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A digital correlator for low-frequency signals Turner, Ross Maclean 1964

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A DIGITAL CORRELATOR FOR LOY-FREQUENCY SIGNALS by ROSS MACLEAN TURNER B.A.Sc, University of B r i t i s h Columbia, 1961 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 Engineering We accept this thesis as conforming to the standards required from candidates for the degree of Master of Applied Science Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA December 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t . c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r mission. The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Department of Date /jtA"- ai £ , / ty£ S~ ABSTRACT A d i g i t a l c o r r e l a t o r o p e r a t i n g by p o l a r i t y - c o i n c i d e n c e i s d e s p r i b e d i n t h i s t h e s i s . Because the comparison l e v e l s f o r p o l a r i t y d e t e r m i n a t i o n are random v a r i a b l e s w i t h r e c t a n g u l a r p r o b a b i l i t y — d e n s i t y d i s t r i b u t i o n , t h e output of the c o r r e l a t o r i s d i r e c t l y p r o p o r t i o n a l t o the c o r r e l a t i o n f u n c t i o n of the i n p u t s i g n a l s f o r a l l c l a s s e s of s i g n a l s . The computer c a l c u l a t e s 32 u n i f o r m l y d i s t r i b u t e d p o i n t s of the c o r r e l o g r a m s i m u l t a n e o u s l y and i s capable 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 or f o r c a l c u l a t i n g p r o b a b i l i t y - d e n s i t y and c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n s . The computer memory i s p r o v i d e d by a 32 x 32 w o r d - s e l e c t memory i core a r r a y . A t h e o r e t i c a l study of the computer o p e r a t i o n as a c o r r e l a t o r i s g i v e n . Two modes of sampling are considered;' ( l ) the s y m m e t r i c a l — s a m p l i n g mode which r e q u i r e s a s h i f t - r e g i s t e r d e l a y l i n e and samples t a k e n a t i n t e r v a l , A T , where A T i s the s p a c i n g between p o i n t s of the c o r r e l o g r a m , and (2) the a s y m m e t r i c a l — s a m p l i n g mode where d e l a y i s o b t a i n e d by sampling the two i n p u t c h a nnels a t d i f f e r e n t r a t e s . R e s u l t s of t h i s s t udy i n d i c a t e the s u p e r i o r i t y of the sym m e t r i c a l sampling system w i t h r e g a r d t o sampling f l u c t u a t i o n s . A comparison i s made between the UBC c o r r e l a t o r and a c o n v e n t i o n a l c o r r e l a t o r on the b a s i s of sampling f l u c t u a t i o n s , w i t h r e s u l t s dependent on the type of s i g n a l i n p u t and the method of comparison. I t i s shown t h a t under a c e r t a i n c o n d i t i o n the UBC c o r r e l a t o r i s s u p e r i o r t o the c o n v e n t i o n a l d e v i c e and i n a l l cases the a v a i l a b i l i t y of the symmetrical sampling mode wholly or p a r t i a l l y offsets the disadvantage of a high sampling-noise l e v e l which i s inherently present because of the quantization process. i i i ACKNOWLEDGEMENT I would l i k e t o thank Dr. A. D. Moore, the s u p e r v i s i n g p r o f e s s o r , f o r h i s h e l p and encouragement d u r i n g the course of t h i s p r o j e c t . I am a l s o i n d e b t e d t o the t e c h n i c i a n s of the E l e c t r i c a l E n g i n e e r i n g Shop f o r t h e i r a s s i s t a n c e . G r a t e f u l acknowledgement i s g i v e n t o the N a t i o n a l R e search C o u n c i l of Canada f o r a s s i s t a n c e r e c e i v e d under B l o c k Term Grant A - 6 8 . v i i i TABLES OF CONTENTS Page LIST OF ILLUSTRATIONS . . • v i ACKNOWLEDGEMENT • v i i i 1 • INTRODUCTION . . . . . . . . . . o . o . . . . « « > . . . . . . > • • . . . » > 1 2. PRINCIPLE OF OPERATION . . . . . . . o o o . . . . . . . . . 4 2.1 Theory . . • • • . • • • • o . o . . . « . . . . o ° e ° * * . e o ° * * * * 4 2.2 Impl ementati on . • 15 2.3 The UBC C o r r e l a t o r 19 2.4 Computer of Average T r a n s i e n t s 25 2.5 Cumulative P r o b a b i l i t y F u n c t i o n 25 2.6 P r o b a b i l i t y D e n s i t y F u n c t i o n 26 3. DETAILS OF THE BLOCK DIAGRAM AND CIRCUITS 28 3.1 S u b d i v i s i o n of the Computer 28 3.2 Timing C i r c u i t s • 30 3.3 Comparator and Sampling C i r c u i t s ......... 38 3.4 The S h i f t R e g i s t e r and Scanning C i r c u i t s . 41 3.5 The P o l a r i t y - C o m p a r i s o n C i r c u i t ... . 44 3.6 The R e v e r s i b l e Counter 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 . . . o c e o . « o o . o o « . . . . . » o . » « 48 3—6.1 The c o u n t e r ».«....««.... . 0 . 0 . 0 0 . 0 . 48 3-6.2 The c o u n t e r c o n t r o l c i r c u i t s 52 3.7 The Memory Address System 56 3-7.1 The memory core a r r a y 56 3-7.2 The memory address c i r c u i t ........ 58 3-7.3 The address system 61 3.8 The A u x i l i a r y N oise Generator and 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 64 4. SAMPLING FLUCTUATIONS 72 i v Page 4.1 Ge n e r a l 72 4.2 B a n d - L i m i t e d G a u s s i a n Noise 73 4.3 B a n d - L i m i t e d G a u s s i a n Noise P l u s a Sine 4.4 Sampling F l u c t u a t i o n s Due to P e r i o d i c Sampling of a P e r i o d i c F u n c t i o n .......... 93 4.5 Comparison of the C o n v e n t i o n a l Sampling C o r r e l a t o r and the UBC C o r r e l a t o r on the B a s i s of Sampling F l u c t u a t i o n s 101 4.6 The E f f e c t of Q u a n t i z a t i o n on Sampling F l u c t u a t i o n s and B i a s 107 3 • C O N C X J U S I O N • • • • * » * * * * « o o o « o « 6 « o o o 6 » « e o o « « o « c o * « 1 18 REFERENCES . 120 APPENDIX I 121 v LIST OP ILLUSTRATIONS F i g u r e Page 2.1 Range of Input V a r i a b l e s t o the C o r r e l a t o r . 6 2.2 Q u a n t i z a t i o n L e v e l s of the A u x i l i a r y N o i s e , y i ? and S i g n a l , x i 10 2.3 Simple C o r r e l a t o r .• 17 2.4 J e s p e r s C o r r e l a t o r 20 2.5 UBC C o r r e l a t o r - Symmetrical Sampling Mode • 21 3.1 Timing C i r c u i t B l o c k Diagram 31 3.2 Th r e e - S t a t e Counter . 33 3.3 AND Gate C o n n e c t i o n s t o the 32-State Counter 35 3.4 Timing Waveforms 37 3.5 D i f f e r e n t i a l A m p l i f i e r 39 3.6 Comparator and Sample Gates 40 3.7 The S h i f t R e g i s t e r and Scanning C i r c u i t .... 42 3.8 32-Input OR Gate 43 3.9 P o l a r i t y - C o m p a r i s o n C i r c u i t 45 3.10 The R e v e r s i b l e Counter \ 50 j 3.11 R e v e r s i b l e Counter C o n t r o l C i r c u i t s 54 3.12 18-Input AND Gate 55 3.13 The Core H y s t e r e s i s Loop 57 3.14 Word-Select A r r a y W i r i n g 57 3.15 Word-Select System 60 3.J.6 D i g i t D r i v e r and Sense C i r c u i t 62 3.17 The Memory Address System 63 3.18 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 65 3.19 The A u x i l i a r y Noise Generator 67 3.20 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 System ...... 71 4.1 E x p o n e n t i a l C o r r e l o g r a m 80 v i F i g u r e Page 4.2 F i g u r e of M e r i t F o r T = 0 89 4.3 Comparison Table For T / 0 90 4.4 Upper Bound to the V a r i a n c e of the E s t i m a t e d A u t o c o r r e l a t i o n of G a u s s i a n Noise P l u s a Sine Wave of Frequency = b rad./sec 91 4.5 Upper Bound to the V a r i a n c e of the E s t i m a t e d A u t o c o r r e l a t i o n of G a u s s i a n Noise P l u s a Sine Wave of Frequency tt = b/2 r a d . / s e c . ... 92 4*6 " S y n c h r o n i z a t i o n " F u n c t i o n 97 4.7 C o r r e l a t o r F i g u r e s of M e r i t For E x p o n e n t i a l C o r r e l a t i o n F u n c t i 4.8 Input-Output R e l a t i o n s h i p f o r the General Q u a n t i z e r „ 110 v i i 1. INTRODUCTION T h i s t h e s i s d e s c r i b e s a g e n e r a l - p u r p o s e s t a t i s t i c a l ana-l y z e r capable 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 y and d i s t r i b u t i o n f u n c t i o n s , or o p e r a t i n g as a computer of average t r a n s i e n t s . The p r i m a r y emphasis i n the t h e s i s i s on o p e r a t i o n as a c o r r e l a t o r . C o r r e l a t i o n f u n c t i o n s are of prime importance i n s t u d i e s i n v o l v i n g random phenomena e i t h e r from a t h e o r e t i c a l or ex-p e r i m e n t a l v i e w p o i n t . The reasons f o r t h i s are s e v e r a l ; ( l ) the c o r r e l a t i o n f u n c t i o n or i t s t r a n s f o r m , the power s p e c t r a l d e n s i t y f u n c t i o n , are the o n l y e a s i l y measurable q u a n t i t i e s of a random p r o c e s s ; (2) the power spectrum can o f t e n be more a c c u r a t e l y e s t i m a t e d by t r a n s f o r m i n g the s u i t a b l y m o d i f i e d c o r r e l a t i o n f u n c t i o n t h a n by use of a h e t e r o d y n i n g and f i l t e r i n g system; (3) the G a u s s i a n random p r o c e s s i s c o m p l e t e l y s p e c i f i e d once the c o r r e l a t i o n f u n c t i o n and mean are known; (4) the response of l i n e a r networks t o random i n p u t s can o n l y be e x p r e s s e d i n terms of the c o r r e l a t i o n f u n c t i o n or the power s p e c t r a l d e n s i t y . A l a r g e number of analogue and d i g i t a l d e v i c e s which compute c o r r e l a t i o n f u n c t i o n s have been b u i l t i n the p a s t . Among the e a r l i e r c o r r e l a t o r s are two d e v i c e s developed a t M.I.T. The f i r s t o f these was an analogue c o r r e l a t o r d e s c r i b e d by L e v i n and R e i n t j e s , ^ a sam p l i n g c o r r e l a t o r w hich used c a p a c i t o r s t o r a g e and performed m u l t i p l i c a t i o n by s i m u l t a n e o u s p u l s e -a m p l i t u d e and p u l s e — w i d t h m o d u l a t i o n . F i v e p o i n t s of the c o r -r e l o g r a m were computed s i m u l t a n e o u s l y . The second* a d i g i t a l 2 2 correlator described by Singleton , was a vacuum—tube device which performed analogue-to-digital conversion by width-modulating a pulse i n proportion to the input sample and using this pulse to gate clock pulses to a counter. Delays i n both devices were obtained by introducing delay between the sample times of the various channels. Recently* considerable interest has been shown i n the cor-r e l a t i o n functions of quantized signals. The objective has been to f i n d a quantization scheme which w i l l introduce a r e l a t i v e l y small error and yet decrease s i g n i f i c a n t l y the data-handling 3 4 ' requirements of the correlator. Widrow and Watts haVe both studied the effects of rough quantization on the correlation functions of random signals. Watts has developed a general method for dealing with quantized signals, applicable to the two best-known quantization correlators, the relay correlator and the polarity-coincidence correlator. The relay correlator has one input X^ quantized into sgn X-^ . The polarity-coincidence cor-relator quantizes the two inputs, X-^  and X£» into sgn X-^  and sgn X£ respectively. In addition, Watts has proposed several models of quantization correlators, and these he has referred to as S t i e l t j e s correlators. The disadvantages of these devices are as follows; the output correlation function of the relay correlator i s d i r e c t l y proportional to the desired correlation function only i f the input signals belong to a p a r t i c u l a r class of functions (one of which i s the Gaussian random process), and considerable data-handling c a p a b i l i t y i s required; the output of the polarity-coincidence correlator has a non-linear r e l a t i o n -ship to the desired correlation function and th i s r e l a t i o n -ship depends upon the s t a t i s t i c s of the input signals; the S t i e l t j e s c o r r e l a t o r r e q u i r e s a l a r g e amount of d a t a p r o c e s s i n g . The e a s i e s t type of c o r r e l a t o r t o i n s t r u m e n t i s the p o l a r i t y — c o i n c i d e n c e c o r r e l a t o r . M u l t i p l i c a t i o n i s performed w i t h r e l a t i v e l y u n c o m p l i c a t e d p o l a r i t y - c o i n c i d e n c e c i r c u i t s and d i g i t a l a c c u r a c y i s o b t a i n e d w i t h o u t the n e c e s s i t y of c o m p l i c a -t e d a n a l o g u e — t o - d i g i t a l c o n v e r t e r s , A p o l a r i t y c o i n c i d e n c e c o r r e l a t o r u s i n g a s h i f t r e g i s t e r d e l a y l i n e has been d e s c r i b e d by Greene.^ An e l e g a n t e x t e n s i o n t o the p o l a r i t y - c o i n c i d e n c e scheme 7 has been d e s c r i b e d by Ikebe and Sato , and a l s o by J e s p e r s , g Chu and F e t t w e i s . T h i s e x t e n s i o n c o n s i s t s of adding an independent random v o l t a g e , w i t h a u n i f o r m a m p l i t u d e p r o b a b i l i t y d e n s i t y over a range e x c e e d i n g t h a t of the i n p u t s i g n a l s , t o each of the s i g n a l i n p u t s t o the p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r , 7 8 They ' have shown t h a t the c o r r e l a t i o n f u n c t i o n so o b t a i n e d a t the output of the p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r , bears a l i n e a r r e l a t i o n s h i p t o the c o r r e l a t i o n f u n c t i o n of the i n p u t s i g n a l s f o r a l l c l a s s e s of s i g n a l s . T h i s t h e s i s d e s c r i b e s a p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r w i t h random comparison l e v e l s somewhat s i m i l a r t o the d e v i c e s d e s c r i b e d i n 7»8„ A s t a t i s t i c a l a n a l y s i s of the new p o l a r i t y -c o i n c i d e n c e c o r r e l a t o r i s c a r r i e d out and a sampling system i s proposed which has advantages i n the c o r r e l a t i o n of low-f r e q u e n c y s i g n a l s . 2. PRINCIPLE OF OPERATION 4 2.1 Theory, This chapter describes the computer i n general terms. The basic p r i n c i p l e i s contained i n a theorem derived by Jespers et a l . Because th i s p r i n c i p l e i s not widely known and may be useful for purposes other than correlation, the proof of Jespers 1 theorem i s reproduced. A modified theorem i s also derived which i s applicable to the case where the a u x i l i a r y noise comparison levels are quantized or the input signals are quantized. The object of Theorem I i s to show that i f a u x i l i a r y noise sources having a uniform amplitude probability density over a range exceeding that of the input signals are used as comparison l e v e l s for the polarity-coincidence correlator, the c o r r e l a t i o n function so determined i s d i r e c t l y proportional to the co r r e l a t i o n function of the input signals^ Theorem Is Let x,, x~,....x be continuous random variables, with x.-1 * 2 q i having upper and lower bounds A^ and -A^ respectively, A^ > 0. Let y^, y2'°*** Vq ^ e continuous random variables, inde-pendent of x^'s as well as of each other, and l e t the probability density P-^(y^) he defined by P i W - 217 ' y i " A i ; i p i ( y i ) = o , |y i Then, i f we write z- = x.- y. and z = z, z~ ....z , ' i 1 J 1 1 2 q x,x_ x = A,A 0 A .sgn z 1 2 q 1 2 q. t h That i s , a q -order moment for the random signals x^ can be found from the mean value of the sign of the product of the modified waveforms x^- y^, under the given conditions on y^. Proof t Let u = sgn z, A u± = sgn z i . 1 Then u = u. . l i=l u = (+l)P(u=l) + (-l)P(u= -1) = 2P(u=l) - 1 where P i s the prob a b i l i t y . Therefore P(u=l) = £(l+u). To f i n d u, i t i s only necessary to determine P(u=l)= P(( Uj)=l). i=l Now consider the components of z^ = x^- y^. In the plane of x^ and y^,(x^,y^) may l i e anywhere within a square 2A^x 2A^ centred at the o r i g i n . If the square i s divided into smaller squares, there i s equal l i k e l i h o o d of finding (x^,y^) i n any row, but an unspecified d i s t r i b u t i o n of l i k e l i h o o d among the columns. 6 x. = v. P i g . 2.1 Range of Input V a r i a b l e s t o C o r r e l a t o r The l i n e x^— y^ i s the boundary between r e g i o n s where u^= +1 = - 1 . The p r o b a b i l i t y t h a t , g i v e n x ^ , u i = 1 ( i . e . , v i < x i ) i s and u^= . P(u.= 1 x.) or P(u.=l -A-X0 = i ( l + 7^ ") x. p ( y i ) d y i = J -A-2A" d v i S i m i l a r l y , ?(u.= -1 p ( y i ) d y i = x i H i - ^) A i Alternatively, the probability that u^ w i l l be as prescribed, given x^, i s P(u. x.) = 1 ( 1 + Now l e t x ^ , X 2 » » * . .x^ be given; l e t u-j^, » . • »u^ be chosen a r b i t r a r i l y . Because the y^ 1s are independent, the probability that t h i s a r b i t r a r i l y chosen set w i l l occur i s P(u 1»u 2,....u q X 1 ' X 2 * * * • * Xq^ P(u. i=l q 2 i=l U . X . Then P(u=l x l fx 2,..„x^) = ^ u. x. . ( i + - i - i ) 4 P A i c-u=l i=l where u=l means "over a l l choices of u.'s for which u=l", j x Similarly, P(u= «-l KV 2 1 ..x ) £ P q q 2 q u. x. (1+ ^ r - ^ ) . Next, A i u= -1 i=l consider the relationship between P , P , , P' and P' , , i q q-i q q- 1 • e, P = 1 ( 1 + i - ) p + 1 ( 1 - ^L )pt q ^ A . q-1 • A q-1 and 8 X-. X/j • • • • X -. Let us assume that P , = \ (l+ . . . q ~ -) q-1 * A , A 0 . . . . A -i 1 2 q-1 d-1 - 2 V-L A A A > • q A 1 A 2 q - 1 Then 1 + x l x 2 # ' » • X -i X ^ + -3- + 1 2 q - f q = f +1 -x l x 2 ' A1 A2' X ^ « » • *X 1 q A-, A ^ » » « » A 1 <L q X X , x „ » » « » x -3. + 1 2 q A A , A 0 » » * . A q 1 2 q j Therefore, X1 X2* * * ** x P q = A 1 A 0 . . . . . A A ) * u 1 2 q S i m i l a r l y , X-j^ X^* • • • »x L A 0 A 1 2 q p« = i d — T M r3-) q ^ A Therefore, i f the solutions for P , and P' , have th i s form, q-1 q-1 » those for P and P' do also. But q q X x p i = ? ( 1 + a n d p i = A7} Thus, by i n d u c t i o n , i t has been e s t a b l i s h e d t h a t 9 X 1 X 2 ' P = H l + A A q * A T A 1"2' , A But P(u=l) = P(u=l X l * •• xq^P( xl» x2' * * * X q ^ d x l d x 2 d x q -A -A A l A q - A . x, x 1~2' p (x1 , x 2 , . . .x H ( l + A A ^ . . . A q ) d x l d x 2 * • - d x 1"2' x x • x T h e r e f o r e P ( u = l ) = \ + 4 A 1 A 2 " " A q = \ + i u 1 2 q \ + % sgn z , and hence x, x n.....x = A 1 A 0 . . . . A sgn z 1 <d q 1 <i 0. where z = z,z~....z and z.= x. - v . . 1 2 q I I I An a l t e r n a t i v e theorem a p p l i c a b l e t o systems i n which x^ and/ or y^ are q u a n t i z e d i n t o d i s c r e t e e q u i - s p a c e d l e v e l s w i l l now be d e r i v e d f o l l o w i n g c l o s e l y the method used i n Theorem I . Theorem I I ; L e t 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 equal 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 random 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 10 i n t e r v a l s w i t h equal 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 - T T ~ ) < x . < A - ( l - 4—), where the bounds are l e v e l s of y.. 1 2n y i ^ l 2n' ' J l F i g . 2.2 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. l 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 . 2.2)» Then X l X 2 * * * * X q = A l A 2 * * * * A q S ^ n Z w ^ - e r e z = z l z 2 * * * * Z q a n ^ Z i = X i ~ ^ i as b e f o r e . P r o o f ; As i n theorem I , l e t u = sgn z and l e t u^ = sgn z^. Then q u = | I u^ i = l 11 and u = (+l)P(u=l)+(-l)P(u= -1) = 2P(u= + l ) - l where P i s the p r o b a b i l i t y . To f i n d u i t i s only necessary to determine P(u=l)=P(( u.)= 1). i = l F i r s t c o n s i d e r the c o n d i t i o n a l p r o b a b i l i t y of y ^ < x i t h f o r g i v e n x^ w h e r e x^ l i e s i n t h e m i n t e r v a l d e f i n e d by /2m.