A REPRODUCEABLE NOISE GENERATOR by DONALD GEORGE WATTS B . A . S c , University of B r i t i s h Columbia, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Electrieal Engineering We accept t h i s 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 B r i t i s h Columbia April, 1958 Abstract This thesis describes the design of a device for generating a reproduceable noise signals The noise signal i s generated by adding three periodic waveforms having nonmultiple periods. Pulse techniques are used i n the generation of the member functions so that the output may be reproduced exactly. Theoretical and experimental determinations of the amplitude p r o b a b i l i t y d i s t r i b u t i o n and of the autoc o r r e l a t i o n function of the signal were made. On the basis of tests and observations made, i t i s concluded that the signal generated may be considered a noise signal having a near-Gaussian amplitude p r o b a b i l i t y d i s t r i b u t i o n , very l i t t l e c o r r e l a t i o n for time-shifts greater than 30 seconds, and a bandwidth of about 60 cps. ii In presenting the this thesis i n partial requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree that it fulfilment of freely agree t h a t for available the Library f o r r e f e r e n c e and s t u d y . permission f o r extensive I further copying o f t h i s thesis s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e . I t i s understood that copying or p u b l i c a t i o n o f t h i s thesis gain s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n Department o f Electrical The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a . Date s h a l l make April 15, 1958 Engineering, Columbia, for financial permission. Acknowledgment. The author i s indebted to the Defence fesearch Board, Department of National Defence, Canada, for sponsoring the research project under Grant Number DRB C-9931-02 (550-GC) o Acknowledgment is gratefully given Dr. E. V. Bonn, under whose guidance t h i s work was performed, and to Dr. Noakes, grantee of the project. to the U . B . C . F. Thanks, too, are extended Computing Centre for computer time and to the Van de Graaff Section of the U . B . C . Physics Department for granting time on the kicksorters. The author would l i k e to thank Mr. ft. M. Pye for writing the programme for the autocorrelation determination, and would also l i k e to express his appreciation for the helpful suggestions r e ceived from other members and Graduate Students of the E l e c t r i c a l Engineering Department. The author i s indebted to the National Research Council of Canada for the assistance received through a postgraduate bursary granted i n 1956 and to the Defence Research Board of Canada for assistance received while on a Research Assistantship. vi Table of Contents page Abstract . . . • • • • Acknowledgment . . . . . . . . s o . V 1. Introduction . . . . . . . 2. Generation of a Noise Signal Using Periodic Functions 3. 1 . . . . . . 5 Introduction 5 2-2. Theory of Generation of the Noise Signal . . . . . . 2-2-1. The Period of a Composite Function . . . . . 2-2-2. The Near-period of a Composite Function . . 5 6 7 2- 3. Designing the Noise Generator 2-3-1. Generation of a Member Function . . . . . . 2-3-2. Determination of the Periods of the Member Functions . . . . . . . 2-3-3. Summing of the Member Functions . . . . . . 8 9 P r a c t i c a l Design Considerations Introduction 12 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2. Electronic C i r c u i t r y 3- 3. Scaling and Addition of Component Waveforms . . . . . . . . . . . . . . . Analysis of the Noise Signal . . . . . . . . . . . . . 15 . 1 5 15 15 18 4- 1. Introduction 18 4-2. Theoretical Analysis 4-2-1. Theoretical Amplitude P r o b a b i l i t y Distribution . . . . . . . . . . . . . . . . 4-2-2. Theoretical Autocorrelation Function . . . . 18 4-3. 4-4. 5. . . . . . . . . . . . . 1 2-1. 3— 1. 4. ii Conclusions Experimental Analysis . . . . . . . . . 4-3-1. Experimental Amplitude P r o b a b i l i t y Distribution . < 4-3-2. Experimental Autocorrelation Function . . . 18 20 24 24 26 Comparison of Theoretical and Experimental Results 28 . . . . . . . . . . . 31 APPENDIX A. A-l. A-2. Programme for Determining the Near-period of a Function . . . . . . . . . . . . . . . 33 Computer Results 34 . . . . . * . . . . . . . . . . . . • iii APPENDIX B. B-l. Progranmie for Determining the Theoretical P r o b a b i l i t y D i s t r i b u t i o n of Amplitudes „ „ . < , » . . . . . . < > . B-2. Computer Results . . « . « . • • • • • • • • • • • • • • • 36 39 APPENDIX C. C-l. C—2» APPENDIX D. Programme for Determining the Cardinal Points of the Theoretical Autocorrelation Function „ 43 Computer Results 44 References o o o » , > < > « o o o < > o « o o iv 46 List of Illustrations Figure P a ge 1—1. Quasi—linearization . . . . . . . . 3 1-2. Correlation of Signals . . . . . . . . . . . . . . . . . 4 2- 1. Example Output Waveform from M u l t i v i b r a t o r Chain . . . . 3- 1. Block Diagram of Noise 3-4. Generator 15 o . . o . . o o . o o . . . . . o . o » 15 Phantastron Divider C i r c u i t to follow . . . . . . . . . . . . . . . . . . . . 15 »•-»•-. o o o . o o . a e o . lOOKc Pulse Generator to follOW 3-3. 11 . . . . . . -tO-follOW 3-2. o f » a SerVoraechanism Multivibrator and Gating C i r c u i t s to follow . . . . . . . . . . . . . . . . . . . . 15 3— 5. Output Network . . . . . . . . . . . . . . . . . . . . . 16 4- 1. Amplitude P r o b a b i l i t y Distributions to follow . . . . . . . . . . . . . . . . . . . . Theoretical Normalized Amplitude P r o b a b i l i t y Distribution 18 4-2. to follow . . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . . . . . . . 21 4—3. Sample Function 4-4. Theoretical Normalized Autocorrelation 4-5. 4-6. to follow . . . . . . . . . . Sampling System Schematic to follOW o e . o o o o . o Chopper and Differentiator C i r c u i t t0f0ll0W 4-7. . O O O O O O O O . 9 O Functions . . O . O . . . . f t o . o » e O . . . . O . . . O O O O O O O O O O O . O O 0 O . . O 25 25 Experimental Normalized Amplitude P r o b a b i l i t y Distribution tO follOW 23 . 26 4-8. Experimental Normalized Autocorrelation Function . . . . 27 4-9. Block Diagram of Correlator . . . . . . . . . . . . . . 28 4-10. Integrator C i r c u i t Diagram to follOW • » • • • • • • 28 0 4-11. 0 0 0 Normalized P r o b a b i l i t y to follOW . . . 0 0 0 . 9 0 0 . Distributions o o o o o o . a . . . o a o . . . 28 4-12. Alternate Form of Dissociation of Noise Signal . . . . . 29 4-13. B-1. Theoretical and Experimental Autocorrelation Functions . Theoretical Normalized Amplitude P r o b a b i l i t y Distributions for two Member Function Sets to follOW 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 C~l« Theoretical Normalized Autocorrelation Functions t O f*011OW o o o e « o o 0 0 O 0 o o o « 0 * V o o « 41 44 A REPRODUCEABLE NOISE GENERATOR. 1. Introduction. Noise generators are finding extensive applications i n ana7,19 logue computer systems and i n servomechanism a n a l y s i s . 5,13,22 In most a p p l i c a t i o n s , the signal i s derived from conventional noise 2,3,23, sources for instance, the fluctuating component of the plate voltage of a thyratron kept i n a condition of constant discharge, or the fluctuating component of the current i n a conducting diode. The most important c h a r a c t e r i s t i c of these signals and of any noise s i g nal i s that the signal i s random - that i s , for a l l i t should be impossible T to specify the value of the function at a time T + T , knowing completely the behaviour of the function from time 0 to T. The noise signal i s usually described i n terms of i t s p r o b a b i l i t y d i s t r i b u t i o n functions. The most important of these are the first p r o b a b i l i t y function which determines the d i s t r i b u t i o n of signal amplitudes and the second p r o b a b i l i t y d i s t r i b u t i o n which determines the autocorrelation function of the s i g n a l . subject to s t a t i s t i c a l In physical systems fluctuations, the f i r s t p r o b a b i l i t y d i s t r i - bution of the fluctuations i s usually Gaussian and the autocorrel a t i o n function ft ( T ) decreases r a p i d l y to 0 as X increases. In d i g i t a l computers random numbers can be generated by 10,11 various methods and t h i s would be the equivalent of the noise 24 generator i n an analogue computer . Since the d i g i t a l computer has only a f i n i t e number of states the sequence of random numbers 2 must have a f i n i t e period and therefore cannot be t r u l y random. This pseudo-random sequence i s reproduceable since i t i s generated by a predetermined programme. This reproduceability has c e r t a i n a t t r a c t i v e features associated with i t which the physical noise generator does not possess. For c e r t a i n analogue computer studies i t would be d e s i r - able to have a reproduceable pseudo-noise signal s i m i l a r to the pseudorandom number sequence generated by a d i g i t a l computer. This thesis is concerned with the design and development of an electronic device for generating a reproduceable noise s i g n a l . The applications of t h i s noise generator and conventional noise generators i n servomechanism design are discussed i n the following paragraphs. The design of a control system depends upon the nature of the input functions, random disturbances, (such as uncontrolled load disturbance or noise i n amplifiers) and the nature of the desired 5,13 s 15,22 response. Usually these functions can only be described s t a t i s t i c a l l y , and hence the mechanism i s designed according to 4j 5j13p15j 2S s t a t i s t i c a l design theory. This requires a deter- mination of the average c h a r a c t e r i s t i c s of the s i g n a l , the choice of the1 measure of the error, and the design of the system i n accordance with the conditions of error minimization. Unfortunately, a general mathematical analysis of control systems seems limited to l i n e a r 5 • ; systems, and to minimization of the mean-square e r r o r . Thus, i f another c r i t e r i o n for minimum error i s chosen, or i f a nonlinear 5,14 system is analyzed, the,.,recourse i s to experimental methods. As an example, consider the q u a s i - l i n e a r i z a t i o n of a nonlinear servomechanism. 