Extracting analogue signals from noise using a digital computer.

N PS ARCHIVE1966BARRETT, N.

SIGNALS FROM NOISEISING A DIGITAL COMPUTER

NEIL A. BARRETT

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOL

LIBRARY MONTEREY, CA 93943-6101

NAVAL POSTGRADUATE SCHOOL

MONTEREY, CALIF. 93940

This document has bee wed for publicrelease and sale; its < bution is unlim:

< '3 2~

EXTRACTING ANALOGUE SIGNALS FROM NOISE

USING A DIGITAL, COMPUTER

by

Neil A. J3arrett

Lieutenant, Royal Canadian NavyB. S.E.E. , Nova Scotia Technical College, 1958

Submitted in partial fulfillment

for the degree of

MASTER OF SCIENCE IN ELECTRONICS ENGINEERING

from the

UNITED STATES NAVAL POSTGRADUATE SCHOOLMay 1966

Abstract

It is frequently convenient in data processing to convert analogue

to digital data for computer assimilation. A convenient method of such

conversion has been developed and used in the study of correlation detec-

tion of audio signals corrupted by noise.

A method to use apriori knowledge of the corrupting noise to increase

processing gain has been studied. In the case of detection of a sinusoid in

noise, an additional gain over conventional auto-correlation of up to 14. 5

db has been achieved.

Finally, a signal source located in an unknown random noise field

was detected, classified and located in relative bearing by the cross-cor-

relation of the signals received from two spatially separated sensors.