- 1\ /2m. + IN A i l - i d < x i < A i H d (2.1) T h i i s p r o b a b i l i t y i s P(y^<x^ x^) = P(u^= 1 |x^) = f r a c t i on of d i s c r e t e l e v e l s of y. below x. J I l '2m. + l \ r - k - 7 V - A i ( 1 " y n 1_ 2n m. u. m. or P(u. = 1 x.) = £(1+ — ) = • i x * n y 2 n 7 The p r o b a b i l i t y t h a t y^>x^ f o r x^ given as i n (2»l) i s i s P(y^ >x^|x^) = P(u^= -1 x^) = f r a c t i o n of d i s c r e t e l e v e l s of y. above x. J I I , 2m. - 1 v*- y - ^ i i % A ± 2n _ 2 U n ' n o r P(u, = -1 x.) •= H l + " " " i — ) u m. j i n and hence P(u. u. m. X . ) = + - i - i ) i ' ^ n 1 (2.2) r : i A i Now c o n s i d e r a q u a n t i z a t i o n of x^ such t h a t x | = - n (2.3) where x^ i s g i v e n by ( 2 . 1 ) , i . e . , x^ i s a q u a n t i z a t i o n t o the m i d - p o i n t s between the y^ l e v e l s . S u b s t i t u t i n g (2.3) i n t o (2.2) y i e l d s the r e s u l t P ( u . x., „ i ( 1 + V i , (2.4) Now l e t x,,x /,,...x be g i v e n : l e t u, fu-, #».u be chosen 1 d q 1 Z q a r b i t r a r i l y . Because the y^ are independent, the p r o b a b i l i t y t h a t t h i s a r b i t r a r i l y chosen s e t w i l l occur i s q P ( u l ' u 2 ' * * , U q X l ' X 2 ' * * * X q ^ = P ^ U i i = l u.x: l l • Then P(u=l 2 q i = l A I 7 2' q/ q 2* -i q ,TT<" u.x.' 1 1 A- ) where u=l means u=l i = l a sum over a l l u. such t h a t u=u, u-., ; % . ,u = l i ' q S i m i l a r l y , P(u=-1|x^,x 2,....x ) = P k P . _ L _ u.x." 1 1 • A i u= -1 i = l - i • e t c • , q q-i 9 f Now c o n s i d e r the r e l a t i o n s h i p between P and P v i ( 1 +5> V i +  i { 1 - i ] K - i and 13 x ' P* = 1(1- P n + i ( l + J L ) P 1 , a 2 A q-1 2 A q-1 q Let us assume that P q-1 X-! x i » » • » x ' -. 1 ( 1 + 1 2 q-1) A-i A ^ • • • * A 1 2. q and q-1 ^l x2* 11=1) A-, A 0 . . » » A 1 2 q Then i+ +i-X T X ,1 #• • * » X ' -, X ' ••1 2 q-1 + _g_ + A l A 2 * ' * , A q - l A q x ^ X2 * • • * x A-i A f t • • ft A 1 1 2 q-1 1 x 1 A + q x l x 2 , # A1 A2*- ..A q x l x 2 ' * A 1 A 2 " » » A q Therefore, P = 1(1 + . x A-i A^-j • • • • A l c q Similarly, P» q = i d -L l x 2 ' A-. A o • • • • A 1 2 q t Therefore, i f the solutions for P , and P' , have this form, 7 q-1 q-1 7 those for P and P' do also. q q 1 x x But P 1 = H l + j - ) and P^ = i ( l - - i ) . Therefore, by induction, we have established that x J x J L » » * » x ' q ^ A n A ~ . • . .A 1 d. q 14 But v 1- y r P(u= +1) = v1- y r pCx-j^  ,x2,x3. ..x )P(u y x^  f x2 • * • x ) dx^  • • • dx^  h^-y \j}-y - A i ( 1 - k> • • • x p ( x 1 , x 2 , ...x Hfi+ A i t M t n d x i 1*2' ,dx h^-y \^-y X-J XX • e » • X ' 1 2 - A l ( l - - A (.1- |j) 1 2 q ^ • dx but 1 2n V 2n P(x^,x 2, . • »X * ) — pCx-j^  ,x2, . .x Jdx-j^  ». . . ....dx and x'-1 2n H 2n .1x2. t <1 xl x2**** Xq ^^X1*X2' * * " *Xq^ all x.' I 15 a l l x! x l x 2 # A A. ,) d x l dx, • dx I •••• / x^x^.^.^x^ p(X-^ • x 2• • • • x^)dx^dx 2• ««*dx^« T h e r e f o r e , we have shown t h a t 2P(u=l)= 1+1—7 A = 1 + U , U - , . . . U 7 A 1 A 0 f » « « A „ 1 2 q 1 2 q r L and hence xl'xU* ...x 1 = A nA~«...A u,u~....u =«'A 1A^...A sgn z 1 2 q 1 2 q 1 2 q 1 2 1 T h i s theorem shows t h a t , i n s o f a r as x| i s a good estimate of x^, sgn z pro v i d e s an estimate 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^ to 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 both of x^ and y^ are quantized, provided the l e v e l s are i n t e r -l a c e d i f both are quantized. 2.2 Implementation A p p l i c a t i o n of Theorem I to the determination of a c o r r e l a t i o n f u n c t i o n of a s t a t i o n a r y " random p r o c e s s y i e l d s the r e s u l t , T 0 1 2 ( r ) = l i m i t | Y x 1 ( t ) x 2 ( t + T ) d t = X ; L ( t ) x 2 ( t + T ) T — > o o J = x ^ ( t - T ) x 2 ( t ) = A-^A2u where u = u ^ u 2 and u x = s g n U j - y ^ , -k1<y1<Xl u 2 = s g n ( x 2 - y 2 ) , - A 2 < y 2 < A 2 » L e t C ^ 2 ( T ) be an e s t i m a t e of 0-^2^ T ^ o b t a i n e d by f i n i t e - t i m e i n t e g r a t i o n . T C 1 2 ( T ) = ^ j X i ( t ) x 2 ( t + T ) d t = A x A 2 ( u ) T . 0 The d e t e r m i n a t i o n of ( u ) ^ f o r a s i n g l e v a l u e of T can be a c c o m p l i s h e d w i t h r e a s o n a b l y simple i n s t r u m e n t a t i o n . (See P i g . 2.2). 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 n o i s e s o u r c e , y^ or y 2 , i n a l e v e l comparator which s h o u l d be a l o w - h y s t e r e s i s comparator such as a h i g h - g a i n d i f f e r e n t i a l a m p l i f i e r f o l l o w e d by a Sc h m i t t t r i g g e r . The com-p a r a t o r o u t p u t s have t w o * 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 two p o s s i b l e s i g n s of x^-y^* A f t e r d e l a y i n g the output of channel-one comparator by time T , the two p o l a r i t y s i g n a l s u^ and u 2 are compared i n a p o l a r i t y - c o m p a r i s o n c i r c u i t w h i c h 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 . The o u t p u t , u, t h e r e f o r e r e p r e s e n t s sgn z, and i n t e g r a t i o n f o r time T produces 17 an output p r o p o r t i o n a l t o ( u ) ^ . x 2 ( t ) Comparator F T — Noise Source x ^ t ) N o i s e Source Comparator P o l a r i t y -O o i n cidence C i r c u i t • ^ " T — ^ I n t e g r a t e D elay T P i g . 2.3 Simple C o r r e l a t o r The c o r r e l a t o r b u i l t by J e s p e r s , Chu and F e t t w e i s i s of t h i s b a s i c form, w i t h y^ and y^ b e i n g " c o n t i n u o u s " sawtooth waves of randomly v a r y i n g s l o p e . The d e l a y i s a c h i e v e d by sampling and h o l d i n g s g n ( x 1 - y.^) u n t i l s g n ( x 2 ~ y^) ±s sampled T seconds l a t e r . The s i g n s are t h e n compared and used t o i n c r e a s e or decrease the count C on a r e v e r s i b l e c o u n t e r . A f t e r making N C C such comparisons i n a time T, (u),j, = ^ and C^ 2( T ) = A^A 2» (JJ) • On the o t h e r hand, the c o r r e l a t o r d e s c r i b e d by Ikebe and Sato uses a magnetic-tape system f o r d e l a y , w i t h waveforms 18 sgn ( x ^ - y^) and s g n ( x 2 ~ y 2 ) r e c o r d e d s i m u l t a n e o u s l y on the tape and read w i t h time s e p a r a t i o n T • P e r i o d i c s i g n comparison causes a c o u n t e r t o step as b e f o r e . However, the r e f e r e n c e n o i s e waveforms f o r the comparators are q u a n t i z e d ^ so Theorem I I i s a p p l i c a b l e r a t h e r t h a n Theorem I . I n t h i s case the n o i s e i s o b t a i n e d from a 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 w i t h i t s n i n p u t s c o n t r o l l e d by n o i s e p u l s e s from n separate t h y r a t r o n s . Because the c o n v e r t e r i n p u t l o g i c l e v e l s are zero or p o s i t i v e , the n o i s e i s d i s t r i b u t e d over the range 0<!y<2A, b u t , w i t h 0<^x<.2A, the t h e o r y developed p r e v i o u s l y can be shown t o a p p l y * Each of the systems above has c e r t a i n advantages. The Jkebe system i s p r a c t i c a b l e o n l y f o r a s i n g l e ( a l t h o u g h v a r i a b l e ) v a l u e of T , but has the advantage t h a t the sampling i n t e r v a l i s not r e s t r i c t e d t o b e i n g g r e a t e r than T , as i t must be i n the J e s p e r s system. The Ikebe system c o u l d be used on an analogue b a s i s i f an i n t e g r a t o r r e p l a c e d the c o u n t e r . I t s h o u l d be noted t h a t e i t h e r system w i l l work w i t h q u a n t i z e d or w i t h c o n t i n u o u s r e f e r e n c e n o i s e . The p r i n c i p a l advantage of the J e s p e r s system i s t h a t i t i s w e l l s u i t e d t o the simu l t a n e o u s d e t e r m i n a t i o n of a m u l t i p l i c i t y •of p o i n t s on the graph of the c o r r e l a t i o n f u n c t i o n . R e f e r r i n g t p F i g u r e 2.4» J e s p e r s ' c o r r e l a t o r , a s a c t u a l l y b u i l t , i n c o r p o r a t e s a magn e t i c - c o r e memory a r r a y t o a c t as a b u f f e r s t o r a g e u n i t . To each b i t of the r e v e r s i b l e c o u n t e r t h e r e corresponds one column of the a r r a y . To each v a l u e of time d e l a y , T » t h e r e corresponds one row of the a r r a y . The s i g n a l i n channel—two i s sampled as i n the b a s i c system by sample p u l s e P 2 and a f l i p - f l o p h o l d s a v a l u e c o r r e s p o n d i n g t o s g n ( x 9 - y ~ ) . I n channel—one, sgn(x,- y, ) 19 i s sampled every A T seconds by the p u l s e t r a i n P-^ , w i t h the f l i p - f l o p h o l d i n g between samples. I f t h e r e are p v a l u e s of T , 0 <T= r AT < IT , w i t h 0 < r < p , and T = ( p - l ) A T , t h e n max 7 ^' max * 7 t h e r e w i l l be p p u l s e s i n the t r a i n P^ f o r each p u l s e i n P^, Thus r corresponds t o a row number i n the a r r a y . At the i n s t a n t when the r ^ * 1 p u l s e i n P^ o c c u r s , the c o n t e n t s of the r^* 1 row of the a r r a y are c o p i e d i n t o the r e v e r s i b l e c o u n t e r , which has p r e v i o u s l y been c l e a r e d . B e f o r e the ( r + l ) s t p u l s e , the s i g n comparison between f l i p - f l o p s t a t e s i s made, i n c r e a s i n g or t h d e c r e a s i n g the co u n t , the new count i s t r a n s f e r r e d t o the r row of the a r r a y , and the co u n t e r i s c l e a r e d a g a i n . T h i s p r o -cess i s r e p e a t e d f o r each v a l u e of r up t o p-1, and the n p u l s e i?2 o c c u r s , and the c y c l e b e g i n s a g a i n a t row zero of the a r r a y . Because of the sequence f o l l o w e d , o n l y one sample i s t a k e n i n a p p r o x i m a t e l y T seconds,. T h i s i m p l i e s l o n g a n a l y s i s time f o r lo w - f r e q u e n c y waveforms. F or example, i f o n l y -one c y c l e of a p e r i o d i c component of ^ a * f r e < l u e n c y f i s t o be d i s p l a y e d , one sample can be o b t a i n e d i n a time seconds. That i s * t o o b t a i n 10,000 samples a t one c y c l e per second w i l l r e q u i r e 10,000 seconds. To overcome t h i s drawback of J e s p e r s ' c o r r e l a t o r , a m o d i f i e d v e r s i o n p a r t i c u l a r l y s u i t e d f o r lo w - f r e q u e n c y work has been d e s i g n e d . 2.3 The U.B.C. C o r r e l a t o r The b a s i c i n n o v a t i o n i n the UBC c o r r e l a t o r i s a p - b i t s h i f t r e g i s t e r p r o v i d i n g temporary storage i n channel-one t o 20 Compar- AND FF2 a t o r 1 Compar-AND FF1 a t o r -> P o l a r i t y -C o i n c i d e n c e C i r c u i t R e v e r s i b l e Counter F i g . 2.4 J e s p e r s ' C o r r e l a t o r p e r m i t channel-one and two t o be sampled a t the same r a t e (see F i g u r e 2»5), The f i r s t f l i p - f l o p of t h i s r e g i s t e r r e p l a c e s the f l i p - f l o p i n channel-one of the J e s p e r s - c o r r e l a t o r . Samples c o r r e s p o n d i n g t o s g n ( x 1 ~ y^) are s t o r e d i n the f i r s t f l i p - f l o p and s h i f t e d s u c c e s s i v e l y from the f i r s t t o the. p f l i p - f l o p of a > yz > C O M P A R -A T O K > A N D F L I P -F L O P •* S Y M M E T R I C A L S f t M P L I N I G X , C O M P A R -A T O R -'S A N D S H I F T a E & I S T E R , S H I F T K , E G - | S T E R S C A N N I N & C I K C U I T P O L A R I T Y -C O I N C I D E N C E C I R C U I T R E V E R S I B L E C O U N T E R i M E M O R Y — A R R A Y F i g , 2.5 UBC C o r r e l a t o r - S y m m e t r i c a l Sampling Mode 22 the r e g i s t e r w i t h the i n t e r v a l A T per s t e p * At the same i n s t a n t as the sample b i t from channel-one e n t e r s the f i r s t f l i p - f l o p of the r e g i s t e r , the f l i p - f l o p i n channel—two i s s e t by the b i t r e p r e s e n t i n g sgn(x£- y^)' I n the f o l l o w i n g i n t e r v a l , A T , a l l rows i n ! t h e a r r a y have t h e i r count changed by s i g n comparison between the channel-two f l i p - f l o p and the f l i p - f l o p s 0 t o p-1 i n the s h i f t r e g i s t e r * The sequence i s as f o l l o w s : 1 (a) Row (or word) r = 0 i n a r r a y i s r e a d i n t o r e v e r s i b l e c o u n t e r . (b) S i g n comparison between PFO i n r e g i s t e r and s t o r a g e FF i n channel-two changes c o u n t . (c) New count i s w r i t t e n i n t o word zero of a r r a y . 2 (a) Word r = 1 i n a r r a y i s r e a d i n t o r e v e r s i b l e c o u n t e r . (b) S i g n comparison between F F l i n r e g i s t e r and s t o r a g e FF i n channel-two changes count, e t c . e t c . p . ( c ) New count i s w r i t t e n i n t o word ( p — l ) of a r r a y . p+1 — Contents of s h i f t r e g i s t e r are s h i f t e d one b i t . Afte'.r s t e p ( p + l ) , the next sample p u l s e s o c c u r , s e t t i n g FFO i n s h i f t r e g i s t e r and s t o r a g e FF i n channel-two. T h i s s t e p i s a c t u a l l y synchronous w i t h s t e p l ( a ) . T h e r e f o r e * t h e r e are 3p+l o p e r a t i o n s t o be performed i n time AT . S i n c e T = max ( p - l ) A T ^ ( 3 p + l ) ( p - l ) = 3p o p e r a t i o n s must be performed i n the i n t e r v a l T + AT . max 23 Thus f o r any r e a s o n a b l e v a l u e f o r p. t h i s system i s q u i t e i m p r a c t i c a l f o r c o r r e l a t i o n of h i g h - f r e q u e n c y waveforms, where T ^ ^ i i i s s m a l l . However, i n l o w - f r e q u e n c y a n a l y s i s , f o r a g i v e n v a l u e of T , the s a m p l i n g r a t e i s p t i m e s as g r e a t as i n the J e s p e r s " c o r r e l a t o r ^ w i t h a c o r r e s p o n d i n g r e d u c t i o n i n a n a l y s i s t i m e . W i t h the above c o n s i d e r a t i o n s i n mind, the UBC c o r r e l a t o r has been d e s i g n e d t o p r o v i d e 32 e q u a l l y - s p a c e d p o i n t s on the graph of the c o r r e l a t i o n f u n c t i o n . The maximum count t h a t can 18 be r e c o r d e d i s 2 = 264,144. C o u n t i n g the s i g n b i t , t h i s r e q u i r e s 19 x 32 cores of a 32 x 32 a r r a y . A p p r o x i m a t e l y 3,000 o p e r a t i o n s must be performed i n the maximum d e l a y t i m e , T . nicix U s i n g 200-Kc l o g i c u n i t s , we cannot expect T t o be l e s s t h a n 15 m i l l i s e c o n d s . Thus, the u s e f u l range might extend t o about 250 cps, t a k i n g i n t o account the s m a l l number (p=32) of p o i n t s a v a i l a b l e . At 250 c p s , a p p r o x i m a t e l y 8 samples are t a k e n per c y c l e . Because of t h i s upper f r e q u e n c y l i m i t j i t was d e c i d e d t h a t the c o r r e l a t o r s h o u l d be a b l e t o operate i n e i t h e r a s y m m e t r i c a l (synchronous sampling or l o w - f r e q u e n c y ) mode or an a s y m m e t r i c a l (asynchronous s a m p l i n g , Jespers', or h i g h - f r e q u e n c y ) mode» I n the a s y m m e t r i c a l mode t h e r e are o n l y 3p o p e r a t i o n s i n the p e r i o d T . U s i n g 200-Kc l o g i c elements and p=32, ^max m a v ^ e a s s m a i ± a s 500 m i c r o s e c o n d s . An upper f r e q u e n c y of 10 k i l o c y c l e s / s e c o n d s h o u l d t h e r e f o r e be a t t a i n a b l e . The s y m m e t r i c a l mode might r e a s o n a b l y be used from dc t o 250 cps and the a s y m m e t r i c a l mode from 25 t o 10,000 c p s . 24 W i t h r e f e r e n c e t o the a u x i l i a r y n o i s e sources g e n e r a t i n g the r e f e r e n c e waveforms and y^i i t was d e c i d e d t o use two d i g i t a l - a n a l o g u e c o n v e r t e r s c o n t r o l l e d by Zener diode n o i s e so as t o p r o v i d e 256 l e v e l s of q u a n t i z a t i o n between + 6 v o l t s . The c l o c k p u l s e s c o n t r o l l i n g the t i m i n g c i r c u i t s w i l l be d e r i v e d from an e x t e r n a l v a r i a b l e - f r e q u e n c y o s c i l l a t o r (see P i g . 2.5)* A p r e s e t b i n a r y c o u n t e r w i l l be used t o determine the number of samples N. A f t e r N samples, the count s t o p s , and non-d e s t r u c t i v e readout i s ac c o m p l i s h e d u s i n g one of the d-a con-v e r t e r s t o r e a d the 32 words i n the a r r a y i n sequence. T h i s p r o v i d e s a maximum of 8 b i t s , w hich i s adequate f o r analogue r e c o r d i n g * A f u r t h e r N samples may the n be t a k e n by u s i n g the next s t e p on the p r e s e t c o u n t e r i f i t i s d e s i r e d t o check convergence• The r e m a i n i n g u n i t s i n the c o r r e l a t o r , the r e v e r s i b l e c o u n t e r , s h i f t r e g i s t e r , t i m i n g c i r c u i t s , comparator and c o i n -c i d e n c e l o g i c u n i t s are d e s c r i b e d i n d e t a i l i n c h a p t e r t h r e e . I n a d d i t i o n a d e t a i l e d d e s c r i p t i o n of the d-a c o n v e r t e r and a u x i l i a r y n o i s e source i s g i v e n . As f a r as p o s s i b l e , W y l e p r i n t e d c i r c u i t s are employed; s p e c i a l purpose ca r d s are made i n the same l a y o u t as the Wyle c a r d s . W ith minor m o d i f i c a t i o n s , the c o r r e l a t o r d e s c r i b e d p r e -v i o u s l y can be use-d., (1) as a computer of average t r a n s i e n t s , (2) t o determine c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s , and (3) t o determine p r o b a b i l i t y d e n s i t y f u n c t i o n s . The l a t t e r two f u n c t i o n s were suggested by J e s p e r s , and 25 i n a l l t h r e e cases the i n s t r u m e n t operates i n the h i g h - f r e q u e n c y mode. 2.4 Computer of Average T r a n s i e n t s The o b j e c t i v e i s t o f i n d the average of a r e p e a t e d s e t of " n o i s y " f u n c t i o n s . The t i m i n g waveform must be o b t a i n e d from a source s y n c h r o n i z e d w i t h the s i g n a l b e i n g observed. I n most i n s t a n c e s , the s i g n a l i s the response t o a s t i m u l u s and t h e r e f o r e i t i s l o g i c a l t o s y n c h r o n i z e w i t h the s t i m u l u s . The waveform b e i n g observed i s a p p l i e d t o the i n p u t of ch a n n e l — one, w i t h the r e f e r e n c e f l i p - f l o p i n channel—two h e l d i n the one s t a t e . Because y^ has u n i f o r m p r o b a b i l i t y d e n s i t y , the count i n each of the 32 rows of the memory w i l l r e p r e s e n t the average v a l u e of the t r a n s i e n t a t the c o r r e s p o n d i n g time a f t e r the a p p l i c a t i o n of the s t i m u l u s . 2.5 Cumulative P r o b a b i l i t y F u n c t i o n The c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n i s e s t i m a t e d by comparing the i n p u t s i g n a l t o a number of d i s c r e t e v o l t a g e l e v e l s and d e t e r m i n i n g the p o r t i o n of the time the s i g n a l i s l e s s t h a n the p r e s c r i b e d l e v e l s . D i s c r e t e comparison l e v e l s are g e n e r a t e d by d r i v i n g a d-a c o n v e r t e r w i t h the outputs of a c o u n t e r w h i c h c o n t i n u o u s l y c y c l e s through e v e r y number from 0 t o 255. Sampling i s s y n c h r o n i z e d w i t h e v e r y e i g h t h l e v e l of the s t a i r c a s e waveform so g e n e r a t e d , t o g i v e 32 comparison t h l e v e l s . A count i s added t o the c o n t e n t s of the r word, w h i l e s t o r e d i n the r e v e r s i b l e c o u n t e r , on the occ u r r e n c e of the r comparison l e v e l , y, = rA v - v , i f sgn(x,-rAv+v) = - 1 , 26 The comparison l e v e l s g e n e rated by the d-a c o n v e r t e r l i e i n the range —v t o —v+32Av=v where v i s a p p r o x i m a t e l y s i x v o l t s . The computer reads each of the r words of the memory i n t o the r e v e r s i b l e c o u n t e r i n synchronism w i t h the sampling of sgn(x^ - y^) and each of the new cumulants i s w r i t t e n i n t o the memory b e f o r e the occurrence of the next sample p u l s e . The l i m i t i n g v a l u e of ^  f o r each comparison l e v e l r e p r e -s e n t s the p r o b a b i l i t y t h a t an i n s t a n t a n e o u s v a l u e of the i n -put s i g n a l l i e s below t h a t l e v e l , where C i s the count s t o r e d i n the memory f o r the p a r t i c u l a r l e v e l and N i s the sample s i z e . 