5 element. of The nonlinear element i s approximated by a quasi-linear The l i n e a r i z e d system i s then designed so that the the l i n e a r system to an input ^ ^ ( f ) y4(t) = JM y4(t+T ) , j * i 0 ( ± y0( t + T is ) >a of the nonlinear system to an input y . ( t ) . fiiQ( x) n d fi^i' ) ° 1 where y0(t) i s response t h e output The process i s shown diagrammatically i n Figure 1-1. y (tk A y Nonlinear System ^ Ly (t) 0 t a) Nonlinear System. Quasilinear System b) Figure 1-1. The Quasi-linearized System. Quasi-linearization of a Servomechanism. equivalent system can then be analyzed for behaviour with respect to changes i n system parameters or changes i n input signal istics, result. character- and hence a better understanding of the nonlinear system may In order to design the equivalent l i n e a r system, the input autocorrelation function t) and the input-output c r o s s - c o r r e l a t i o n function ft. ( l) must be determined. With a-conventional noise generator 1 21 22 as the signal source, complex autocorrelators ' ' or variable delay lines would have to be used i n order to be able to c a l c u l a t e the functions p,. ( T) and p. ( T ) O With two of the reproduceable noise generators, howevever, only simple multiplying and averaging devices would be needed to determine these functions, since J ^ ( t ) and y^(t+T ) could be generated simply by turning one of the generators on a time X before the other one. (See Figure 1-2) Generator One > Nonlinear System v (t)v (t+x) . i Generator Two 0 > y.(t)y.(t+T) Figure 1-2. Correlation of Signals. A l s o , i f the input signal were obtained from a conventional noise generator, the effect of varying a parameter i n the l i n e a r system would have to be evaluated s t a t i s t i c a l l y whereas i f a reproduceable noise generator were used a fast comparison of the responses of the altered and unaltered system to the same random signal would be possible. Thus the design of both l i n e a r and nonlinear servomechanisms subject to minimization of various error c r i t e r i a could be f a c i l i t a t e d using the reproduceable noise generator. 5 2. Generation of a Noise Signal Using Periodic Functions 2-1. Introduction. Since a noise signal i s specified only by i t s p r o b a b i l i t y 3,23 d i s t r i b u t i o n functions, there i s no d e f i n i t e approach to the problem of generating a reproduceable noise s i g n a l „ For the probable a p p l i - cations of t h i s generator i t i s more important to have the signal amplitude fluctuate according to a d e f i n i t e p r o b a b i l i t y d i s t r i b u t i o n than i t i s to have an autocorrelation function which f a l l s increasing x . off r a p i d l y with The signal output w i l l therefore be required to have a f i r s t p r o b a b i l i t y d i s t r i b u t i o n which i s approximately Gaussian, and an autocorrelation function which f a l l s with increasing x , off as rapidly as possible due consideration being given to achieving t h i s result with simple c i r c u i t r y , , 2—2„ Theory of Generation of the Noise S i g n a l . Tg meet these requirements i t was proposed that the signal be generated as the sum of a number of periodic functions, under the assumption that the signal so produced would fluctuate manner. i n a haphazard Because periodic member functions are used, the composite function i s also p e r i o d i c I f the noise signal i s to be random i n a time i n t e r v a l , however, the signal must not repeat i n that i n t e r v a l ; hence, the composite function must have a long period. There i s also a time of recurrence or near-period associated with a composite function of t h i s form which may be very much shorter than the period of the function. Consequently, a further r e s t r i c t i o n is made on the function; that i s , the near-period should be as long as possible. 2-2-1„ The Period of a Composite Function. A function F(t) composed of a number of periodic m f.(t) ( i = 1,2, . . . , m), such that F(t) = functions ^> f . ( t ) , has a period which i s dependent upon the periods of the component functions. To determine the relationship between the period of F(t) and the periods of f . ( t ) consider the function l F(t) = ^>*.fo) fx(t) » + f2(t) + ... + fm(t) i«l where F(t) = F(t + T ) , and ^i^^ That i s , + T^o = i s the period of F(t) and x^ i s the period of f\,(t). x Then F(t) =* f.-(t) + f „ ( t ) + . . . + f (t) » F(t + x ) and so F(t + x ) Since f-(t) « f.(t 1 we must have = f,(t 1 n. l + T, l n 1 o + X )+..,+ T „ t = <s 3 0 . . . ^n x =» T. m m d, . (n^, n^, o.o I f we l e t the largest common d i v i s o r of T , 1 , n m integers) T> ^ =^ are dimensionless That i s , T, ^ X ^ T, . . . , T f f l be T, then we T _ , ... , x JL may write f (t + X ) T . ) = f, (t + n T . ) where n i s an integer, then 1 a +X ) + f g ( t ID » X. T , where ...,X integral m u l t i p l i e r s whose largest common d i v i s o r i s T defines a unit period and the periods of the functions 1. are multiples of t h i s u n i t period. Then, w r i t i n g x = XT =n^XjT = XgT** • • • = n X T we have X = n 1 l X = n 0 X . = i = n X . In the general case X mm 1 1 2 2 mm 0 * 0 0 The largest common d i v i s o r of the numbers a , b , c , . . . k, i s the largest positive integer which divides a.b.c, . . . k. 1 2 m 7 of ^ ^, \ , , «.o e q u a l s t h e l e a s t common m u l t i p l e s p e c i a l case where X , X X , ... , A > ^ a r e a l l coprime b • m * n * n e , the least ID common m u l t i p l e o f X p X g , .00 , X f f l The period i n t h i s case i s i s simply the product T = XT » X^Xg";"?* X m A, ^X^••*?•• X To Thus i t is seen that i f the periods of the member functions are chosen so that t h e X ^ ' s are coprime, the composite function w i l l have a very long period. 2-2-2. The Near-period of a Composite Function. A function qbtained by summing two or more periodic may exhibit what s h a l l be termed n e a r - p e r i o d i c i t y . functions To i l l u s t r a t e phenomena of near-periodicity and to establish a c r i t e r i o n for the deter- mining the near-period of a function, consider the function F(t) = f x ( t ) where + f2(t), fj(t) '« f x ( t + X l T), f2(t) +X2 - f2(t ' T), and X^, X g , and T are as defined i n the preceding section. The m u l t i p l i e r s Xj andXg F(t) i s XT ^ X j X g T . n^( < a r © necessarily coprime and the period of Suppose, however, that there are integers and n^ (< X^) such that n^ XjSrf-iig X g . The least common multiple of the numbers a , b , c , . . . , k i s the 12 least positive number which i s d i v i s i b l e by a , b , c , . . . , k. Two integers a and b are said to be coprime i f the largest which divides both a and b i s 1. integer The numbers a , b , c , . . . k, are to be coprime i f every two of them are coprime. said m That i s n l X l n = 2 2 X + 6 2 (| 2l « h 6 (6 ^ o r X 2> 0) 2 Then, F(t + n XjT) x = fj(t + nt = f^t + n t XjT) + f 2 fx(t + nJ X J T ) + f S: X ^ ) + fg (t + ^ X jT) (t + n 2 \ 2 T 2 = +6 (t + n 2 \ 2 T ) 2 T) = « (neglecting6 g T w . r . t . X gT) or, F(t + ^ X j T ) - f^t) 25 F(t). + fg(t). We see that the function 4 almost repeats i n the time n^ X j T . We s h a l l c a l l t h i s timeiT^, the near-period of the function F ( t ) . course, the value of of <§ g. Of i s dependent upon 5 , the maximum allowed value I f the value of 6 i s chosen to be 0, then T^ i s simply the period of the function, X ^ XgT = X T . In the general case where F(t) = f.(t) m i=l f^ (t) = f.