m e , a f t e r d i s p l a y of the t h i r t y - s e c o n d word, f o r c a r r i a g e r e t u r n of an X-Y r e c o r d e r . V was chosen ' c a r t o have a l o n g e r d e l a y than ^° a v o i d ambiguous l o g i c - s t a t e c o i n c i d e n c e s i n v a r i o u s g a t i n g schemes c o n s i d e r e d f o r the con-t r o l u n i t . 3 2 . A " s t a r t i n g " f l i p - f l o p g e n e r a t i n g the t i m i n g waveform S_j. i n i t i a t e s the o p e r a t i o n of the c o n t r o l u n i t and i s s e t upon s w i t c h i n g T^ from the 0 - s t a t e t o the 1 - s t a t e ( o r w i t h R^). enables V„„ and V t o be p r o c e s s e d , and i t s l e a d i n g edge 32 c a r * r e s e t s the V f l i p - f l o p i n the n o n - r e p e t i t i v e m o d e — i t s e t s the l a t t e r i n the r e p e t i t i v e mode ( o r w i t h V-g) . The s t a r t i n g f l i p -f l o p i s r e s e t when T^ i s i n the 0 - s t a t e p o s i t i o n and a t the end of the d e l a y V . T h i s e f f e c t i v e l y s t o p s the computer a u t o -m a t i c a l l y a t the end of the l a s t d i s p l a y of i n t e r e s t . I f the l a s t d i s p l a y i s t o be r e p e a t e d , T^ i s a g a i n s w i t c h e d t o the 1 - s t a t e p o s i t i o n a f t e r 1^ has been p l a c e d i n the " R e p e t i t i v e " p o s i t i o n . R e p e t i t i v e readout can be governed by the d i s p l a y or compute c l o c k r a t e s depending on the p o s i t i o n of T,. 3 F ^ (= 2¥^), when enabled by a c o n t r o l f l i p - f l o p , produces the v o l t a g e waveform Vg. T h i s f l i p - f l o p i s s e t by F *M. d e l a y e d w i t h an RC i n t e g r a t o r and r e s e t by W, = F + IYL. The l e a d i n g edge of M-^  y i e l d s V^. One can b e s t understand the b e h a v i o u r of the whole system by s i m u l t a n e o u s use o f F i g u r e s 1 - 2 , 2 - 4 , 2-8, 2 - 9(a) and 2 - 9 ( b ) . F i g u r e s 2 - 9(a) and (b) i l l u s t r a t e the time sequence of the c o n t r o l l o g i c by means of p e r t i n e n t t i m i n g waveforms o b t a i n e d w i t h an 8-channel Brush r e c o r d e r . Note.' how the d e l a y l i n e i s f i l l e d (32 " b l i p s " i n S Q waveform) b e f o r e a c t u a l compu-t a t i o n b e g i n s . F i g u r e 2 - 9(a) shows the o p e r a t i o n of the system f o r N = 2^, and F i g u r e 2 - 9(b) shows the t r a n s i t i o n from N = 2^. t o N = 2 2 . 5. F. F i g u r e 2-9(a) Timing Waveforms a t B e g i n n i n g of Computation 34. F i g u r e 2-9(b) Tim i n g Waveforms D u r i n g T r a n s i t i o n from N = 2 1 t o N = 2 2 35. 3. DESIGN AMD IMPLEMENTATION Because of s i z e r e q u i r e m e n t s a l l the l o g i c c i r c u i t r y i s implemented w i t h F a i r c h i l d RTL i n t e g r a t e d c i r c u i t s . The f o l l o w i n g l i s t i n c l u d e s s i x d i f f e r e n t types of m i c r o l o g i c modules used: Type F u n c t i o n uL 900 I n v e r t i n g B u f f e r (B) uL 914 D u a l two-input NOR gate uL 915 Dual t h r e e - i n p u t NOR gate uL 923 JK f l i p - f l o p w i t h " p r e s e t " f e a t u r e uL 926 JK f l i p - f l o p w i t h " p r e s e t " and " p r e c l e a r " uL 927 Expander (4 i n v e r t e r s ) ( I ) The uL 914 and uL 915 modules are i n t e r c o n n e c t e d a p p r o p r i a t e l y where t h e r e i s need f o r more than t h r e e i n p u t s i n a p a r t i c u l a r NOR g a t e . The uL 914 i s a l s o used t o make RS f l i p - f l o p s r e q u i r e d i n the c o n t r o l l o g i c . The s h i f t r e g i s t e r of the d i s p l a y system c o n s i s t s of twenty-one uL 926 modules. Other components used f o r the d r i v e r s and the c o n t r o l are o u t l i n e d i n the s e c t i o n s t h a t f o l l o w . The power-supply v o l t a g e f o r the i n t e g r a t e d c i r c u i t r y i s +3.3 VDC (the c u r r e n t d r a i n i s a p p r o x i m a t e l y 1.8 amperes), s i n c e the uL 926 (type UX5992629X) r e q u i r e s +3.0 VDC +10$ whereas the r e m a i n i n g modules r e q u i r e +3.6 VDC +10$. (Because the system i s t o operate a t room temperature, t h e r e s h o u l d be no problems i n u s i n g +3.6 VDC.) A l s o , a v a i l a b l e from the power s u p p l i e s i n the c o r r e l a t o r p r o p e r are the v o l t a g e s +12 VDC and +6 VDC. 36. 3.1 L e v e l C o n v e r t e r s I n the c o n t e x t of t h i s t h e s i s , p o s i t i v e l o g i c (PL) e x i s t s when the 1 - s t a t e i s r e p r e s e n t e d by a p o s i t i v e v o l t a g e .'. l e v e l and the 0 - s t a t e by ground. N e g a t i v e l o g i c (NL) a p p l i e s t o a system which has the 1 - s t a t e r e p r e s e n t e d by a n e g a t i v e v o l t a g e l e v e l and the 0 - s t a t e by ground. The d i s p l a y system operates w i t h p o s i t i v e l o g i c ( l o g i c l e v e l s : <"^ +3 VDC, G-ND) and the c o r r e l a t o r o p e r a t e s w i t h n e g a t i v e l o g i c ( l o g i c l e v e l s : -^-12 VDC, G-ND). Lev e l " . c o n v e r t e r s are thus n e c e s s a r y i n an i n t e r f a c e between the two. Both c u r -r e n t and v o l t a g e l e v e l s have t o be c o n s i d e r e d i n the d e s i g n of such c o n v e r t e r s , and these s h o u l d s a t i s f y p r e s e t s p e c i f i c a -t i o n s as t o the f a n - i n and f a n - o u t of each i n p u t and output r e s p e c t i v e l y . 3.1.1 Input L e v e l C o n v e r t e r s A w o r s t - c a s e d e s i g n a n a l y s i s f o r the i n p u t l e v e l con-v e r t e r s i s g i v e n i n Appendix A. T h i s Appendix a l s o s e r v e s as an example o f the wo r s t - c a s e d e s i g n t e c h n i q u e s employed i n o t h e r p a r t s o f the system. One t r a n s i s t o r and two r e s i s t o r s a r e r e q u i r e d f o r each c o n v e r t e r (see F i g u r e A - l ) . Each output i s t o be l o a d e d w i t h as many as f o u r i n t e g r a t e d - c i r c u i t i n p u t s , each w i t h a f a n - i n o f ~ 0 . 