2.6 The P r o b a b i l i t y D e n s i t y F u n c t i o n The p r o b a b i l i t y d e n s i t y f u n c t i o n , f ( x ) , i s e s t i m a t e d as the d e r i v a t i v e of the c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n , F ( x ) . A S i n 2»5, the d-a c o n v e r t e r produces a s t a i r c a s e v o l t -age which p r o v i d e s 32 comparison l e v e l s . A t any g i v e n sample t i m e , t , the f i r s t f l i p - f l o p of the s h i f t r e g i s t e r i n channel—one ?. c o n t a i n s the sample s g n ( x ( t ) - (r+l)Av+v) and the second con-t a i n s the sample s g n ( x ( t - A t ) -rAv+v) where At i s the sampling i n t e r v a l and the comparison l e v e l s are y ^ ( t ) = ( r + l ) A v - v ;and y ^ ( t - A t ) = r A v - v . The outputs of these two f l i p — f l o p s are com-b i n e d i n a p o l a r i t y - c o m p a r i s o n c i r c u i t t o add one toy s u b t r a c t one from, or l e a v e unchanged, the count s t o r e d i n the r e v e r s i b l e c o u n t e r . The e s t i m a t e d v a l u e of the p r o b a b i l i t y d e n s i t y f u n c t i o n i s f ( ( r + l ) A v - v ) = * ((r+l)Av-v)^ - F ( r A v - v ) 27 where the count estimating f ^(r+i)Av-v) i s stored i n the r^* 1 word of the memory. Hence one i s added to the count stored i n the reversible counter i f sgn (x(t)-(r+1)Av+v)=-l and sgn ^c(t-At)-rAv+vj = +1; one i s subtracted i f sgn ^ c(t)-(r+l )Av+v) = +1 and sgn ^c(t-At)-rAv+vj = -1; and the count---is unchanged i f sgn ^ t(t)-(r+l )Av+v) = -1 and sgn (x(t-At)-rAv+v) = -1. Q The l i m i t i n g value, ^ , for each pair of comparison levels represents the probability that an instantaneous value of the input signal l i e s between these l e v e l s . 3. DETAILS OF THE BLOCK DIAGRAM AND CIRCUITS 28 3.1 S u b d i v i s i o n of the Computer The computer can be d i v i d e d i n t o e i g h t s e c t i o n s , each of whic h w i l l be d i s c u s s e d i n d e t a i l . These a r e : 1. t i m i n g c i r c u i t s ; 2. the comparator and sampling c i r c u i t s ; 3. the s h i f t r e g i s t e r and sc a n n i n g c i r c u i t s ; 4. the p o l a r i t y - c o m p a r i s o n c i r c u i t ; 5. the r e v e r s i b l e c o u n t e r 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 ; 6. the memory address system; 7. the a u x i l i a r y n o i s e g e n e r a t o r s ; 8. 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 . I n a d d i t i o n t h e r e i s the m o d e - s e l e c t i o n system which i s not d e s c r i b e d i n the t h e s i s . The computer has f i v e modes of o p e r a t i o n , namely; 1. s y m m e t r i c a l sampling c o r r e l a t o r , 2. a s y m m e t r i c a l s a m p l i n g c o r r e l a t o r , 3. computer of average t r a n s i e n t s , 4. p r o b a b i l i t y d e n s i t y c a l c u l a t i o n , 5. c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n c a l c u l a t i o n . Each mode of o p e r a t i o n r e q u i r e s c e r t a i n m o d i f i c a t i o n s w h i c h are d e s c r i b e d i n t h i s c h a p t e r . S e l e c t i o n of a p a r t i c u l a r mode i n v o l v e s m e c h a n i c a l s w i t c h i n g which i s a c c o m p l i s h e d , i n many cases,,,by a s t e p p i n g s w i t c h . Mode s e l e c t i o n i s i n d i c a t e d s y m b o l i c a l l y on the diagrams by means of two- or t h r e e - p o s i t i o n s w i t c h e s . 29 I n the diagrams t o f o l l o w , the symbol A denotes "And g a t e " , the symbol 0 "Or g a t e " , and the symbol I denotes " I n v e r t e r " . These symbols w i l l be used w i t h o u t f u r t h e r e x p l a n a t i o n . Whenever And gates and Or gates are connected i n cascade an e m i t t e r - f o l l o w e r b u f f e r c i r c u i t must be p l a c e d between them. The presence of the symbol, B. , f o l l o w i n g a b l o c k i n d i c a t e s the J presence of a b u f f e r c i r c u i t where the number j i d e n t i f i e s the p a r t i c u l a r e m i t t e r - f o l l o w e r . The f l i p - f l o p s used are of two k i n d s ; ( l ) a c o u n t e r f l i p - f l o p w h i c h has a s i n g l e p u l s e i n p u t ( t r i g g e r i n g occurs on the p o s i t i v e g o i n g edge of the i n p u t p u l s e ) and a dc " s e t 0" i n p u t ; (2) a f l i p - f l o p w i t h s e p a r a t e p u l s e " s e t 1" and p u l s e " s e t 0" i n p u t s as w e l l as a dc " s e t 0'' i n p u t . The output of an Or g a t e , a p p l i e d t o the p u l s e i n p u t of a f l i p - f l o p or monostable, has too l a r g e a r i s e time t o cause t r i g g e r i n g . Hence Or gate o u t p u t s t r i g g e r i n g c a p a c i t a n c e - c o u p l e d f l i p - f l o p s must f i r s t pass t h r o u g h s q u a r i n g a m p l i f i e r s denoted SA• where j i d e n t i f i e s the p a r t i c u l a r a m p l i f i e r . And gate outputs d r i v i n g c a p a c i t a n c e -c o u p l e d l o g i c elements such as f l i p - f l o p s or monostables may have b u f f e r i n g ( b u f f e r c i r c u i t s r e q u i r e d f o r 200-Kc unites but not f o r 1-MC u n i t s ) . I n p u t s t o v a r i o u s s e c t i o n s of the computer are o f t e n w r i t t e n i n the n o t a t i o n of B o o l e a n a l g e b r a . I n t h i s n o t a t i o n , the e q u a t i o n F = DEG means F i s t r u e (F=l) o n l y i f each of E,D, and G are t r u e (=l) and F i s f a l s e (F=0) o t h e r w i s e . The e q u a t i o n F = E+D+G means t h a t F i s t r u e (F=l) i f any one of the i n p u t s E,D, or G i s t r u e and F i s f a l s e o n l y i f each of E,D, and 30 G i s f a l s e . I n Boolean a l g e b r a , the complement of P i s denoted by F where, i f F i s t r u e (F=l) t h e n F=0, or i f F i s f a l s e (F=0) the n F = l . E q u a t i o n s u s i n g t h i s n o t a t i o n w i l l h e n c e f o r t h be w r i t t e n w i t h o u t f u r t h e r e x p l a n a t i o n . 3.2 Timing C i r c u i t The t i m i n g p u l s e t r a i n s r e q u i r e d t o c o n t r o l the o p e r a t i o n of the computer are generated by c o n v e n t i o n a l d i g i t a l c o u n t i n g c i r c u i t s used i n c o n j u n c t i o n w i t h a c l o c k o s c i l l a t o r . The ne c e s s a r y o u t p u t s are o b t a i n e d from these c o u n t i n g c i r c u i t s by a system of d i g i t a l gates and used to c o n t r o l the o p e r a t i o n of the computer d i r e c t l y , o r , i n some c a s e s , t o t r i g g e r mono-s t a b l e c i r c u i t s w hich are r e q u i r e d f o r c o n t r o l . The e s s e n t i a l s of the t i m i n g c i r c u i t are an e x t e r n a l v a r i a b l e - f r e q u e n c y o s c i l l a t o r , a f r e q u e n c y d i v i d e r (two f l i p -f l o p s i n c a s c a d e ) , a t h r e e — s t a t e c o u n t e r , a 3 2 - s t a t e c o u n t e r j a 18 sample c o u n t e r capable of c o u n t i n g up t o 2 samples and num-« erous d i g i t a l gates and mpnstable c i r c u i t s (see F i g u r e 3 . 1). As shown i n F i g u r e 3*1, the t h r e e - s t a t e c o u n t e r d r i v e p u l s e can be d e r i v e d from e i t h e r of two sources depending on the s e t t i n g of the Clock s e l e c t s w i t c h . T h i s s w i t c h has t h r e e p o s i t i o n s ; the c a l c u l a t e p o s i t i o n where the h i g h - f r e q u e n c y c l o c k i s used; the standby p o s i t i o n w i t h no c l o c k i n p u t ; and the readout p o s i t i o n where a l o w - f r e q u e n c y c l o c k i s used f o r con-v e r s i o n of the e s t i m a t e d c o r r e l o g r a m t o an analogue v o l t a g e . W i t h the c l o c k s e l e c t s w i t c h s e t a t the c a l c u l a t e p o s i -t i o n , the d r i v e p u l s e reaches the t h r e e - s t a t e c o u n t e r t h r o u g h a t w o - i n p u t And g a t e . The a l g e b r a i c e x p r e s s i o n f o r the d r i v e C L O C K FREQUENCY DIVIDER T C L O C K S E L E C T S W I T C H LOVY FREO CLOCK A 8, 3-STATE COUNTER TTT DC SET O M A . RT DC SET O 32-STATE COUNTER MON0-STftBLE I MONO-STABLE Ms RT M SI S A M P L E C O U N T E R 3 2 5-INPUTAND &ATES FIT TTTT .SAMPLE SI7 MONO STABLE DC RESET DC RESET e4 SET I REAOOUT C O N T R O L F U P - F L O P DC RESET j '! ' SAMPLE S E T I WRITE CONTROL FLIP-FLOP T MONO-STABLE Fw V R C O N T R O L FLIP- FLOP M O N O -STABLE MOMO-M 3 MONO-1 — > STABLE — > STABlE F i g . 3.1 Timing C i r c u i t Block Diagram 32 p u l s e , D^, i s D^= F^rt^where t ^ i s the output of the f r e q u e n c y d i v i d e r andF„ i s an output of the sample c o n t r o l f l i p - f l o p . The sample c o n t r o l o utput i s n o r m a l l y i n the s t a t e F = 1 but i s s s e t t o the s t a t e F g= 0 when the sample c o u n t e r reaches a p r e s e t count* The oc c u r r e n c e of the F = 0 s t a t e i n h i b i t s the d r i v e s p u l s e t o the t h r e e - s t a t e c o u n t e r , thus b r i n g i n g the computer o p e r a t i o n t o a h a l t * B e f o r e the c a l c u l a t i o n can b e g i n , the sample c o n t r o l f l i p - f l o p i s s e t t o the s t a t e F = 1 by a dc s r e s e t p u l s e , R^. Three c o n t r o l o u t p u t s are d e r i v e d from the t h r e e - s t a t e c o u n t e r by means of the g a t i n g arrangement i l l u s t r a t e d i n F i g u r e 3.2. These p u l s e t r a i n s are d e s i g n a t e d t ^ , the readout 1 2 c l o c k , t , the operate c l o c k , and t , the w r i t e c l o c k * The co u n t e r a c t u a l l y has a f o u r t h s t a t e of v e r y s h o r t d u r a t i o n c o r r e s p o n d i n g to the time r e q u i r e d by the r e s e t monostable, 0 . M 0 , t o r e s e t the c o u n t e r t o the t s t a t e ( l e s s than 2 m i c r o — S d, seconds)« W i t h the c l o c k s e l e c t s w i t c h s e t a t the readout p o s i t i o n , a l o w - f r e q u e n c y d r i v e p u l s e reaches the t h r e e - s t a t e c o u n t e r through another t w o - i n p u t And g a t e . The a l g e b r a i c e q u a t i o n f o r the d r i v e p u l s e i s now D^= ^ g ' ^ j j where t j j i s the lo w - f r e q u e n c y c l o c k and FJJ i s an output of the readout c o n t r o l f l i p - f l o p w h ich g e n e r a t e s an i n h i b i t o u t p u t , FJJ= 0» a f t e r a l l 32 words of the memory have been r e a d out and r e c o r d e d . The s e p a r a t e l o w - f r e q u e n c y c l o c k , t j j , i s r e q u i r e d f o r readout so t h a t a slow-speed analogue r e c o r d e r can be d r i v e n by 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 . The readout c o n t r o l f l i p - f l o p i s s e t t p the s t a t e FLIP-FLOP 3 A K'1 "fii OC SET O » I MONO-STABLE F i g . 3.2 Three-State Counter PJJ= 1 by means of the dc r e s e t s w i t c h b e f o r e s w i t c h i n g from the standby p o s i t i o n t o the readout p o s i t i o n (the dc r e s e t s w i t c h s h o u l d o n l y be used when the computer i s i n the standby p o s i -t i o n ) . When the 3 2 - s t a t e c o u n t e r reaches a count of 31, a 31 t r i g g e r pulse,M^.S , i s a p p l i e d t o the readout c o n t r o l f l i p -— 31 f l o p s e t t i n g i t t o the s t a t e F^= 0. The t r i g g e r p u l s e , M^.S , i s o b t a i n e d by g a t i n g the monostable p u l s e , M^, w i t h the 32 31 s t a t e of the c o u n t e r , S « The monostable i s one of a s e r i e s of c o n t r o l m o n s t a b l e s . The t r i g g e r p u l s e f o r the monostable i s o b t a i n e d from the M^monostable which i s i n t u r n —2 t r i g g e r e d by the occ u r r e n c e of t . The output p r o v i d e s a d r i v e p u l s e f o r the 3 2 - s t a t e c o u n t e r which c o n s i s t s of f i v e f l i p - f l o p s . Outputs S ^ j S 1 , . . . . S 3 1 c o r r e s p o n d i n g t o the 32 s t a t e s of t h i s c o u n t e r are o b t a i n e d by means of 32 f i v e - i n p u t And g a t e s . The And gate c o n n e c t i o n s are g i v e n i n terms of the b i n a r y number system i n F i g u r e 3.3* The t i m i n g c i r c u i t c o n t a i n s a number of c o n t r o l mono-s t a b l e s . The monostable, , t r i g g e r e d by t ^ , se r v e s as a r e a d -out p u l s e and as a sampl i n g p u l s e f o r a l l modes of o p e r a t i o n except the s y m m e t r i c a l s a m p l i n g mode. The monostable p u l s e j t r i g g e r e d by the monostable, i s r e q u i r e d t o i n i t i a t e the p o l a r i t y - c o m p a r i s o n o u t p u t s ( e x p l a i n e d i n s e c t i o n 3.5)* The monostable, M,-, a l s o t r i g g e r e d by , i s of s h o r t e r d u r a t i o n t h a n M.^ and i s r e q u i r e d f o r c o n t r o l of the r e v e r s i b l e c o u n t e r ( s e c t i o n 3.6). The monostable o u t p u t , M^, t r i g g e r e d by t ^ , ser v e s as a t r i g g e r p u l s e f o r the r e v e r s i b l e c o u n t e r . 31 The s h i f t p u l s e , Pg = M-^ .S , f o r the s h i f t r e g i s t e r , and the r e s e t p u l s e , M.,= M 4.S 3 1, f o r the f i r s t f l i p - f l o p of the s h i f t r e g i s t e r are ge n e r a t e d i n the t i m i n g c i r c u i t and used when 35 AND Gate Inputs AND Input Input Input Input Input Outpu-iate 1 2 3 4 5 1 0 0 0 0 0 0 s 2 1 0 0 0 0 1 s 3 0 1 0 0 0 2 s 4 1 1 0 0 0 3 s 5 0 0 1 0 0 4 s 6 1 0 1 0 0 s 5 7 0 1 1 0 0 6 s 8 1 1 1 0 0 7 s 9 0 0 0 1 0 8 s 10 1 0 0 1 0 9 s 11 0 1 0 1 0 10 s 12 1 1 0 1 0 11 s 13 0 0 1 1 0 12 s 14 1 0 1 1 0 13 s 15 0 1 1 1 0 14 s 16 1 1 1 1 0 s 1 5 17 0 0 0 0 1 16 s 18 1 0 0 0 1 s 1 7 19 0 1 0 0 1 18 s 20 1 1 0 0 1 19 s 21 0 0 1 0 1 20 s 22 1 0 1 0 1 21 s 23 0 1 1 0 1 22 s 24 1 1 1 0 1 23 s 25 0 0 0 1 1 24 s 26 1 0 0 1 1 25 s 27 0 1 0 1 1 26 s 28 1 1 0 1 1 27 s 29 0 0 1 1 1 28 s 30 1 0 1 1 1 29 s 31 0 1 1 1 1 30 s 32 1 1 1 1 1 31 s Pig. 3.3 AND Gate Connections to the 32-State Counter the computer i s operating i n the symmetrical sampling mode. A strobe pulse for the sense c i r c u i t i s provided by the R ' monostable, M„, which i s triggered by t^, (section 3.7). h A high-frequency clock, t , i s provided for the genera-ti o n of a staircase voltage which i s necessary for determination of the probability density and cumulative probability function (section 3.8). The timing sequence of the various control outputs i s i l l u s t r a t e d i n Figure 3*4. The remaining section of the timing c i r c u i t to be d i s -cussed i s the sample counter and control outputs derived from i t . The drive pulse for the sample counter i s obtained from the l a s t f l i p - f l o p of the 32-state counter. Hence, a trigger pulse i s received for each sample product formed, whatever the mode of operation. The two control outputs obtained from the sample counter are the write control output, F^, and the sample control out-put, F g. The output, F^, i s obtained from the write control f l i p - f l o p which i s i n i t i a l l y i n the state F = 0 and i s t r i g g -ered to the state F =1 when the sample counter has received w r a t o t a l of 32 trigger pulses corresponding to 32 samples. The function of the F output i s to prevent information from being written into the buffer memory u n t i l the s h i f t register has received 32 samples. It i s evident that the F^ output also performs an automatic clearing action at the beginning of the cal c u l a t i o n as each word of the memory i s read out a number of times during the interval F^ = 0. The t r a n s i t i o n of the write control f l i p - f l o p from the state F - 1 to the state F = 0 w ! w n. ~ mm iniuiniinni e M 2 M0 M S' -31 r r ir IT u U L f U — .11 u . u u u I u ~1 F i g . 3 . 4 Timing Waveforms triggers the monostable, , which resets the sample counter to zero. The function of the sample control output, F , has been s e x p l a i n e d previously* The trigger input for this f l i p - f l o p i s obtained by gating the pulse with the output of the j ^ * 1 f l i p - f l o p of the sample counter. If the output of the j ^ * 1 f l i p - f l o p i s selected as the gate input, the trigger pulse, M..F-, occurs when the counter has received a to t a l of 2^  ^  4 J sample s. 3.3 Comparator and Sampling C i r c u i t s The comparator forms a quantization of the sum of the signal and au x i l i a r y noise, x^+ y^, into the function u^ = sgn (x^+ y ^ ) . The sum i s formed by a simple r e s i s t i v e adder and compared to zero i n a two-stage d i f f e r e n t i a l amplifier (see Figure 3.5). The output of the d i f f e r e n t i a l amplifier drives a Schmitt-trigger c i r c u i t which i s i n the one state i f sgn(x^+ y^) =1 and the zero state i f sgn(x^+ y^) = -1. Theorems I and II can apply equally well to sgn(x^+ y^) or sgn(x^- y ^ ) . Hence the sign of the sum, u^ = sgn(x^+ y ^ ) , has been chosen for convenience, because the required dynamic range of the d i f f e r e n t i a l amplifier i s smaller i f one input i s grounded. The Schmitt-trigger outputs of channel-one and channel-two, u-^  and U £ , respectively, are sampled by means of two-input And gates (Figure 3.6). The channel-two output, u^, i s controlled by the mono-stable, MQ, SO that the output of the sample gate i s U^.MQ* yi AAA J 4 - 7 K ISTK 2 N 4 0 4 4 - . 7 K *3 < 2 N 4 - 0 4 -3- 5CHM ITT T R I & & E R ¥ith 0 and y\ = O r adjust ^ so that point T-^ i s at zero v o l t s . Then adjust potentiometers and R^  so that P 2 and are at same potential and P^ and P^ are at same potential. Fig. 3 . 5 D i f f e r e n t i a l Amplifier 40 M Comparator u l _ A B8 T -M, ^ 1 o or u^.M^ Comparator T U 2 A J B, 'se' t xif* Flip-Flop dc Reset T R T set 0" F i g . 3 . 6 Comparator and Sample Gates The sampling rate of channel-one depends on the mode of operation. In the symmetrical-sampling mode, the output of the channel-one sample gate i s U^.MQ. For a l l other modes of operation the channel-one output, u^, i s controlled by the monostable, M^, The accuracy of p o l a r i t y determination i s increased by amplifying the sum, y^, before going to the Schmitt-trigger c i r c u i t . If the gain of the amplifier i s G and the trigger point of the Schmitt c i r c u i t i s displaced from the zero l e v e l by an amount AV, the effect i s exactly the same as i f a dc le v e l of AV/G were added to the input signal. If AVm i s the larger of the offsets i n the two channels, the correlator (AV \ 2 output due to these offsets i s less than e = \~Tj~y • If w e con-sider the worst case imaginable, say, AV = 10 v o l t s , and G = 200, /AV \2 1 0 Q 2 >n = = — — 4 = •0025 volts . This error i s compared 41 / " ' m i " inn o thei 4x10" to the maximum value of the autocorrelation function (the mean-2 2 2 A 36 square value), CT , for a t y p i c a l case. UsingO" = —g = — = 4 v o l t s 2 , the r e l a t i v e error i s ^  x 100% = °0025 x iQ0f0 - .06$. Hence error i n the triggering level of the Schmitt trigger has a negligible e f f e c t . Another source of error i s the hysteresis of the Schmitt trigger. The Schmitt c i r c u i t used has a hysteresis of less than 2.5 v o l t s , so that the hysteresis referred to the input i s 2 5 2QQ = 13 m i l l i v o l t s , which appears to be adequate for signal levels i n the range one to six v o l t s . 