(t and +X.T) we s h a l l define the near-period T^ of F ( t ) to be the value n^ X j T . The n^ ( i = 1,2, . . . , m) are integers which s a t i s f y the conditions 6 . . <f<C6 where 6.. = ( n . X . — n . X.) i> j = * » 2 » ...,111 m and where 6 i s small compared t o X ^, X g , . . . . X^ 2-3. Designing the Noise Generator. The design of the noise generator consisted of a choice of the form of the member functions and the determination of the number and periods of the member functions so as to produce a long near-period. 9 The form of the member functions was f i r s t decided upon from considerations of ease and r e l i a b i l i t y of generation with electronic c i r c u i t r y . Because reproduceability was a prime consideration, a combination of pulse and d i g i t a l techniques was used. Once the form of the member functions had been adopted, computations were made to determine the number and the periods of these functions so as to y i e l d a long nearperiod of the composite function. The form of the member functions is best described by considering the manner i n which the member functions are generated and the r e s u l t s of t h i s method of generation. 2-3-1. Generation of a Member Function. Each member function i s generated as the r e s u l t of the continuous c y c l i n g of a multivibrator chain by a precision-timed pulse train. A 1 0 0 Kc c r y s t a l o s c i l l a t o r serves as the source from which are derived the actuating pulses for the multivibrator chains. Consider the generation of one of these member functions. The output of the c r y s t a l o s c i l l a t o r i s used to i n i t i a t e a continuous pulse output with a pulse-recurrence-frequency per second (pps). (PRF) of 100,000 pulses This output PRF i s divided down successively by three phantastron divider units whose d i v i s i o n r a t i o s - that i s the r a t i o s of the input PRF's to the output PRF's - are d^, dg, and dg. The PRF of the output of the l a s t phantastron i s then , 1 " ^ ^ pps. a ia2 3 This output i s used to actuate a multivibrator chain consisting of k bistable multivibrators i n s e r i e s . The multivibrator chain i s designed so that one cycle of the chain i s equivalent to N pulses, ^ N d^dgdg where N = (2 — l ) . The period of the chain i s then i n n n n n seconds. Each multivibrator of the chain actuates a gate so that a voltage V. (j = 1,2, . . . , k) i s transmitted to the output of the generator when J the multivibrator i s i n one state and a voltage 0 when the multivibrator i s i n the other state. The voltages V. are either p o s i t i v e or negative. The output voltage at any instant i s the sum of the voltages transmitted to the output of the generator. To i l l u s t r a t e the waveform obtained at the output of the generator from one of the multivibrator chains, consider the output of a chain consisting of three multivibrators connected so that Vj = + 3V, Vg = — 2V, and Vg = - V, and the chain i s connected so that i t recycles after (2 - l ) = 7 pulses. such that the output voltage V q The i n i t i a l configuration i s i s 0. After one pulse, V Q = VJ= two pulses, v e s V 0 i + V 2 + V three pulses, V =Vg= +3V 3 = •• + o v -IV Q four pulses, V =V.+V„= +2V f i v e pulses, V ^ V ^ -2V six pulses, V =V.+V= .....+1V 0 seven pulses, v = 0+0+0 =».*»#-0V o . •• -4 ' The cycle repeats after the seventh pulse. •; • • . The sequence of operation depends e n t i r e l y upon the i n i t i a l configuration of the m u l t i v i b r a t o r s . The output waveform is shown i n Figure 2-1. 11 feedback pulse to recycle chain. Input pulses I IT~T f r r r r Output from first +3V multivibrator 0 Output from second multivibrator A 6 -2V Output from f third Mult ivibrator 0 -V Output from i Chain +3VI -3V-r Figure 2-1. Example Output Waveform from M u l t i v i b r a t o r Chain. The waveform described above i l l u s t r a t e s the form of the member function adopted for use i n the noise generator. The remaining step i n the design was to obtain a long near-period for the composite function. 2-3-2. Determination of the Periods of the Member Functions. The f i n a l step i n the design was to determine the number of member functions and the periods of the member functions so as to obtain a long near-^period of the output s i g n a l . (At t h i s point, no idea of what constituted a " l o n g " near-period had been formulated.) of the composite function was t r i e d , namely F(t) » f j ( t ) The simplest form + fg(t) where the member functions were of the form described i n the preceding s e c t i o n . The period of f j ( t ) N i s T g «» 2 N 'd-jdjgd^g ia x ^ =» —100 000—'— 21 d 22 d 23 100 000— sees, and that of fg(t) d s e c s » The problem was to determine values of the N's and d's which would give a long near-period. Because phantastron dividers were to be used, the d's were constrained to l i e i n the range 7 to 20 to ensure stable operation. A l s o , the product djdgdg was to be about 1000 so that the input PRF to the multivibrator chains would be about 100 pps. The N f s were chosen to be of the form (2 -1) so the factors Nj and Ng would be coprime. minimize the biasing effect The k*s were kept small so as to inherent i n the binary scaling action of the multivibrator chains, ( v i z . the l a s t multivibrator i n a chain i s " o f f " k k 2 • 2 for ( g- — l ) successive pulses and "on" for •g successive pulses: t h i s merely produces a s h i f t i n the output dc l e v e l every h a l f c y c l e , and hence i f the k's are too large t h i s may be considered a sort of "biasing" action as opposed to the "switching" action of the first multivibrators.) In order to determine values of the d's and k ' s for the twofunction case, a programme was written for the Alwac III £ computer to solve the problem: Given 6 , K and M Compute integers x and z so that the difference (i.e. \ o ^ 1 6 ^ = (xK-zM) i n absolute value i s less than or equal to 6 6 ) Various values of K = ( 2 k i l ) d 1 1 d 1 2 d 1 3 and M = (2 2 -l)dgjdggdgg were t r i e d , but the length of the near-periods lld12d13 tained (as determined for 6 = go ) ob- d a few seconds. w e r e a * l only of the order of This was considered too short. The next simplest form of the composite functions was t r i e d , that of F(t) equal to the sum of three member functions. Again, a pro- gramme was written for the Alwac III E computer to solve the problems Given 6 , K, L, and M Compute integers x, y , and z, so that the erences 6^= (xK-zM) and 6 g = (yL-zM) are, diff- i n absolute value, less than fcl or equal to 6 . Various values of K = (2 " * l ) n < i i 2 d 1 3 ' / fc2 \ i k3 \ L » (2 - 1 / d £ i d 2 2 d 2 3 a n d M " ( 2 ~1'd3]d32d33 w e r e t r i e d » d ranging from a few seconds to s i x minutes were obtained. Near periods The values which gave a near-period of six minutes are the values used i n the noise generator. They ares d d 12 d 13 10 d 21 = 13 d 22 m 8 d 23 - 7 a ll (* - l) k l (2 -l) k 2 - 10 d 31 = 11 d 32 - d 33 9 = 15 (2 k 3 a 10 - 12 -D - 8 a 31 case, the values of the \ ' s are \ x » \ m 728 = 7(13)8 = 1485 a 15(11)9 » 2976 = 31(12)8 and that of T, T = 1 Q Q 1 O O O — seconds = 0 . 1 milliseconds. The period of the function, given by T times the least common multiple of Xj,, Xg and Xg iss 728(1485)2976(0.1) 3(8) T T = .... . 1 Q .n_ „QO n milliseconds = 13,405,392.0 msecs., which i s approximately 223 minutes. for c d 6 = T lld12d13 - n " " lX1T = iss 5 0 6 9 ( 7 2 8 )(°»1) msecs = 369,023.2 msecs., which i s approximately six minutes. that The near-period, T q , as determined (The value of 6 chosen i s such 6T i s about one-half the time between pulses into the chains.) The programme for the three-function case i s described i n Appendix A . 2-3-3. Summing of the Member Functions. Addition of the three member functions i s effected through an operational amplifier c i r c u i t using a P h i l b r i c k K2-X operational 20 amplifier unit » The member functions are summed and passed through a smoothing network simultaneously and i n t h i s way the output smoothed into a continuous s i g n a l . is V"'"' 3, "'.'.'.'V 1 P r a c t i c a l Design Considerations 3-1. Introduction. This section presents the c i r c u i t s used i n the noise gener- ator and a discussion of the manner i n which the member function waveform voltages and the output network parameters were decided upon. 3-2. Electronic Circuitry. A l l the c i r c u i t s used i n the noise generator are of standard design. For t h i s reason, no descriptions of the c i r c u i t actions are given and instead the reader i s referred to standard texts and ijournals. (References 6, 17, 18, and 19.) A block diagram of the noise generator and diagrams of the c i r c u i t s used i n the noise generator are given i n Figures ( 3-1. to 3-4 inclusive. Component values are included i n the c i r c u i t diagrams as well as some of the pertinent waveforms. 