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5101

TABLE OF CONTENTS

Section Page

1. Introduction 13

2. Statement of the Problem 14

3. Conversion ofAnalogue Signals to Digital Form 17

for Computer Analysis

4. Noise Removal in Auto-Correlation 20

5. Location of a Source by Cross-Correlation 27

6. Conclusion 29

7. Bibliography 31

APPENDIX

A. Analogue to Digital Data Conversion 32

B. Noise Removal in Auto-Correlation Using a Priori 77

Knowledge of Noise

C. Detection and Location of Source in Noise. 130

LIST OF FIGURES

Figure Text Page

1. Flow Diagram of Digitizing Process 19

2. Geometry of Lunch-room Problem 28

Appendix A

1. Block Diagram of Digitizing Process 36

2. Clock. Pulse Development 36

3. The Digitizing Process 37

4. Continuous Sampling Percent vs. Frequency 38

5. 163 Magnetic Tape Capacity vs. Frequency 38

6. Timer Control Word vs. Intersample Delay 39

7. 253 cps. Square Wave Digitally Described 65

8. Digital Auto-Correlation of 253 cps Square Wave 66

9. 253 cps. Triangular Wave Digitally Described 67

10. Digital Auto-Correlation of Triangular Wave 68

11. 253 cps. Sinusoidal Wave Digitally Described 69

12. Digital Auto-Correlation of Sine Wave 70

13. Fluctuations About the Mean of a D.C. Voltage 71

14. Digital Auto-Correlation of D.C. Fluctuation 72

15. Band-limited Noise Digitally Described 73

16. Digital Auto-Correlation of Band-limited Noise 74

17. Noise Generator Output Digitally Described 7 5

18. Digital Auto-Correlation of Noise Generator Output 76

Figure Appendix B, Page

1. F?(oO vs. <* 79

2. Processing Gain vs. (SNR) in 86

3. Processing Advantage of Cross Correlation over 86Auto-correlation vs. (SNR) in

4. Pure Signal Auto-Correlation 112

5. Pure Signal Power Spectral Density, 113

6. Power Spectral Density of Noise, 3750 Samples 114

7. Power Spectral Density of Noise, 18, 750 Samples 115

8. Power Spectral Density of Noise, 37, 500ft Samples H6

9. Auto-Correlation of O DB SNR, 3750 Samples 117

10. Power Spectral Density of ODB SNR After Noise Removal H8

11. ODB Signal After Noise Removal 119

12. Power Spectral Density of -10 DB SNR Before Noise 120Removal. 37 50 Samples

13o Power Spectral Density of -10 DB SNR After Noise Removal 12

1

in Correlation Domain. 3750 Samples

14. Power Spectral Density of -10 DB SNR After Noise Removal 122

in Correlation Domain. 11, 250 Samples

15. Power Spectral Density of -10 DB SNR after Noise Removal 123

in Correlation Domain. 18, 750 Samples

16. -10 DB SNR Singal After Noise Removal in Correlation 124

Domain. 18, 750 Samples

17. -10 DB SNR Auto-Correlation Before Noise Removal. 125

22, 500 Samples

18. Auto-Correlation of Noise. 37, 500 Samples 126

19. Power Spectral Density of -10 DB SNR Before Noise 127

Removal. 22, 500 Samples

Figure Page

20. -10 DB SNR Signal After Noise Removal in Frequency 128

Domain. 22, 500 Samples

21. Power Spectral Density of -10 DB SNR After Noise 129

Removal in Frequency Domain. 22, 500 Samples

Appendix C

1. (a) The Basic D. F. Problem 131

(b) Cross -Correlation of M 1# M, 131

2o Geometry of the Lunch-room Problem 139

3» Cross -Correlation of "O" DB Signal. 3, 950 Samples 140

4. Cross -Correlation of "O" DB Signal. 11, 850 Samples 141

5. Cross -Correlation of "O" DB Signal . 19, 750 Samples 142

6. Auto-Correlation of "O" DB Signah 11, 850 Samples 143

7. Auto-Correlation of Noise. 11, 850 Samples 144

8. Relative Power Spectral Densities of Noise Alone, 145

and Signal Plus Noise

LIST OF TABLES

Table Appendix A Page

I. 160 Program Digitize 49

II Modification to Program Digitize 52

III Program Test 160 53

IV Summary of Test Results 44

Appendix C

I Details of Data Digitized 136

II Frequency Reduction Process 136

LIST OF FORTRAN PROGRAMS

Appendix A Page

1. Subroutine DATA 54

2. Subroutine FINDIT 55

3. Subroutine UNPACK 56

4. Program TEST 58

5. Program SHUFFLE 62

Appendix B

1. Program SIMSIG 90

2. Subroutine PSD 97

3. Program PROCES 100Appendix C

1. Program LOCATE 137

11

1. Introduction .

During the summer of 1965, the author visited the U.S. Naval Elecr,

tronics Laboratory, San Diego, for six weeks. The. experience gained there

suggested several problem areas which would profit from further study.

It appeared that there was a need to analyze large quantities of ana-

logue data collected during various experiments. The volume of the data

almost precluded "hand" analysis and yet no facilities existed to enable a

computer to assimilate the data. This facility was also lacking at the U. S. N.

Postgraduate School Monterey.

The data collected during these experiments at San Diego was often

corrupted by unavoidable locally generated noise. It was felt that if more

could be learned about this self-noise some better method of data extraction

might be possible.

One of the most powerful methods of data analysis is the correlation

technique which may reveal information masked by corrupting noise. These

techniques have been thoroughly investigated over the past 25 years, and a

wealth of literature exists on this subject.

Although continuous time integrations and correlations are possible

mathematically on known explicit functions, physical data often does not

fit a known model. In this case approximations to infinite integrals etc.

,

must be made using either analogue or digital equipment. Analogue equip-

ment tends to be limited in versatility and is expensive, while if a general

purpose digital computer is available, it can be made to simulate the physical

13

situation. Therefore attention was directed toward describing the analogue

data by a digital computer. The use of a digital computer to compute various

correlation functions is the subject of this report.

2. Statement of the Problem.

The correlation function of two continuous time variables is given by

T/2Rs + n<t ) = lim i f sW N (t + T) dt (1)

T-x*>l

-T/2

where s(t), N(t) are defined for -e©<t< oo

When the functional form of s(t) and N(t) are known it may be possible

to evaluate Eq. (l)mathematically. When the form of s(t) and N(t) are known

only as analogue signals Eq.(l) may be instrumented by a multiplier and an

integrator.

The same effect may be achieved by digital means, subject to errors to

be discussed.

The sampling theorem states that a continuous function can be repre-

sented in a finite interval by a finite number of samples of the function. The

rate of sampling must exceed twice the highest frequency component of the

Fourier transform (spectrum) of the function.

By suitably band-limiting a function, samples may be taken at a prac-

tical rate„ The samples if digitally described can then be processed. Equa-

tion (l) implies a memory extending into the infinite past which cannot be

achieved physically. As an approximation the "short-time" correlation may

be formed with some resulting error.

14

If the "short-time" correlation is formed digitally, errors arise from

both the limited number of samples and by the finite accuracy of the indivi-

dual samples. If "short-time" is defined to be some period much longer than

the period of the lowest frequency in the signal, errors of the first kind above

tend to zero as the length of record grows. Measurement error is negligible

if the samples are described to four place accuracy.

The "short-time" correlation is formed as

N-K

Vft (k*T) = TAr^ si

N *+K (2)

i = 1

where

AT - the sampling interval

st , Nl* = sample values of the continuous function iaken at

intervals of T

k = o, 1, 2 K

N = the number of samples available.