6 ma a t ~ 1 VDC (3 F a i r c h i l d u n i t s ) . The main r e s t r i c t i o n i n the d e s i g n l i e s i n the f a c t t h a t Rg, the Thevenin e q u i v a l e n t r e s i s t a n c e o f s o u r c e s t o the i n p u t s of the c o n v e r t e r s , i s not n e g l i g i b l e and cannot be o v e r l o a d e d . (With R q n e g l i g i b l e , no t r a n s i s t o r would be needed.) Complete con-37. v e r s i o n from n e g a t i v e l o g i c t o p o s i t i v e l o g i c e x i s t s o n l y when t h e r e i s an i n v e r t e r a t the o u t p u t , V . W i t h no i n v e r t e r , the L output i s de s i g n e d to behave as the complement of the p a r t i c u l a r p o s i t i v e l o g i c f u n c t i o n i n q u e s t i o n . S i x t y such c o n v e r t e r s are r e q u i r e d w i t h a v a r i e t y of i n p u t and output s p e c i f i c a t i o n s (see Appendix A ) . F o r reasons of economy, one was des i g n e d which i s s u i t a b l e f o r a l l i n p u t s : the o u t p u t s - o f the r e v e r s i b l e c o u n t e r , the ou t p u t s of the sample c o u n t e r , the t i m i n g and c o n t r o l waveforms, R^, M ^ , i M ^ , M ^ , S Q , S , S , and the e x t e r n a l c l o c k waveform, a l l of which o r i g i n a t e i n n e g a t i v e l o g i c . 3 .1 .2 Output L e v e l C o n v e r t e r s and D r i v e r s I n g o i n g from the p o s i t i v e l o g i c l e v e l s of the d i s -p l a y system t o the n e g a t i v e l o g i c l e v e l s r e q u i r e d by the c o r -r e l a t o r , the c i r c u i t shown i n F i g u r e 3-l(a) i s employed. T h i s c i r c u i t c o n v e r t s p o s i t i v e l o g i c t o n e g a t i v e l o g i c i n i t s comple-mentary s t a t e , s i n c e t h e r e i s an even number of v o l t a g e i n v e r -s i o n s . The minimum v o l t a g e l e v e l s o f + 1 VDC and 0 VDC a s s u r e s p r o p e r o p e r a t i o n of the c o n v e r t e r . I t i s used as an i n t e r f a c e module t o d r i v e V t o NL-AND gates a t the i n p u t s of the r e v e r -s i b l e and sample c o u n t e r s , the s h i f t p u l s e , P s , t o the d e l a y l i n e , and the c l o c k , PQ2» ^° ^ e c o r r e l a t o r . The c i r c u i t i n F i g u r e 3 - 1 ( b ) can be used t o d r i v e a r e c o r d i n g d e v i c e w i t h V . F o r example, an XY r e c o r d e r might r e q u i r e remote c o n t r o l t o d r i v e one or more r e l a y c o i l s , depending on whether one has t o c o n t r o l the c a r r i a g e r e t u r n , s t o p - s t a r t s i g n a l , e t c . F o r v e r s a t i l i t y the 2N3638 t r a n s i s t o r , 38. +12 V OV 1-\<2.2K n / W (PL Lfl/FLS) , , o - M A / * — 2N3€t6 1 ^ -o r O V -I - /2V 4 -/2/ (PL LEVELS) o - / VW V ov 3 . 9 K T I I {RELAY ! -/2 V o -25TV AM* F i g u r e 3-1 Output L e v e l C o n v e r t e r s 3 9 . ( I Q Zzt 500 ma, ^CEO ~ c o u l d d r i v e a r e l a y w i t h max max a s e t of normally-opened and n o r m a l l y - c l o s e d c o n t a c t s . The a l t e r n a t i v e i s t o d r i v e the a p p r o p r i a t e r e l a y s i n the r e c o r d e r d i r e c t l y . Both c i r c u i t s can be des i g n e d t o handle the w o r s t -case s i t u a t i o n s i n a s i m i l a r manner t o t h a t o u t l i n e d i n Appendix A. 3 . 2 NOR-logic Implementation Fo r reasons of s i m p l i c i t y , f l e x i b i l i t y and economy, NOR l o g i c components are- used i n the d i s p l a y system. T h i s f a c t thus had t o be kept i n mind i n d e s i g n i n g the o r i g i n a l AND/OR l o g i c v e r s i o n i n o r d e r t o mi n i m i z e the number of a d d i t i o n a l i n v e r s i o n s n e c e s s a r y f o r p r o p e r c o n v e r s i o n , (For example, one r e c a l l s the two a l t e r n a t i v e c o n f i g u r a t i o n s f o r the s i g n -p r o c e s s i n g g a t es i n s e c t i o n 2 . 3 . 3 . ) With the s e t of r u l e s o u t l i n e d i n Appendix B, the d i s p l a y system i s implemented as i n F i g u r e s 3 - 2 , 3 - 3 and 3 - 4 . These diagrams a re i n c l u d e d t o i l l u s t r a t e how the c o n v e r s i o n from AND/OR l o g i c t o NOR l o g i c i s a p p l i e d , and t o a c t as a guide f o r t r o u b l e - s h o o t i n g purposes. I n d i c a t e d a t the bottom of each diagram i n par e n t h e s e s a re the l o g i c v a r i a b l e s t h a t behave as n e g a t i v e l o g i c v a r i a b l e s ; the r e m a i n i n g v o l t a g e s behave as p o s i t i v e l o g i c v a r i a b l e s . F o r s i m p l i c i t y , the i n p u t l e v e l c o n v e r t e r s a re not shown. Four a d d i t i o n a l i n v e r t e r s a r e shown i n F i g u r e 3 - 3 f o r and I g Q — t h e s e a re n e c e s s a r y because of the f a n - o u t l i m i t a t i o n s of the uL 926 module. The component v a l u e s of the waveform-shaping c i r c u i t s a r e g i v e n i n t h e s e c t i o n t h a t f o l l o w s . 40. F i g u r e 3-2 Input Gates t o D i s p l a y - S y s t e m S h i f t R e g i s t e r ( N I i : D., G., R T) F i g u r e 3-4 C o n t r o l L o g i c Gates (UL: S Q, M 3, Gk^, P s , F C 2 ? V and V a f t e r conv.) n n 43. A v e r y u s e f u l a l t e r n a t i v e s et of l o g i c symbols to represent an implemented l o g i c diagram i s proposed by '. P. M. K i n t n e r ^ \ and i s b r i e f l y d e s c r i b e d i n Appendix B f o r RTL c i r c u i t s . 3.3 Waveform-Shaping C i r c u i t s Various waveform-shaping c i r c u i t s appear i n c e r t a i n p a r t s of the system. These are necessary f o r proper t i m i n g and to prevent spurious l o g i c c o incidences from o c c u r r i n g . A l l such operations are done at the i n t e g r a t e d - c i r c u i t l o g i c l e v e l s . 3-3.1 The Astable and Monostable M u l t i v i b r a t o r s This s u b - s e c t i o n i l l u s t r a t e s how the choice of a s u i t a b l e d i s p l a y c l o c k r a t e and corresponding delays V ^ and V can be made f o r a p a r t i c u l a r r e c o r d e r , the E l e c t r o -car r Instruments X-Y Recorder (M0-500). To have the f u l l t h i r t y - t w o words of the core memory d i s p l a y e d on an 8-5-" x 11" sheet, i t may be d e s i r a b l e t o have approximately f o u r normalized estimates per i n c h . With the XY re c o r d e r , the d i s p l a y s can be superimposed one upon the other, a l l o w i n g the observer to compare the l a s t two or three d i s p l a y s . (A storage o s c i l l o s c o p e can be used q u i t e e f f i c i e n t l y i n t h i s regard, e s p e c i a l l y i f the d i s p l a y time i s to be kept to a m i n i -mum. ) The r e c o r d e r gave best response when running at 0.05 inches per second; t h i s d i c t a t e s the need to d i s p l a y a p p r o x i -mately one estimate every f i v e seconds, r e q u i r i n g a d i s p l a y c l o c