3-3. Scaling and Addition of Component Waveforms. The form of the member functions having been decided, the next step was to choose values of the output waveform voltages. In 1 order to further minimize the biasing effects of the binary s c a l i n g action of the multivibrator chains (Section 2-3-2.) i t was decided to weight the output voltages gated by the m u l t i v i b r a t o r s . For example, for a chain of k multivibrators, the weighting factors are such that Vj « -j- v « Vg 85 o.. , V^ = ~ V. A l s o , the voltages were chosen so that the sum of the p o s i t i v e voltages gated by the chain was equal to the sum of the negative voltages gated by the c h a i n , (i.e., 5 1 Multivibrators and Gates Phantastrons Recycle Loop Pulse Generator -r 10 Phantastron Summing and Smoothing Network Common Driver Unit for Two Generators. _ UM.... . T- 12 —^ T- Phantastrons 8 i 1 "" ^ ' > Multivibrators and Gates Recycle Loop Figure 3 - 1 . Block Diagram of Noise Generator. x Crystal h Oscillator +300v.-- Diode Clipper l R R R 0 4 R K e R -300v.-- R R Output of Crystal +5v. 0 -5v. Oscillator r yr Output of Diode Clipper R. • ...150K • ...1MEG • . . . 39K • ...270K 1 watt £ ti n tt tt ,. „ ... 5K pot. n tt • ...470Kf tt ti ...470A 2 n • . . . 15K' « . . . ! 22K •...250pf •. . O . O t y f + = •,.4000pf • . . . 25pf * . . . lOmh . i lOOKc c r y s t a l • 4•.6AN8 • i,.6AL5 m,.1N191 * -20v; Differentiated Output of Amplifier Figure 3-2. Amplifier lOOKc Pulse Generator. TTTTTTT~rt~ R l R Input Pulses o o••220K • «••6•8K R R • «« « e +300 v o l t s ....IOOK ....aooK . . . . 10K K o » • • 9K R 10 • »« »4» 7K ....0.05uf ....0.25Uf . . . . 2 5 0 pf ....6AL5 ....6AS6 -300 v o l t s R R Plate Waveform Input PRF (pps) Division Ratio Cathode Waveform . R 3 100,000 10,000 10 11,12,13 8,9 39K 68E 9.4MEG 9.4MEG 10MEG 50pf 250pf lOOOpf T Differentiated Cathode Waveform Figure 3-3. 1,000 Phantastron Divider C i r c u i t . 9 100K Bistable M u l t i v i b r a t o r R R l 5 ....270K ...470K ....100K ....150K ....330K ^-watt " " II it ii II " " + = +300 v o l t s Cathode Follower R„ . . . . 6 ,. ,. 8 «Q . . . . R Gating C i r c u i t s 10K £ w a t t 68K 2 " f 27K 1 " 68K £ " See Section 3-3. - = -300 v o l t s Figure 3-4. Multivibrator and Gating C i r c u i t s . ...250pf ...0.15 f ...1N191 ...5751 (12AX7) ...12AT7 ...5651 Output Stage 16 r + v 2 + . . . + v k = o). 1 A schematic of the output stage i s shown i n Figure 3 - 5 . The response of t h i s network at time t to a step input of V\ v o l t s at time 0 R is e „ ( t ) = - V. Xi - RQC. Q g £ (l-e x ) n where S i m i l a r l y , the response at time t to a step input of ^ at time 0 i s e Q (t) Regrouping gives, •.<*' " " v " V i - ~ O volts R e0(t) R o ( jp)(l-e i _^ R °) Figure 3 - 5 . Output Network, i < BC> ( > - ~ ° > 1 That i s , by choosing values for V ^ , R ^ , R q , and x Q t then the input r e s i s t o r value for an output scaled down (weighted) by a factor just k times the unit r e s i s t o r value R ^ . rjr i s In the noise generator, the V^'s were chosen to be i 4 5 v o l t s , and the R ^ ' s were chosen to give output voltages weighted as described i n the preceding paragraph. The time-constant T Q was chosen to be approximately equal to the duration of the voltage steps from the multivibrator chains. The value of the r e s i s t o r R was chosen so that the maximum excursions of the o smoothed waveform are - 100 v o l t s . The values of the components are tabulated below with the steady-state output voltages and the output voltages at time T • Resistance R u R12 R21 R 22 Steady-state voltage Voltage at = 60K - 51.0 - 32.5 m 90K + 34.0 .+• 21.7 « 180K +17.0 + 10.8 - 75K + 40.8 + 26.0 = 1 0 0 - 30.6 K - 19.5 R23 « 150K - 20.4 - 13.0 Rg9 - 300K + 10.2 + R31 = 84K + 36.5 + 23.2 Ron m 120K - 25.5 - 16.3 R33 - 160K - 19.1 - 12.2 « 210K +'l4.8 + 9.3 - 480K =6.4 - 4.1 Roc OO R o = 68K C « 0.15 rlit, T 10.2 msecs. o 6.5 4. Analysis of the Noise Signal. 4-1. Introduction. The analysis of the noise signal consisted of a t h e o r e t i c a l and an experimental determination- of the p r o b a b i l i t y d i s t r i b u t i o n of amplitudes and of the autocorrelation function of the s i g n a l . The methods of determination and the ^results so obtained are discussed i n the following sections, as well as,a comparison of the experimental and theoretical r e s u l t s . 4=2. Theoretical Analysis The theoretical analysis was not of the actual output s i g n a l , but rather of the unsmoothed output s i g n a l . Consequently, only a rough agreement with the experimental r e s u l t s was expected. The un- smoothed waveform was used as an approximation to the actual signal because i t i s composed of t r u l y periodic functions and hence lends i t s e l f more r e a d i l y to t h e o r e t i c a l analysis than does the actual sig- nal which is composed of aperiodic functions. 4—2-1. Theoretical Amplitude P r o b a b i l i t y D i s t r i b u t i o n . The p r o b a b i l i t y d i s t r i b u t i o n of amplitudes was determined by considering the d i s t r i b u t i o n obtained by sampling the waveform over a long time i n t e r v a l . To determine t h i s d i s t r i b u t i o n i t i s necessary to dissociate the unsmoothed signal into i t s component parts. Consider f i r s t the waveform obtained vibrator chain. (Figure 4-1 (a)) from the f i r s t multi- I f t h i s waveform were sampled ran- 1 (Amplitude k x unit +560 -amplitude) + 2 g 0 a) Waveform from Chain One. 2 7 P^kV) 1 T -800 + -600 4- -400 b) •+- -200 J 200 1- 400 600 800 Amplitude P r o b a b i l i t y D i s t r i b u t i o n for Chain One Waveform. 2 15 P 2 (kV) 1_ 15 -L -L. -800 -600 -400 c) -4_ -200 0 _l_ 4 200 L —V- J 400 600 800 Amplitude P r o b a b i l i t y D i s t r i b u t i o n for Chain Two Waveform. P 3 (kV) _2_ 31 31 4- -800 -600 -400 d) -200 0 200 400 600 -r 800 Amplitude P r o b a b i l i t y D i s t r i b u t i o n for Chain Three Waveform. Figure 4-1. Amplitude P r o b a b i l i t y Distributions. domly over a large number of cycles of the waveform, the p r o b a b i l i t y d i s t r i b u t i o n of amplitudes would be as i n Figure 4-1 (b). S i m i l a r l y , for the second and t h i r d chains, the p r o b a b i l i t y d i s t r i b u t i o n s of amplitudes would be as i n Figure 4-1 (c) and Figure 4-1 (d) respectively. Thus, there i s an average p r o b a b i l i t y pj(kjV) that a voltage kjV i s delivered by chain 1 at a time t , an average p r o b a b i l i t y p 2 ( k 2 V ) that a voltage kgV i s delivered by chain 2 at a time t , and an average p r o b a b i l i t y p 3 (kgV) that a voltage kgV i s delivered by chain 3 at a time t . Because the waveforms have non-multiple periods, i n the aver- age the cross—correlation between any two of the member functions is zero, and hence the average p r o b a b i l i t i e s associated with the waveforms are independent. V » (kj + kg + kg)V * ties, That i s , the p r o b a b i l i t y of occurrence of 0 U S * * h e product of the i n d i v i d u a l p r o b a b i l i - s ( i . e . p(V o ) = P 1 ( k 1 V ) p 2 ( k 2 V ) p 3 ( k s V ) . ) I f we associate with each member function a generating function G . ( s ) , we may easily determine the p r o b a b i l i t y of occurrence of a certain output amplitude. G^S) y ^ p ^ k ^ S * 560 x a = s j 280 1 0 + "• + j % For member function 1 we have ? j s + T -66© -280 S + 7 8 -840 x •* 7 8 k The coefficient of s i s the p r o b a b i l i t y of the voltage kV occurring. There are s i m i l a r generating functions G 2 (s) t^rid Gg(s) for member functions 2 and 3 respectively. (The powers of s are high because the unit voltage V was chosen so that a l l the k's would be integers.) Because the average p r o b a b i l i t i e s associated with the member functions p(v) Amplitude -100 _80 ~l—i-* | " ' -60 l j—i—r -40 Figure 4—2, Theoretical Normalized Amplitude P r o b a b i l i t y D i s t r i b u t i o n . 80 100 v are independent, the generating function for the sum of the waveforms is equal to the product of the generating functions of the i n d i v i d u a l waveforms, i . e . CrQ(s) = Gj^sjGgfsjGj^s),, The product of the generating functions consists of about 3000 terms, and so a programme was written for the Alwac III E computer, (See Appendix B), to group the terms according to the values of the exponents and to count the number of terms whose exponents l i e i n the range (n)l68 to (n+l)l68. (n)l68 <^ (n+l)l68, n = -15, -14, . . . , 0, . . . , f 5 . ) (i.e. Dividing the count for each range by 7(15)31 gives the average p r o b a b i l i t y that the output amplitude l i e s i n the range (n)l68 and (n+l)l68, since the t (n)168 t i o n so formed i s the c o e f f i c i e n t of the term s* corresponds to an increment of ' v o l t s = 6.45 v . ) frac- . ( The range 168V A plot of the normalized t h e o r e t i c a l amplitude 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 shown i n Figure 4-2. 