In the limit as N goes to»o , andA T ->0

V+ N* (kAT) = RS+N (T) (3)

For finite N

Rs*+N *(k*T) = R&+n(?) (4)

Equation (2) holds only if the data has zero mean. If in Eq. (l) we

assume that s(t) is the desired signal, of zero mean and that N(t) is random

Gaussian noise and that the observed V(t) = s(t) + N(t), then we may form

Ryy(T). This function may in turn be broken down into components:

15

RVvm = RSS^) + %N^ + Wl") (5)

Since the noise and signal are assumed independent the third number

vanishes in the limit T -^°o,

We may approximate Eq. (5) as

Rs *+N *(k*T) = Rg*s*(kAT) + RN *N*(kAT) (6)

in which we wish to determine Rg*g*(k^T).

For very large k = K the second member of Eq. (6) will tend toward

zero because of independence leaving

RS *+N *(K + |at, = RS *S *(Ko +lAT W

J = o, 1, 2, J

This restriction is not desirables since if Rj,j*N*(k4t) were known for

all k, then in Eq. (6)

RS *S *(kAT)= Rs*+N*(kAT) " %*N*(kAT) (8 )

However R^*j^*(k^T) is never known exactly. It can only be known,

since it is a random variable, in an average sense.

If we assume that the normalized correlation function Rj^fcA.T) is

known a priori

RNN(k*T) = ± Rrfrf(kdT) (9)

where R^]^(kAT) is formed in the absence of signal

°^ is to be determined where £_ °< 3r 1

then in Eq. (8) we may write

Rs *s

*(kAT) = Rs *+N *{kAT) - C* RNN (kAT)

Methods of determining <* have been studied for the very limited case

16

of a single sinusoid signal masked by noise.

To gain some experience with the technique of cross -correlation, a

low level signal was placed in a noise field. The output of two spatially

separated sensors was processed to detect and locate the signal. The re-

sults are presented in section 5.

3. Conversion of Analogue Signals to Digital Form for Computer Analysis .

Many analyses of real data may be performed on a digital computer by

first manually sampling the data a sufficient number of times, and entering

the sampled data into the computer.

When the number of data samples desired exceeds a few thousand, some

automatic means of data conversion must be used.

The method should preferably be "off-line, " that is, use auxiliary

equipment, for efficient employment of the main high speed computer. The

Computer Laboratory of the Electrical Engineering Department was equipped

with the necessary items of hardware and these facilities were used to sample

the analog data and form the digital record.

The three main items used were a 12 bit analogue to digital convertor

a Control Data Corporation (C. D. C.) Model 160 Computer having a capacity

of 4096, 12 bit words, and a C. D. C. Model 163 magnetic tape unit.

As now constituted the analogue data is consecutively sampled by the

analogue to digital converter working under control of the CDC 160 computer.

When 4000 samples have been taken, the computer writes the samples

on magnetic tape in a format that is recognized by the main CDC 1604 comf

17

puter. After a delay of 0. 4 second another 4000 samples may be collected,

and so on. Each group of 4000 samples when written on tape is accompanied

by an identifying word so that any block of 4000 samples may later be located.

The maximum sampling rate is 5 Kcs. implying that all signals should be

band-limited to less than 2500 cps. The sampling rate may be varied at will

by manual program entries, as can other convenient parameters.

The system is capable of digitizing only one source at a time. To allow

cross -correlation of two or more sources, a clock pulse input is used to in-

itiate a round of sampling.

At present a simple AND gate is used to sense the presence of a clock

pulse. A more permanent arrangement should make provision for the detection

of a simple clock code, to eliminate the occasional false starts due to noise

pulses, which have occured under present arrangements.

Once the data has been recorded digitally on tape, the data may be

read into the main computer using standard programming techniques. The

subroutines to accomplish this are available as Subroutine Data from the

Computer Facility.

The data when read into the computer is delivered in integer format,

i.e., without a decimal. To convert it to units of volts each data point is

then divided by 409. 6, a factor inherent in the particular A/D converter used.

The diagram below shows the essential steps.

18

SOURCE *\ A/D CONVERTER

CLOCK BIAS

*iCDC 1601* CDC 163

TAPE REELOUTPUT

COMPUTER LABORATORY

TAPE REEL INPUT

CDC 163

EQUIVALENTS/R DATA *—FIND BLOCKUNPACK DATARETURN

MAIN PROGRAM

- CALL DATA-J-REMOVE MEANCONVERT TO VOLTSPROCESS DATA

CDC 1604 COMPUTER

COMPUTER FACILITY

Figure 1. Flow Diagram of Digitizing Process.

The system is presently adequate to describe many physical processes.

Certain signals, such as speech, are beyond the capabilities of the system.

However modifications to the Computer Laboratory hardware to improve per-

formance were not attempted because of pending contractual arrangements to

completely replace the hardware with more sophisticated equipment. The new

equipment will enable two records to be digitized simultaneously. The memory

capacity will be increased by nearly a factor of four, and the sampling rate

will be approximately 20 Kcs. This equipment will.be available for use in

mid 1967.