3.4 The Shift Register and Scanning C i r c u i t s The s h i f t register serves as a temporary storage element when the computer i s operating as a correlator i n the symmet-rical-sampling mode. This allows both channels to be sampled at the same rate. The s h i f t register i s composed of f l i p -flops with separate pulse "set 1" and "set 0" inputs as well as a dc reset input. Information i s shifted from one f l i p -f l o p to the next by means of And gates i n a conventional manner 31 (see Figure 3.7) on the occurrence of a s h i f t pulse, P = M-.S s _? The s h i f t pulse, P , serves as an input to every And gate. s The reset pulse for the f i r s t f l i p - f l o p must occur after the s h i f t pulse and before the next sample i s taken. This reset 31 pulse i s provided by the monostable output, Mg = M^ .S . The output of each f l i p - f l o p of the s h i f t register i s sequentially scanned by means of 32 two-input And gates and a th 32-input Or gate. The j f l i p - f l o p output reaches the P-ROM C H A N N E L - O N E S f t M P L E <3-ATE Del R.ESET S E T 1 S H I F T P U L S E A b b r e v i a t i o n s SS - Symmetrical Sampling AM - A l l Other Modes TO POL.A.R.ITY COINCIDENCE C l R . C U IT F i g . 3.7 The S h i f t R e g i s t e r and Scanning C i r c u i t s p o l a r i t y - c o m p a r i s o n c i r c u i t d u r i n g the i n t e r v a l d e f i n e d by the t i m i n g p u l s e . The B q o l e a n - a l g e b r a i c e x p r e s s i o n f o r the channel-one i n p u t t o the p o l a r i t y - c o m p a r i s o n c i r c u i t i s u± = S°.u 1(t+na) + S 1 , ^ (t+na-AT ) +... + S 3 1 , ( t + n o C r - 3 I A T ) . The sequence of occ u r r e n c e of the t i m i n g p u l s e s i s i l l u s -t r a t e d i n F i g . 3.4. The 3 2 - i n p u t Or gate i s b u i l t up of V y l e modules. E i g h t f i v e - t e r m i n a l Or gates are r e q u i r e d and connected i n the con-f i g u r a t i o n of F i g . 3.8, T h i s Or gate i s f o l l o w e d by a double i n v e r t e r f o r l e v e l r e s t o r a t i o n . | 111 y 11 1 11 0 0 0 l M \J \t J i l l " J 0 0 i L 0 0 \ 1 \ ' \ 1 \ / 0 Double I n v e r t e r F i g . 3.8 32-Input Or Gate Only the f i r s t f l i p - f l o p of the s h i f t r e g i s t e r i s used when o p e r a t i n g i n the a s y m m e t r i c a l - s a m p l i n g mode or f o r the c a l c u l a t i o n of the c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n . I n these two c a s e s , the r e s e t p u l s e f o r the f i r s t f l i p - f 3 . o p i s now p r o v i d e d by the monostable, M^. The f i r s t two f l i p - f l o p s of the s h i f t r e g i s t e r are r e -q u i r e d when the computer i s used f o r e s t i m a t i o n of the p r o -b a b i l i t y - d e n s i t y f u n c t i o n . C o n t r o l p u l s e s t h e n c o n s i s t of a s h i f t p u l s e , M^, and a r e s e t p u l s e , M^, f o r the f i r s t f l i p - f l o p . Outputs are t a k e n d i r e c t l y from these two f l i p - f l o p s t o the p o l a r i t y - c o m p a r i s o n c i r c u i t . 3.5 The P o l a r i t y - C o m p a r i s o n C i r c u i t P o l a r i t y - c o i n c i d e n c e d e t e r m i n a t i o n i s r e q u i r e d when the computer i s o p e r a t i n g i n a l l modes except the c u m u l a t i v e - p r o -b a b i l i t y mode. The c i r c u i t ( F i g u r e 3.9) i s b a s i c a l l y u n a l t e r -ed w i t h the computer o p e r a t i n g as a c o r r e l a t o r or computer of average t r a n s i e n t s . A c o n s i d e r a b l e m o d i f i c a t i o n i s r e q u i r -ed f o r the p r o b a b i l i t y - d e n s i t y mode. Vhen o p e r a t i n g as a c o r r e l a t o r i n the sym m e t r i c a l samp-l i n g mode, the a l g e b r a i c e x p r e s s i o n s f o r the c o i n c i d e n c e o u t -p u t , C, and the a n t i c o i n c i d e n c e o u t p u t , A , are as f o l l o w s : C = S + ^ oC^H"«oo o +S "^31 where °P= T2,4' u 2(t+na).u^(t+na-pAT )+ u 2 ( t + n a ) ( t + n a - p A T ) and A = S ^ . A Q + S ' ' " . A-^ + o . . . + S 3 " ' ' . A ^ ^ -> TO FIRST FLIP-FLOP OF S R. T O 5R SCANNING OR AS SS •»> =» AM A r T O C H A N N E L ' S F L l P - P L O P AT L L ( Y & \ / - V ) -A u u M i I A T " r -B45 A * PD F L I P -- F L O P SETll SET Q D C R E S E T C O R STET 1 F L I P -F L O P T~ PD DC RESET SET 0 A M I" S w i t c h P o s i t i o n A b b r e v i a t i o n s AS - As y m m e t r i c a l Sampling SS - Symmetrical Sampling COR - - C o r r e l a t i o n PD - P r o b a b i l i t y D e n s i t y CP - Cumulative P r o b a b i l i t y AM - A l l Other Modes CAT - Computer of Average T r a n s i e n t s SR - S h i f t R e g i s t e r F i g . 3 . 9 P o l a r i t y - C o m p a r i s o n C i r c u i t 46 where A P = T2,4 (t+na) .u^ (t+na-pAT ) + xa^ (t+na) .u^ (t+na-pAT ) where the bar ( ) denotes the complement and the gating function, T 0 A , s i g n i f i e s a time interval defined as the int e r v a l between the occurrence of the positive-going edge of the monostable pulse, H^t which sets the required information into the storage f l i p - f l o p , and the positive-going edge of the monostable, , which resets the f l i p - f l o p s to the C=0 and A=0 0 31 state. The control outputs S , . . S , perform a scanning function; for example the output, u^(t+na-pAT ), of the p^*1 f l i p - f l o p of the s h i f t register i s compared to the output, U2(t+nAT ), of the single f l i p - f l o p i n channel-two when = 1. When the computer i s operating as a correlator i n the asymmetrical-sampling mode, the algebraic: expressions for the C and A outputs are: u 2(t+na) .u-^  (t+na+pAT ) + (t+na).u^ (t+na +pAT ) .T 2,4 and A = u^ (t+na) .u^ (t+na+pAT ) (t+na) .u^ (t+na +pAT ) .T 2,4 where u 2(t+na) forms a constant input to the p o l a r i t y - c o i n c i -dent c i r c u i t during the interval a, whereas the u-^  samples are taken at intervals of AT for a t o t a l of 32 samples i n the i n t e r v a l . The channel-two sample pulse i s s t i l l MQ; however, the channel-one sample pulse i s now . When the computer i s operating as a computer of average 4 7 transients (CAT), the channel-two input i s switched to -12 volts (CAT switch) and the po l a r i t y of the channel-one input i s then compared to the "1" input on channel-two. The com-puter i s operated i n the high-frequency mode i n this case (M^ i s the sample pulse). The c i r c u i t i s modified for determination of the cumu-l a t i v e p robability function by connecting the monostable, M2, d i r e c t l y into the "set 1" input of the storage f l i p - f l o p , thus always generating a C pulse. The A output i s grounded by means of a switch. In thi s case, p o l a r i t i e s are not compared. As mentioned previously, considerable modification i s required when operating i n the probability-density mode. The probability-density function, f ( x ) , i s estimated from the cum-ulative d i s t r i b u t i o n , F(x), as follows: f((rn4)AV-V) = F((r+1)AV-Vj - F«rAV)-V) where r goes from 0 to 31. The method of estimation i s to form the functions u=sgn (x+y) where -y i s a le v e l of a staircase voltage produced by the d-a converter. The sampled values are fed to the s h i f t register where only the f i r s t two f l i p - f l o p s are used. The f i r s t f l i p - f l o p contains the function u((r+1)AV-V), which results from comparison of x and -y=(r+l)AV-V. The second f l i p - f l o p contains u(rAV-V). The output of the f i r s t i s considered to be an estimate of F((r+1)AV-V) and the second, an estimate of F(rAV). Hence a count of 1 i s added to the estimate of f((r+ i)AV-V) i f M2-u((r+l)AV-V)„u(rAV-V) = U + = 1 4 8 and a count of 1 s u b t r a c t e d i f M 2.u((r+1)AV-V).u (rAV -V) = U" = 1 No count i s added i f u((r+1)AV-V).u(rAV-V) = U° = 1. I n t h i s case a b l o c k i n g p u l s e , *U°, i s generated which p r e v e n t s the t r i g g e r p u l s e from r e a c h i n g the r e v e r s i b l e c o u n t e r . The s w i t c h e s , S, , and S~ , must be s e t t o connect i n U + and U t o ' l p ' 2p' the s t o r a g e f l i p - f l o p s i f the p r o b a b i l i t y - d e n s i t y mode of o p e r a t i i s r e q u i r e d . The monostable p u l s e , M 2, i s i n c l u d e d so t h a t the i n f o r m a t i o n i s i n s e r t e d i n t o the f l i p - f l o p a t the d e s i r e d t i m e . 3.6 The R e v e r s i b l e Counter 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 3.6.1 The co u n t e r The r e v e r s i b l e c o u n t e r performs the f o l l o w i n g two f u n c t i o n s ; ( l ) i t s e r v e s as a means of a l t e r i n g any of the p a r t i a l sums s t o r e d i n the b u f f e r memory and (2) i t serves as a temporary s t o r e f o r each word of the b u f f e r memory d u r i n g the c o n v e r s i o n of the c a l c u l a t e d c o r r e l o g r a m from d i g i t a l t o analogue form. The r e v e r s i b l e f e a t u r e of the co u n t e r i s r e q u i r e d so t h a t b o t h a d d i t i o n and s u b t r a c t i o n may be c a r r i e d out. The oper-a t i o n of the r e v e r s i b l e c o u n t e r i s unchanged f o r a l l modes of o p e r a t i o n except the c u m u l a t i v e - p r o b a b i l i t y mode where the co u n t e r i s c o n s t r a i n e d t o count up by making the c o i n c i d e n c e o u t p u t , C = l , the o n l y p o s s i b l e output of the p o l a r i t y - c o m -p a r i s o n c i r c u i t . I n t h i s case a count change of 1 occurs o n l y i f the sampled v a l u e of sgn(x+y) i s - 1 , c o r r e s p o n d i n g t o x < - y = rAV-V where -y i s the comparison l e v e l g enerated by the d-a s t a i r c a s e g e n e r a t o r ( e x p l a i n e d i n s e c t i o n 3.8). The t r i g g e r - p u l s e i n p u t , t , t o the c o u n t e r when o p e r a t i n g i n the c u m u l a t i v e - p r o b a b i l i t y mode, i s g i v e n by 1 1 t = u, (rAV-V).M/-+ S where S i s the output of sense c i r c u i t 1, r 1 6 e e r Mg i s a monostable p u l s e o c c u r r i n g i n the operate c l o c k i n t e r v a l , and u^(rAV-V) i s an output of the f i r s t f l i p - f l o p of the s h i f t r e g i s t e r where u^(rAV-V) = 1 i f sgn (x+y) = - 1 . For a l l o t h e r modes, the c o u n t e r i s capable of c o u n t i n g up br down one co u n t . T h i s i s a c c o m p l i s h e d by t a k i n g the t r i g g e r p u l s e f o r a c o u n t e r f l i p - f l o p from the r i g h t - h a n d or "1" s i d e of the p r e c e d i n g f l i p - f l o p i f a d d i t i o n i s r e q u i r e d or from the l e f t - h a n d or "0" s i d e i f s u b t r a c t i o n i s r e q u i r e d * To a c c o m p l i s h t h i s , the output of each s i d e of every f l i p - f l o p i s t a k e n t h r o u g h a s e p a r a t e And gate which can be i n h i b i t e d a t w i l l (see F i g . 3.10) and these outputs are combined w i t h the sense c i r c u i t output i n a t h r e e - i n p u t Or gate which feeds t h r o u g h a s q u a r i n g a m p l i f i e r (not shown) t o the t r i g g e r i n p u t of the n e x t f l i p - f l o p . When the computer i s o p e r a t i n g as a c o r r e l a t o r or computer of average t r a n s i e n t s , the t r i g g e r i n -put t o the c o u n t e r i s t = M, + S"''. r r o e When o p e r a t i n g i n the p r o b a b i l i t y - d e n s i t y mode, the t r i g g e r p u l s e i n p u t i s t = M^.U^+ S"*" where i s an i n h i b i t i n g e output as e x p l a i n e d i n s e c t i o n 3.5. When the computer i s s e t to the readout mode t h e r e i s no t r i g g e r p u l s e i n p u t . The t r i g g e r i n p u t , t . , of the j ^ * 1 f l i p - f l o p of the r e v e r -s i b l e c o u n t e r i s g i v e n by t.= H, .F._, + H~.F._,+ where F. , J i J i J J- ® J l i s the "1" output of the ( j - l ) t h f l i p - f l o p , i s the "0" output of the ( j - l ) ^ f l i p - f l o p and i s the output of the j ^ * 1 A O - i T s i SA2 -FLIP-FLOP < A b b r e v i a t i o n s CAL - C a l c u l a t e RO-- Readout PD - P r o b a b i l i t y D e n s i t y AM - A l l Other Modes CP - Cumulative P r o b a b i l i t y P i g . 3.10 The R e v e r s i b l e Counter 51 sense c i r c u i t . This equation does not apply to the f i r s t f l i p -f l o p of the reversible counter. The control functions and E^ are normally zero but become H^ = 1 for subtraction and 1^= 1 for addition. The termination of the and E^ outputs, brought about by the occurrence of the positive-going edge of the monostable pulse, , causes spurious triggering of each f l i p - f l o p except the f i r s t i n the counter* Hence the contents of the reversible counter are written into the buffer memory before the termin-ation of H^  and ^  by using the pulse to activate the write And gates. The reversible counter i s reset to zero immediately after the termination of H^  or E^ D v " t n e application of the mono-stable pulse, M^, to the dc "set 0" input.. The dc "set. 0" input to the counter i s given by B^H-M^, i . e . the counter i s also reset by the main dc reset, R^ ,, of the computer. The monostable pulse, , reaches the write And gates through a two-input And gate and may be inhibited by the auto-matic clear switch, thus preventing information from being written into the buffer memory. The output D., of the j ^ 1 * 1 write And gate i s given by D.= M-.F...F. where F„ i s the output 3 j w j w of the write control f l i p - f l o p and F. i s the output of the j ^ * 1 f l i p - f l o p of the reversible counter. The output, D., activates the j d i g i t driver which provides a hal f - s e l e c t current pulse on the j ^ * 1 d i g i t l i n e of the core array i f D. = 1. The clearing operation i s automatic. If an estimate of the correlogram has been made and i s stored i n the buffer mem-ory and i t i s desired to discard this information and make a new calcu l a t i o n , the computer i s reset to the zero state while i n the standby position (no clock input) thus setting P^ . = 0, When the calcu l a t i o n i s started, the write control f l i p -f lop w i l l remain i n the state F^ = 0 u n t i l a t o t a l of 32 samples have been taken. During t h i s time, each of the 32 words w i l l have been read out of the buffer memory and no information written i n . Hence, when the write control f l i p - f l o p i s set to the state F = 1, the memory w i l l be completely cleared. The clearing operation can be eliminated by setting the clear switch, S c, to connect -12 volts into the i n h i b i t And gate while simultaneously disconnecting the F^ . input. This cap-a b i l i t y i s required when i t i s desired to estimate the c o r r e l -ogram by means of a small sample and i f the resultant estimate i s too "noisy", to continue the calcu l a t i o n without destroying the previously accumulated sample product sums. When the computer i s operating i n the readout mode i t i s desired that the readout be non-destructive; hence the i n h i b i t And gate i s bypassed by means of the operate mode selection switch, Sp2« 3.6.2 The counter control c i r c u i t s The addition of a count of 1 to the contents of the re-versible counter i s required i f , after readout, the sign b i t i s S = 1 (signifying a positive sign) and the output of the pol-arity-comparison c i r c u i t i s C=l, A=0, or i f the sign b i t i s S=0, corresponding to a negative p a r t i a l sum, and C=0, A=l indicating an anticoincidence. Subtraction of a count of 1 i s required i f the opposite set of conditions occurs, that i s , i f S=l and A=l, C=0, or i f S=0, A=0, C=l. The counter control outputs, H^  and H 2, are generated to accomplish t h i s (Figure 3*11) and are given by, H, = (S.C+S.A).F JL Z H 2 = (S.A+'S.C).Fz In the above equations F i s an i n h i b i t i n g output which i s required i f the p a r t i a l sum stored i n the counter i s zero. The d i f f i c u l t y with the zero value i s that i t can have either a positive or negative sign. Error due to propagation delay; may result i f the sign i s positive and an anticoincidence pulse A=l occurs or i f the sign i s negative and a coincidence pulse C=l occurs, since both these situations result i n a change of state of the sign-indicator f l i p - f l o p . If the blocking pulse, F , i s omitted and there i s a delay between the occurrence of z J the C or A pulse and the t r a n s i t i o n of the sign indicator binary, the wrong output, either or H 2 > could occur momentarily and cause spurious triggering of the f l i p - f l o p s i n the reversible counter. When the p a r t i a l sum i s zero, only the f i r s t f l i p - f l o p of the counter can change state. Hence,if a l l interconnections between f l i p - f l o p s are inh i b i t e d , the correct value (+ l ) w i l l result as the f i r s t f l i p - f l o p always changes i t s state at the occurrence of the trigger pulse, t . If the p a r t i a l sum i s zero, the i n h i b i t i n g output, F must be generated before z the C or A outputs can occur. Since the C or A outputs are i n i t i a t e d by the M 2 monostable pulse which i s triggered by the readout monostable, , the zero control f l i p - f l o p i s set to the state F = 0 by the occurrence of M,- which i s also triggered by £ L TfT" S I G N I N D I C A T O R F L I P - F L O P ~ T 7T~ 0 .. I SET1 DC RESET A SET O Z E P O CONTROL F H P - F L O P ~~7R * ' • . SET i SET 0 DC R.E.SET Fig. 3.11 Reversible Counter Control C i r c u i t s but i s of shorter duration than The trigger pulse for the "set 1" input of the zero control f l i p - f l o p i s obtained by gating the monostable pulse, M^ , with an And gate output which i s i n the state D=l only i f a l l f l i p - f l o p s of the reversible counter are i n the zero state. The trigger pulse i s then D.Mj-. The waveform D i s generated by an 18-input And gate which i s realized by a combination of three six-input And gates and a three-input And gate (Figure 3.12). 18 Inputs From Reversible Counter r - . • * '• 1 11 1 1 1 1 11111 1 1 1 1 I 1 A D Fig . 3.12 18-Input AND Gate The "set 0" pulse for the zero-control f l i p - f l o p i s pro-vided by the monostable pulse., This f l i p - f l o p i s also reset to zero by the dc reset switch (separate dc reset input). The situation where the p a r t i a l sum read out of the mem-ory and into the reversible counter i s zero i s that where sign change can occur. The sign indicator f l i p - f l o p i s always i n the state S=0, corresponding to a negative sign, before readout. If the part-56 i a l sum read out of the memory has a positive sign, a pulse reaches the "set 1" input of the f l i p - f l o p through sense c i r -cuit 19, thus setting the f l i p - f l o p into the S=l state. If the sum i s negative, no pulse i s applied to the f l i p - f l o p which remains i n the state S=0* The sign of the p a r t i a l sum read out of the memory i s di f f e r e n t from the sign of the altered p a r t i a l sum written into the memory only i f (l) the p a r t i a l sum i s zero when readout occurs and (2) S i s i n i t i a l l y S=l and an a n t i -coincidence, A=l, occurs, or S i s i n i t i a l l y S=0 and a coin-cidence, C=l, occurs* The equations for the "set 1" and "set 0" inputs of the sign-indicator f l i p - f l o p are set 1 = (C,D+Sg9), set 0 = A 7 D . + W~, where triggering occurs on the positive-going edge of these inputs corresponding to a t r a n s i t i o n from a "1" (-12 volts) to a "0" (0 v o l t s ) . The "set 0" input includes the monostable pulse, M^ , which resets the f l i p - f l o p to the S=0 state after write. 