k r a t e of ~4.8 cps. The slewing speed f o r c a r r i a g e r e t u r n i s ' - ' 2 5 i n c h e s / s e c . Delays of over one second and two seconds 44. are therefore used f o r and V r e s p e c t i v e l y , l e a v i n g 32 car r J 9 b approximately one second f o r the carriage r e t u r n . The c i r c u i t of Figure 3-5(a) i s employed to generate v_ 0 and V w i t h values of T dependent on the type of recorder jc- car m used. The astable m u l t i v i b r a t o r of Figure 3-5(b) serves as a l o c a l o s c i l l a t o r f o r the d i s p l a y clock with, v a r i a b l e frequency adjustment. Because both these c i r c u i t s have a very low power-supply v o l t a g e , the formulas f o r T^ and T^ i n Figure 3-5 are (5) only approximate. As given by Millman and Taub , the more exact formula f o r the monostable c i r c u i t i s T = R C m m m v _ V C E ( s a t ) + V B E ( s a t ) ln2 + I n (-22 _ 1 ) VCC " h where i s the cut i n v o l t a g e . Now, ^Qj]( s a-t) + ^ B E ( s a t ) ~ 2 ^ at room temperature so that T^~ 2" l n 2 , i f ^QQ~^\• However, = 3.3 vDC i n t h i s a p p l i c a t i o n and the approximation can thus be s l i g h t l y i n e r r o r . A s i m i l a r argument holds f o r T . There i s an upper l i m i t of lOKn. on R and R to m a ensure s a t u r a t i o n of the t r a n s i s t o r s of the uL 914 module. The two monostable c i r c u i t s are s e n s i t i v e to spurious pick-up as t h i s upper l i m i t i s reached. A l l three m u l t i v i b r a t o r s are s t a b l e w i t h R = 5.6 Ka and R = 6.8KH . The a d d i t i o n of the 1N456 m a diode and the 680-ohm r e s i s t o r i n Figure 3-5(b) r e s u l t s i n an output c o l l e c t o r waveform w i t h v e r t i c a l edges. 3.3.2 D i f f e r e n t i a t i n g and I n t e g r a t i n g C i r c u i t s Only the p o s i t i v e - g o i n g t r a n s i t i o n s of the outputs C. ( j = 1 , 2, 18) are to be detected to generate the Figure 5-5 The uL 914 Module Connected as a M u l t i v i b r a t o r (a) Monostable (b) Astable voltage V-g. This i s done w i t h simple CR d i f f e r e n t i a t o r s as shown i n Figure 3-2. The minimum time between two C. pulses i s that taken to switch from C^ to C^ corresponding to the 1 2 a r r i v a l of N = 2 and N = 2 counts r e s p e c t i v e l y . Since the c o r r e l a t o r can acquire samples at a maximum rat e of~20 K cps, the minimum time d u r a t i o n of two sampling i n t e r v a l s i s about 3.2 msec, Hence, i f the' d i f f e r e n t i a t e d s i g n a l i s ;to decay'to <2$ of i t s maximum value i n 2 msec,,, the time constant should 46. be l e s s than~0.5 msec. The value of the c a p a c i t o r was there-fore chosen to be 0.1 uf, and the r e s i s t o r , 3.3K. By s i m i l a r reasoning, the i n t e g r a t o r and d i f f e r e n t i a t o r , producing ¥^ and r e s p e c t i v e l y i n Figure 3-3, have c a p a c i t o r values of 0.001 uf and r e s i s t o r values of IK. The same values are used to d i f f e r e n t i a t e i n Figure 3-4° In Figure 3-4, P g and F g are d i f f e r e n t i a t e d w i t h 0.1 (if and 3-3K; s i m i l a r l y , R^ = 3.3K and C^ = 0.1 uf. F i n a l l y , 10 uf and 10K are used to d i f f e r e n t i a t e V_ 0 and V because of ^ 32 car t h e i r poor f a l l times. 3.4 The Digital-to-Analogue Converter Both the b i t d r i v e r s and the r e s i s t i v e elements i n t r o -duce e r r o r s at the output of a D/A converter. An estimate f o r these e r r o r s i s derived here f o r the ladder-type converter used i n the d i s p l a y system. The b i t d r i v e r shown i n Figure 3-6 i s an extension of that proposed by C. R. Pearmann ^ \ The ladder input v o l t -age i s E-g = +E, depending on whether the 2N1304 t r a n s i s t o r i s i n c u t o f f or i n s a t u r a t i o n . The 2N3646 t r a n s i s t o r and r e s i s t o r s R,, R„ and R_ e s s e n t i a l l y convert the m i c r o l o g i c l e v e l s to those required by the b i t d r i v e r . R^, Rg, and R^ were s e l e c t e d by a worst-case a n a l y s i s s i m i l a r to that f o r the l e v e l converter i n Appendix A. One must compromise between making R^ and R,_ («-g"^) small to ensure s u f f i c i e n t conduction i n both diodes and making R^ l a r g e so as not to overload the +12- and - 6 - v o l t l i n e s . With R. % 820H and Rr-» 390n , the s a t u r a t i o n current of the 4 o 820 LADDER ! 2R 47, D 5; 4.7* •590 5% IN4I£+-PMRS MATCHED WITHIN S>V AT € ma. F i g u r e 5-6 B i t D r i v e r cc fc •a 4 -M/V^— ft (a) (hor. s c a l e : .1 V / d i v . ; v e r t , s c a l e : 1 ma/div.) F i g u r e 3-7 (a) Model f o r Upper Reference V o l t a g e (b) T y p i c a l C h a r a c t e r i s t i c of IN4154 2N1304 i s ~22 ma, and the di o d e c u r r e n t s are a t ~ 6 ma d u r i n g c u t o f f . 48. There are three f a c t o r s which w i l l cause an e r r o r i n E-g when Eg ~ -E. For R^ z: 8 2 0 A , an output impedance of 0.1 A . i n the lower reference power supply would cause ~ r e g u l a t i o n f o r a 200 ma load (9 b i t d r i v e r s ) . Secondly, i t was found by measure-ment that Vp-g ~ 45 +10 mv f o r a l l nine d r i v e r s . L a s t l y , the sat ladder w i l l load one d r i v e r by l e s s than 0.8 ma when a l l other d r i v e r s have opposite p o l a r i t y f o r E^. However, t h i s causes V_,g to change by ^ 2 mv ( n e g l i g i b l e ) because of the low sat s a t u r a t i o n r e s i s t a n c e . The expected inaccuracy i n the lower reference voltage i s thus <+0.5$ + 45 mv. -E^ can be set to -6.045 v o l t s to compensate f o r the 45 mv o f f s e t voltage. When both diodes are conducting and n e g l e c t i n g the 2N1504 leakage current (~2 ua), one can make a piece-wise l i n e a r approximation to the 1N4154 diode c h a r a c t e r i s t i c of Figure 3-7(b) and consider the model of Figure 3-7(a) f o r an estimate of the e r r o r produced with the upper reference voltage. Assuming r ^ ||R^  » r ^ and r ^ + R^ « R^, the Thevenin equivalent v o l t a g e , E p , i s the f o l l o w i n g : •"o.c. E ~ __ kk . + a + ± B 0 . C 1 + _ A _ P d 1 + r d 2 ^1 + ^ 2 1 + R 4HR5 1 + R 4 d 2 i — + R 4ll R 5 And, f o r r d « r d 2 ~ r d = 9 ± l s i > RA = Q 2 0 ^ ± ^ ° > R 5 = 390fl+5%, 49. AE B O.C, AV CCl AE. AV _ 46.6 + Hi d n AV 1.034 max "7 2mv (a l lows VQQ to be +1%) 6mv (a l lows Eg to be + .1%) + 1.022 + 1.034 + N v ' <5 mv + 14.6 \ _ Ar, 1 + 8.1 Ar. + 0.24 AR '4 + 0.08 ARr where AE u 2 A / 32.7 mv and changes i n vo l tage are i n m i l l i v o l t s ; changes max J0.C. i n r e s i s t a n c e s , i n ohms. Hence, AE B O.C, <46 mv. The nominal v o l t a g e Eg ~ 6.06 v o l t s . There fore , the e r r o r i n the upper re ference v o l t a g e i s + 0.76% + 60 mv w i t h no l o a d . The Thevenin e q u i v a l e n t c i r c u i t f o r the upper re ference vo l tage can now be represented by ->• V B O.C. 6.06 v +0.76% so t h a t w i t h <0.8 ma l o a d there i s l e s s than 16 mv r e g u l a t i o n . There fore , i n the worst case , Vg could be 6.06 v ^ ° 0 6 % O R 6 0 0 5 7 V ± 0 ° 8 2 ^ ° + E H c a n compensate f o r the 57 mv o f f s e t by making i t equal to +5.941 v o l t s . I f the upper re ference power supply has an output impedance of 0.1 - Q , there w i l l be ~ 5 mv r e g u l a t i o n ( n e g l i g i b l e ) to accommodate ~ 5 5 m^a change f o r the d i f f e r e n t s t a t e s of the b i t d r i v e r s . A l l o w i n g +1% on both the upper and lower re ference v o l t a g e s would produce a maximum of +1% e r r o r of f u l l s ca le at the output of the D/A conver ter (see Appendix C)„ Al though 5 0 . t h i s i s ~ 2 . 5 L S B , one should r e c a l l that i t i s a worst-case e r r o r estimate. As i s evident from the a n a l y s i s of Appendix C, the ladder sections c l o s e s t to the output of the converter have most s i g n i f i c a n c e i n c o n t r i b u t i n g to the output e r r o r . In t h i s a p p l i c a t i o n , the four most s i g n i f i c a n t b i t d r i v e r s have c l o s e l y matched 2 N 1 3 0 4 t r a n s i s t o r s a t u r a t i o n voltages (+5 mv), and the 1 N 4 1 5 4 diodes have low dynamic impedances ( ~ 8 . n . ) . Weighted-tolerance r e s i s t o r s , as prescribed i n Appen-d i x C, were used f o r a 5 K - 1 0 K ladder network, the most s i g n i f i -AV , cant being + 0 . 0 0 5 % . This makes the r a t i o — ° ^ — l e s s than ~ 0 . 0 5 % due to the r e s i s t i v e elements only (an unattainable upper bound). This e r r o r i s ~ T L S B and n e g l i g i b l e compared to the e r r o r produced by the b i t d r i v e r s . The D/A converter produced s a t i s f a c t o r y d i s p l a y s of correlograms at as high a rate as 2 0 , 0 0 0 correlogram points per second, which i s the maximum rat e at which the c o r r e l a t o r can f u n c t i o n properly. Chapter 4 i l l u s t r a t e s how the i n a c c u r a c i e s produced by the D/A converter c o n t r i b u t e to the o v e r a l l inaccuracy i n the s t a t i s t i c a l estimates measured from the d i g i t a l c o r r e l a t o r . 3 . 5 The D i s p l a y System Layout Figure 3 - 8 i l l u s t r a t e s the various sections of the d i s p l a y system. One n o t i c e s that there are over eighty inputs to be processed and there are only fourteen leads i n t e r c o n -n e c t i n g the two boards. Proper planning f o r layout of the components was necessary to meet p h y s i c a l space requirements on the u n i t . 51. F i g u r e 5-8 The D i s p l a y System Layout ( s c a l e ~2-£-:l) 4. MEASUREMENT OF STATISTICAL ESTIMATES 52. The m o n i t o r i n g system d i s p l a y s the s t a t i s t i c a l e s t i -mate (s) computed by the d i g i t a l c o r r e l a t o r , as the number of samples, N, i n c r e a s e s , u n t i l s a t i s f a c t o r y convergence t o a f i n a l v a l u e o f a p a r t i c u l a r e s t i m a t e has been reached. The f o l l o w i n g s e c t i o n o u t l i n e s the v a r i o u s s o u r c e s of e r r o r i n these e s t i m a t e s , and i n c l u d e s a q u a n t i t a t i v e a n a l y s i s f o r a p a r t i c u -l a r c ase, the a u t o c o r r e l a t i o n of a s i n u s o i d a l waveform. Sec-t i o n 4.2 i l l u s t r a t e s how the measurement of these e s t i m a t e s can be c a r r i e d out. 4.1 Sampling F l u c t u a t i o n s and Measurement E r r o r s There are s e v e r a l f a c t o r s c o n t r i b u t i n g t o the e r r o r i n the s t a t i s t i c a l e s t i m a t e s measured from the d i g i t a l c o r -r e l a t o r . These are d e p i c t e d i n F i g u r e 4.1. The q u a n t i z a t i o n of the i n p u t s i g n a l s causes both a b i a s e r r o r and an a d d i t i o n a l term i n the v a r i a n c e of the c o r r e l a t i o n e s t i m a t e . I t i s shown i n Turner's t h e s i s t h a t these e r r o r s due t o q u a n t i z a t i o n are n e g l i g i b l e r e l a t i v e t o those due t o s a m p l i n g f l u c t u a t i o n s . The major source of s t a t i s t i c a l n o i s e ( o r s a m p l i n g f l u c t u a t i o n s ) i n the c o r r e l o g r a m i s the e s t i m a t i o n of the c o r r e l a t i o n f u n c t i o n from a f i n i t e sample. The s a m p l i n g f l u c -t u a t i o n s a l s o i n c l u d e the i n t e r a c t i o n between p e r i o d i c s a m p l i n g and p e r i o d i c components i n the i n p u t s i g n a l s . However, the l a t t e r i s shown, i n Turner's t h e s i s , t o have n e g l i g i b l e c o n t r i -b u t i o n t o the v a r i a n c e of the e s t i m a t e , except when the sa m p l i n g f r e q u e n c y i s near synchronism w i t h the i n p u t s i g n a l . I n p a r t i c u l a r , the v a r i a n c e of e s t i m a t e d p o i n t s on the c o r r e l o -BIAS ERROR DUE TO QUANTIZATION OF INPUT SIGNALS VARIANCE DOE TO QUANTIZATION OF INPUT SIGNALS ERROR DUE TO DECODING RESISTORS ERROR DUE TO LOADING OF DECODING NETWORK LESS SIGNIFICANT ERRORS ANALOGUE INPUT SIGNALS QUANTIZATION QUANTIZED INPUT SlGNUS DIGITAL CORRELATION CORRELATION ESTIMATE DIGITAL TO ANALOGUE CONVERSION ANALOGUE OUTPUT ANALOGUE VOLTAGE MEASUREMENT 53. VARIANCE DUE TO SAMPLING FLUCTUATIONS ERROR DUE TO ROUND-OFF IN LSB FOR N> 26H ERROR DUE TO BIT PR I VERS ERROR DUE TO MEASURING DEVICE MOST SIGNIFICANT ERRORS F i g u r e 4-1 Sources of E r r o r i n the Measurement of C o r r e l a t i o n E s t i m a t e s 5 4 . gram f o r the a u t o - c o r r e l a t i o n f u n c t i o n of a s i n u s o i d a l waveform, non-synchronous w i t h the s a m p l i n g f r e q u e n c y , can be w e l l approximated as v a r C1±(T) » | ! - |t « | i , (R^Cr) = \- cos (a**)) where "A" i s the peak a m p l i t u d e of the n o i s e comparison l e v e l s , 11N" i s the number of samples, "a" i s the peak a m p l i t u d e of the s i n e wave ( < A ) , and co i s the r a d i a n f r e q u e n c y of the s i n u s o i d . I f one assumes the C e n t r a l L i m i t Theorem t o h o l d f o r a f i r s t - o r d e r a p p r o x i m a t i o n , the p r o b a b i l i t y d i s t r i b u t i o n of the e s t i m a t e s becomes a s y m p t o t i c t o a normal d i s t r i b u t i o n f o r a l a r g e sample s i z e . The s t a n d a r d d e v i a t i o n , o~% can then be r e l a t e d d i r e c t l y t o c o n f i d e n c e l i m i t s on the e s t i m a t e . F o r the a u t o - c o r r e l a t i o n e s t i m a t e of a s i n u s o i d a l i n p u t , we have o- =± A' 2 i - k^)A A 2 [Z 1.