19

The author has recorded the mixed output of a signal and a noise gen-

erator to provide a vehicle for the study of correlation. Several other students

Q 6j, L 7J have used the system for their own particular needs with no difficulty

experienced to date. Additional details on the digiting process are presented

in Appendix A.

4. Noise Removal in Auto-Correlation .

Correlation techniques have gained considerable favour in recent years

as a method of signal enhancement. Theoretical derivations of the processing

gain to be expected have been stated by Lee£ I J, for both auto and cross-

correlation.

Auto and cross -correlation are defined as follows:

Mr) = lim 4 s T/2

v(t). v(t + r)dt (i)

T^-o -T/2

where V(t) = s(t) + N(t)

s(t) = the signal to be recovered

N(t) = noise,, here assumed Gaussian

i r V1RcCT) = lim -i- J V(t). sMt + rJdt

T->-° -T/2

where

(2)

s*(t) is chosen apriori to have the form of s(t) and is locally

generated

.

Approximations to Eq. (l) and (2) may be generated using sampled

data in which case.

Ra (T)

N /S A±£ ViU). vx a + T) = R

a (r)i = 1

(3)

20

Rc(T") = =£• § Vi (i) - V2 U + T)

.£ ^(T) (4)

i = 1

where Vj = V(t) for t = i^t

It may be shown that Eq. (3) and (4) are of the form

R(£) = Rss(r) + RNdr) + RsN(t) + RNs(r) (5)

Of the four terms in Eq. (5) it is desired to determine only the first,

ARSS (T) which represents the signal contribution. The remaining terms are

undersirabie and success is measured by the extent to which they can be

eliminated.

A AThe cross-terms R

gjyR^ will vanish as N—>«*>, because of the as-

sumed independence of the signal and the noise.

Since the correlation is to be done using a finite number N of samples,

these terms will become small, but may be significant.

Let us assume that a signal is of the form

s(t) = TTE cos wt

N(t) = Gaussian noise of zero mean and variance 6'r,

The variance of the output of Eq. (3) and (4) has been evaluated by

Lee LlL who clearly shows the superiority of cross -correlation over auto

correlation.

In the case of auto-correlation

«l = -fV^'a + <r« (6)

while for cross -correlation

21

This improvement is achieved only when the form of the desired signal

is known apriori. In many cases of interest particularly in the passive sonar

situation, the desired signal form is not known apriori, and consequently

auto-correlation must be used.

In addition to the noise input supplied by the environment, local dis-

turbances may cause noise effects to appear as part of the correlator output.

If these two sources of noise, taken together, are known apriori, and can be

considered to remain stationary over the interval under consideration, then

perhaps some means can be found to eliminate their corrupting effect on the

output of the auto-correlator.

This problem has been studied for many years. In an early work

Lee [_A] states

". . .the correlation of the noise, once measured

can be compensated, (for)"

The details of the implementation of this were not specified.

In a much more recent work Van Trees \_ZJ considers an optimum

receiver in which he considers that the noise is known apriori as a norm-

alized correlation / (T). After correlation of the received data some

constant K times / ( T) is substracted form the correlator output. Deter-

mination of K rests upon knowledge of signal strength and noise strength.

22

Again it is not clear how to instrument the calculation of K.

We will assume that the sample auto-correlation of the signal,

Rs+nC^*) ^as been formed in the presence of corrupting noise. We shall also

assume that the statistics of the corrupting noise are known apriori and are

represented by the normalized correlation function R^N^"^ formed in the

absence of signal.

The object of the processing is to determine a linear combination of

RS+n(T) and R^nM which will enhance the signal and tend to remove the

noise. No claims are made for the optimality of the method presented here,

other than it does improve the signal to noise ratio of the output.

In the following we will be dealing with sampled data functions, not

continuous functions of time. To simplify notation however the symbol T

will be used to denote a sample data function.

The power spectral density P(w), and the auto-correlation functions

are transform pairs, that is

r(7) =J p(w ) e gwT dw (8)

P(w) = \— f R(r) e"Jwr dT (9)

Given one member of the pair, the other may be found. Therefore the

processor could attempt to operate on either the power spectral density or

the correlation function.

From Eq. (5) we get (neglecting cross terms),

23

RSs< T ^ - Ks+nCH-W^) do)

or equivalently by Eq. (9)

Pss (w) = Ps+N (w) - PNN (w) (11)

We have assumed that as apriori information estimates of the norm-

*alized power spectral density function P^^wO and normalized auto-correla-

tion function R^n^ formec* in the absence of signal, are available. At

any given instant there is no assurance that the apriori known noise estimate

represents the true noise in other than an expected value sense.