4-2-2. Theoretical Autocorrelation Function, The autocorrelation function fi (t) of a function F(t) - f (t) + f (t) + f ( t ) , where f ^ ( t ) , f 2 (t)> a n d ^ f a ) a r e periodic functions 2 3 with coprime periods, i s merely the sum of the autocorrelation functions 22 of the i n d i v i d u a l functions. components of different This i s because the m u l t i p l i c a t i o n of frequencies r e s u l t s i n a zero average value. In the case of a periodic waveform, the basic form of the autocorrelation function -+T 0 (*) O&f^loijr / f(t) f(t + x) d t may be replaced by the 0( ) T expression i exprea O _A_ / j f(t) f(t + x) dt where f ( t ) i s periodic of period x .. J Consider a function f ( t ) f (t) where f n , f^, - of period Nx ^ fQ 0 <t * 1 < * • " f l = f k " f N-l k <Tj < 2 T 1 + i H i l T such t h a t : (N-l)^<t<NT . . . . f^ . . . , fjj ^ are constants. 1 ( Se^ Figure 4-3), f(t>> N-1 _I f 0 0 N-2 2 " l 2 1 r 3Tj i 1 t-S/ l T ) 3 ( N - 2 ) T 1 (Nj-ljTjN^ t Figure 4-3. Sample- Function. " The autocorrelation ^((nXj) « ^ function ^(n x^ ), i s then, f(t) I f(t+nT x ) dt = . j i - ( / 1 /r. NT, 2T, + / f(t) fft+nx^ dt + . . . + / ! 1 f(t+ n x ^ dt 1 f(t) f(t+nXj) dt) (N-lhj NX, A N X T f(t) < o/ + f y X X 0 ) ^(N-l)x x 0 Sjj- ( f f f ( t + nXj) dt + n / Nx, 2T dt • f j f ^ dt + . . . . + f ^ fN_1+n (N-l)x = i '< f O f n *1 + f n+l — + + f N - l r H-l+n > S i m i l a r l y we may write 0((n + l) x ^ i;( - f0fn+1 —+*SZi)^ + Suppose, however, the time-shift i s not an i n t e g r a l number times, x ^ butj say (n+a) % 1 where O ^ a ^ l . Then, ^'((n+tt)^) (f J /Xl f(t+(n+a)^) dt / / ( l ) T M (*+(n+a) x ) f x ° dt = /(k+1)^ + . . . + fk / f(t+(n+a)x ) dt + . . . i dt) . ) tMwjtj) " - f / 1 0 + f(N t ) Fx = l ,&»!>... Consider the general term i n the bracket, f^ / f(t+(n+a)xj) dt Ac X j Now, )x^) assumes'two f (t+(n-ta values i n the range k ^ ^ ^ t ^(k+ljt^. For k i j ^ t ^ k X j + (l-a) % 1 we have t(t+(n+a)n; ) = f f c + n and for kx x + (l-ajxj ^t fore, the integral ^(k+l)x / k T k+n 'k^ we have (n+o) V i s fk+Q+1 . There- l d t + /(k+lJx,^ / W l « d t /(k+U-a))^ and so the function ft ( (n+oc) x^) H f(t+(n^)x ) f(t+(n-KX)Xj) dt equals ,(k+(l-a)^ 1 f 1 ( f0fn(l-^\ W^V'k+n+l a l becomes + f0fn+1 (a) \ + ... Vkw + + F N - l N+n F + <a> f (a) k k+n+l 1 f } + + fN-1 ( l - a ) ^ .•. ' Regrouping and cancelling givess a <(n+a)^) + N (f Ofn+l . i + f <f0fn lfn 2 + — + + f l + fn+1 + . . .+ ^ W l ) N - l N+n - 0 f n f O f n F f F Thus.$-the autocorrelation function of t h i s + l + " — rectangular-shaped waveform i s a triangular-shaped function having the same period as the o r i g i n a l function. The autocorrelation function i s an even function and hence 0(t) = and also •= ^ ( N T J - T ) . ). But, fi(l ) = ^ ( N T + T ) X Therefore, 0( T ) » 0 ( N ^ - x ) . That i s , the autocorrelation function i s symmetric about the h a l f NT^ period point - r — . A plot of the normalized t h e o r e t i c a l auto- c o r r e l a t i o n functions 0n(x) ^—rQ-y , * <T) and jg|—jfoj ' - ^ (X) ^ (o) 3 3 * ne member functions f^, f g , and fg are shown i n Figures 4—4 (a), (b) and (c) respectively. (4. programme was written to determine the cardinal points ^ ( n X ) . This programme i s described i n Appendix C.) Because the i n d i v i d u a l waveforms have independent probab i l i t y functions ( i . e . are uncorrelated) the autocorrelation funct i o n of the sum of the waveforms i s equal to the sum of the autoc o r r e l a t i o n functions of the individual waveforms - that i s , 0(T ) - fi ( u +0 2 2 ( T > + 033 ( t ) ' v h e r e P ( t ) = f i( t ) + f 2 ( t ) + f 3 ( t ) a) Normalized Theoretical Autocorrelation Function for Waveform One. b) Normalized Theoretical Autocorrelation Function for Waveform Two. -0.5^ c) d) Normalized Theoretical Autocorrelation Function for Waveform Three. Normalized Theoretical Autocorrelation Function for Composite Waveform. Figure 4-4. Theoretical Normalized Autocorrelation Functions. and ^ ( T ) i s the autocorrelation function of F ( t ) . The normalized autocorrelation function of the composite waveform i s irn = j (o) n + ^ (o) 2 2 + ^ (o) 3 A plot of the normalized autocorrelation function ^ | Q | is shown i n Figure 4 - 4 (d). 4-3, Experimental A n a l y s i s . A laboratory determination was made of the amplitude proba- b i l i t y d i s t r i b u t i o n and of the autocorrelation function of the actual output s i g n a l . The p r o b a b i l i t y d i s t r i b u t i o n of amplitudes was de- termined using sampling techniques and the autocorrelation function was determined using two of the noise generators and a m u l t i p l y i n g and averaging device. These methods are described i n the following sections. 4-3-1. Experimental Amplitude P r o b a b i l i t y D i s t r i b u t i o n . The method by which the amplitude p r o b a b i l i t y d i s t r i b u t i o n was obtained was to sample the waveform 500 times per second for a time T q minutes, and to count the number of times the ampli- tude of the pulse so formed was i n the range v to v + / \ v. Dividing t h i s number by the t o t a l number of pulses then gives the p r o b a b i l i t y of the amplitude of the waveform l y i n g i n the range v to v + Av. The time T was chosen because t h i s i s the nearn period of the signal and hence should give an accurate p r o b a b i l i t y d i s t r i b u t i o n for a l l time i n t e r v a l s . The sampling frequency (500 times per second) was chosen to be very much higher than the h a l f power bandwidth of the noise signal (approximately 16 cps as determined by ^ , the smoothing-network time-constant) so that, i n effect, the waveform was sampled continually and consequently the count for a part i c u l a r amplitude is a very accurate measure of the time the waveform was at that amplitude during the time T q . A block diagram of the sampling system i s shown i n Figure 4-5 (a), and the waveforms at various points i n the sampling process are shown i n Figure 4—5 (b). The chopper system consisted of a gating c i r c u i t activated by a b i stable m u l t i v i b r a t o r . The d i f f e r e n t i a t o r and inverter system con20 sisted of a P h i l b r i c k K2-X unit which inverted the chopped wave- form, followed by a simple RC d i f f e r e n t i a t i n g network. The K2-X unit was used so as to provide a high-impedance r e s i s t i v e load for the gating c i r c u i t and to allow a gain adjustment to offset the s l i g h t attenuation introduced by the d i f f e r e n t i a t i n g network. Thus, the amplitude of the pulse delivered at time t by the d i f f e r e n t i a t i n g network may be made exactly equal to the instantaneous amplitude of the noise signal at the time t . The c i r c u i t diagrams for the chopper and d i f f e r e n t i a t o r units are shown i n Figure 4-6. The pulse-height discriminators (kicksorters) used were Marconi type 115-935, and were the property of the section of the U . B . C . Physics Department. Van de Graaff T h i r t y kicksorters were available and so i t was possible to divide the noise signal into 3 0 discrete amplitude ranges, 0 to V ^ , Vj to Vg, . . . , Vgg to V ^ Q , and so determine the number of counts for each range* The maximum to kicksorters Noise Generator Bias and Scaler Chopper Differentiator and inverter Output of Bias and Scaler Output of Noise Generator +45 -f + 5, 0 -5 rr -451 Output of D i f f e r e n t i a t o r and inverter Output of Chopper b) Waveforms i n Sampling System. Figure 4-5. Sampling System Schematic. Bistable R, B 2 5 1 6 Cathode Multivibrator • • • • 33K i^rvatt ....100K n n .... 1SK 1 .... 47K I " ....4.7K ; w n w R, 8 *; R 10 1 " + - +300 v o l t s ...2.2MEG ...150K .. .. •• Diode Gates Foilovers 50K ...500pf ...O.Olflf •Jrw&tt n it ii ti ii ti pot. R - = -300 v o l t s ii Figure 4-6. Chopper and D i f f e r e n t i a t o r C i r c u i t Differentiating Circuit ,...250pf ....12AT7 ....12AX7 ....6AL5 ....1N191 .... IK pot. pulse amplitude accepted by the kicksorters was 40 v o l t s (negative- going) and the maximum counting rate per channel was 50 pps for sus- 16 tained counting. For t h i s reason, the noise signal was scaled down and biased so that the maximum signal excursions were from +5 to +45 volts. The pulses delivered from the inverter and d i f f e r e n t i a t o r system thus had an amplitude of from -5 to -45 v o l t s . The noise signal was made to vary from +5 to +45 v o l t s so that the minimum pulse amplitude due to the signal ( i . e . 5 v o l t s ) was greater than the maximum pulse amplitude caused by switching transients chopper system, (approximately 2 v o l t s ) . counting rate was only 50 pps, rangesAVj => V - V ^ + ^ ( i = 1 , i n the Because the maximum allowed i t was necessary to make the voltage 2, . . . , 30) of different magnitudes in order to obtain an accurate count of the number of pulses of each amplitude. Using the t h e o r e t i c a l p r o b a b i l i t y d i s t r i b u t i o n of ampli- tudes as a guide, a set of values was determined for the V \ ' s so as to ensure the counting of a l l the pulses. The normalized p r o b a b i l - i t y d i s t r i b u t i o n obtained from the kicksorter data i s shown i n Figure 4-7. 