3.7 The Memory Address System 3.7.1 The memory core array The computer memory i s a 32x32-core word-select magnetic-core array, using core type 51-114B made by Electronic Memories Inc. Each core i s threaded by four wires, two i n the v e r t i c a l d i r e c t i o n , one of which i s a sense wire and one a write d i g i t wire, and two i n the horizontal d i r e c t i o n , one of which i s a read wire and one a write wire* The memory cores have a nearly square hysteresis loop (Figure 3.13) with one saturation state corresponding to a binary "1" and the opposite saturation state corresponding to a binary "0". B l . B s • 1 I 1 or , / . ^ " Bs F i g . 3.13 The Core Hysteresis Loop For the sake of i l l u s t r a t i o n , the positive state B g i s defined as a "1" state and the state -B i s defined as the "0" state. s The arrangement of wires i s i l l u s t r a t e d for a small sec-t i o n of the array i n Figure 3.14* Sense Winding Read Winding .Write Winding R D i g i t Winding F i g . 3.14 Word-Select Array Wiring 58 A p a r t i c u l a r core of the array can be set i n the one state by the coincidence of the currents 1^ and 1^. If the core i s already i n the one state i t i s simply driven further into saturation when 1^ .+ 1^ i s present and the remanent flux density returns to the state, B t after the current pulse has terminated. s The occurrence of 1^ . or 1-^  by themselves i s not s u f f i c i e n t to switch a core. A core i s set to the zero state ( _B g) by the occurrence of the read current I D . If the core i s already i n the zero state i t i s only driven further into saturation and returns to the state, -B , after the read current pulse has terminated. S If the core i s driven from the one state to the zero state, a dB voltage pulse proportional to the rate of flux density change, ^ r , i s generated on the sense winding. The current driver c i r c u i t s are designed to produce current pulses of value 1^ . = 120 m.a. (one microsecond duration) 1-p = 120 m.a. (one microsecond duration) Ig = 415 m.a. (two microseconds duration) where the current values are accurate to + 10$ and the durations of the pulses are a r b i t r a r i l y chosen. With these values* a ty p i c a l one output i s 75 m i l l i v o l t s and a t y p i c a l zdro output i s 16 m i l l i v o l t s . The pulse width of the one output i s approximately 0.25 microseconds. 3.7.2 The memory address c i r c u i t The memory address system i s designed so that each word i s read out of a pa r t i c u l a r row of the array and new information i s written into this row, with rows addressed i n a sequential 59 manner. The selection of a par t i c u l a r word for readout or write i s accomplished by means of word-select switches. The read current driver i s connected to each of the read word wires i n p a r a l l e l on one side of the array and current i s steered into the selected read word wire by shorting this wire to ground (by means of a read word-select switch) on the other side of the array. Since a l l other wires are connected to a very high impedance (the co l l e c t o r of a cut-off transistor) almost a l l the current flows through the selected word l i n e . Exactly the same method i s used for write word-selection (see Figure 3.15), except that the write-current driver i s on the side opposite to the read-current driver and the write word-select switch i s on the opposite side to the read word-select switch. The word-select switches consist of an emitter follower and a current switch. The appearance of a negative pulse at the input to the emitter follower drives the current-switch transistor into saturation, allowing the driver output current to flow from the col l e c t o r to ground. If the input to the emitter follower i s held at ground potential, the current-switch transistor i s cut off and no current can flow through the word l i n e . The write- and digit-current drivers are id e n t i c a l and the read-current driver d i f f e r s from them only i n the size of the emitter r e s i s t o r , Rg, which sets the current output l e v e l . Each of the current drivers consist of a 2N1613 s i l i c o n planar tra n s i s t o r connected i n a current-source configuration (common-base connection), a tran s i s t o r switch and an emitter follower. A current pulse i s produced by the application of a negative 60 EN404 -12 WRITE > 470X2. CUKRENT DRIVER 3^12. i z . a > W O O L READ _ / w\_^2N404 300 PF j •2 NI6I3 C. (JR-RtWT DPUvER Z N 4 0 4 Mi +IZ R E A D W O R D S E L E C T S W I T C H - I Z ZN404 1 ^ IK 3 0 0 P F J_ -6 3 0 0 PF r < 3.3 K S E N S E W I N D I N G S + iz + 1 2 W P U T F W O R D SELECT '. S W I T C H 2N404 ^  IK & ^ N 4 0 4 A V - y v \ A / y — j + 12 R-EAD W O R D S E L E C T S W I T C H +12. WRITE WORD SELECT SWITCH Pig. 3.15 Word-Select System 61 p u l s e t o the i n p u t of the d r i v e r c a u s i n g the t r a n s i s t o r s w i t c h t o connect the -6 v o l t s u p p l y t o the base of the output t r a n -s i s t o r . T h i s produces a f i x e d v o l t a g e drop of 5 v o l t s a c r o s s the e m i t t e r r e s i s t o r of the output t r a n s i s t o r and hence t h i s r e s i s t o r s e t s the c u r r e n t l e v e l out. The r e a d - c u r r e n t l e v e l i s s e t a t 415 m.a. and the w r i t e and d i g i t - c u r r e n t outputs are se t a t 120 m.a. The d i g i t d r i v e r and sense c i r c u i t are i l l u s t r a t e d i n F i g u r e 3.16. The d i g i t l i n e i s grounded on one s i d e of the a r r a y and connected d i r e c t l y i n t o the d i g i t d r i v e r on the o p p o s i t e s i d e . The i n p u t s t o the d i g i t d r i v e r are o b t a i n e d from the w r i t e And gates of the r e v e r s i b l e c o u n t e r . The sense c i r c u i t c o n s i s t s of a s i n g l e - s t a g e c a p a c i t o r -c o u p l e d a m p l i f i e r f o l l o w e d by an e m i t t e r f o l l o w e r and a d i s -c r i m i n a t o r c i r c u i t . The d i s c r i m i n a t o r c i r c u i t c o n s i s t s of a S c h m i t t - t r i g g e r c i r c u i t and s t r o b e g a t e . The o c c urrence of the w r i t e - c u r r e n t p u l s e s produces l a r g e n o i s e p u l s e s on the sense w i n d i n g which are l a r g e r t h a n the "1" o u t p u t . To p r e v e n t s p u r i o u s t r i g g e r i n g of the r e v e r s i b l e c o u n t e r , a s t r o b i n g gate has been d e s i g n e d which o n l y g i v e s an output d u r i n g the time i n t e r v a l d e f i n e d by the st r o b e mono-s t a b l e , MJJ, which s t a r t s a t the same time as the readout mono-s t a b l e but i s of s h o r t e r d u r a t i o n . 3.7.3 The address system The memory c o n s i s t s of 32 b i n a r y words each c o n t a i n i n g 19 b i t s . A sep a r a t e d i g i t d r i v e r and sense c i r c u i t are r e q u i r e d f o r each of the 19 b i t s . Only one w r i t e - c u r r e n t d r i v e r and one r e a d -62 SENSE CIRCUIT DISCRIMINATOR 3 0 0 P F SVK.C8£ G-ATE S C H M I T T T R I G & E R ^ N % 4 SENSE AMPLIFIER SENSE BUFFER IMISi <L 10012-IOK STROBE. MR J L -12 DIGIT 4 D R I V E R < H H 30OPF 2NI6.I3 ^ D I & I T LINE - 6 -12. DIGIT LINE T O R E V E R S I B L E . C O U N T E R 5ENSE L I N E READ L I N E -U2. m W R I T E L I N E Pig. 3.16 Digi t Driver and Sense C i r c u i t D, DIGIT DRIVER W R I T E CURRENT DRIVER M i READ WORD-S E L E C T S W I T C H R READ WORD-S E L E C T S W I T C H ^ DIGIT DRIVER C O R E M E M O R Y ARRAY S T R O B f Se,« S E N S E CIRCUIT is Se. SENSE G-ROUND READ CURRENT D R I V E R. W W R I T C WORD-S E L E C T S W I T C H W W R I T E W O R D -S E L E C T S W I T C H S E N S E C I R C U I T DIG-IT G - R O U N D F i g , 3.17 The Memory Address System 64 current driver are required; however, a t o t a l of 32 write word-select switches are necessary. The system of gates and drivers used to address the memory i s i l l u s t r a t e d i n block diagram form i n Figure 3.17. 3,8 The Auxiliary Noise Generator and Digital-to-Analogue Con-verter The a u x i l i a r y noise generator produces a randomly vary-ing voltage which i s equally l i k e l y to be at any one of 256 possible discrete levels at the time of sampling. . Noise gen-eration i s accomplished by using a digital-to-analogue (d-a) converter with random number inputs. The (d-a) converter has eight separate inputs corresponding to the eight b i t s of the binary number to be decoded. A zero b i t results i n -E volts applied to the corresponding input and a one b i t results i n +E volts applied to the input. The d-a converter i s of the ladder-network type (Figure . 9 3.18) described by Lovas . The output of the ladder is given by V = (\ 2^- \ ' 2X)where \ ' ^  = p, the binary o ^n+1 _ 3 | . 7 k g number to be decoded, and ^ '21 = ^ 12 - p = 2 .- i_p Therefore V = E o k=0 2£±1 _ i 2 n + l j (3.1) where E = 6 volts and n-t-l =•:. 8, the number of b i t s . Random-number generation i s accomplished by driving each of the eight inputs to the d-a converter with an independent noise source having probability \ of being either +E or -E at the D E C O D I N G N E T W O R K 5K- 0.5 % A A A 5 K - 0.51' A A A 2.5 K.- .01% -AAA-Fig. 3.18 The Digital-to-Analogue Converter 66 time of sampling. A p a r t i c u l a r random-number input ( b ^ b 2 « • • . b g ) , where b. has only two possible values, +E or -E, has a prob-a b i l i t y of occurrence p (b-^ ) .p ^ 2 ) • . • • »p(bg)= - J . - J . . . . = , where the j o i n t p r obability i s the product of the individual b i t p r o b a b i l i t i e s because of the independence of the noise sources. Hence there are 256 possible random number inputs, each with l i k e l i h o o d 1/256, which are decoded into 256 equally l i k e l y analogue voltage l e v e l s . The independent noise sources are obtained by amplification of zener-diode noise into a random square wave. This random square wave i s applied to the input of a Schmitt-trigger c i r c u i t which i n turn drives a f l i p - f l o p . The output of the f l i p - f l o p i s applied to one of the inputs of the d-a converter (Fig. 3.19). It i s assumed that the zero crossings of the random square wave are Poisson-distributed with the probability of r p o s i t i v e -going zero crossings i n time At given by P(r) = ^— e . Each positive-going zero crossing results i n a change of state of the f l i p - f l o p . Hence the probability that the f l i p - f l o p has made an even number of transitions i n time At i s given by (XAt) 2 U A t ) 4 21 41 T . . . P = ^ p ( r ) = e ~A A t even ' ' r even (3.2) P i f t , —2XAt\ = i ( l + e ) even ^ ' S i m i l a r l y , the probability of an odd number of zero crossings i s given by podd = 2 p ( r ) = e _ A A t r odd x * t + ^ + . . . . Abbreviations PD & CP - Probability Density and Cumulative Probability RO - Readout C A L - Calculate AM - A l l Other Modes M . NIOIS E SOUR.CE SCUM ITT TR. I4SER NOISE S O U R C E SCHMlTT TR.I6&ir5; 9-* A f ' PD4-CP F L I P -F L O P „ PD+CP AM -=S FL\ P-F L O P F R O M O U T P U T GATE-1 1 RO C A L B IX DRAVER F R O n O U T P U T G-ATE-2. f RO CAL B I X DRIVER NOISE SCHM ITT A SOURCE TR.1&G-ER P D + C P A M F L I P -F L O P F R O M O U T P U T ( j f t T e -8 R O CftL e i x DRIVER. Pig. 3.19 Auxiliary Noise Generator P o d d = * ( l - e ~ 2 A A t ) (3.3) Hence 3? o d£ and P e v e n each t e n d t o \ f o r s u f f i c i e n t l y l o n g A t , T h i s r e s u l t i s due t o Ikebe and Sato.. A 1N753 zener diode was used as a n o i s e source and the v a l u e of X e s t i m a t e d on the b a s i s of 18 measurements made u s i n g 4 a s t o r a g e o s c i l l o s c o p e was X = 9°28 x 10 p o s i t i v e - g o i n g zero c r o s s i n g s per second., I t i s d e s i r a b l e t h a t the n o i s e l e v e l used i n f o r m i n g a sample p r o d u c t c o r r e s p o n d i n g t o a p a r t i c u l a r d e l a y , T , s h o u l d be independent from one sample p r o d u c t t o the n e x t . With the computer o p e r a t i n g i n the h i g h - f r e q u e n c y mode, the minimum e f f e c t i v e sampling i n t e r v a l , T + AT , i s 500 microseconds. rrictx W i t h At = 500 jxs. and X = 9.28 x 1 0 4 , , -92.80. _ 1 P o d d = *U + e ) = * P = *(1 - e- 9 2* 8°) * \ even ' J Hence the c o r r e l a t i o n between n o i s e l e v e l s s e p a r a t e d by time i n t e r v a l s of At i s negligible„ There are two a u x i l i a r y n o i s e g e n e r a t o r s of the type p r e v i o u s l y described,, One of these can be a l t e r e d t o operate e i t h e r as a s t a i r c a s e g e n e r a t o r or as a d-a c o n v e r t e r f o r con-v e r s i o n of the d i g i t a l c o r r e l o g r a m t o analogue 'f.e^.. The modif-i c a t i o n s r e q u i r e d are q u i t e s i m p l e . The u n i t i s c o n v e r t e d to a s t a i r c a s e g e n e r a t o r by c o n v e r t i n g the f l i p - f l o p s p r e c e d i n g the b i t d r i v e r s i n t o a co u n t e r which i s d r i v e n by a h i g h - f r e -quency c l o c k , t * 1 . The c o u n t e r c o n t i n u o u s l y c y c l e s t h r o u g h b i n a r y numbers 69 representing decimal equivalents from 0 to 255, resulting i n a staircase function centered about zero and -with a negative slope at the output of the decoder. The staircase voltage has 256 discrete levels and only 32 are required. Therefore, the high-frequency clock, t* 1, provides eight, positive-going trigger pulses for the counter between each sample. Comparison of sgn(x+y) with zero i s only made at every eighth l e v e l of the staircase, -y. The clock input, t* 1, to the counter i s blocked during the readout clock interval defined by t^.. Conversion to staircase generation i s accomplished by the system of switches Sp^ to Spg which are symbolic because the actual switching w i l l be done with stepping switches. To convert the d i g i t a l correlogram to an analogue voltage which can be displayed by a recorder the d-a converter used with the a u x i l i a r y noise generator must be modified. The d-a con-verter can only accept eight b i t s and the memory words are 19 bi t s i n length. Hence a switching system has been designed so that any group of 8 b i t s of the 11 adjacent groups which make up the 19-bit word can be selepted for readout* The b i t group i s selected so that the most s i g n i f i c a n t b i t of the word corresponding to the maximum count i s contained i n preferably the most s i g n i f i c a n t b i t of the selected b i t group. Selection of the b i t group i s by t r i a l and error, however, a "rule of thumb" can be given which relates the selected b i t M group to the sample size. If the sample size i s 2 , the rever-M sible counter i s capable of reaching a count of 2 - 1. A count (M-l) M of 2 i s quite possible with a sample size of 2 and thi s would produce a "one" i n the most s i g n i f i c a n t b i t of the 7 0 b i t group where m = 19-M and z e r o s i n the most s i g n i f i c a n t b i t s of a l l h i g h e r b i t groups even i f the count reached the maximum M v a l u e of 2 - l o T h e r e f o r e b i t group m = 19—M i s a re a s o n a b l e f i r s t c h o i c e and can be s e l e c t e d by means of a s t e p p i n g s w i t c h c o n t r o l l e d by push b u t t o n s e l e c t i o n s w i t c h e s (one push b u t t o n f o r each b i t g r o u p ) e The w i r i n g of t h i s s w i t c h i s not shown. The s e l e c t i o n s w i t c h o b t a i n s i t s i n p u t s from the r e v e r s i b l e c o u n t e r which s e r v e s as a temporary s t o r e f o r each word r e a d out of the b u f f e r memory 0 The group of e i g h t b i t s so s e l e c t e d i s f e d t o the b i t d r i v e r s of the d-a c o n v e r t e r t h r o u g h the output gates (see F i g o 3 o 2 0 ) o The e q u a t i o n f o r the output v o l t a g e of the d e c o d i n g n e t -work g i v e n by ( 3 d ) i s V + E = E ( 2 p + l ) « e o 2 The complementary output i s V q = E where p i s the complement of p = Because of the symmetry of the d e c o d i n g system (the " 0 " v o l t a g e i s the n e g a t i v e of the "1" v o l t a g e ) = -V"o«> I f the s i g n of the b i n a r y number t o be decoded i s p o s i t i v e (S = 19 S = 0 ) , t h e n the d e s i r e d output i s (V q+ E) = E.^?* 1) In;,*fchis case the number, p, i s f e d t o the d—a c o n v e r t e r and +E v o l ^ s i s added t o the o u t p u t b e f o r e g o i n g t o a recorder,. I f the s i g n i s n e g a t i v e (S = 0 , S = l ) t h e n the d e s i r e d output i s V - E = -V - E = - E ^ 2 p t 1 ^ o The complement* p, i s decoded and o o 2 -E added t o the output of the d e c o d i n g network b e f o r e g o i n g t o the recorder.. The c i r c u i t s n e c e s s a r y t o a c c o m p l i s h the above o p e r a t i o n s are i l l u s t r a t e d i n F i g u r e 3 o 2 0 o (2p+l)  2 n + l / Jj * A OUT PUT G A T E S O F R O M N O I S E S O U R C E " " b • * J R E A D O U T B I T DRIVER. FRoi^i N O I S E sounce | READOUT SIT DRIVER D E C o D I M & N E T W 0 ft K A b b r e v i a t i o n s CAL - C a l c u l a t e P o s i t i o n of Operate Mode S w i t c h 300 P F 5.4.K 470A J 2K1I304 IO K S^ l IO K. TO RECoRoe(?«-IOK. -AA/V-5fcK Figo 3.20 . 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 System 4. SAMPLING FLUCTUATIONS 72 4.1 General Sampling fluctuations or noise i n the correlogram are the result of estimating the correlation function from a f i n i t e sample. The magnitude of the fluctuations or noise can be expressed i n terms of the mean-square value of the fluctuation which i s the variance of the estimated correlogram. Each e s t i -mated point, 0^2^) > °f "the correlogram i s made up of the average of a large number of sample products which can be considered to be random variables. Hence, by the Central Limit Theorem, as the sample size becomes large, the d i s t r i b u t i o n of the estimate, C^2^) * tends to a normal d i s t r i b u t i o n . The variance of the estimate can thus be related d i r e c t l y to confidence l i m i t s on the estimate; for example, the prob a b i l i t y i s 0.95 that the true cor r e l a t i o n function l i e s within + 1 .96"^ Var C-^^) °f ^i2^T^' In p r i n c i p l e , the variance can be made a r b i t r a r i l y small by increasing the sample size, N (in this device a sample size 18 of N = 2 = 262,144 i s available); however, the analysis time increases i n direc t proportion to the sample size, and can become intolerably long i n the estimation of the correlation function of low-frequency waveforms. The object of thi s chapter i s to derive expressions for the variance of the estimated correlation function as a function of the sampling i n t e r v a l a and the sample size N. A comparison i s then made between the variance of estimates made while operating i n the symmetrical and asymmetrical modes on the basis of equal analysis times. Because of the dif f e r e n t sampling 73 i n t e r v a l s ( f o r s y m m e t r i c a l sampling a = = AT and f o r a s y m m e t r i c a l sampling a = oc^ = 32AT), the number of samples t a k e n u s i n g the sym m e t r i c a l mode i s N=N = 32N A, where N. i s the 5 A A number of samples t a k e n u s i n g the a s y m m e t r i c a l mode. The v a r i a n c e of the e s t i m a t e d c o r r e l o g r a m i s c a l c u l a t e d f o r the f o l l o w i n g two c a s e s : (1) a u t o c o r r e l a t i o n of b a n d - l i m i t e d G a u s s i a n n o i s e ; (2) a u t o c o r r e l a t i o n of b a n d - l i m i t e d G a u s s i a n n o i s e p l u s a s i n e wave. The r e s u l t s d e r i v e d f o r the above two cases are used as a b a s i s of comparison between the sym m e t r i c a l and as y m m e t r i c a l sampling modes• 4.2 B a n d - L i m i t e d G a u s s i a n Noise The f i r s t example which w i l l be c o n s i d e r e d i s the e s t i -m a t i o n 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 low-pass f i l t e r e d w h i t e n o i s e . I f the low-pass f i l t e r i s a s i n g l e - s t a g e RC i n t e g r a t i n g network and the power s p e c t r a l d e n s i t y of the n o i s e i s f l a t over a l l f r e q u e n c i e s , the c a l c u l a t e d a u t o c o r r e l a t i o n of the f i l t e r _2 —bl7"l 2 output i s R ( T ) = CJ e whereCT i s the mean-square v a l u e of the f i l t e r o utput and b i s the f i l t e r c u t o f f ot h a l f - p o w e r f r e q u e n c y i n r a d i a n s per second. T h i s r e s u l t i s a good approx-i m a t i o n t o the p h y s i c a l l y p o s s i b l e s i t u a t i o n where the n o i s e s p e c t r a l d e n s i t y i s a p p r o x i m a t e l y f l a t over the f i l t e r band-w i d t h but drops o f f a t h i g h e r f r e q u e n c i e s . The e q u a t i o n s t o be d e r i v e d i n t h i s s e c t i o n w i l l a l s o be m o d i f i e d t o i n c l u d e the case of e x p o n e n t i a l - c o s i n e a u t o c o r r e l a t i o n f u n c t i o n s of the t y p e , R(T) = cr2e~k'^' cos (0 T • 74 Exponential and exponential-cosine correlation functions have been chosen as examples because (l) they occur i n a wide variety of natural random processes of which Bendat"^ discusses nine d i f f e r e n t types, and ( 2 ) summations involving exponentials are r e l a t i v e l y easy to handle. The variance of the estimated correlation function, C ( T ) , w i l l now be evaluated for the case where the autocorrelation of the random variable i s exponential. The estimated autocorrelation function i s ,2 N-l C ( T ) = g " u1(ma+t)u2(ma+t-rT) (4.1a) TTT=0 where u^(moc+t) = sgn x(mot+t) - y^(moc+t) and u 0 (moc+t+T) = sgn x(ma+t+T) - y2(ma+t+T) (4.1b) where -y-^(ma+t) and -y 2 (moc+t+T) are outputs of the a u x i l i a r y noise generators and x(t) i s the input signal to the correlator. The quantity, a, i s the sampling interval and i s a = a = A T for the symmetrical sampling mode and a = = 3 2 A T for the asymmetrical sampling mode. The variance of the estimate, C ( T ) , i s Var C ( T ) = ( C 2 ( T ) ) - (c ( T ) ) 2 ( 4 . 