Then in Eq. (10) and (11) we may write

Rss(t) = Rs+n(t) - <* RnN (T) (12)

Pss (w) = PS+N (w) - °^ PNN(W) (I 3 )

where RNn(^ and PNn(w^ are known functions

O $ ex, ^ I / o< a constant to be determined.

Determination of the <*». in Eq. (12) and (13) becomes the central

problem to be considered.

The following methods of calculating <=K hold only for the case of sin-

usoidal signals in noise. Further study is necessary before any general

method can be stated.

Since the correlation function was produced from the data directly and

the power spectral density indirectly from the data via the correlation, it was

thought advisable to operate first on Eq. (12) rather than Eq. (13).

In the case of a sinusoidal signal, it was thought that finding an

24

^ = °<C' such that RSS(°) > Max RSS('f

)

(14)

would be advisable.

However Eq. (14) is only a necessary not a sufficient condition for

the formation of a valid, physically realizable auto-correlation function.

Thus an instrumentation of Eq. (12) to satisfy Eq. (14) may result in

a correlation function which is the output of a physically unrealizable linear

system. In the time domain there seem to be no simple tests for realizabi-

lity.

Several signals were analyzed using Eq. (12) and (14) at input signal

to noise ratios of and -10 db. No trouble was experienced with the db

case; the signal was almost completely recovered and the noise suppressed.

The -10 db input signal after similar processing was transformed in-fo the

frequency domain. The signal was clearly evidenced by a large spike in the

power spectral density at the signal frequency. The amplitude of the signal

spike after processing showed nearly 6 db greater discrimination between

signal and noise peaks than it did before processing.

The above gain would have been encouraging had not the .power spect-

ral density.' showed some negative contributions.

Negative contributions in a power spectral density, are evidence of, and

in fact are a sufficient condition for, physical unrealizability.

Because of this difficulty no further efforts were made to suitably deter-

mine <X on the basis of the correlation functions. Efforts were then directed

to a determination via Eq. (13).

25

If the signal is a pure sinusoid with no noise present, the power

spectral density shows a delta -function cf (w) at w = wg, and is identically

zero for all other values of w. The perfect processor would give this output

even in the presence of noise.

There are two conditions one may then place on Eq. (13). These are

(a) There must be no negative contributions 0*^ ws= wMax

(b) At least one point must be zero.

This allows an °s to be determined directly in the frequency domain.

Condition;(a) above ensures realizability, while (b) ensures that for any

c*>c\ , condition (a) is violated. On the other hand choice of anc

<* ^ °<c results in the removal of less noise than one might achieve other-

wise, but at least ensures physical realizability.

A trial and error search routine was carried out to find an ©< which

satisfies conditions (a) and (b) above. The search routine is halted when

ano< is found which results in at least one point of Eq. (I3)having a value

£, 0i€ f= .00001, and no negative points. The time to complete a search

is in the order of . 5 seconds, which is small compared to the time taken to

form the functions originally (70 seconds).

Having formed Pss(w) in ^ <3« (13) the results were renormalized to make

the area under the curve unity, and then plotted.

The o< determined by the search was then used in Eq. (12) to formc

RggCf) and the result plotted. No claims for validity are made for this step

except that having formed RgsC") in this manner:its transform had no negative

26

component..

The noise removal proceedure based on an «k determined from the

power spectral density resulted in an additional 14. 5 db processing gain.

This figure was obtained by considering the ratio of energy under the 50 cps.

comb- width to the total energy under all other comb-widths, both before and

after the noise removal. Before noise removal the calculated input signal-

to-noise ratio was -8. 84 db. After noise removal the equivalent calculation

yielded + 5.62 db.

Another measure of effectiveness should be mentioned. In many cases

the ability to distinguish a signal from noise in the spectral density is of

interest. In the power spectrum the ratio of the amplitude due to signal to

that due to a noise "spike", may be expressed in db. In the -8. 84 db

(nominally -10 db) case, the ratio of signal to highest noise peak was 5.62

db. before noise removal, and 9.05 db. after noise removal, or an improve-

ment of 3. 4 db.

Details of the processing are included as Appendix B,

5. Location of a Source by Cross -Correlation .

The bearing of a signal with respect to an abserver may be determined

by measuring the difference in arrival time of the signal at two or more sep-

arated observation stations.

As an experiment a 200 cps. sinusoidal signal generator was set in

one corner of a cafeteria. Two microphones spaced a distance d units apart

were located symmetrically with respect to the center line of the room, at

the far wall of the cafeteria. Interfering noise was supplied by the noon-day

27

clientele.

The maximum level of signal was limited by the intolerance of the din-

ers. This was determined to be 38 milli-watts measured at the loudspeaker

voice -coil. At this level the signal was inaudible at the microphones.