4-3-2. Experimental Autocorrelation Function. The experimental autocorrelation function was obtained by using two of the noise generators and a m u l t i p l i e r and an integrator. Generator one was turned on a time T before generator two, and the signals N(t) and N(t+ x) were m u l t i p l i e d and integrated for a period of s i x minutes and f i f t e e n seconds (almost the near-period duration). Dividing the value of the integrator count obtained for the time 4p(v) Amplitude P r o b a b i l i t y 0.018 0,016 0.0141 -0.012 0.010 •0.008 .0.006 0.004 .. 0.002 j -100 i » i i—|—i—i—i—I -75 | -50 i—i—i—r—|—i—i—i—r -25 1 i—I—i—i—i—i—|—i—i—i—»—(—i—i—i—i—[—»" 25 Figure 4-7. Experimental Normalized Amplitude P r o b a b i l i t y D i s t r i b u t i o n . 50 75 100 v shift t by the value of the integrator count obtained for the time s h i f t of 0 seconds gives the normalized autocorrelation function for that p a r t i c u l a r value of x , Shifts of 30 seconds were made for the range t =0 to t => six minutes, and the r e s u l t i n g values of the autoc o r r e l a t i o n function are shown plotted i n Figure 4-8. 1.0 0.8 0.6 0.4 0.2 0 60 Figure 4-8. 120 180 240 300 X Experimental Normalized Autocorrelation Function. The m u l t i p l i e r used was a P h i l b r i c k Model MU/DV Duplex 20 Multiplier/Divider. The output from each generator was scaled down so that the maximum signal excursions at the input and output of the m u l t i p l i e r were i n the allowed range of - 50 v o l t s . The output of the m u l t i p l i e r was then scaled down and fed into an ac tachometerfeedback servo motor. 360 The speed of the motor was proportional to the voltage i n and hence the count of the number of revolutions of the motor for a given, time T^ represents the integral of the speed 28 for that time ( i . e . , the count i s proportional to the value of the integral of the input voltage for the time T ^ . ) . autocorrelation function was desired, Since a normalized i t was only necessary to ascer- t a i n over what range of input voltages the integrator was l i n e a r , and then keep within that range.. The constant of p r o p o r t i o n a l i t y out i n the normalization procedure,, cancels A block diagram of the c o r r e l a - tor i s shown i n Figure 4-9 and the c i r c u i t diagram for the integrator i s shown i n Figure 4-10. Generator One N(t) /lJ(t)N(t+T) dt Multiplier Generator Two ->• N(t+x) Figure 4=9. 4-4. ' N(t)N(t+'Q Block Diagram of Correlator 0 Comparison of Theoretical and Experimental Results, The theoretical and experimental amplitude 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 a computed Gaussian d i s t r i b u t i o n are shown together i n Figure 4—11. The Gaussian curve was computed using the mean and mean-square values' calculated frpm the experimental d i s t r i b u t i o n . A very good f i t was obtained. The t h e o r e t i c a l d i s t r i b u t i o n was shifted to the r i g h t (13.5 - 2 . l ) v o l t s = 11.4 v o l t s so as to make the t h e o r e t i c a l and experimental means coincide. As was expected, the experimental d i s t r i b u t i o n was more peaked near zero amplitude than was the t h e o r e t i c a l d i s t r i b u t i o n . This i s due to the action CM CM o ,o o O a • 1 "5 O r-H OJ CO TJ* o o o -p -p ni: : u tt tt © O © © 113 H eH a P*- o « o -p o (1) =C e • 1—I o -p o o a o cj i-l U Q uw © —i t—f" pCJ (ti S3 O bD 03 •H s o a OT S3 o u O •H O a} 00 M oo o -p d bD 0) -P -P aj o (4 • H u -p tt tt tt rH © O O IO t - t r H ^* O J o o o • I o © - o i—I 4 0) 1—I o o e. u S o efl o bDi-t fl o dS 00 C i 03 pg CS •H fa OT 00 fl> g o r | i r r• s> CD fl a} fad W W O O lO 0 t» CO o -P w o » o • t0 3 H n s o e o 6 e • • • « « • o 03 S i S i 0Z^ U o o> at OT 03 r H a, u -P O P< O EH H a> p(v) Amplitude P r o b a b i l i t y fO.018 Figure 4—11„ Normalized P r o b a b i l i t y Distributions of the smoothing network i n the fr(t) output stage of the generator. The good Gaussian f i t may be j u s t i f i e d by viewing the generation process of the noise signal i n a s l i g h t l y different manner - viz<>, consider the noise signal as being generated by adding many short pulses of various amplitudes and of various du' ^tions,, (See Figure 4-12 o ) . , The f3(t) Central Limit Theorem of .... r g statistics provides the means for j u s t i f i c a t i o n . This theorem states that the d i s t r i b u t i o n of pul s,e 1 f(t) the sum of an i n d e f i n i t e l y large pulse 2 number of other independently d i s t r i b u t e d quantities must ^ approach the Gaussian d i s t r i b u t i o n , no matter what the individual d i s t r i b u t i o n s may be 0 '•—'pulse 4 pulse 3 Figure 4 - 1 2 „ Alternate Form of Dissociation of Noise S i g n a l * The t h e o r e t i c a l and experimental autocorrelation functions for the noise signal are shown i n Figure 4-13* Poor agreement between the two results was expected because no account was taken of the smoothing network effects i n the t h e o r e t i c a l determination, and also the accuracy of the integrator is questionable. b u i l d a p r e c i s i o n dc tachometer-feedback ( i t is proposed to servo integrator i n order to more accurately determine the autocorrelation function.) The experimental and t h e o r e t i c a l r e s u l t s agree very w e l l , however, under the circumstances. 1.0-O 0.8-1 Figure 4-13. Theoretical and Experimental Autocorrelation Functions. 5. Conclusions. Examination of the data obtained f a i l s to reveal a precise relationship between the t h e o r e t i c a l and experimental p r o b a b i l i t y distributions. The general good agreement between the t h e o r e t i c a l and experimental r e s u l t s does, however, a t t e s t to the v a l i d i t y of the t h e o r e t i c a l analyses of the waveforms. On t h i s basis, i t i s suggested that the noise signal produced using the i n i t i a l configurations of the multivibrators as described i n Case 2 , Appendix B, would have a more nearly Gaussian p r o b a b i l i t y d i s t r i b u t i o n of amplitudes. This is because the signal so generated would be composed of member functions which are balanced ( i . e . have equal p o s i t i v e and negative voltage steps. A determination of the power spectrum of the signal would be of help i n ascertaining i f the signal is a true noise signal (or nearly so) and also would specify the bandwidth of the noise signal. The signal probably has a bandwidth of 0 to 60 cps (approximately). This figure was arrived at from v i s u a l examination of the output of the sampling system when the sampling frequency was varied from 500 cps down to 20 cps. At a sampling frequency of 100 cps there were very abrupt changes i n the output of the sampling system. This i n d i - cates that considerable change i n the noise signal had taken place i n the time between samples. This would occur i f the noise signal had a frequency component of around 50 cps. On the basis of tests and observations made, i t i s con— eluded that the signal generated may be considered a noise signal having a near-Gaussian amplitude p r o b a b i l i t y d i s t r i b u t i o n , very l i t t l e l a t i o n for time-shifts about 60 cps. corre- greater than 30 seconds, and a bandwidth of Appendix A. A-l. Programme for Determining the Near-periods of a Function. As developed i n the text (Section 2 - 3 - 2 ) the problem involved i n determining the near-periods of a composite function may be stated mathematically ass Given integers mine integers x, y , and z so that |xK-yL | ^ |zM-xK|^ 6, K, L, and M, deter&9 | z M - y L | ^ 6, and A b r i e f explanation of the programme used to deter- ^a mine the integers x, y , and z for a given set of K, L, and M i s presented i n the next paragraph. An augmenting procedure is used i n t h i s programme. The com- puter takes a value of z (say z ) and calculates the two integers x^j (below) and x^g (above) nearest to the value X machine then tests to see i f | z Q M-x n jK j q or i f = . The Jz^M-x^gKJ^ 6 I f neither of X J nor x & g s a t i s f y the i n e q u a l i t i e s above, the cycle Q i s repeated with and increased by 1. The i n i t i a l value of Z so a l l values of z from 1 up are t r i e d . q is 1 This procedure continues u n t i l values of x and z are found (say x' and z ' ) such that |Z'M-X'K|^6 The machine then takes t h i s value of z and computes the two integers y^ 1 (below) and y g ' (above) nearest to y ' « tests to see i f either Iz'M-y^'L | ^ 6 or . The machine then jz'M-yg'LJ ^ 6. If neither y j ' nor yg* s a t i s f y these i n e q u a l i t i e s , the computer returns to the beginning of the programme and starts to determine new values of x and z, t h i s time with the i n i t i a l value of z = z ' + l . * n This series of calculations and decisions i s continued values of x, y , and z are found which s a t i s f y I zM-xK I ?C 6 and until zM-yL . The machine then outputs (zM-xK), ^zM-yL), (xK-yL), x, y , and z and then returns to the beginning of the programme where i t s t a r t s to determine new values of x and z, t h i s time with the i n i t i a l value of Z q = z+,1. The programme continues u n t i l the operator stops the machine. A copy of the programme tape is given below. aOOh 83al5717 28ooflc6 ^82^1107 tofe723 eb21c528 ^9256720 M157925 6721&925 2c006720 ld967926 112c0000 Id8d791f 6722^926 793^f78l 6llb^91f 2c006720 170lO10c 110c7928 Idl21l85 00000000 6m>k9ZQ 79296U-b 00000000 lnife723 ^929791f 000*40000 eb22c529 !