2 ) where the symbol^ ^denotes expected value or an average over an i n f i n i t e population of estimated correlation functions which might have been measured. The particular estimate, C ( T ) , measured by the correlator, i s one member of the population for which i t i s desired to establish confidence l i m i t s * Using equations (4.1a) and ( 4 . 2 ) , the expression for the variance can be written 75 N-l N-l Var C(T) = ^ ~ ^ ^ (ma+t)u2(ma+t+T)u1 (pa+t )u2;(pa+t+T^ m=o p=o A 2 : N-l . \ if y | ^t 1 (ma+t)u 2 (ma+t+T^ ra=o (4.3) An exact evaluation- of the above equation requires the use of Theorem II because the a u x i l i a r y noise generators produce d i s -crete-level noise. Instead, Theorem I w i l l be used to derive an approximate result where i t i s assumed y^(t) and x^(t) are both random variables with continuous probability densities. The error introduced by thi s assumption i s quite small and w i l l be discussed in, section 4.6. Using Theorem I, the following result i s obtained: ^ (mot+t)u2(ma+t+T)u^ (pa+t)u 2 (pa+t+T^ = ^c(ma+t)x(ma+t+T)x(pa+t)x(pa+t+T^ for p ^ m (4.4a) 1 for p m and ^ (moc+t)u2(ma+t+T^ = ^ (x(ma+t)x(ma+t+T)) A (4.4b) If x(t) i s a stationary Gaussian random variable, the fourth—order moment of the random process can be written i n terms of second-order moments or correlation functions, R(T), as 7 6 c^(moc+t )x(ma+t+T)x(pa+t )x(poc+t+T^ = R 2 ( T ) + R 2((p - m)a) + R((p - m)a - T ) R ( ( p - ra)a+r) ( 4 . 5 ) where R ( T ) = (x(t) x(t+ r j ) A derivation of the above result i s given by Laning and B a t t i n ^ . If v = p - m, then terms involving v > 0 occur N - v times i n the double sum i n equation ( 4 . 3 ) and terms involving v < 0 occur N + v tiroes. Using equations ( 4 . 5 ) and ( 4 . 3 ) , and noting that R ( T ) i s an even function of T , Var C ( T ) can be written A 4 o Var C ( T ) = % + f N-1 v=l R 2 ( v o c ) + R ( v a - T ) R ( V O C + T ) R 2 ( T ) N ( 4 . 6 ) 2 b!7~l If R ( T ) =CT e , the following result i s obtained: N-1 Var C ( T ) = N + N v=l -2WT| /, v\ -2bav Q~ e (1 - jf) e - — ( 4 . 7 ) The above expression i s evaluated by use of the following summation formulas: 77 and N - l I v V = ¥ - WN '+ 1 + N V N ( V - 1) (1 - V) where V = e v=l -2ba (4.8b) Hence 4 4 Var C(T) = ^ + 4 f -- 2 b a -2bNcc 1-e -2ba 1-e -2ba N 2 -2ba - 2 b a ( N + l ) , ,T -2bNa, -2ba e - e ' + Ne ^e - 1) (1 -2bou' e ) 4 -2blT| CT e N Terms i n v o l v i n g e~2hNa a n ( j y o r d i v i d e d by N 2 are n e g l i g i b l e and can be dropped w i t h o u t n o t i c e a b l e e r r o r . Hence, Var C(T) i s c l o s e l y approximated by the e x p r e s s i o n •2blTj (4.9) Var C(T) = ^ + 2bcc N ( l - e - 2 b a ) N I f 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 the f i l t e r output i s of the e x p o n e n t i a l - c o s i n e t y p e , R ( T ) = Cye~ cos ttJT , the v a r i a n c e of the e s t i m a t e , C ( T ) , i s then g i v e n by the e q u a t i o n , -2blT| A" r r ' e 2. Var C(T) - — - °" 6 cos to T N - l 204 V " N (1 - | ) e -2bva v=l cos toT + costt (voc-T)costt (va+T) o o o (4.10a) A f t e r s u i t a b l e m a n i p u l a t i o n of t r i g o n o m e t r i c i d e n t i t i e s , equation (4.10a) can be written as 78 ,4 _ 4 N - X ,r n/_x A . 2 CT 2_ _ \ /, v N -2bva Var C(T) = -Jj + - J J — cos ttT ^ ) ^ (1 - jj) e v=l 4 N-1 4 -2UTI + 321 J " 1 ( 1 _ v } e-2bva c Q s ^ _ cr_e ^ 2 ^ ( 4 > 1 Q b ) The f i r s t sum i n the above equation has been evaluated previously, The second sum can be evaluated by using equations (4.8a) and (4.8b) i n the following manner: N-1 N-1 n, A U 1 -2boc _ -4ba •2bva \ „v _ e cos2ft>n<x - e  cos2tt va = Be > V — T — ° ^ — T - 1 i <, ~2ba _ ^ -4ba v=l v=l 1- 2 e cos2tt a + e o (4.10c) where ¥ = e 2a(b+jWQ) a n a "Re" means "real part of" and N-l | N-1 | ye-2bvoc 2o> va = Re \ v ¥ V o / v=l v=l . _6ba -2bou - „ 0 -4ba (e - e cos2tt a - 2e _ o -8ba, . -4ba , -, , ~ -4ba A t J k „ / -6ba -2ba\ - n e + 4e + 1 + 2e cos4ft> a-4(e + e )cos2« a o o (4.10d) where terms involving e ~ b N a have been neglected. 2 If terms divided by N are neglected, equation (4.10d) does not enter into the expression for the variance. Using equation (4.10b), (4.10c), and the previous results i n (4.9), the 79 v a r i a n c e i s . . . -2ba _ -4ba .4 0<± 0 Q-2ba e cos2to a - e Var L IT) = - r r + >T c o s to / — + X T 7, x ' N N o , -2ba N , ~ -2ba _ , -4ba 1-e l-2e cos2to a+e 0 4 -2blT| - QLe c o s 2 t o T (4.11) Since the magnitude of the variance given by equation (4.11) contains terms which fluctuate p e r i o d i c a l l y with T , care must be taken i n the interpretation of the correlogram. If the sample size i s too small, fluctuations i n the variance may appear as a small steady periodic component to the unwary observer. One of the main reasons for evaluation of the variance has been to enable a comparison to be made between symmetrical and asymmetrical sampling. The sampling interval for asymmetrical sampling i s T O A T -7- X A T A -ha A -b32AT a. - 32AT = T + A/ and e A = e A max A 4 i s very small. Hence the dominant term of the variance i s JJ— A where i s the sample size i n asymmetrical sampling. The sampling interval for symmetrical sampling i s given by O g = AT = m a x and e~^^~ i s not ne g l i g i b l e . The variance, 2 as given by equation (4.1l), i s a maximum when cos <*0T = 1 and cos2« oa = 1. In this case equation (4.1l) reduces to equation (4.9). Hence i f a comparison i s made between sampling systems using equation (4.9),the results obtained w i l l be conservative when applied to the case of exponential-cosine correlation 8 0 functions. A comparison w i l l now be made between sampling systems on the basis of equal analysis time. If the analysis time i s T seconds, the number of samples T T taken using asymmetrical sampling i s = — = 3 2 AT ' w n e r e a s A the number of samples taken using symmetrical sampling i s T T Ng = — = -j^j. , and hence Ng = ^2N^. Because of the larger sample S size, one would i n t u i t i v e l y expect the variance of the estimate formed by symmetrical sampling to be less than the variance of the estimate formed by asymmetrical sampling, and this i s borne out by theory. It i s evident from the form of equations (4.9) and (4.1l) that the improvement i s a function of the magnitude of the mean square value of the signal input, the functional form of the autocorrelation of the signal input and the choice of AT . The example chosen for comparison i s a random process with 2 —b|T| an autocorrelation described by CT e (see F i g . 4.1). 1.0 0.1 O.g 0.7 0.6 OS 0,4 03 0.2 01 0-0 FUT J/CT' = e _fc>|T| 'AT T _7Tnax -=3IAr F i g . 4.1 Exponential Correlogram 81 The 32 points of the correlogram are equally spaced so as to display the exponential down to p r a c t i c a l l y ten per cent of i t s maximum value. For simplicity, the correlation interval i s chosen as AT = l n 10 and hence e~ b o cA = 0.1. Vith this value of AT , the variance of the two estimates, Cg(T), formed by symmetrical sampling, and C ^ ( T ) , formed by asymmetrical sampling are obtained from equation (4.9), and are as follows: Var C q(T) * ^ + ^ ^ C T 4 (4.12a) S S a n 4 4 4 4 - 2 K T I Var C A(T) a $- + 2 ^ - ^  (4.12b) 4 -2b!T| where the term - ^ r p has been neglected i n (4.12a). S The two estimates cannot be compared u n t i l the mean square 2 noise power, (j , i s specified. Two f a i r l y t y p i c a l cases are chosen corresponding to clipping approximately fiv e per cent of the time and clipping less than one per cent of the time. Clipping i s a nonlinear operation which changes the probability density d i s t r i b u t i o n of the input signals. Clipping fiv e per cent of the time i s rather a r b i t r a r i l y chosen as the maximum amount of cli p p i n g for which one can safely apply the previously derived equations neglecting the modification of the probability density function due to the nonlinearity. The probability of the signal input (absolute value) exceeding 1.96CT at a given instant i s 0.05, provided that the signal input i s a Gaussian random variable. Hence a value of (X = — means that the signal exceeds the clipping levels + A approximately f i v e per cent of the time. For this value of CT, the 82 v a r i a n c e s of the e s t i m a t e s a r e , A Var Cg(T) ^  ±- (2.61) and A4 Var C A ( T ) (0.94) A Var C (T) 0.94N g Hence the f i g u r e of m e r i t i s K = V a r Q ( T) = 2 t 6 l N = S A 11.53. T h i s means t h a t i n the a t t a i n m e n t of e q u i v a l e n t n o i s e l e v e l s , the a s y m m e t r i c a l sampling system would r e q u i r e 11.53 times more a n a l y s i s time t h a n the s y m m e t r i c a l s a m p l i n g system. I f CT = A/3, c l i p p i n g o c c u r s l e s s than one p e r c e n t of the t i m e . F o r t h i s v a l u e of CT the v a r i a n c e s of the two e s t i m a t e s a r e : A4 Var CAT) ^ 1.32 B -S N s A4 Var CAT) = f- (0.99) A Hence Var C.(T) K = Var Cg(T) = 2 4 ' ° ° I t i s e v i d e n t from these r e s u l t s t h a t as CT d e c r e a s e s , s y m m e t r i c a l sampling becomes more f a v o u r a b l e . 4.3 B a n d - L i m i t e d G a u s s i a n N o i s e P l u s a Sine Wave The v a r i a n c e of the e s t i m a t e d c o r r e l o g r a m w i l l now be d e r i v e d f o r the case of a s i n e wave p l u s f i l t e r e d wide-band n o i s e . I n t h i s s e c t i o n , because of c o m p u t a t i o n a l d i f f i c u l t i e s , o n l y the case where the n o i s e i n p u t to the c o r r e l a t o r has an 83 exponential autocorrelation function of the type R ( T ) = CT^ e"^ '^ "' w i l l be considered. The signal input to the correlator i s x(t) = f ( t ) + B cos (ttt+0) where f ( t ) i s a Gaussian random variable. The variance of the estimated correlation function i s obtained by using equation (4.3) and Theorem I as i n section 4.2. Before Theorem I can be used, the fourth order moment must be evaluated as follows: (x(t+moc)x (t +moc+T)x (t+poc)x (t+poc+T")) = (f (t+ma) f (t+ma+T)f (t+pa) f (t+ poc+T)) + (f (t+ma) f (t+pa )^B2 (cos (tt (ma+t+T) + 0) c o s (tt (pa+t+T) + 0)^) + (f (t+ma+T)f (t+pa^ B 2 (cos (tt (ma +t) +0) cos (tt (pa+t+T) +0)^  + (f (t+ma+T) f (t+pa+T)} B 2 (c o s (tt (ma+t) +0) co s (tt (pa+t) +0)^  + (f (t+ma)f (t+ma+T)^ B 2 (cos (tt (pa+t)+0) cos (tt (pa +t +T) +0)^  + (f (t+pa)f (t+pa+T)^ B2(cos(w(ma+t)+0) cos (tt (ma+t+T)+0^ ) + (f (t+ma)f (t+pa+T}) B 2 ^cos(tt( ma+t+T)+0) cos (tt(pa+t)+0)^  + B 4 (cos (tt (ma+t)+0) cos (tt (ma+t+T)+0) cos (tt (pa+t) +0) cos («(pa+t+T)+0^ (4.13) The random variable, f ( t ) , i s assumed to be stationary and Gaussian with zero mean value. As a consequence a l l odd-order moments vanish and are absent from equation (4.13). The fourth-order moment i n equation (4.13) i s assumed to occur as the resu l t of an average over an i n f i n i t e population of correlation functions 84 which might have been measured. This average i s then an average over an ensemble of records, x ( t ) , where the starting time, t, of each record, i s independent of 0 and hence the phase angle can be considered a random variable with a rectangular probability density, = ^- , for 0 ^ 0 ^ 2TI and = 0 elsewhere. Hence ^cos 0^  = ^sin 0^ =0 and consequently ^cos (a (ma+t) +0)^  = ^cos (<o(pa+t)+0)^  = ^os(<a(ma4-t+T)+0)^= ^cos (tt (pa+t+7~) +0)^  = 0 Using the above results and l e t t i n g ^f ( t ) f ( t + T ) ^ = R ( T ) and v = p-m, the fourth-order moment can be written, ^x(t+ma)x(t+ma+T)x(t+pa)x(t+pa+T")^ = R 2 ( T ) + R 2(va)+R(va+T)R(va-T) P 0 R 2 +R(va)B cos«va+R(T)B cosa>T+ ^ R(va-T")costt (va+T) R 2 + R(va+T)cosa(va-T) 4 4 B B 2 + — cos 2wva + cos » T (4.14) where a i s the sampling i n t e r v a l . 2 The second-order moment i s ^x(ma+t )x(ma+t+T)^ = R ( T ) + ^ costtT (4.15) Using equations (4.14), (4.15) and (4.3), the following equation for the variance i s obtained: 4 ^ Var C ( T ) = ^ + | \ (1- V j ) ( R 2 ( v a ) + R ( v a + T ) R ( v a - T ) ) v^ r " N-l + JN "v^T 2 2 B B R(va)cosO)va+ —^ R(va-T)cosa) (va+T) B 2 B 4 + — R(voc+T)cosft>(va-T) + ^g cos2ttva 1 N 85 4 ~ R 2 (T) +R ( T ) B 2 C o sttT+^c o s 2 « T (4.16) The f i r s t sum i n equation (4.16) i s exactly the same as that evaluated i n section'(4.2). The method of evaluating the second sum w i l l now be indicated. If the autocorrelation of the the second sum i n equation 4.16 i s noise* f ( t ) , i s R(T) = CT2e""b,T1 a ^ B 2 N-1 N B _ 4N ) v=T N-1 " -bvoc -b |va-T| e-b|va+T| e coswva + ^ co sft) (va+T) + ^ cos»(va-T) v=l S 1 ( a , a ) -S 2 ( a , a ) S 3 ( « , a ) N (4.17) S4(<o,a) S5(<o,a) S 6 ( » f a ) 2N + 2 2N 11 4N Sg(a),a) S7(<o,a) - - — The evaluation of the series S-^  (<o,oc),.. .Sg(<o,oc) i s accomplished by the use of the two series summation formulas, equations (4.8a) and (4.8b) and the results obtained are given i n Appendix 1. The equation for the variance of the estimated correlation function formed with the computer operating i n the asymmetrical sampling mode i s obtained by substituting the results obtained i n Appendix 1, along with the results obtained i n section 4.2, into equation (4.17) with a = = 32AT . Hence 4 4 -2ba. Y a r c (T) _ Al + 4CJ1 e * V a r O \T) - N + N _ 2 b<x I d * •2ba, A A 1 - e N 2 -2ba 2 A iNA (1 - e A ) B h2 r -ba. bT -bT e (2costta.+e cosa (a.+T) +e cosa(.a.-T) 86 N, -ba, -2ba, 1 - 2e cos<oa^+ e Bh2 " -2ba bT -bT e (2+e cos«T+ e co s«T) N A -ba. 1 - 2e cos»a.+ L A -2ba. e A t2 -3ba. -ba. ""^ha. bT - b a A 2(e +e A)cos»a A- 4e A+e (e cos<o(T+a A)) -4ba -2ba, -2b a, -3ba, -ba, A+4e A+ 2e Acos2<oa A- 4(e A " A +e )cos<oa^+ 1 ^-3ba. le cosw(T-aA) -2ba bT - 2e cos«T)e -4ba. ' -2ba. _ e A+4e A+ -2ba 2e cos2coa.- 4(e A -3ba. - b a . A+e A ) c o s t o a A + l (-2e -2ba, -3ba -ba, c o s (AT + e Acos<o(a A+T)+e A c o s t t (a^-T))e - b T -4ba. -2ba e A+4e -2ba, -3ba. -ba A+2e Acos2«a A- 4(e A+e A )cos »a^ + 1 B 8N. A where h(wa A) - 1 R 2 ( T ) + B ( T ) B 2 C O S » T + ^ c o s 2 » T h(a»aA) = 1+2 = 1+2 S 7 ( t t , a A ) - S («,a A) 8 N (4.18) N~l ( 1 - JJ ) cos 2a>a^ . (4.19) I t w i l l be shown i n s e c t i o n 4.4 t h a t h(»a A) i s v e r y l a r g e a t coa A = nu where n i s any i n t e g e r and i s n e g l i g i b l y s m a l l elsewhere f o r a r e a s o n a b l e sample s i z e , N A > (say N A_>4096). —ba,. —b32AT Si n c e e = e — i s v e r y s m a l l , the dominant term which i s v e r y c l o s e to the t r u e v a l u e of the v a r i a n c e f o r a s y m m e t r i c a l sampling i s A A 1 Var C A ( T ) * ^ -• § J - - N 4 _2blT[ 2^2 -b|T| _ B 2 ^ CT e T C T B e costtT + = - r cos » T (4.20) The e q u a t i o n f o r the v a r i a n c e of the e s t i m a t e d c o r r e l a t i o n f u n c t i o n ) 87 with the: computer operating i n the symmetrical sampling mode, i s given i n equation (4.2l) on the following page. Equation ( 4 « 2 l ) i obtained by substitution of the results obtained i n Appendix 1 i n t equation (4.17) along with the results obtained i n section 4.2. —bin Equation (4.20) with e~" and cosioT set equal to one, and equation (4.21) w i l l be used to compare the two sampling systems. Because of the complexity of equation ( 4 . 2 l ) , i t i s d i f f i c u l t to obtain an upper bound to Var Cg(7") which i s not unduly high. In order: not to obtain too conservative a res u l t , the following procedure has been adopted , F i r s t , the maximum mean-square value of the signal input has been specified as A 2 2 — = C T which gives roughly f i v e percent c l i p p i n g (not exactly f i v e percent clipping since the sine wave does not have a 2 2 Gaussian d i s t r i b u t i o n ) . In addition,CT and B have been chosen p> p B 2 2 A 2 such that B CJ i s a maximum which means that -z— =CT = . This 2 Z o JD 2 choice of =CT means that the effect of the sine wave on the 2 2 variance i s maximized. The alternative choice of CT and B . A2 which would produce a maximum of the variance i s CJ = — and B = 0 but thi s case has already been dealt with i n section 4.2. Secondly, «T i s allowed to vary over the range of values, 2TX 2TC 0, -y^ , j-g »••»» 2it i n the trigonometric terms only. The upper bound for each of the points, i s determined by evaluation of equation (4.21) subject to the condition that e i s set equal to either zero or e~ according as the —bl" quantity multiplied by e~ subtracts or adds to the variance. This i s done for a l l terms of order (^ ) with the exception of V 2 the term - C K('k)x(t+T^— ^ g a w o r s t case situation forlTl >0. Var C G ( T ) A4 . 4c^e C + -2b AT 4C7 4e- 2 b A T N s ( l - e - 2 b A T ) N s 2 ( l - e " 2 b A T ) 2 Bjcx! N S e"bATcosa>(2T+AT) - e~ 2 b A Tos 2ttT /, 0 -bAT -2bATv 11 - 2e - costs AT + e ) • b A T(2costtAT + cosft>(2T+AT)- e" 2 b A T(2+cos2<oT) + e ~ b T ( e ~ b A T c o s f t ) (T-AT)- e" 2 b A Tcos«>T) i 0 -bAT t A _ ^ —2bAT 1 - 2e costtAT + e + e" b T(e b A Tcosfti(T+AT) - e 2 b A Tcosa)T)- ebATcosa>(2T+AT)+ e 2 b A T c o s 2 « T 0 bAT .