(^4 2 V N 7 ' s i n c e a<A. The "dashed" s t r a i g h t l i n e i n F i g u r e 4 - 2 shows the d e v i a t i o n of the e s t i m a t e , C - Q * from the t r u e c o r r e l a t i o n f u n c -t i o n , R-|_i' w i t h a p p r o x i m a t e l y 95% c o n f i d e n c e l i m i t s , assuming a normal p r o b a b i l i t y d i s t r i b u t i o n f o r C -. . As i s o u t l i n e d f o r A 4 s e v e r a l examples i n Turner's t h e s i s , — i s u s u a l l y the dominant term i n the v a r i a n c e of the c o r r e l a t i o n e s t i m a t e . T h i s p l o t can t h e r e f o r e be used f o r o t h e r i n s t a n c e s where an e s t i m a t e of the o r d e r of magnitude of e r r o r i s d e s i r e d . Another major c o n t r i b u t i o n t o the e r r o r i n the o u t p u t , V-Q, i s a r o u n d - o f f e r r o r a s s o c i a t e d w i t h the b - b i t D/A c o n v e r s i o n f o r sample s i z e s , N ^ 20+^~ ( t h i s i s shown as a " d o t t e d " p l o t i n Figure 4 - 2 ) . A t h i r d p r i n c i p a l source of e r r o r i s t h a t due t o AO 4-0.3 + 0.1 (~9S% CONFIDENCE \ LIMITS) N 55. TOTAL £ +0.05 0.0/ ERROR DUE TO BIT DRIVERS \ ROUND-OFF ERROR ±0.003 \ \ \ -\-\ g8 2 / 0 2 / 2 2'* 2 is 2,s N Figure 4-2 Output E r r o r Due to Inherent S t a t i s t i c a l Noise and D/A Conversion 56. the b i t d r i v e r s of the D/A converter; t h i s i s present f o r a l l values of N and was assumed to be ~ 1 % of f u l l scale i n Figure 4-2. In the worst case, the aforementioned e r r o r s are a d d i t i v e , y i e l d i n g a t o t a l e r r o r c o n t r i b u t i o n , A 1 2CT+ 0.01A2 + 2 b*A 2 • U(N - 2 b + 1 ) where U(N - 2 b + 1 ) k 0 , N < 2 b + 1 1 , N ^ 2 b + 1 . e i s therefore an e r r o r normalized to f u l l scale, which i s j u s t i -f i e d , since i t i s constant f o r a given value of N. Because the e r r o r due to the b i t d r i v e r s i s s l i g h t l y p e s s i m i s t i c , the 95$ c o n f i d e n c e - l i m i t curve i s used f o r better.comparison between the two. An e r r o r of l e s s than 2$ of f u l l scale i s expected 18 f o r N = 2 wit h ~95$ confidence. This e r r o r i s quite small considering that the analogue output i s to be measured wi t h a l i n e a r s c a l e . Figure 4-2 al s o i l l u s t r a t e s how the higher values of c o r r e l a t i o n estimates w i l l e x h i b i t b e t t e r accuracy than lower ones, when the e r r o r i s normalized to F i n a l l y , there i s an e r r o r due to the measuring device i t s e l f , which i s again a d d i t i v e i n the worst case. The lo a d i n g of the decoding network by the measuring device i s u s u a l l y small i n comparison. 4.2 Some Examples of Output Displays A storage o s c i l l o s c o p e (Tektronix, type 564) can be used quite conveniently to d i s p l a y the t h i r t y - t w o estimates stored i n the core memory of the computer f o r N = 2 n (n = 1, 57. 2, where "L" i s the l a s t binary i n t e g e r exponent at which s a t i s f a c t o r y convergence i s e s t a b l i s h e d ) . I f the trace i s t r i g g e r e d w i t h v"n and the d i s p l a y clock i s adjusted to produce an appropriate rate of d i s p l a y (governed by R C i n the astable c i r c u i t of Figure 3 - 5 ( b ) ) , the d i s p l a y s can be super-imposed f o r comparison. Previous d i s p l a y s can be erased on the storage o s c i l l o s c o p e when comparing the l a s t two or three. When convergence has been determined, a more accurate i n s t r u -ment can be used to a c t u a l l y measure the t h i r t y - t w o estimates (e.g., Electro-Instruments X-Y Recorder (MO-500) gives ~0.2% of f u l l s c a l e f o r speeds l e s s than 10 in/sec and can be used w i t h 11" x 17" paper). Figure 4 - 3 i l l u s t r a t e s how the d i s p l a y system can be used to determine convergence to a f i n a l v a l u e — t h e auto-c o r r e l a t i o n estimation of a s i n u s o i d a l waveform i s considered. One n o t i c e s how the frequency of the correlogram i s quite apparent from the d i s p l a y a f t e r only two samples were taken. A f o u r -trace p l u g - i n u n i t was used f o r i l l u s t r a t i v e purposes. Figure 4-4 i s a s i m i l a r i l l u s t r a t i o n w i t h a d d i t i v e S noise on the s i n u s o i d a l s i g n a l (^ ~ l ) . This r e s u l t was ob-tained i n the asymmetrical mode of the computer to achieve a sampling r a t e whose frequency exceeded the bandwidth of the band-limited noise source. The amplitude cumulative p r o b a b i l i t y d i s t r i b u t i o n and p r o b a b i l i t y d ensity estimation of t h i s noise source i s shown i n Figure 4-5. The- peak-to-peak amplitude of the noise was set at approximately eighty percent of the ampli-tude l i m i t s of the d i s t r i b u t i o n f o r i l l u s t r a t i v e purposes. Figure 4 - 3 A u t o - C o r r e l a t i o n E s t i m a t i o n of a S i n u s o i d a l Waveform a f t e r 2 n (n = 1 , 2 , 1 2 ) Samples—Symmetrical Sampling Mode Figure 4-4 A u t o - C o r r e l a t i o n E s t i m a t i o n of a S i n u s o i d a l S i g n a l Plus A d d i t i v e Noise a f t e r 2 n (n = 1, 2, ...,12) Samples—Asymmetrical Sampling Mode ui 6 0 . F i g u r e 4 - 5(b) Amplitude P r o b a b i l i t y D e n s i t y E s t i m a t i o n of Noise used f o r F i g u r e 4 - 4 6 1 . The tw o - p o i n t a p p r o x i m a t i o n i n d e t e r m i n i n g the p r o b a b i l i t y d e n s i t y from the c u m u l a t i v e d i s t r i b u t i o n p r o b a b l y accounts f o r the s l o w e r convergence of the l a t t e r . A c c u r a t e measurements of the f o r e g o i n g e s t i m a t i o n s are not i n c l u d e d as t h e y are not d i r e c t l y r e l e v a n t t o the purposes of t h i s t h e s i s . 