+92a7926 00000001 ^9266720 6725U927 00000001 elOh 00000000 00000000 00000000 00000000 00000000 00000000 793^f78l 173^1185 573287if 00000000 5b36782b 00000000 1160792e 00000000 f78lll2d 00000000 2822 aO 6l750ec2 a l a8b993^5 A-2. 00000000 00000000 00000000 00000000 00000000 00000000 58OOOOOO 00000000 00060000 00000000 050000b8 00000000 00000000 00000000 00000000 00000000 Computer Results A p a r t i a l l i s t of the results for the case 6= 600, K = 14850, L = 7280, and M = 29760 i s presented below. (zM-xK) (zM-yL) -570 -510 150 210 -360 360 -210 -150 510 -060 000 -570 150 (xK~yL) -560 080 -160 480 -080 320 -240 400 160 -400 240 -320 080 010 590 -310 270 280 -040 -030 550 -350 -340 240 250 -070 y z 477 479 501 503 980 1004 1481 1483 1505 1982 1984 2461 2485 973 977 1022 1026 1999 2048 3021 3025 3070 4043 4047 5020 5069 238 239 250 251 489 501 739 740 751 989 990 1228 1240 used i n t h i s case corresponds to, a time of The value of 6 §T a 6.0 msecs. X The time between pulses into the chains i s approxi- mately the same for a l l the chains and i s about 10 msecs. Thus, the value of 6 T corresponds to about h a l f the time between pulses into the chains. A l l the values of the differences to cause s u f f i c i e n t x = 2485. were f e l t to be great enough change i n the composite waveform up to the value of At t h i s point, the maximum s h i f t of any one of the member functions with respect to the others is only 1.5 msecs which was considered too small to cause s u f f i c i e n t change i n the composite waveform. Thus, t h i s value of x determines the near-period of the waveform. . J • m near-period i s T n (14850)2485 *» \„:n n An 100,000 approximately s i x minutes. „„„ sees » 369.0225 sees . . . . which i s The Appendix B B-l. Programme for Determining the Theoretical P r o b a b i l i t y D i s t r i b u t i o n of Amplitudes. To determine the t h e o r e t i c a l p r o b a b i l i t y d i s t r i b u t i o n of amplitudes, i t was necessary to find the product of the generating functions of the i n d i v i d u a l waveforms. Each generating function i s a polynomial i n s and i s of the general form G(s) = ^,p(kV)s^ , where p(wV) i s the p r o b a b i l i t y of the amplitude kV occurring. example, the generating, function Gj(s) for waveform 1 iss i \ i 560 ^ 1 880 2 0 1 -280 ^ 1 GjVsj = j , s -+Y § +j s +|S Y V r + consists of 6 terms. S , \ G^sJ 1 / 560 , = y (s +s -560 ^ 1 +| 280 , 0 , 0 , + S + S + S -280 -840 . . . > which 8 This may be written as 7 terms, J coefficient y ' , which may be factored out. r For each having the i This gives, + s -56© ^ -840x +s ) . S i m i l a r l y , G«,(s) and Gg(s) may be w r i t t e n . Gg(s) a s I f now the product of and Gg(s) = g-|"~( ..o31 terms i n s Y5 ( • • • 1 5 terms i n these polynomials i s formed, the result would be of the form G(s) • <y(15)31 ( , . 7 ( l 5 ) 3 1 terms i n s ^ . . . ) . 0 The values of the exponents i n t h i s ex- pression are obtained by taking the sum of a l l possible combinations of the 7 exponents from G^(s) plus the 15 exponents from Gg(s) P ^ u a * 31 exponents from Gg(s), a t o t a l of 7(15)31 = 3255 sums. n e The generating function so formed is of the same form as the modified version of G^(s) (above). M u l t i p l y i n g each term by the common factor gggg a n d grouping together the terms having the same value of the exponent restores the generating function to the o r i g i n a l form. (i.e. G(s) » p(kV)s^ ). k 37 The coefficient of s k (some multiple of a b i l i t y of the amplitude kV occurring. 3255 ) i s then p(kV), the prob- Thus, when the number of times the exponent k occurs i s known, the p r o b a b i l i t y p(kV) for the amplitude kV occurring i s simply t h i s number m u l t i p l i e d by 3255 p r o b a b i l i t y d i s t r i b u t i o n was desired, " Because a quantized i t was s u f f i c i e n t to determine the number of times (say n^) the exponent k was i n the range k^ to to get the p r o b a b i l i t y that the amplitude was i n the range k^V to k ^ + j V » A programme was written which calculated the sums of a l l possible combinations of 7 plus 15 plus 31 numbers and counted the t o t a l number of times the sums lay i n each of 31 equal ranges. 7, 15, and 31 numbers to be summed were a l l p o s i t i v e , The (This merely —840 means removing the factor s from the generating functions and hence increasing the exponents i n the brackets by 840.) The range of kV from —100 to +100 v o l t s i n the physical system corresponds to the range of k from 0 to 5040 i n the computer. This range was divided into 31 equal increments of 168 units each, (which corresponds to d i v i d i n g the range -100 to +100 v into 6.45 v o l t increments). The count for the range (n(l68)) to (n(l68) + 167 ) n= 0, 1, . . . , 30 was stored i n word location n i n the computer. Thus, i t was only necessary to divide the sum for a p a r t i c u l a r combination by 168 to determine the location of the count which should be increased by 1. A copy of the programme tape i s given on the following page. &OQk 57if2800 a510J+927 ^ 6 7 9 0 0 78lfll6o flckk&kl 793^a510 6 l 2 f J + 9 0 0 0 0 0 0 0 0 0 0 k 17a0573'b i*91f79 7 17ML791f 0 U 0 0 0 0 5 9 2800flc% 6lUoU97f 5737797? 672fU91f 19587927 00000001 0000Q0a8 5 7 3 7 2 8 0 0 6 0 7 f 3 0 0 0 672^927 2800elb33 I9565737 1 7 b ^ 7 9 3 f 3 0 0 0 l k 3 & 2 871f5b2b OOlfOOOO OOOfOOOO 1+856172C fleW^f 00070000 alO>4 00000000 00000000 00000000 00000000 00000000 792fa710 6l2dlf92d >492921a9 00000000 00000000 00000000 00000000 00000000 00000000 795f±78l 17bell20 a20*+ 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 8la283a0 11200000 00000000 00000000 00000000 00000000 792f6l29 2822 aO d5l65'8d9 a l d3e3Ucf a2 00000000 00000000 00000000 00000000 00000000 00000000 Uda9H25 00000000 6daUlL57 58OOOOOO 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 B-2. Computer Results. Two determinations of p r o b a b i l i t y d i s t r i b u t i o n s were made, one corresponding to a different corresponds to a different set of member functions. i n i t i a l configurations i n the multivibrator chains.) each (Which i n turn of the multivibrators The waveform voltages for the two cases are tabulated for each member function set, where V ( n X i ) i s the voltage "til step from the i Case 1. chain after n pulses. ( A l l multivibrators output voltage i s i n i t i a l l y 0 o ) i n i t i a l l y i n the state where the This was the case used i n the generator. Waveform 1. V(nT )= 1 -840V 000V +280V -560V +560V 1 2 3 4 5 » n -280V 000V 6 7 Waveform 2. V(nx2)= +168V -504V +336V -336V -168V -840V +168V +336V 1 2 3 4 5 6 7 8 -336V +504V -168V 000V -672V +672V 9 10 11 12 +180V -420V +285V 1 2 +315V -285V 9 10 = n V(n T ) « 2 n - 000V 13 14 15 -315V -135V -735V +735V +135V 3 4 5 6 7 8 +420V -180V 000V -600V +495V -105V Waveform 3. V (n T ) = 3 n = V(n T;3)= n 11 12 13 14 15 16 Waveform 3. V(n Tg) m (Continued) +075V -525V 17 V(n x ) 3 18 -180V +525V 25 26 Case 2. multivibrators +180V -420V 26 19 -075V +105V 27 28 -240V 21 -495V 29 -840V +240V +420V 22 23 24 +600V 000V 30 31 (Multivibrators gating p o s i t i v e voltages i n 0 state, gating negative voltages i n 1 state. This r e s u l t s i n a symmetric waveform from each chain.) Waveform 1, V (n Tj) = -840V +560V -280V +280V -560V +840V 000V 7 n Waveform 2„ V(n Xg) +672V -504V +168V -336V +336V -840V -168V +168V 7 8 n V(n Tg) » +840V n -336V +336V 10 11 -420V +180V -168V +504V -672V 13 14 -315V +285V -735V -135V +240V 3 4 5 6 7 8 +420V -075V +525V -495V +105V -105V 11 12 13 14 12 000V 15 Waveform 3. V(n T 3 ) n V(n tg) = +600V 1 +840V 2 -180V 10 15 16 41 Waveform 3. (Continued) V(n T ) = +495V o 0 n = -525V 17 V(n T g ) = +735V 25 n +075V 18 19 -285V 26 -420V +180V 20 +315VJ 27 -180V 28 -840V 21 22 +420V 29 -600V 30 -240V +135V 23 24 000V 31 A plot of the p r o b a b i l i t y d i s t r i b u t i o n s obtained for each case i s shown i n Figure B - l . The s h i f t i n the mean values r e s u l t s because the member functions for the f i r s t case do not have equal numbers of positive and negative voltage steps whereas i n the second case they do. This i s also the reason why the curve for the second case i s symmetric and that for the f i r s t case is not. The r e l a t i o n s h i p between the normalized p r o b a b i l i t y d i s t r i b u t i o n and the number of counts per range is given by the formula p(kV) = —TT=T where p(kV) i s the p r o b a b i l i t y that the amplitude l i e s i n the range k / V to (k+l)Av, C k i s the number of counts for the range (k(l68) to ( k + l ) l 6 8 - l ) , and Cy i s the t o t a l number of counts ( i . e . C^ = ^fc)' The counts for each range are tabulated below for each case. Case 1. c - 2 5 9 21 32 53 79 107 149 176 220 n = 0 1 2 3 4 5 6 7 8 9 10 C = n 244 275 286 282 278 242 220 178 142 101 69 n = 11 12 13 14 15 16 17 18 19 20 21 C = n 42 25 12 5 1 0 0 0 0 n = 22 23 24 25 26 27 28 29 30 n ! P(v) A Amplitude P r o b a b i l i t y --0o014 -80 Figure B - l s Theoretical Normalized Amplitude P r o b a b i l i t y Distributions for two Member Function Sets, Case 2. 