- 2bAT 1 - 2e co so) AT + e B 2 < ^ 2e~ 2 b A T(2+cos 2t6T) - e b A T(2cosft>AT + cosft>(2T+AT))- e ~ 3 b A T ( 2 c o s ft)AT+cos<o (2T-AT)) -4bAT . -2bAT/_ " . a t V ^ n . , -3bAT -bAT. e + 2e (2 + cos2ft)AT) + 1 - 4(e + e ) cos ft) AT e ~ b T ( 2 e " 2 b A T c o s ft)T - e" b A Tcos(Q(AT-T)- e" 3 b A Tcosfl)(T+ AT) ) -4b AT --2bAT r_ _ ' * . _ ., -3b AT -bAT* . e + 2e (2+ cos2ft)AT) + 1 - 4(e + e Jcos ft)AT + | ^-pe 4 b A Tcos2ft)T + e 3 b A T((p+l)cos«(2T-AT)+ 2p cosft)(2T+AT)) - e 2 b A T(pcos2tt (T+ A T ) + 2(p+l)cos2ft)T) + e b A T(p+l)cosfl>(2T+AT)-e b^ 3 A T~ T^cos ( O ( T - A T ) + 2 e b ^ 2 A T - T ^ cos « T + fo(h(.AT)- 1) - \ R 2(T)+R(T)B 2cos«T + |- cos 2«T - e b< A T- T> cosa,(T+AT) where p = — and D i s given by equation (A1.6). The above equation holds f o r T ^ 0 . For T < 0 , use |TI i n place of T • (4.21) 00 CO 89 The e v a l u a t i o n i s c a r r i e d out f o r f o u r v a l u e s of AT, namely, A T = ffb ' A T = 16b f A T = I f ' a n d A T = 4b ' T e r m s d i v i d e d by 2 N are n e g l e c t e d . A v e r y c o n s e r v a t i v e upper bound has been 2 12 c a l c u l a t e d f o r the terms d i v i d e d by N and i f 4096 = 2 , 2 terms d i v i d e d by N amount to no more th a n 0.02$ of the dominant b 2 term f o r the u n r e a l i s t i c w o r s t c a s e , « = — a n d hAT = ^ . The case of T = 0 i s d e a l t w i t h s e p a r a t e l y and the V a r C ( T ) v a r i a n c e of two e s t i m a t e s and f i g u r e of m e r i t , K = Var C (T) S are t a b u l a t e d i n P i g . 4.2. bAT Var C A ( t ) V a r C S ( T ) / |-S K = V a r C A (T)/Var Cg(T) to = b (0 = b/2 co = b (0 = b/2 % 16 29.75 S 1.325 1.516 22.45 19.62 IT 8 1! 1.090 1.186 27.28 25.08 4 II 0.979 1.027 30.38 28.97 2 II 0.935 0.958 31.81 31.07 F i g . 4.2 F i g u r e of M e r i t For T = 0 The r e s u l t s f o r T ^ 0 are d i s p l a y e d i n two ways, ( l ) The maximum v a l u e of V a r C Q ( T ) f o r each A T i s t a b u l -a t e d i n F i g . 4.3 t o g e t h e r w i t h the c o r r e s p o n d i n g V a r CAT) f i g u r e of m e r i t , K = V a r C g ( r ) (2) Graphs of the upper bound of Var Cg(T) as a f u n c t i o n of 0)T where « T i s o n l y a l l o w e d to v a r y i n the t r i g o n o -m e t r i c terms, are g i v e n i n F i g . 4.4 and F i g . 4.5. 90 b AT Var C A ( T ) Var C g A4 K = V a r C A ( T ) / V a r C g ( T ) 0) = b ft) = b/2 ft) = b ft) = b/2 71 16 29.75 f f - 1.430 1.621 20.80 18.35 Tl 8 1.191 1.287 24.97 23.12 71 4 » 1.073 1.121 27.73 26.5 5 71 2 it 1.021 1.043 29.14 2 8 . 5 1 P i g . 4.3 Comparison Table F o r T ^ 0 I t i s e v i d e n t from the v a l u e s of K g i v e n i n P i g . 4.2 and P i g . 4.3 t h a t the s y m m e t r i c a l s a m p l i n g mode i s a s i g n i f i c a n t improvement over the a s y m m e t r i c a l sampling mode. An e x p l a n a t i o n of the c h o i c e of A T w i l l now be g i v e n . T h i s c h o i c e has been governed by the r e s t r i c t i o n on ft), i . e . , 0 ^ ft) £ b, s i n c e the s i g n a l i s the ou t p u t of a low-pass f i l t e r w i t h bandwidth 27t b. A v a l u e of A T = would r e s u l t i n a u n i f o r m s p a c i n g of the e n t i r e 32 p o i n t s of the c o r r e l o g r a m over a s i n g l e c y c l e of the s i n e wave p r o v i d e d ft) = b and over o n l y a f r a c t i o n of a c y c l e i f 2TT ft) < b. Hence A T = -—- would o n l y be chosen i f i t were known beforehand t h a t ft> = b. A c h o i c e of A T = ^2L_ w o u l d r e s u l t i n 16 p o i n t s per c y c l e i f 6) = b and 32 p o i n t s per c y c l e i f ft) = !| . A s i n u s o i d of unknown f r e q u e n c y c o u l d t h e n be d e t e c t e d i f ^  ^ ft) ^  b, 2TI S i m i l a r l y a c h o i c e of A T = ^ would p r o v i d e e i g h t p o i n t s per c y c l e i f ft) = b and 32 p o i n t s per c y c l e i f ft) = |- . T h i s i s p r o b a b l y the b e s t c h o i c e of A T f o r d e t e c t i o n of a s i n u s o i d of unknown f r e q u e n c y i n the range j ^ ft) ^ b. A c h o i c e of AT = |S. would p r o v i d e f o u r p o i n t s per c y c l e i f ft) = b and 32 p o i n t s per c y c l e b h i f w = g- , thus a l l o w i n g d e t e c t i o n i n the range g- ^  ft) ^ b a l t h o u g h F i g 4.4 Upper Bound to the V a r i a n c e of the E s t i m a t e d A u t o c o r r e l a t i o n of G a u s s i a n Noise P l u s a Sine Wave of Frequency tt = b r a d . / s e c . F i g . 4.5 Upper Bound to the Variance of the Estimated Autocorrelation of Gaussi Noise Plus a Sine Wave of Frequency 6) = ^ rad./sec. 9 3 the number of points per cycle near » = b would be rather low. 4.4 Sampling Fluctuations Due to Periodic Sampling of a Periodic  Function The sampling fluctuations considered i n this section are the r e s u l t of an inte r a c t i o n between the periodic sampling and the periodic function. The effect of this interaction i s the addition to the variance of the estimated correlogram of terms which are dependent on the Fourier c o e f f i c i e n t s of the periodic function, the sampling i n t e r v a l a, and the sample size N. It w i l l be shown that i n almost a l l cases the effect of this i n t e r -action i s negligible except at certain values of a where the variance becomes very large. Two cases w i l l be considered: ( l ) the ordinary sampling correlator, and (2) the polarity-coincidence correlator with random comparison l e v e l s . The ordinary sampling correlator gives an estimate of the autocorrelation function, 0 ( T ) , by calculating SS C g g ( T ) = N" y tg(t+ma)g(t-fma+T) (4.22) m=o' where g(t) i s a periodic function of period T and zero mean value. The variance of the above estimate i s Var Cgg(T) = ( C G G 2 ( T ) ) - (cgg(r)) 2 (4.23) where ^ g g ( T ) ) = 0gg(r) (4.24) The mean-square value of the estimated correlation function i s 94 N-1 _ N-1 < (cgg 2(T)^ = ^ y ' ^> ' (t-t-ma) g ( t +ma+T)g (t+pa )g (t+poc+T)^ N m=o' p=0' (4.25) The ensemble average i n e q u a t i o n (4.25) w i l l be r e p l a c e d by an e q u i v a l e n t time average assuming e r g o d i c i t y ; i . e . , by g (t+moc) g ( t +ma+T) g (t+pct ) g ( t +poc+T) T/_2 g(t+ma)g(t+ma+T)g(t+pa)g(t+pa+T)dt (4.26) P/2 1 T Si n c e g ( t ) i s a f u n c t i o n of p e r i o d T w i t h zero mean v a l u e , i t can be r e p r e s e n t e d as a F o u r i e r s e r i e s where O O g ( t ) = \ ( a ^ c o s k i o t + b j ^ s i n k t o t ) (4.27) k= l » = , a,. = £ | g ( t ) cos kttt d t T/2  k =f J g(t)-T/2 T/2 i | J g ( t ) s i -T/2 and b k = „ I g ( t ) s i n kttt d t The time average i n e q u a t i o n (4.26) can now be e v a l u a t e d by s u b s t i t u t i o n of e q u a t i o n (4.27) f o r g ( t ) i n e q u a t i o n ( 4 , 2 6 ) . The r e s u l t a n t e x p r e s s i o n f o r the f o u r t h - o r d e r time average i s g(t+ma)g(t+ma+T)g(t+poc)g(t+pa +T) O O i , 2 . , 2 \2 (a t + b .) k" " k ; i k=l 2 cos k«T cos 2(p-m)ttkoc 95 O O D O ( a k 2 + b k 2 ) ( a k 2 + b k 2 ) K l K l K2 2 k 1 = 1 k 2= 1 k l * k2 cosk^wTcos kg*7" + cos(p-m)«k^a cos (p-nOttkgCx + costt( (m-p)k^a-T)costt( (m-p)k2a+7") (4.28) The evaluation of the variance can be completed after writing (T)\ 2 = 0~ff('7") i n terms of the Fourier components of g ( t ) . Again, assuming ergodicity, 0__(T) i s gg 0 g g(T) = g(t+m<x)g(t+ma+T) T/2 i r = f J g(t+ma)g(t+maT»-T)dt (4.29) -T/2 Substitution of equation (4.27) into equation (4.29) yie l d s the result , O Q , ' 0 * <, ( a k + bk> 0__(7~) = / 5 cos kwT and hence gg L i ^ k=l o o k=l C O o o + i ^ | ^ ( (a| + b 2 ) ( a 2 + b 2 ) cos k ^ T cos k 2«T (4.30) k 1 = l k 2= 1 k l ^ k2 Substitution of equation (4.30) and (4.28) into equation (4.23) results i n the following expression for Var C (T): gg 96 Var Q g g ( T ) p=© m=©> ^ (a,,+ b.) k y k • cos 2ft)k(p-m)a k=l o o o o ( a * + b 2 ) ( a 2 + b 2 ) K l K l K 2 2 k 1 = 1 k 2 = 1 k l ^ k 2 [cos (m-p)a)ky cos (ra-pjttk^a + cos(«(m-p),kj<x-T) cos«( (ra-p)k 2a+T) (4.31) I f v = p-m i s s u b s t i t u t e d i n e q u a t i o n (4.31) and c e r t a i n m a n i p u l a t i o n s of t r i g o n o m e t r i c i d e n t i t i e s c a r r i e d o u t , the e q u a t i o n f o r Var C (T) may be w r i t t e n : Var 0 g g ( T ) . o o k = l o o o o , ( 2 2 s 2 V ' V ' ( a J + b j M • V V h(*k«) + X H l k 8N 4N 1 ^ k 1 = 1 k 2 = 1 + b k > k 2 k l * k 2 -h(«(k 1+ k 2 ) |)+(cos2«7)h(<o(k1- k 2 ) |) N-1 (4.32) where h ( x ) = 1 + ^ ( l - fy cos2ttx v=l i s g i v e n by e q u a t i o n ( 4 , 3 3 ) . As was s t a t e d i n s e c t i o n 4,3 t h ( x ) has sharp maxima due t o s y n c h r o n i z a t i o n a t x = nil and i s n e g l i g i b l y s m a l l e l s e w h e r e , A p l o t of h ( x ) i s g i v e n i n F i g , 4,6 where • h(x ) = 1 + c o s 2x- cos 2Nx -1 + cos 2 ( N - l ) x 1-cos x _ 1 ( 3 N - l ) c o s 2 ( N - l ) x - N c o s 2(N-2)x+(N-l)cos 2(N+1)x-(3N-2)cos 2Nx ~ N 3+ cos 4x - 4 cos x 1 2cos 2x-2 N 3+ cos 4x - 4 cos x (4.33) * x 0 . 0 0 . 5 1.0 I .S 2 . 0 a s 3 . 0 3 S -+S S . O F i g . 4.6 " S y n c h r o n i z a t i o n " F u n c t i o n vO 98 E q u a t i o n (4.33) i s o b t a i n e d by s u b s t i t u t i o n of e q u a t i o n ( A L I O ) and ( A l . l l ) i n t o e q u a t i o n (4.19) w i t h x = <DAT. The f u n c t i o n , h ( x ) , i s i n d e t e r m i n a t e a t x=nfT;however, l i m i t x—*nrt h ( x ) = N where t h i s l i m i t i s e v a l u a t e d by a p p l i c a t i o n of l ' H d p i t a l s r u l e . The ze r o s of h( x ) oc c u r a t x = ^  where k i s an i n t e g e r and not a m u l t i p l e of f o u r and the f u n c t i o n i s e s s e n t i a l l y 1 "DTI zero ( o n l y terms of o r d e r do not v a n i s h ) a t x = where p i s an odd i n t e g e r . I t i s e v i d e n t from F i g . 4.6, t h a t h ( x ) i s 13 n e g l i g i b l y s m a l l except a t x = n%. For a sample s i z e of N=2 = 8192, maximum h e i g h t of the i n t e r m e d i a t e peaks i s a p p r o x i m a t e l y 0.0005. I f ttpoc = % t h e n p i s the l o w e s t v a l u e of k f o r which " s y n c h r o n i z a t i o n " can oc c u r and the v a r i a n c e , g i v e n by e q u a t i o n ( 4 . 3 2 ) , becomes v e r y l a r g e and the dominant term i s o o o o o o Hi Var C G G ( T ) « \[_^ ( a ^ + b j | p ) 2 + i / ^ / j ^ b*«p> <*Lnp k=l n=l k=np+I + b k - n p > c o s 2 * T oo o o V - 1 T I + * /_> Z_J(anp-k+ bnp-k> < ak+np + b k + n p > < 4- 3 4> n=l k=np-l The r e s u l t s d e r i v e d so f a r i n t h i s s e c t i o n a p p l y t o the c o n v e n t i o n a l sampling c o r r e l a t o r . The m o d i f i c a t i o n s r e q u i r e d f o r the p o l a r i t y - c o i n c i d e n c e c o r r e l a t o r w i t h random comparison l e v e l s are not v e r y g r e a t . I n t h i s case 2 ^ C(T) = ^  N u 1(t+ma)u 2(t+ma+T) (4.35) m=.(J 99 where u^(t+moc) = sgn ^ g(t+ma)-y^ (t+ma)^ , u 2(t4ma+T) = sgn(g(t+ma+T)-y 2(t+ma+T)] and -y^(t+moc) and -y 2(t+ma+T) are independent a u x i l i a r y n o i s e i n p u t s w i t h r e c t a n g u l a r p r o b a b i l i t y d e n s i t y f u n c t i o n s as e x p l a i n e d i n s e c t i o n 4.2. The f o u r t h - o r d e r moment, found by a p p l i c a t i o n of Theorem I , i s u^ (t+ma)u 2(t+ma+T)u^ (t+pa)u 2(t+pa+T) = g(t+ma)g(t+ma+T)g(t+pa)g(t+pa+T) f o r p ^ m (4.36a) A = 1 f o r p = m The second-order moment, a l s o found by a p p l i c a t i o n of Theorem I , i s _ _ _ _ _ _ _ _ _ 0 (T) (t+moc)u 2 (t+ma+T) = ^ g(t+ma)g(t+ma+T) = (4.36b) A A I f e q u a t i o n (4.28) i s s u b s t i t u t e d i n t o e q u a t i o n (4.36a) and the r e s u l t s o b t a i n e d s u b s t i t u t e d i n e q u a t i o n (4.23) a l o n g w i t h e q u a t i o n (4.36b), the f o l l o w i n g e x p r e s s i o n i s o b t a i n e d f o r the v a r i a n c e : Var C(T) = ^ + l j ^ (a|+b 2) 2(h(«ka)-l) + / _ _ , k l ^ k 2 + b 2 ) ( a 2 + b 2 ) K l K 2 K 2 h(<o(k1-k2)|)-l+cos2«T(h(fi)(k1+k2)|)-l) (4.37) 100 where terms of order have been neglected, N For nearly a l l values of a, h(ttka) i s very small and the 2 x 2 4 oo dominant term i n (4.37). 'is Var C(T) * ^ - |^ >^ (a 2+ b 2) CX) o o - I 1 > > (a? + b 2 ) ( a 2 * b 2 )(l+cos 2«T) MN / , Z j ^k-, k, ' vu,k,/ k k 1 = 1 k 2= 1 L l ~ i ~2 ~2 k l ^ k2 On the other hand, i f a i s chosen such that ttpoc = % where p i s an integer, the dominant term i s given by equation (4.34). A comparison can now be made between symmetrical and asymmetrical sampling on the basis of the results obtained i n this section. In a symmetrical sampling system, a = AT, whereas i n an asymmetrical system, a = 32AT. This means that the maxima of h(ktta A) = h(ttk32AT), corresponding to asymmetrical sampling, are spaced at intervals of AT = , whereas the maxima of r 3 2ktt h(kttoc ) = h(kttAT) corresponding to symmetrical sampling, are o spaced at intervals of AT = S_ # Hence, i f tt i s not known exactly and AT i s chosen to display the autocorrelation of the periodic function, g ( t ) , there i s a much greater l i k e l i h o o d of choosing AT so that h(x) i s at or near a peak i f asymmetrical, rather than symmetrical, sampling i s used. Because of the narrow-X 3 ness of the peaks of h(x) for a reasonable size of N (say N 2 = 8192) the l i k e l i h o o d of landing close enough to a peak to cause any s i g n i f i c a n t increase i n the variance i s extremely small unless one deliberately t r i e s to synchronize the sampling with the periodic function. Hence the advantage of the symmetrical sampling system over the asymmetrical system i s not very great on 101 this account except possibly at quite small sample sizes* However, for h(x) = 0 the dominant term i n the variance Var C A(T) „ Ng i s A_ and hence, the figure of merit, K = ^—r^j = ^- = 32 for the UBC correlator. Therefore the symmetrical sampling mode is preferable for cal c u l a t i o n of the correlation function of periodic signals i n the UBC correlator but provides l i t t l e advantage i n the conventional sampling correlator. 4.5 Comparison of the Conventional Sampling Correlator and the  UBC Polarity-Coincidence Correlator on the Basis of  Sampling Fluctuations The conventional sampling correlator which produces results roughly equivalent to those obtained by the UBC p o l a r i t y -coincidence correlator i s a multichannel device which computes k points of the correlogram simultaneously. The signal x-^(t) i s connected into the f i r s t channel and sampled at time t+moc; the signal x 2 ( t ) i s connected into the remaining k-1 channels and i s sampled at time t+ma i n the f i r s t and at time t+moc+(k-l) AT i n the k^*1 channel. The (q+l)^* 1 point of the correlogram i s estimated by forming the following sum of sample products: N-1 C (T) (4.38) m=0 The variance of the estimate so formed by the conventional correlator i s Var C (qAT) = ( x 2 ( t ) x 2 ( t + q A T ^ x l x 2 102 1 ^ N_I_ + —^ ) ^ ) (t+ma)x2(t+ma+qAT)x1 (t+pa)x 2(t+pa+qAT^-^ ( t ) x 2 ( t N' m=0 p=0 m ^ P +qAT)) 2 (4.39) The estimate of the correlation function formed by the polarity-coincidence correlator with random comparison levels.such as i s described i n th i s thesis i s N-l 2 r — i C(qAT) = ^ ) u^ (t+ma)u2(t+moc+qAT) m=0 where u^(t-fma) and u2(t+ma+qAT) are sgn functions defined previously by equation (4.1b). The variance of the estimate, C(qAT), i s N-l N-l Var C(qAT) = ^ + ^ Y^l^l ( ( ^ (t+ma)x2(t+ma+qAT)Xi (t ^ m=o' p=o' m^ p +pa)x2(t+pa+qAT)^ - (x± (t)x 2(t+qAT^ 2 (4,40) The difference between noise levels i s A4 ( x 1 2 ( t ) x 2 2 ( t - , q A 4 U A 1 , If x-^(t) = ^ 2 ( t ) = x(t) and x(t) i s a stationary Gaussian random variable with ^ x 2 ( t ) ^ = (T 2, then ^x 2(t)x 2(t+qATJ> = < ^ 2 ( t ) 2 +2 (x(t)x(t+qAT)) 2 =<T4 + 204 p 2(r) (4.42) where T = qAT and p(7~) i s the correlation c o e f f i c i e n t . Equation 103 (4.42) i s obtained by setting p=m i n equation (4.5) and noting that R(T) = ^x(t)x(t+T)^ . For Gaussian x(t) and hence and d = 4 - ^  (O(T) = 0) . max N N r 4 4 If O" = 4 , d . = ^ (fl) and d = ^ (fl) . ^ 2 ' mm N V16' max N 16 Values for d and d . can also be calculated for the max min case of CT = j for Gaussian x(t) or for a sine wave plus Gaussian noise. In the case of the sine wave plus noise, ( T ) X 2 ( t + T ^ i s obtained by setting v=0 i n equation (4.14). Numerical Values for the variances of the conventional correlator can then be obtained by subtracting the d values from the variances of the UBC correlator given i n sections 4.2 and 4.3. Figures of merit for the symmetrical and asymmetrical sampling modes can then be calculated for the conventional correlator i f desired. The symmetrical sampling mode i s quite e a s i l y instrumented i n the polarity-coincidence correlator. A l l ithat i s required i s a s h i f t - r e g i s t e r delay l i n e . On the other hand, implementation of the symmetrical sampling mode i n the conventional sampling correlator i s almost impossible due to the complexity* lack of f l e x i b i l i t y and narrow frequency range of non-digital delay l i n e s . Hence a comparison between the conventional sampling correlator operating i n the asymmetrical mode and the UBC 104 polarity-coincidence correlator operating i n the symmetrical sampling mode i s j u s t i f i e d * The minimum value of the variance, Var C ( T ) , for the A ordinary sampling correlator operating i n the asymmetrical mode 4 i s approximately for p(r) = 0 neglecting terms where m ^ p A i n equation (4.39). The: maximum value of Var C x x ( T ) i s 4 A approximately >fr with p(r) = 1, again neglecting terms where m ^ p. A figure of merit for comparing the ordinary sampling correlator and the UBC correlator i s Var C (r) X X A Var CQ(T) x . X = — , (4.43) where Var ^ ^ ( T ) i s the variance of the estimate formed by the UBC correlator while operating i n the symmetrical sampling mode. If x(t) i s Gaussian noise, the minimum and maximum values of X are ; c-4 4 . = and X = & m i n ' Var C S ( T ) m a x ^ Var C G ( T ) If the correlation c o e f f i c i e n t , p(T), i s always positive and i s a monotonic decreasing function of T , for example p(T) = e , there i s a range of T including T = 0 for which X. —'1 and the noise l e v e l of the ordinary sampling c o r r e l -ator i s higher than the noise l e v e l of the UBC correlator, provided<T i s large enough. If the correlogram i s uniformly distributed over the exponential curve, 0~2e~b'T' down to 0.1O" 2 , 105 i . e . , D (T ) = 0.1 and X. = 1 f o r T = T T t h e n the f r a c t i o n of ~ max' L the range f o r which the UBC c o r r e l a t o r g i v e s a lower n o i s e l e v e l t han the o r d i n a r y c o r r e l a t o r i s a p p r o x i m a t e l y L / T • I f Q(7~) i s of e x p o n e n t i a l - c o s i n e t y p e , z e r o s of j O ( T ) can occur b e f o r e T and the f r a c t i o n L / T has l i t t l e meaning. The minima L max.. & and maxima of the f i g u r e of m e r i t , X min and X max,, are A A T / t a b u l a t e d f o r CT = -r and CT = — t o g e t h e r w i t h L / T m d J IT1Q/X c P i g . 4.7. cr x„ • „ min,»;. X max, \ T 'max. A 2 0.766 1.533 0.258 A 3 0.300 0.600 0 F i g . 4.7 C o r r e l a t o r F i g u r e s of M e r i t F o r E x p o n e n t i a l C o r r e l a t i o n F u n c t i o n s I t i s e v i d e n t from the r e s u l t s t a b u l a t e d above t h a t , i f / A\ CTis l a r g e enough (say CT = -^j , the UBC c o r r e l a t o r i s an improve-ment over the c o n v e n t i o n a l c o r r e l a t o r . F o r s m a l l v a l u e s of (T, the c o n v e n t i o n a l c o r r e l a t o r i s s u p e r i o r t o the UBC d e v i c e i n n o i s e performance and the h i g h e r n o i s e l e v e l i s the p r i c e w h i c h must be p a i d f o r s i m p l i c i t y of o p e r a t i o n . I f the i n p u t s i g n a l t o the c o r r e l a t o r i s a c o s i n e wave, B cos(ft)t+0) p l u s G a u s s i a n n o i s e , f ( t ) , and the a s y m m e t r i c a l s a m p l i n g mode i s used, the dominant term i n Var C (T) i s A 106 Var C (T) 3f A 4 R 2(T) + R 2(0) + R(0)B 2 + B 2R(T)coswT + B — '.