A stereo recording was made of the signal plus noise, and after slow-

ing the tape by a factor of four, selected portions of data were digitized

using techniques detailed in Appendix A. The resulting digital data was

cross -correlated in the CDC 1604 computer and the correlation function plot-

ted.

The geometry of the problem is shown below.

source

/"

Nioise FIELD

70. 7S

130

Figure 2. Geometry of Lunch-room Problem.

Assuming plane wave propagation, and in the absence of noise, we

may associate with A D in Figure 1, an equivalent amount of time; that is,

28

A T = AD/C where C is the velocity of sound in air.

If the two sensors are cross -correlated a maximum in the function will

occur whenever the time variable of correlation is equal to a multiple, of the

period of the signal.

The signal was located successfully to an accuracy of + 0. 17°. The

frequency of the observed correlation function was nearly 150 cps. , or three

times the expected value of 50 cps. This would indicate that detection was

accomplished on the third harmonic of the source oscillator.

A study of the noise characteristics supplied by the diners was not

made.

Provided that accurate time synchronization between the two data

sources to be correlated can be maintained during the digitizing process,

correlation will provide both bearing and frequency information.

Details of the problem are presented as Appendix C.

6. Conclusion.

A convenient method of digitizing analogue data has been developed

and this digital data used to detect signals masked by noise.

The digitizing system is now operational.

Many interesting areas of research remain to be pursued in the field

of recovery of noisy signals. In particular additional effort is needed in the

area of removal of apriori known noise effects.

The facilities to perform the necessary calculation on a digital computer

are now available at the U.S. Naval Postgraduate School.

29

ACKNOWLEGEMENT

The author extends his thanks to Dr. G. Myers of the Electrical

Engineering Department for his thoughtful guidance and help.

Dr. L. V. Schmidt of the Aeronautical Engineering Department has been

vitally interested in the development of the digitizing process and has lent

his time and his energy to help accomplish this goal.

Mr. E. Jangen of the Programming Staff of the Computer Facility has

been extemely helpful with details of programming and has been of great

assistance.

Finally, I would like to thank the Staff of the U. S. Naval Electronics

Laboratory, San Diego, in particular Mr. W. Littrell and his staff, for the

courtesies extended to me during my visit of June 1965. The experience

gained there largely motivated this work.

30

BIBLIOGRAPHY

1. Lee, Y. W. , Statistical Theory of Communication , John Wiley andSons, New York, London, 1960.

2. Lindgen, B. W. , and McElrath, G. W. , Introduction to Probability

and Statistics , MacMillan Co„ , New York, 1959.

3. Van Trees, H. L. , "Optimum Signal Design and Processing for

Reverberation - Limited Environments ", A. D. Little Inc. Report

No. 1501064, Oct. 1964 a p. 7.

4. Lee, Y. W. , Cheathom, and Wiesner. "The Application of Correlation

Functions in the Detection of Small Signals in Noise " , Mass

.

Inst, of Tech./ Research Lab of Electronics, Technical Report 141,

Oct. 1964. p 7.

5. Fano, R. M. , On the Slgnal-to-Noise Ratio in Correlation Detectors,Mass. Inst of Tech. , Technical Report No. 186, February, 1951.

6. Pope, J. W. , Real Time Estimation and Random Signal Detection ,

Ph. D. Thesis, U.S. Naval Postgraduate School, Monterey, May,1966.

7. Jesberg, R. H. , Data Handling System in Support of Antenna Vibration

Project, M.S. Thesis, U.S. Naval Postgraduate School, Monterey,May, 1966.

31

APPENDIX A

ANALOGUE TO DIGITAL DATA CONVERSION

1. INTRODUCTION.

The process of translating real, analogue data into computer digitized

form has been accomplished in two separate stages.

In the first stage, the analogue data on magnetic tape is sampled at

discrete points of time. These samples are digitized and written in bit form

on magnetic tape.

In the second stage, the digital magnetic tape is loaded on the CDC

1604 peripheral tape units and then read into the computer

Earlier work at USNPG School in computer analysis of analogue data

utilized the "on line" approach, in which the data was fed to the main com-

puter directly from the digitizing computer. This mode of operation is now

impractical because of high computer facility utilization.

The method detailed here results in data being reduced to a form suit-

able for submission as a standard program input.

2. DIGITIZING THE DATA.

Basically the method takes samples until the Core storage of the dig-

itizing computer is full, and then writes this information on magnetic tape

in digital form.

The digitizing computer is a CDC 160 located in the Computer Labor-

otory of the Electrical Engineering Department.

This computer has a 4096, 12 bit memory length, and has the ability

to select various external equipments through the proper external function

32

codes. The lower 4000 cells of the computer were reserved for data storage

and the top 96 cells for programming. Thus the core storage available sets

one quite restrictive limitation on the system. In practice however, 4000

samples were adequate to describe many common processes.