5. CONCLUSIONS 62. A convenient d i s p l a y system has been b u i l t f o r a general-purpose d i g i t a l c o r r e l a t o r . I t allows one to determine convergence to a f i n a l value of s t a t i s t i c a l estimates, by v i s u a l comparison of superimposed d i s p l a y s , during t h e i r computation. This i s p a r t i c u l a r l y u s e f u l when processing low-frequency s i g -n a l s . Computation can be i n t e r r u p t e d at any sample s i z e N = 2 n (n = 1, 2, 18) f o r measurement of the t h i r t y - t w o estimates stored i n the core memory of the computer. An a l t e r n a t i v e scheme could c o n s i s t of an a u x i l i a r y memory u n i t to store past estimates f o r e l e c t r o n i c comparison wi t h newer estimates. This more complex method would therefore determine convergence e l e c t r o n i c a l l y and the computation could be stopped a f t e r a preset maximum average comparison d i f f e r e n c e had been reached. Although the c o r r e l a t o r can be used to determine time-average estimates f o r slowly v a r y i n g non-stationary processes, the d i s p l a y system w i l l create time "gaps" at N = 2 n (n = 1, 2, ...) unless i t s binary exponent detector i s i n h i b i t e d u n t i l a preassigned sample s i z e i s reached. However, t h i s should not cause any d i f f i c u l t y f o r processes which are q u a s i - s t a t i o n a r y w i t h i n the time taken to accumulate information from'the pre-assigned sample s i z e . B e t t e r s t a t i s t i c a l estimates could be obtained w i t h a more accurate D/A conversion and a l a r g e r maximum sample s i z e . Nevertheless, the r e s u l t s shown i n Figures 4-3 to 4-5 i n d i c a t e 63. 18 the existence of f a i r l y good estimates for N<2 , the maximum sample size of the correlator. An error of less than 2% of I R f u l l scale i s expected for N = 2 (with ^  95% confidence l i m i t s ) for the auto-correlation estimation of a sinusoidal waveform. The operation of the system allows estimates which are a s u f f i c i e n t percentage of the maximum analogue output to be meaningfully displayed (^ =p of f u l l scale minimum 2° for an ide a l 8-bit D/A conversion). Due regard for the inherent s t a t i s t i c a l noise present at a given sample size would s t i l l be necessary i n scaling the values of the estimates. The display system described i n t h i s thesis w i l l c e r t a i nly determine any correlation between two signals greater than ^ 5 % of f u l l scale at N = 2 1 8. 64. APPENDIX A A worst-case design a n a l y s i s i s o u t l i n e d below f o r the l e v e l converter used at the inputs of the d i s p l a y system. Prom t h i s a n a l y s i s , components are chosen to make the converter v e r s a t i l e f o r various d r i v i n g source and output r e s t r i c t i o n s . Six types of sources are to feed the input. Let the Thevenin equivalent voltage and impedance of these sources be Vg and Rg r e s p e c t i v e l y . A l s o , l e t the fan-out of the con-v e r t e r be represented by a voltage V ^ . and current 1^. The f o l l o w i n g two stat e s are to be s a t i s f i e d : State A: -2<V g<0 v o l t s , Rg =s 0 1 < V L < 4 v o l t s , 0.6<I L<2.4 ma. State B: -13 <V g <-10 v o l t s , 0<Rg<2.2Kil -4<V L < 0 v o l t s , I L = 0. S i x t y such converters are required to feed negative l o g i c from Turner's c o r r e l a t o r i n t o p a r a l l e l e d inputs of F a i r c h i l d micro-l o g i c modules as i s shown i n Figure A - l . The lo a d i n g r e q u i r e -ment on Rg i s that | Rg Ig| <1 v o l t . From Figure A - l , V L = V p p - R 1 ( I L + I R ) and I = B \ S R 2 + p where V-g-g and |3 r e f e r to the t r a n s i s t o r , Q. *The h o r i z o n t a l bar placed above or below a given quantity i s to r e f e r to the maximum or minimum of that q u a n t i t y , respec-t i v e l y . i f , I i TV i ft 9. L 1° LOAD I SOURCE SB 65. '/////////////// y 777777777777777777. ~& ~ 7777777777777777 //FSTA TEA.'/// //////////////// '///STATE BA .TT- 7///,///////////// %B ///////////////// — '//STATE &/. -V, 58 ^ 8 Figure A - l Input L e v e l Converter and Associated States Therefore, V P P " R l VS + VBE R + ^ 2 + ~ J 1 + L l R 2 + — I t i s r equired to determine R^, R 2 and J3. For State A: (A-l) -v S A<v s<o R S A ~ ° ! L A < V L < V L A ' ^ L A < I L < I L A • We note from the c i r c u i t diagram that V-^  decreases as I-^ increases. Hence, using (A-l) we obtain the f o l l o w i n g i n e q u a l i t i e s : 66. V S A " V B E ZjT - R 1 ( I L + B 2 E. — > Z M = V + V and w.——<\A 1 + 4 /from w h i c h Vo,. + V T . - V,,-V T A " VT<T? v Ro SA _LA BE  L ^ _ B E = ===== (A-2) VPP " V L A " R l h 1 V P P " V L A " R l h F o r S t a t e B: " VSB< VS<2!SB ' 0 < R S < R S B . \ B < V L < \ B " > 1LB = ° -F o r t h i s s t a t e , a low v a l u e of Rg and/or h i g h a b s o l u t e v a l u e of Vg w i l l l o w e r the v a l u e of V-^ . W i t h ( A - l ) we thus have and ( R 2 R S B +• _ )vpp -8 — V V S B - V R S B R-L + R 2 + — ~ — P ( R 2 , _ S B N V + ^ ;vp  V V S B R 1 + R 2 R S B > V L B < V L B = 0 • 67 For Rg-g = 0 and = 0, we have V S B " V B E RQ-T, R 0 V S B + V L B V B E - SB_ > _2 > — — ^ R l R l VPP " VLB (A-3) Now, the i n e q u a l i t i e s (A -2) and (A -3) can be rearranged on the basis of two cons i d e r a t i o n s . For the c i r c u i t to operate r e l i a b l y , we must have VSA + VLA " VBE VPP " VLA " R l h El R 0 "^ SB "^ BE Rnr, (A-4) PP Upon a choice of , the device Q-, ( F a i r c h i l d module) w i l l be 1 protected i f V - V ™ R ?  L — B E > R f > V L A < + 4 v o l t s (A-5a) VPP " VLA " R l h VSB + VLB " VBE R, and VPP " VLB < ; V T 1 3 > -4 v o l t s LB (A-5b) R 2 The bounds on — i n (A-4) produce another i n e q u a l i t y , from which f 2 ( R x ) t R± + 1 ( 7 ^ - v )^ vpp R g B 1_ Rn VPP " VLA VPP< VSA + VLA " VBE> 68, d f ( I L ) and, ^ ( R ^ ) has a minimum when — ^ = 0 , f o r which <ZPP - I L A ) V P P R S B < V S B " ^ h , & (A - 6 ) Sub s t i t u t i n g t h i s value of i n f^(R^)<C y i e l d s l i m i t s on permissible values f o r SB. A subsequent choice of SB w i t h i n these l i m i t s produces an allowable range f o r R^, and hence, Rg. R e c a l l i n g that the above ranges s p e c i f y a working range f o r the converter, a f i n a l choice of R^, Rg, and S_ r e s t s on whether V^ A < 4 v o l t s and V-Fjg> -4 v o l t s , by i n e q u a l i t i e s ( A - 5 ) . Loading on Rg-g must a l s o be checked. F i n a l l y , depending on Rg-g, the n e c e s s i t y of the t r a n s i s t o r , Q, i s determined from the range of permissible SB. 1 For V p p = 13 v, V p p = 11 v, V g B = 10 v, V g A = 2 v, V ^ = 1 v, Vg-g = 1 v, V-g-g = 0.7 v, 1-^= 2.4 ma, and s u b s t i t u t i n g (A - 6 ) i n f^(R^) <C, a f t e r s i m p l i f i c a t i o n we get R 2 Because — must be greater than 0 i n ( A - 4 ) , t h i s r e s u l t y i e l d s 1 an upper l i m i t on SB, that i s , R s B - - 0 . 