(4.44) NA where x(t) = f (t) + Bcos(cot+0) and R(T) = (f (t) f (t+T)^ =0^ p(T) Equation (4.44) i s obtained by substituting equation (4.14) into equation (4.39) and dropping terms involving v = p-m >0 as was done to obtain equation (4.20). The minimum value of (4.44) occurs when p(7~) = 0 and therefore (Var 0 ( T ) ) _ B * {j- (o* + C r V + ^ ) (4.45) A A The maximum value of (4.4 4) occurs when p d " ) =1 and therefore C x x < T > ' M x = ST ( 2 < ^ + 2 ^ 2 b 2 + 4) <4'46> A A The ordinary correlator and the UBC correlator w i l l be compared under the conditions discussed i n section 4.3, namely, B 2 2 —-• = CT (equal signal and noise power). Again, the r e s t r i c t i o n 2 2 on the mean square of the correlator input i s A_ _ B_ f—2 4 _ 2 + U which corresponds to roughly fiv e percent c l i p p i n g . 2 2 With the above values of CT and B , ~ 128NA 107 4 Hence 7 A 128N. X m i n Var C C ( T ) S 4 and 13 A 128N. ~ , A X max Var C (T) S For a comparison of the two c o r r e l a t o r s a case i s chosen 2it which i s the w o r s t t h a t might occur i n p r a c t i c e ; say AT = / \ A 4 and ft) = b. I n t h i s c a s e , Var Cg(T) = 1.430 j j — , where t h i s v a l u e i s o b t a i n e d from the t a b l e i n F i g u r e 4.3. Hence and X . = 1.223 m m Xmov = 2.272 max Fo r a s i n e wave p l u s G a u s s i a n n o i s e , the UBC c o r r e l a t o r i s c o n s i d e r a b l y b e t t e r t h a n the o r d i n a r y c o r r e l a t o r as shown by X . = 1.223 > 1 where X = 1 i n d i c a t e s e q u i v a l e n t n o i s e o u t p u t s min ^ r of b o t h c o r r e l a t o r s . I t s h o u l d be noted t h a t i n the example chosen the p o i n t s of the c o r r e l o g r a m are u n i f o r m l y spaced over one c y c l e of the s i n e wave of the h i g h e s t p o s s i b l e f r e q u e n c y , ft) = b. T h i s means t h a t AT i s a minimum w h i c h g i v e s the h i g h e s t v a l u e of V a r Cg(T) and hence the p o o r e s t performance. I n p r a c t i c e , AT w i l l u s u a l l y be l a r g e r and much b e t t e r r e s u l t s can be e x p e c t e d . 4.6 The E f f e c t of Q u a n t i z a t i o n on Sampling F l u c t u a t i o n s and B i a s I n s e c t i o n s 4.2 and 4.3, the a u x i l i a r y n o i s e sources were 108 considered to be unquantized for the purpose of deriving the variance of the estimated correlogram. The correlator described i n this thesis uses a u x i l i a r y noise sources with d i s c r e t e — l e v e l outputs. This section i s concerned with establishing an upper bound to the additional noise i n the output of the correlator due to quantization. The noise i s of two types: ( l ) the computer estimates the correlation function of the quantized variables, namely, ' (t+moc)x' (t+moc+T^ rather than the desired correlation function (x(t+ma)x(t+ma+T^ of the un-quantized variables and the difference between these two second-order moments i s a bias error; (2) an additional noise term due to quantization appears i n the expression for the variance of the correlogram, which i s a measure of the amount of sampling f l u c t u a t i o n . The quantization process introduces quantization noise, e^, such that x| = X i + e ^  (4.47) where the prime denotes the quantized variable. The quantization noise, » l i e s i n the range - -x— ^ e . ^ T - because the quantization i n t e r v a l 2n I <sn ^ i s — . n The relationship between the correlation function of the quantized variables and that of the unquantized variables i s &(T) = (x« (t +moc)x 1 (t+ma+T)) = ^x(t+ma) + e (t+moc)) (x(t+ma+T) + e(t+ma+T))^ 109 ^c(t+ma)x(t+ma+T^ + 2 <x(t)^ (e (t) (t+ma)e (t+ma+T")^ where i t i s assumed that x(t) and e(t) are independent stationary random v a r i ables. If (x(t)^ = 0 then R'(T) = R ( r ) + JO (T) (z2^ ( 4 . 4 8 ) where R(T) = <^c(t+moc)x(t+ma+T)^ and |0 ( T ) i s the ; correlation c o e f f i c i e n t of quantization noise. If (^c(t^ ^ 0, equation ( 4 . 4 8 ) i s s t i l l quite a good approximation as <^ e^  = 0 provided the probability density of the input signals, x ( t ) , varies only a small amount over a range of x equal to the quantizati/on i n t e r v a l , — , i . e . the quantization i s r e l a t i v e l y fine« For a stationary random process, ( T ) i s a maximum at T = 0 and hence the maximum bias error i s (e2 ^ . The mean—square output, (^^)* w i l l now be evaluated by a method due to noi se 4 Watts , who has derived an expression for the characteristic function Wx,(oc), the Fourier transform of the probability density of the output of a general quantizer, which i s as follows oo V x . ( . ) - e - J w t c - ) . - J 2 * « V x(0£ _ k= - c o r where Wx,(a) = w'(x') e"^*' dx' Here W (a) = x ' J J - o o o o w(x) e""^ax dx i s the characteristic function of o o the input to the quantizer and w'(x') and w(x) are the prob-a b i l i t y densities of x' and x respectively. The quantities r, c, a, are defined by the input-output diagram i n Figure 4.8. T r c x F i g . 4.8 Input—Output Relationship for the General Quantizer In the case of the correlator described i n th i s thesis, A r = q = — , the quantization i n t e r v a l , and c = a = 1« Hence o o ¥ (« . Sank, s i n j ' n ~ 2 A ' X. A k = - oo - 2uk) (4.49) and thus ^c' 2^ = <x2 >^ + (e 6a a=0 /a k 0 0 o o 2-nnk ) k = - OO da cos kit kfi a=0 I l l 2 ° ° 2 + 2 ( ^ __' *x ^  + * ^ < 4 ' 5 0 ) k = - 0 0 where the prime after the summation sign means omit the value zero. If x(t) i s a Gaussian random variable, ¥ ( S ) a3 .2 x - 2 = <x 2) = e 2 where CT = (  ) . Hence 0 0 „ _ 2 ( T j n _ T k ) 2 (x' 2>=cr 2 + c r 2 > <-Dke = - O O 0 0 ' T c n q k \ 2 2 + 2<tn"> ) ^ 2 6 + I ^2n^ (4.51) k = - 0 0 The i n f i n i t e series i n equation (4.5l) are alternating series which can be truncated after the f i r s t term with an error which l i e s between zero and the l a s t term neglected. However, even the f i r s t term of each series i s negligibly small since, for <y = — y „ 2 L 2 2 2 2TC n<T _ Tt n e A = e = e _ l x ( 8 1 9 2 ) _ 0 for a l l p r a c t i c a l purposes. 2 Hence (x« 2) £• CT2 + i - (_^ ) and ( e 2) « i <f-) 2 A If CT = — , the maximum or f u l l - s c a l e value of the 2 2 A correlation function i s (T = — and the re l a t i v e bias error i s 112 ( 2 ) 1 1 —\ £ ' = — — = ^2 which i s n e g l i g i b l y s m a l l . <X 3n I f , i n s t e a d of pure G a u s s i a n n o i s e , t h e r e i s a s i n e wave superimposed on the n o i s e , the r e s u l t s are not g r e a t l y changed. I f B cos (_»t+0) i s superimposed on Ga u s s i a n n o i s e , f ( t ) , w i t h <^f(t)^ = 0, Rice"'' 2 has shown the c h a r a c t e r i s t i c f u n c t i o n t o V X ( S ) = J 0 ( B S ) e 2 where x = B cos (fl)t+0)+f(t) and J Q ( B S ) i s the B e s s e l f u n c t i o n of the f i r s t k i n d of or d e r z e r o . I n t h i s c a s e , u s i n g e q u a t i o n (4.49) oo 2 \ __/(7Tinkx a2 ) ( ^ ^ ( ^ D n B k , e k = - o o oo , 2 " k 2 ( c n t n k A _ _ B _ \ I_llt J A Z Z ^ ) E '  A 2 n * V ^ - O O o o 2  A 2 \ ( - l ) k | T / 2TcnBkN - 2 ( < 2 | n - ) i _ 2 iA?2^ ^  ° ~t_) <4-52) k = - O O where the prime a f t e r the summation s i g n means omit the v a l u e , k = 0. The i n f i n i t e s e r i e s are n e g l i g i b l e i n e q u a t i o n (4.52) 1 / A \ 2 and the dominant term i s a g a i n — • T h i s i s e v i d e n t from k. a w o r s t - c a s e a n a l y s i s of e q u a t i o n (4.52) where J~(~  |T[;—) k 2 0 A \k and * j ^ ' ^—2TcqBk^ a r e g e^. e ( j U a j o n e ^ n l a s t two sums of (4.52) and ( - l ) k ^ o^'^A^^ ""'S s e ^ e ( l u a l ^° o n e ^ n ^ n e f i r s t sum of ( 4 . 5 2 ) , The o n l y i n f i n i t e s e r i e s w hich appears i n the upper bound t o e q u a t i o n (4.52) i s o o , 2 2 o o 2 — i _2 (OTtnk ^  _2 (CTftny r ^/CDtnk e A < e A + e A dk« U s i n g the t a b u l a t e d f u n c t i o o 2 x on, 0(x) = 7 = . f V 2 F J e d t and n o t i n g t h a t e-* k dk -C O * - 0(V~2X) i t can be shown t h a t k = l ~ 2 ^ A ^ - I T 2(8192) -9 j e A < e n W**> + 5 x l 0 y f o r n = 128, and G~ 2 = 4 • I n t h i s case i t i s e v i d e n t t h a t the i n f i n i t e s e r i e s c o n t r i b u t e almost n o t h i n g i n e q u a t i o n (4.52) and /e \ S' ' j ( ^ ) f o r a s i n e wave p l u s G a u s s i a n n o i s e . The m o d i f i c a t i o n of the v a r i a n c e of the e s t i m a t e d c o r r e l o g r a m i n d u c e d by q u a n t i z a t i o n can a l s o be e x p r e s s e d i n terms of (^^^ • •"•n o r < i e r to f i n d an e x a c t e x p r e s s i o n f o r the v a r i a n c e , i t i s n e c e s s a r y t o e v a l u a t e the e x p r e s s i o n Var C(T) = + i -p=0 m=0 P ^ m 2 ^x' (ma+t)x» (ma+t+T)x' (pa+t)x» (pa+t+T)^ - (ma+tjx*(ma+t+T^ (4.53) An upper bound to the a d d i t i o n a l n o i s e power i n t r o d u c e d by q u a n t i z a t i o n w i l l now be f o u n d . L e t x^ = x^+ where i s a random v a r i a b l e independent of x^ and i s i n the range - ~ < < A A - e • = -T— w i t h q u a n t i z a t i o n i n t e r v a l — . 1 <sn ^ n 114 L e t R(T) = (x(ma+t)x(ma+t+T)^ Then (x» (ma+t)x» (ma+t+r)x 1 (pa+t)x' (pa+t+T)^ = ^ ( m a + t j - t e j ^ ) (x(raa+t+T)+e 2) (x(pa+t)+e 3) (x(pa+t+T) +e 4)^ = ( £ 1 ^ 2 ) R ( T ) + ( e n e 4 ^ R(va-T)+ ( j ^ e ^ R(va) + ( e 2 e 3 ) R ( v a + T ) + ^ e 2 e 4 ^ R ( v a ) + ^e3e4^R(r)+ (x(ma+t )x(ma+t+T)x(pa+t ) x ( p a + t+T)^ + ( 6 ^ 2 6 3 6 ^ (4.54) where v = p-m and i t i s assumed t h a t odd—order moments o f x ( t ) are z e r o , e.g., x ( t ) i s a G a u s s i a n random v a r i a b l e or G a u s s i a n n o i s e p l u s a s i n e wave. Now - 2 n ^ E i ^ 2 n a n d h e n c e | ( e l e2 e3 e4^ I ^ 2 n ^ A 4 8 = where 2n = 2 = 256, the number of q u a n t i z a t i o n i n t e r v a l s . 2 ^ I f e-j^  = e(ma+t), E 2 = e (ma+t+T) £3 = e(poc+t) f and = e (pa+t+T) t h e n (^1^2) ~ { E 3 e 4 ^ = ( e 2 ) P ^ w h e r e /e ^  i s t h e c o r r e l a t i o n c o e f f i c i e n t . S i m i l a r l y , ( e ^ ^ = ( e 2 ) /3 (va+T) U4.55) and and I f e q u a t i o n (4.55) i s s u b s t i t u t e d i n t o e q u a t i o n (4«54) and the r e s u l t s o b t a i n e d s u b s t i t u t e d i n t o e q u a t i o n (4«53) a l o n g w i t h e q u a t i o n (4.48) the f o l l o w i n g r e s u l t i s o b t a i n e d : 4 21 2 \ V a r C ( T ) S A _ + 2 C T ^ v } v = l 2 /0(va) + P (va-T) 115 + pe (voc+T) N - l N - l N ^ ' ^  ^x(t+ma)x(t+ma+T)x(t+pa)x(t+pa+T)^ - ^x(ma4-t)x(ma p=0 m=0 m ^ p + t + 7 f - ( e 2 ) P e 2 ( T ) + 2 " 3 2 A 4 (4.56) where R ( T ) , R ( v a ) , R(va - T ) and R(voc-rT) have been s e t equal to a2 i n the f i r s t sum and ( e i e 2 e 3 e 4 ) b a s keen s e " t E A L U a l "to -32 4 2 A , w h i c h g i v e s an upper bound to the e r r o r . E q u a t i o n (4.56) d i f f e r s from e q u a t i o n (4.3) by the a d d i t i o n a l n o i s e term U = V = l 2|0(voc) + ^ ( V O C - T ) + /j>(va+T) - (e 2) p 2 ( T ) + 2 - 3 2 A ^ (4.57) The f u n c t i o n a l form of (T) i s unknown, but ^ (0) = 1 and f3 ( T ) — ^ 0 as T—s» OO . I t i s assumed t h a t 0^ (T) decreases a t l e a s t as r a p i d l y as does the a u t o c o r r e l a t i o n of the G a u s s i a n n o i s e o c c u r r i n g i n the examples of s e c t i o n 4.2 and 4.3. I f t h i s i s so, an upper bound to U can be found by r e p l a c i n g f)(T) by e ~ b | T | where b i s such t h a t e ~ 3 2 b A T = 0.1. The upper bound t o U i s U < 2 o2 <**> N N - l ( 2 + e - b m ) N - l /, v\ -bva , U - if) e + v=l (I ~L\ _ - b | v a - I T l | v = l + 2 - 3 2 A 4 (4.58) The worst case occurs when a = AT corresponding to symmetri-cal sampling. Then the f i r s t sum i n equation (4.58) i s N-1 v x -bvAT e -bAT ( i - j}) -V=l 1-e -bAT where terms of order e -bNAT -1 and N hav been neglected. The second sum of (4.58) i s . N-1 p \ ' /, v v -b|vAT-ITl| -tfrl V V i v ^ b v A T b l T l N-1 v=l v=l I (1 v=p+l v ^ -bvAT 8=1 b|T| Now e v=p+i and pAT = | T| . /, v\ -bvAT H-L-P vs -bkAT (1- |j)e = \ (1- ^)e where v-p = k k=l T - l I f terms of order N are neglected, the following result i s obtained by the use of equations (4.8a) and (4.8b)s y ^ ( 1 _ i ) e - M v * - m i s g ! _ + v=l Hence b(AT+|Tl) bAT -e bAT -b|T! U < 2CT' I f we set e -b32AT N -blTl ( 3 + e - b , T I ) -bAT bAT . —bAT bAT . 1-e e - 1 +2 " 3 2 A 4 = 1 (maximum value) i n the above equation and and (e 2) 3f y ^ ) 2 where n = 128 and 2n . 4 -.4 i s the number of quantization l e v e l s ; then = 0.1, <T2 = --12 4 —4 A 4 -S A 4 + 2 A = 1.73x10 ^jf + 6.1x10 0 ^ — max where N = 2^^. the max 7 maximum possible sample si z e . Since the dominant term i n the variance i s —jj , the additional noise introduced by quantizing the input signals and/or the a u x i l i a r y noise inputs, i s n e g l i g i b l y small provided the input signals have a mean-square value small enough for the given number of quantization l e v e l s , 2n. If CT = A , corresponding to 32$ clipping, U would be increased about four times and would s t i l l be n e g l i g i b l e . 5 . CONCLUSION 1 1 8 A general purpose s t a t i s t i c a l : analyzer has been developed. The design of almost a l l c i r c u i t s has been completed and the c i r c u i t s are contained i n the various printed c i r c u i t cards of the correlator. The assembly and wiring of the device i s not yet complete. The operation of this device as a correlator has been studied from a theoretical viewpoint. The theoretical studies have produced an analysis of the sampling f l u c t u -ations and the results of this analysis indicate the superiority of the symmetrical sampling system over the asymmetrical system. A comparison between the UBC correlator and the con-ventional sampling correlator has been carried out. As might be expected the results indicate that the sampling noise l e v e l of the UBC device i s always higher than that of the conventional correlator provided the sampling systems are the same. However, a conventional correlator operating i n a symmetrical sampling mode (delay-line storage) i s very d i f f i -c u l t to instrument, or even impossible i f AT must be variable over wide l i m i t s . Hence a comparison has been made between the UBC correlator operating i n the symmetrical sampling mode and the conventional correlator operating i n the asymmetrical mode. It has been shown that for a sine wave superimposed on low-pass f i l t e r e d white noise and AT chosen so as to display the sine wave to the best advantage the UBC c o r r e l -ator has a sampling noise l e v e l lower than that of the con-ventional correlator. For a l l types of input signals, the 119 a v a i l a b i l i t y of the symmetrical sampling mode i n the UBC correlator p a r t i a l l y offsets the disadvantage of high noise le v e l which i s inherently present because of the quantization process involved* REFERENCES 120 1. L e v i n , M.J. and R e i n t j e s , J.B., "A F i v e - C h a n n e l E l e c t r o n i c Analog C o r r e l a t o r " , P r o c . NEC, v , 8, p. 647, 1952. 2. S i n g l e t o n , H.E., "A D i g i t a l E l e c t r o n i c C o r r e l a t o r " , P r o c . IRE, 38, No. 12, Dec. 1950. 3. V i d r o w , B., "A Study of Rough Amplitude Q u a n t i z a t i o n By Means of N y q u i s t Sampling Theory", Trans. IRE, PGCT-O, 1956* 4. Watts, D.G., "A General Theory of Amplitude Q u a n t i z a t i o n w i t h A p p l i c a t i o n t o C o r r e l a t i o n D e t e r m i n a t i o n " , P r o c . I E E , P a r t C, March, 1962. 5. B a r r e t , J . F . , and Lampard, D.G., "An E x p a n s i o n F o r Some Second Order P r o b a b i l i t y D i s t r i b u t i o n And I t s A p p l i -c a t i o n t o Noise Problems", Trans. IRE. PGIT, v . I T - 1 , March, 1955. 6. Greene, C.R., "Measurement of P o l a r i t y C o i n c i d e n c e C o r r e l -a t i o n F u n c t i o n s U s i n g S h i f t R e g i s t e r E l e m e n t s " , NAVORD R e p o r t . 4396. 7. Ikebe, J * , and S a t o , T., "A New I n t e g r a t o r U s i n g Random-V o l t a g e " , ETJ of Japan, v. 7, No. 2, 1962. 8. J e s p e r s , P., Chu, P.T., and F e t t w e i s , A., "A New Method to Compute C o r r e l a t i o n F u n c t i o n s " , ( A b s t r a c t o n l y ) , T r a n s . IRE, PGIT, IT-8, No. 5, Sept., 1962. 9. Lo v a s , L.T., A V e r s a t i l e D i g i t a l - t o - A n a l o g F u n c t i o n Gen-e r a t o r and M u l t i p l i e r , M.A.Sc. T h e s i s , Dept.. of E l e c t r i c a l E n g i n e e r i n g , F a c u l t y of A p p l i e d S c i e n c e , U n i v e r s i t y of B r i t i s h Columbia, March, 1964. 10. Bendat, J.S.* P r i n c i p l e s and A p p l i c a t i o n s of Random Noise Theory, John W i l e y and Sons I n c . , New Y o r k , 1958. 11. L a n i n g , J . , and B a t t i n , R.H., Random P r o c e s s e s i n Auto-m a t i c C o n t r o l , McGraw H i l l , New York, 1956. 12. Wax, N., S e l e c t e d Papers on Noise and S t o c h a s t i c P r o c e s s e s , p. 105* Dover P u b l i c a t i o n s , New Yo r k , 1954. 121 APPENDIX I The evaluation of the series (ft) ,oc) ,...Sg (ft) ,a) which occur i n equation (4.17), i s carried out by using equations (4,8a) and (4.8b). Very good approximations are obtained by dropping terms multiplied by e"" b(N-l-p)a a n d g-bNa p _ JTj^ The de t a i l s of the manipulations required are straightforward but long and tedious. The f i n a l results are as follows: N-l S 1 (ft),a) = ^ v=l -bva e cos (ova -ba I A -2ba e cos ft)a - e /._ nv i _ -ba -2ba \AI.i; 1 - 2e cos ft)a + e N-l < S 2(«,a) = ^ ^ v e ~ b v a cos vaa v=l / -3ba, -ba\ ~ -2ba _ _e + e ) cos ft)a-2e -4ba, A-2ba^ , ,^ -2ba _ . , -3ba, -bax e + 4e + }+2e cos 2ft)a-4(e +e jcosaa (A1.2) N-l _ S 3(«,AT) e - b | v A T - p A T | ' c o s ft) (vAT+pAT) v=l r b T ( e b A T c o s ft)(T+AT)-eb^AT)cosft)(2T+AT)+eb(T+2AT)cos _rrl 1 - 2 e b A T cos ft)AT + e 2 b A T 122 -bT -2b AT _ © e cos 6>T  1 - 2 e b A T c o s <oAT + e 2 b A T , e " b A T c o s M(2T+AT)- e " 2 M T c o s 2ttT , • ^. + . - -bAT / ^ -2bAT ( A l - * ) 1 - 2e cos ft)AT + e where T ^ 0. ForT < 0 use |T| instead of T throughout appendix. S 3(«,a A) = ) e - b | v a A " p A T l c o s «(va A+ pAT) v=l b T / ~ b a A , " 2 b a A \ e [e cos tt(T+aA)- e A cos <OT) - b » A ~ -2ba ! (A1.4) 1 - 2e cos wa. + e A where a. = 32 AT A N-1 S 4(«,AT) = v e - b | v A T - P A T l C o s <o(vAT+ p A T ) e " b T + e = u 3bAT p . b < ™ A T ) cos 2*T- ( P + 1 ) e b ^ 3 A T ) _ . cos •(T-AT)+ p e b ( T + 2 A T > c o s 2» (T+AT)-( p +i) e b ( T + A T>cosO> (2T +AT) cos «(T+AT)- 2 p e b ( T + 3 A T > c o s « (2T+AT)+2 (p+1 ) e b ( T + 2 A T \ ^ bAT + e cos 2«T 123 0 2b AT _ -2e cos wT -bAT + cos <o(2T+AT)-e 2 b A T cos 2<tfT 1 - 2 e ~ b A T cos 6)AT + e " 2 b A T + e " b A T cos tt(2T+AT)+e-3bAT cos 0) ( 2 T - A T ) - 2 e " 2 b A T cos 2coT e " 4 b A T +4e-2hLT + l + 2 e ^ 2 b A T cos 2«AT - 4 ( e - 3 b A T + e - b A T ) c o s 0>AT (A1.5) where D = 1 + 2 e 2 b A T cos 2<oAT + e 4 b A T - 4 ( e b A T + e 3 b A T ) c o s <oAT , ^ 2b AT . , + 4 e (A1.6) N-l , . 1 -b|va.-T| v 1 -oI K 'I S 4(w,a A) =\ ve A cos <o(vaA +T) v=l bT , -ba. ""^ba. -2ba. v e (e cos a>(T+aA)+e cos w(T-a A)-2e cos ft)Ty -4ba. -2ba. -2ba. -3ba. -hex. e + 4e +1+ 2e cos 2tta A- 4(e + e )cos tta N-l [ S 5 ( « , a ) = e - b l v a + T l c o s « ( v a - T ) A ~ A (A1.7) v=l ,~- -ba , / \ -2ba , _ _ -bT e cos (o(T~a)-e cos ft)T / A-, Q \ ~ i n "ba _ -2ba 1 - 2e cos »a + e 124 N-l .-b Iva+Tl S f ( a , a ) =Y_] v e ~ b l v a + T l c o s <o(va-T) v=l e-bT e b a cos fl)(a-T)-t-e~3ba cos (o(a+T)-2e 2 b a cos (CT -4ba, . -2ba,. , ~ -2ba ~ ./ -3ba, -bou e + 4e +1+ 2e cos 2tta-4(e +e )cos ffla (A1.9) N-l S7(<o,a) = y> | cos 2_»va v=l cos 2fl>a-cos 2a)Na + cos 2*0 (N-l )a-l ^ 2(1 - cos 2(oa) Sg(w,a) = v cos 2«ova v=l N-l __ (3N-l)cos 2M (N-l )a-N cos 2ft)(N-2)a + (N-l)cos 2a>(N+l)a 2(3+cos 4aa - 4 cos 2_>a) + 2 cos 2tta - 2 - (3N-2) cos 2(oNa ^ ^ 2(3+cos 4<oa - 4cos 2«oa) 

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