The block diagram of the system is shown in Figure 1. The 160 selects

either the A/D converter, for data input, or the 163 tape unit for data output.

When the A/D converter is selected (EXT 14xx) and the INA (IN put to A) in-

struction is executed, the input analogue signal is sampled, and a 12 bit

number representing amplitude is sent to the 160 and stored. Repetition of

the process in quick succession allows blocks of up to 4000 samples to be

stored. The maximum sampling rate achievable is 5000 samples/second.

To control exactly the time at which samples are taken, the execution

of the INA instruction is prevented until two events have occurred; first, an

enabling switch must be closed to ground (EXF 2410), and second, a pulse

(see Figure 2) must be present at the base of the AND Gate* When these

conditions have been satisfied the flip-flop goes to the "1" state placing -3

volts on the INA line and thus executing the INA instruction.

Further samples may now be taken as the flip-flop will stay in the "1"

state until EXF 2400 is called, resetting FFI to the "0" state. The complete

process is illustrated in Figure 3.

The usual mode of operation is to have each block initiated by a pulse,

with the intersample delay under computer control. Under certain circumstan-

ces, it may be desired to have only the runs initiated by a pulse, with both

33

the block count and the sample count under computer control.

The inter-block timing in this mode is not accurate enough for cross-

correlation. The basic program shown in Table 1 may be modified using

Table 2 to give various modes of operation.

Operating Instructions -Digitize Eq. (3).

Features of Digitize Eg. (3) .

1. Manual Entries

a. 0066 - intersample delay. See Figure 6

b. 0067 - any identifier you wish

c. 0070 - initial run number

d. 0071 - number of runs desired

e. 0072 - number of samples/block £ 7640g

= (4000)1Q

f. 0073 - number of blocks/run

Note that (d) x (f) £ 7608

2. Main Points

a. A block of n, n <- 4000 -,q samples is taken each time a clock

pulse is present and EXF 2410 is made.

b. The inter-sample spacing interval is program controlled and is

variable from approx. 190 Msec, minimum to about 50 7nsec. maxi-

mum.

c. The first four (4) words written on the 163, of each block, are

identifying words in the following order:

(l). Run number.

34

(2). Spare (may be preset to anything)

(3). Block number

(4). Number blocks/run

d. The identifying information when digitizing data must be recorded,

as 1604 subroutine DATA uses this information to locate blocks of

data on digital tape.

e. At the end of each run a check is made to ensure that the storage

capacity of a full roll of 163 tape is not being exceeded. If it is,

an End of File is written and the program halts with the total numb-

er of blocks written in the A register.

f . Program may be stopped at any point to allow cueing data tape,

provided that it is not cleared. Progress resumes when 160 placed

in run condition.

g. Analogue, input is at A/D Channel 1. A voltage reference is est-

ablished by an external source at about -5 volts. Correct opera-

tion is assured if the proper test pattern is displayed. See in-

structions for Program Test 160.

h. A voltage of V volts at A/D Channel 1 is converted to a digital

reading by the A/D converter, equal to -409. 6 x V

3. Operating Instructions

a. Load from cell 0000

b. Run from cell 0000

c. When digitizing is complete program stops at 0124 with current

run number in A register.

35

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38

d. Clear and run from cell 0127 to write E. O.F.

4. Modification-'sum

a. See Table II for various possible modifications.

The limitations of the sampling process are graphed in Figures 4 and 5.

Figure 4 shows the effect of sampling speed on the percentage of the data

that may be digitized. The speed of the data gathering is a variable under

program control, while the speed of output is not. Thus at slower sampling

rates, the sampling approaches continuous sampling, and the amount of the

real time data which may be represented on tape, increases, as seen in

Figure 5.

Figure 6 shows the relationship between the number entered as inter-

sample delay timing word, and the resultant inter-sample delay.

The development of a clock pulse from initial, recording to output at

the flip-flop is shown in Figure 2. Experiment has shown that a total delay

of 100 micro-seconds exists between tQ , the time of initiating the pulse on

record and t^, the time the pulse is delivered. This, period remains station-

ary over a large range of level settings, pulse widths and amplifier gain

settings . Thus the clock pulse may be used to synchronize two pieees-of data,

as in cross -correlation.

Experience has shown that since only the first of a series of adjacent

clock pulses in used to initiate sampling, the recording of a train of pulses,

rather than a single pulse is a great aid in cueing the analogue tape. The

length of the train must be less than the length of time taken to sample one

block.