5 2 . 1 I f V p p = Vp, the + 12 VDC l i n e would not be necessary f o r the d i s p l a y system. However, f o r Vpp = 3 v and Vpp = 4 v, SB <0.15. I t i s a l s o of i n t e r e s t to note that the upper l i m i t i s 0.92 f o r V p p - > 0 0 • Analyses were c a r r i e d out f o r Vpp ~ 12 VDC and Vpp = Vp 3-3 VDC. The l a t t e r had t i g h t e r l i m i t s on R^, Ro and j3, and more current d r a i n (~4 times) was required from the power supply, Vpp. Graphical r e s u l t s f o r the former are shown i n Figure A-2. Rearrangement of f-^(R^) <C and numerical s u b s t i t u t i o n f o r Vpp = 11 v and Vpp = 13 v generates the f o l l o w i n g i n e q u a l i t y : f 2 ( R 1 ) H R 2 -(2.78 + 1.445 ^I^R-L + 6.02 <0. The root locus of f 2(R-^) - 0 gives an allowable region of values p f o r R, and SB. From (A-4) we get an upper and lower l i m i t f o r 1 R~~ R„ depending on R, and • SB-. The slope of the upper l i m i t i s 1 in-f i x e d and i t s bias depends on SB (=0.1 i n Figure A-2). The 1 ' lower l i m i t i s f i x e d f o r a given R^. The highest standard values of R^ and R 2 are chosen f o r a minimum d r a i n on Vpp. R. =2. 2K'A and R 9=l. 2Ksi permits R S B <0.3. Hence, J3 « 30 for; .a 2N3640 t r a n s i s t o r i s more than adequate f o r Rg-g = 2 . 2 K i l . These values y i e l d V ^ = 2.06 v, V ^ = -3.43 V, I R — p — = 0.7 v. Experimental r e s u l t s f o r three such converters "~2 gave values w e l l w i t h i n these ranges. A s i m i l a r a n a l y s i s was done r e p l a c i n g R 2 by a zener diode. Results s i m i l a r to those f o r R^ = 2.2K.TL, R 2 = 1.2K.TL, are obtained w i t h R^ = 680SL and an\lN756A i n place of (Vpp = 3.3 v ) . / • + / . a /.0 0,8 0.£ OA 0.2 0 O.Z OA 0.6 F i g u r e A-2 A l l o w a b l e Values of R^, R 0, and £ f o r O p e r a t i o n of C o n v e r t e r APPENDIX B 71. Implementation of a l o g i c system o f t e n r e q u i r e s the use of NOR and/or NAND gates f o r reasons of i s o l a t i o n , r e l i a b i l i t y , f l e x i b i l i t y and i n t e r c h a n g e a b i l i t y . Because of i n h e r e n t n e g a t i o n i n such g a t e s , the f o l l o w i n g s e t of r u l e s can be used, a l o n g w i t h (7) s u g g e s t i o n s made by C. F. H i l l , t o c o n v e r t an e x i s t i n g OR/AND l o g i c system t o a NOR/NAND l o g i c system: (a) S t a r t i n g from the output of a cascade of OR/AND g a t e s , r e p l a c e by or xy x • V-•xy x-y-x+y •x+y x-y--M o »x+y (b) I f t h e r e i s an odd number of gates (OR/AND) a l o n g a p a r t i c u l a r p a t h of t r a n s m i s s i o n , an i n v e r t e r a t the i n p u t or output i s r e q u i r e d i n a d d i t i o n t o the minimum c o m b i n a t i o n of NOR/NAND g a t e s ; no i n v e r t e r i s r e q u i r e d i f t h e r e i s an even number of gates ( d i r e c t g a t e - f o r - g a t e c o n v e r s i o n ) . (c) I f e i t h e r s t r a i g h t NOR or s t r a i g h t NAND l o g i c i s used, (b) i s t r u e o n l y i f the OR/AND gates a l t e r -n a t e from the output towards an i n p u t . (a) 72. I f two l i k e g a t e s occur i n cascade, an i n v e r t e r r e p l a c e s t h e i r d i r e c t c o n n e c t i o n i n the c o n v e r s i o n t o e i t h e r NOR l o g i c or NAND l o g i c ; r u l e s (a) and (c) t h e n a p p l y from the i n p u t of the i n v . e r t e r towards the r e s t of the OR/AND-logic t r e e . F i g u r e ( B - l ) i l l u s t r a t e s use of the f o r e g o i n g r u l e s f o r c o n v e r s i o n of an OR/AND-logic t r e e t o a NOR-logic t r e e . I n o r d e r t o r e t a i n the sense of the o r i g i n a l Boolean e x p r e s s i o n on the implemented l o g i c diagram, P. M. K i n t n e r s u g g ests t h e use of the f o l l o w i n g A l l - s y m b o l s f o r RTL c i r c u i t s ponds t o the 1 - s t a t e a t t h a t p a r t i c u l a r p o i n t i n the diagram; a dashed l i n e i n d i c a t e s t h a t a low v a l u e r e p r e s e n t s the 1 - s t a t e . T h i s i s r e a l l y a n o ther way of r e p r e s e n t i n g a NOR gate i n i t s use t o p e r f o r m an AND- or OR- o p e r a t i o n . A p p l i -c a t i o n of these symbols t o t h e t r e e of F i g u r e B - l y i e l d s the implemented l o g i c diagram of F i g u r e B-2. where "1" - h i g h "1" - low A s o l i d l i n e means t h a t a h i g h v o l t a g e v a l u e c o r r e s -73. 4 _ t T r A/OT i l NOT wit «or L i T Y A/OR WOf? A/OK /VOT Figure B - l Conversion of an AND/OR-Logic'Tree to a NOR-Logic Tree Figure B-2 AIL Symbols f o r RTL Implementation of Figure B - l APPENDIX C 74. Figure 0-1 A Ladder-Type -D/A Converter I t i s the purpose of t h i s appendix to determine the e r r o r produced at the output of a 2R-R ladder-type D/A converter due to i n a c c u r a c i e s i n the r e s i s t i v e elements and to the h i t d r i v e r s . The a n a l y s i s f o l l o w s very s i m i l a r steps o u t l i n e d by (8) 3 1 1. T. Lovas , except that r = ^R and R N = TTR i n h i s t h e s i s . * * xi 2 n-1 2 I t i s c a r r i e d out using i n f i n i t e s i m a l c a l c u l u s , under the assumption that t h i s f i r s t - o r d e r approximation i s j u s t i f i e d f o r small e r r o r s i n .the network components. In Figure C - l , l e t r = r = r, = ... = r, -o 1 k R = RN o 1 = r _-, = r = 2R) n ± n } nominally. ... = R = R J A n-1 n-2 k "k ~ * * * ~ n-1 x, x . and x r e f e r to the s i g h of I o n & P, where P i s a base-10 i n t e g e r number, x^ . = 0 or 1 (k = 0, 1, ..., n). Binary conversion i s performed by switching the voltage sources E^ (k = 0, 1, n) i n such a way that 75. E, = x, E U - x' E T k k H, k L, k k where E ^ and - E T are the a c t u a l v o l t a g e s r e a c h i n g the j u n c t i o n k k A of t h e r e s i s t o r r, w i t h the source E, . L e t ET, - E ^ - A E ^ J£ J£ n-, Jti i i , A k k and E ^ = E-^ - A E , , where EJJ and - E , are the