2 5 21 0 1 3 C = n 200 220 235 n = 11 12 96 22 n n = n n = 32 48 71 4 5 242 241 235 221 200 177 150 119 13 14 15 16 17 18 19 20 21 71 49 33 21 10 5 2 1 23 24 25 26 27 28 29 30 6 95 7 119 8 149 9 176 10 Appendix C C-l. Programme for Determining the Cardinal Points of the Theoretical Autocorrelation Functions. Because of the simple s t r a i g h t - l i n e r e l a t i o n s h i p between 0(nx. ) and 0((n+ functions, l ) x.) of the member function autocorrelation i t was s u f f i c i e n t to determine the values of the c a r d i - nal points 0(nx^) , n = 0, 1, . . . , j | , where the period of the function i s equal to NX ^ . The value of fj(n % /) i s given by the formula: J*(n .) - | ( V ( O x . ) V (n x . ) + t ( l x . ) V ((n+l)x T + V ( ( N - l ) T . ) v ( ( N - l + n ) x .) The equivalent computer problem i s : compute the values of: a la2 + a a la3 + a V M 2 + (aj)2 2a4 . + V K 2+ + Given N integers, a Na2 " X ' " + V N " 2 a^, ag, . . . , a^ = ^(0), J V = . * ) . + (a,,) 2 + . . . + (a^) 2 2 a 3 +••••+ V l )+ . . . . *f T i ) where a^ corresponds to V(0 x ) a^ corresponds to V ( l x^) etc. A programme was written which did exactly t h i s . the programme tape is given on the following page. A copy of aOO>4 871e5bl+3 1X6OJ+800 6751^ddl OOlOOOcltU6oHdi+7 17dc57'+7 79^f701 OOOOOOOO 6l5fUdU9 7800U820 871r7953 OOOOOOcl ^d51sllO 17+957+7 5b5ell6o 00010000 Udcli^ddl lK)00e600 2800»+953 OOOOOOOO laU^d^f bd53c553 175albUo 00000002 ef57c55b 1751575b c35bllcd OOOOOOOO 57k75blrt) 79l+fa510 05000052 00200000 2822 , I aO 2e57fOdb C-2. Computer Results. Two determinations of the autocorrelation function were made. corresponding to the two sets of member functions mentioned i n Appendix B, Plots of the individual autocorrelation functions for each member function are shown i n Figures C - l . (a), (b), and ( c ) . The results of both determinations for the same member function are shown on the same graph. The computer r e s u l t s for the two determinations are below. In a l l cases, the waveform voltage values fed into the computer were made as low as possible by removing common factors. Case 1., tabulated Waveform 1. would be represented V(nx ) = -3 0 +1 -2 where the factor 280V has been removed. For instance, as: +2 -1 0 S i m i l a r l y the common factors 168V and 15V were removed from waveforms 2 and 3 respectively. Waveform 1 n = 0 1 2 3 4 -1 1 2 Case 1. 0(n Tj) « 19 -8 Case 2. 0(n -tj) - 28 -17 -l.OJ. a) Normalized Theoretical Autocorrelation Functions for Waveform One, Case 1 and Case 2. Figure C - l . Theoretical Normalized Autocorrelation Functions. Waveform 2, 0 1 2 3 4 5 6 7 Case 1. = 95 -•38 19 -34 38 -31 00 11 Case 2. 120 --53 14 -31 43 -34 -25 26 Waveform 3. n = n s 0 1 2 3 4 5 21904 -7813 6246 -9967 10972 -10497 25040 -9157 6246 -9967 11756 -10833 6 7 8 9 10 11 Case 1.)*(n Tg)= 2762 -5203 10799 -6374 -571 -4240 Case 2,^(n T g)= 410 -4195 12143 -9510 -571 -4240 12 13 14 15 Case 1. 0(n Tg)= 4173 -4752 251 4830 Case 2. 0(n T ) = 4509 -5536 -757 7182 Case 1. Case 2. 0(n T ) « 3 n «=> n = 3 46 Appendix D 1. B e l l , H. J r . and Rideout, V . C . , "A High-Speed C o r r e l a t o r , " Transactions of the I . R . E . Professional Group on E l e c t r o n i c Computers, v o l . EC-3. no. 2. (June 1954) pp. 30-36. 2. Bennett, R.R. and Fulton, A ; : ^ , "The Generation and Measurement of Low Frequency Random*Noise," Journal of Applied Physic's^' v o l . 22, no. 2. (September 195l) pp. 1187-1191. 3. Bennett, R.W., "Methods of Solving Noise Problems," Proceedings I . R . E . . v o l . 44, no. 5. (May 1956) pp. 609-638. 4. Biernson, G . A . , "Fundamental Equations for the A p p l i c a t i o n of S t a t i s t i c a l Techniques to Feedback-Control Systems." Transactions of the I . R . E . Professional Group on Automatic Control, v o l . AC-2. (February 1957) pp. 56-78. 5. Booton, R.C. J r . , "The Analysis of Nonlinear Control Systems with Random Inputs, "Proceedings of Symposium on Nonlinear C i r c u i t Analysis, Ann Arbor, Michigan, Edwards Brothers, I n c . , 1953 pp. 369-391. 6. Chance, B . , Hughes, V . , MacNichol, E . F . , Sayre, D. and Williams, F . C . , Waveforms. New York, McGraw-Hill, 1949. (Ridenour, L . N . , e d . , M . I . T . Radiation Laboratory Series, v o l . 19.) 7. Diamantides, N . D . , "Analogue Computer Generation of P r o b a b i l i t y Distributions for Operations Research," Transactions of the A . I . E . E . . v o l . 75,' (1956) part I, pp. 86-91. 8. F e l l e r , W., ;'l'An Introduction to P r o b a b i l i t y Theory and Its cations. New York, John Wiley and Sons, I n c . , 1950. 9S Fink, H . J . , " P r e c i s i o n Frequency Control of a Magnetic Storage Drum," M.A.Sc. Thesis. U n i v e r s i t y of B r i t i s h Columbia, 1956. 10. Forsythe, G . A . , "Generation and Testing of Random D i g i t s at the National Bureau of Standards, Los Angeles," Monte Carlo Method, Washington, U.S. Gavgritment P r i n t i n g Office, 1951. (National Bureau of Standards Applied Mathematics Series}.' v o l . 1 2 , ) pp. 34-35. 11. Hammer, P. C , "The Mid-.square Method of Generating D i g i t s , " Monte Carlo Method. Washington, U . S . Government P r i n t i n g O f f i c e , 1951. (National Bureau of Standards Applied Mathematics SeriesyilVol. 12,) p. 33. 12. Heaslet, M.A. and Uspensky, J . V . , Elementary Number Theory. New York, McGraw-Hill, 1939. of Appli- 47 13. James, H . M . , Nichols, N . B . , and P h i l l i p s , R . S . . T h e o r y of ServoMechanisms, New York, McGraw-Hill, 1947. (Ridenour, L . N . , ed., M . l . T . Radiation Laboratory Series, v o l . 25.) 14. Johnson, E . C , "Sinusoidal Techniques Applied to Nonlinear Feedback Systems," Proceedings of Symposium on Nonlinear C i r c u i t Analysis, Ann Arbor, Michigan, Edwards Brothers I n c . , 1953. 15. Lee Y. W., "Communication Applications of Correlation A n a l y s i s , " Symposium on Applications of Autocorrelation Analysis to Physical Problems. Washington, Publication of ONR, Department of Navy, 1949. pp. 4-23. 16. Marconi Instruction Folder #131-673, I n s t a l l i n g and Operating Instructions, Pulse Amplitude Analyser (Kicksorter) Marconi type 115-935. 1 17. Millman, J . and Puckett, T . H . , "Accurate Linear B i d i r e c t i o n a l Gates," Proceedings of I . R . E . , v o l . 43, no. 1. (January 1955) pp. 29-37. 18. Millman, J . and Taub, H . , Pulse and D i g i t a l C i r c u i t s , New York, McGraw-Hill, 1956. 19. Paynter, H . M . , e d . , A Palimpsest on the Electronic Analog A r t . Boston, Massachusetts, Geo. A . P h i l b r i c k Researches, I n c . , 1955. p. 197. 20. P h i l b r i c k , G . A , , Catalog and Manual, Boston, Massachusetts, Geo. A. P h i l b r i c k Researches, I n c . , 1951. 21. Reintjes, J . F . , "An Analogue Electronic C o r r e l a t o r , " Proceedings of National Electronics Conference, v o l . 7, 1951. pp. 390-400. 22. Truxal, J . G . , Control System Synthesis, New York, McGraw-Hill, 1955. 23. 24. van der Z i e l , A . , Noise, New York, P r e n t i c e - H a l l , 1954. von Neumann, J . , "Various Techniques Used i n Connection with Random D i g i t s , " Monte Carlo Method, Washington, U.S. Government P r i n t i n g Office, 1951. (National Bureau of Standards Applied Mathematics Series v o l . 12.) pp. 36-38.
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A reproduceable noise generator Watts, Donald George 1958
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Title | A reproduceable noise generator |
Creator |
Watts, Donald George |
Publisher | University of British Columbia |
Date Issued | 1958 |
Description | This thesis describes the design of a device for generating a reproduceable noise signals. The noise signal is generated by adding three periodic waveforms having non-multiple periods. Pulse techniques are used in the generation of the member functions so that the output may be reproduced exactly. Theoretical and experimental determinations of the amplitude probability distribution and of the autocorrelation function of the signal were made. On the basis of tests and observations made, it is concluded that the signal generated may be considered a noise signal having a near-Gaussian amplitude probability distribution, very little correlation for time-shifts greater than 30 seconds, and a bandwidth of about 60 cps. |
Subject |
Signals and signaling |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0105067 |
URI | http://hdl.handle.net/2429/40120 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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