power s u p p l y v o l -k k tages and E „ and - E Y are the l a d d e r r e f e r e n c e v o l t a g e s . Hence, i f the power s u p p l y v o l t a g e s EJJ and - E ^ are a d j u s t e d so t h a t E ^ zz. E ^ t E , n o m i n a l l y , t h e n k k E. = E k AE. E A E , x k ( 1 - - r ^ } - x k ( 1 - E 0 The t e r m i n a t i o n r e s i s t o r r i s grounded t o have an e q u a l number of l e v e l s f o r b o t h p o s i t i v e and n e g a t i v e v a l u e s of P, making i t i m p o s s i b l e t o have e x a c t l y 0 v o l t s output under i d e a l c o n d i t i o n s . Another t e r m i n a t i o n R n ( = 2R e x a c t l y ) i s added on the output f o r convenience i n the e r r o r a n a l y s i s , s i n c e i t makes A, B^ = 2R, n o m i n a l l y , f o r a l l k. The output v o l t a g e w i t h the t e r m i n a t i o n , R , s h a l l be r e f e r r e d t o as V ,, so t h a t ' n' out' Hence, 3 ' (9) the n o m i n a l v a l u e of V , = V . out 2 out V out 2 V t ( r ' r i ' R i ' V i = 0 , 1 , n) dV out 2 2 a V o u t 6 r n d r + I i = 0 (iZout d V U t d R > b r . I BR. i 2> v + w i t h dR = 0, n ' and V out n v n , k r i ' R i ' E i ; 1 = °* l f ' " ' n ^ ' w h e r e t h v n ^ i s the output v o l t a g e due t o the k source when R^ = 2R. 7,6. Hence, av out = fit • YL fjJfr*°*i + + ; (k=0 i = 0 k=0 1 1 1 Now, i t can be shown that E, v n,k 1 + r ^ t + k > k k | 1 + i m=l c 1 + m+1 m+1 o u^. = +E ( 2 ^ j 1 ^ 1 " L ) > nominally. Also, a f t e r considerable ana V manipulations and s u b s t i t u t i o n of the nominal values f o r con-venience i n s i m p l i f i c a t i o n s , one obtains n-1 , - i - l dV out 3-2 0 n + l ) dr dr. dR. + C k=0 ^  -1 i=0 Lk=0 n dr. dR. ^-- ( E 1 . 2 1 ) ( 2 ? ^ . + 5^) + k=i+l 0. , dr. dR. E k 2 ) ( r 7 + 2wf] r n - l £Z E k 2 k ) - 2 n + l E n ^ + 5XI2l^4 i=0 J >-k=0 Note that a given summation vanishes i n the above expression i f the lower l i m i t of the running index exceeds the upper l i m i t . Let a signum column vector X have components c o r r e s -dr d r ' d R i ponding to the s i g n of — , ( 1 = 0 , 1, n ) , ^ -^ ( i = 0, i i 1, n - l ) ; a signum column vector Y, to the sig n of E^ (k = 0, 1, n ). Al s o , i f |dEj ^ |dE| ( a l l i ) , the above expression can be r e w r i t t e n as AV- . = AV , + AV , out o u t ; g out-^ 77. where AV out E < ( l - —|dE| ) i n the worst case , and ^ 2 n AV out t Y TAX, AR where A i s a (2n+2) by (n+l) mat r ix of preass igned c o e f f i c i e n t s dr p r e s c r i b e d by the f o r e g o i n g equat ions , and AR (= r e s i s t o r t o l e r a n c e r e f e r e n c e . n 'n ) i s the I t would be of i n t e r e s t to determine what v e c t o r s X and Y y i e l d a maximum A V q u ^ . f o r a g i v e n A. However, f o r the purpose of t h i s t h e s i s , an approximat ion to t h i s upper bound w i l l s u f f i c e . AV" O UH- has been computed f o r three cases , a l l of which assume d r . dR. 1 X r . — R. 1 X , ( i = 0 , 1, n - l ) , and each of which ass igns e i t h e r equal t o l e r a n c e s to a l l r e s i s t o r s ( i . e . , AR, ( i = 0, 1, n - l ) ) or weighted d r ^ dr dr n r . X r r n t o l e r a n c e s to a l l r e s i s t o r s ( i . e . , d r . = 2 n-x dr n n dr 2 n + 1 dr n r r n ( i = 0, 1, . . . , n-1), the f o l l o w i n g : ( i ) E^ = +E, E. = - E ( i = 0, 1, n ' x ( i i ) E ± = (-1)XE ( i = 0, 1, . . . , n) ). The three cases are , n-1) d r . dR. For ( i ) and ( i i ) , sgn ( j r ) = sgn ( E q ) , sgn(^ r - i ) = sgn ( ^ - i ) = - s g n ( E ( i i i ) AV O U t T <Sum of absolute va lue of a l l terms i n AR.(Y iAX) The upper bound of ( i i i ) i s p h y s i c a l l y not r e a l i z a b l e , even AV i n the worst case . The r e s u l t s of computation of the three cases are t a b u l a t e d i n F i g u r e C-2. out R AR f o r 78. Cases Equal Tolerances Weighted Tolerances ( i ) 1 + 6x2 n ( i i ) 42 + ( - D n _ io + ( - D n 45 „r-..„n ^..„n+l 45x4 9x2 f ( n + 1) ( i i i ) 1 _ (2n + 7) 3x2 n+1 A 1 M a f g 0 n r i t n u d e ( i ) = ( i i ) ( i ) = ( i i ) ( i ) > ( i i ) ( i ) = ( i i ) >3 ( i ) < ( i i ) ( i ) < ( i i ) l a r g e ( i i ) ^ ^ ( i i i ] ( i i ) * | ( i i i ) F i g u r e C-2 AV out R AR f o r Three Case Studies AV out R 42 45 2n 3 Ar "n n Ar Prom F i g u r e C-2 we thus have the upper bound of s p e c i f i e d as f o l l o w s f o r l a r g e n: E , f o r equal t o l e r a n c e Ar E ^ AV out-r^ ...<! max n r n n 'n E < AV out R < 4n max Ar n 'n E , f o r weighted t o l e r a n c e . The r e l a t i v e magnitudes of cases ( i i ) and ( i i i ) i n d i c a t e the p r o x i m i t y of case ( i i ) to the worst case. We a l s o note that f o r equal t o l e r a n c e s on a l l r e s i s t o r s , a l l three cases give approximately the same r e s u l t s f o r l a r g e n. 79. REFERENCES 1. Turner, R.M., "A D i g i t a l C o r r e l a t o r f o r Low-Frequency S i g -n a l s " , M.A.Sc. Thesis, Department of E l e c t r i c a l Engineering, F a c u l t y of Applied Science, The U n i v e r s i t y of B r i t i s h Columbia, December, 1964. 2. Jespers, P., Chu, P.T., Fett w e i s , A., "A New Method to Compute C o r r e l a t i o n Functions", Paper presented at the I n t e r n a t i o n a l Symposium on Information Theory, B r u s s e l s , September 3-7, 1962. 3. M i t c h e l l , J.N., "Computer M u l t i p l i c a t i o n and D i v i s i o n Using Binary Logarithms", IRE Trans. E l e c . Conrp. , V o l . EC-11, No. 4, pp. 512-17, August, 1962. 4. K i n t n e r , P.M., "AIL Logic Symbols Standard", IEEE Spectrum, V o l . 4, No. 2, p. 5, February, 1967. 5. Millman, J . , Taub, H., "Pulse, D i g i t a l , and Switching  Waveforms", New York: McGraw-Hill Book Co., p. 415, 1965-6. Pearman, C.R., Popodi, A.E., "How to Design High-Speed D/A Converters", E l e c t r o n i c s . V o l . 37, No. 8, pp. 28-32, February 21, 1964. 7. H i l l , C.F., "NOR/NAND Logic the Easy Way", Control Engineering. V o l . 11, No. 5, pp. 81-3, May, 1964-8. Lovas, L.T., "A V e r s a t i l e D i g i tal-to-Analogue Function. Generator and M u l t i p l i e r " , M.A.Sc. Thesis, Department of E l e c t r i c a l Engineering, F a c u l t y of Applied Science, The U n i v e r s i t y of B r i t i s h Columbia, March, 1964-9. Paumard, A., "Roseau de Traduction Numerique Analogique a. Resistances Non Ponderees", L'Onde E l e c t r i q u e , t . XLIV, No. 450, pp. 906-8, septembre, 1964. 

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