40

In order to set the CDC 160 up properly for digitizing, a short program,

TEST 160, has been devised (see Table III). In this program, the first cells

-0000 to 0017, contain a program to accurately set the bias voltage. With

no signal in and the bias at -5.0 volts, the value of the conversion should

be 0000 or 7777 . By monitoring D/A Channel 1 with a "scope", deviations

from the zero condition may be detected and adjusted using the voltage con-

trol on the bias source. Small errors in zeroing may be expected, due to line

drift, etc. The effect of this may be eliminated by removing the mean of the

data in the 1604.

The cells from 0020 to 0045 contain a program which has identical

timing to that of DIGITIZE. By observing the input request pulses on the

patch box, between the CDC 160 and the A/D converter, one may adjust the

value of the timing control to give the desired inter-sample delay. Chang-

ing the control word by one number changes the sampling interval by 12.8

microseconds.

In order to distinctly label each block of data written on digital tape,

an identifying series of numbers is first written, followed by the data points.

Since it takes four CDC 160 words to equal one 1604 word, it was convenient

to specify four quantities in the identifier.

1. Current run number-indexes once a run

2. Spare-may be sertt to anything

3. Current block number in the run

4. The number of blocks per run

41

For example, at the 5th run, using two blocks per run, and the "spare"

set at 4321, the identifiers would appear in the 1604 as:

0005 4321 0001 00020005 4321 0002 0002

Precise records must be kept of the identifiers when digitizing data,

as data location on tape is by means of these identifiers.

3. DATA RECALL

The data in digital form may be recalled by calling Subroutine DATA.

S/R DATA in turn utilizes two subroutines, FINDIT, which locates the data on

tape, and UNPACK, a machine language routine which unpacks the samples.

The result is to create the original CDC 160 sample list, word for word in

the 1604.

The unpacked data is presented to the main program in integer format

and appears in either KDATA (M, 1) or KDATA (M, 2) as specified by KLIST.

Integer format output was chosen for the sake of generality as this is

the format of the raw unpacked data.

Normal use is to change the data to floating point format, remove the

mean, and divide all readingsJgya factor of -409. 6. The letter factor converts

all readings to volts.

When reading the tape on the CDC 1604, it is convenient to have the

blocks arranged sequentially on tape to prevent tape rewinding and scanning.

However, when digitizing data for cross -correlation, it is convenient to

digitize all the right channel information, and then the left.

Since it was desired to cross -correlate the left with the right channel,

42

some reordering of data on tape was necessary. Program SHUFFLE does this,

reordering the data so that the output appears as right samples, left samples,

right samples, etc. , in consecutive order.

Once the data has been written on tape in the order that it will be re-

called, the average time taken to recall one block, unpack it, remove the

mean, and convert it to floating point form is less than 2.7 5 seconds.

4. TESTING THE SYSTEM

In order to test the capabilities of the digitizing system, a test tape

was prepared with two blocks per run, and each block of 4000 samples on

each of six known signals.

Run * Signal

1

4

8. OVo peak-peak, 250 cps

square wave

5. volt, 250 cpsTriangular wave

6 volt, 250 cps sine

wave

-8 volts DC.(-3 below reference)

20 cps band-limited

white noise, 7. 07

volts rms.

White noise 7. 07 volts

rms.

Identifiers

0001 0000 0001 0002

0001 0000 0002 0002

0002 0000 0001 00020002 0000 0002 0002

0003 0000 0001 0002

0003 0000 0002 0002

0004 0000 0001 00020004 0000 0002 0002

0005 0000 0001 0002

0005 0000 0002 0002

0006 0000 0001 00020005 0000 0002 0002

The data was analyzed by Program TEST and the resulting analysis is

tabulated in Table IV. The data was recorded on Reserve Tape 18.

43

Table IV. Summary of Test Results.

Run Signal Type # MEAN/EXPMEAN <S

Z/EXP<S2 fc/EXP

1 Square Wave 250 cps 0/-. 132 16/16.365

2 Triangular 250 cps 0/+. 086 2.0/2.09

3 Sine Wave 250 cps 0/+. 027 4. 5/4. 56

4 -8 Volts D.C. -3/-3.025 0.0/4.74xl0"6

5 20 cps WhiteNoise

Bandlimited 0/+. 147 -/2.10 20/29.5cps

6 White Noise 0/-. 094 -/2.2 200/199cps

Mean Recovery Time Per Block ^L 2.75 Sees.

Digitizing Noise is 63 db Below a 3 Volt Signal

ERRORANALYSIS

In the analysis of the data it was desired to determine an estimate of

<J2

, the data second moment, and to determine a confidence interval on the

estimator.

It has been shown [2] that

Ns2 = —-— ^ (Xj - X) 2

, is an unbiased estimator

for if"2 s may be conveniently calculated by:

s2 =

N

To obtain an estimate of the confidence we may place on s 2 , we assume

that in the large sample case, as here, the distribution of s 2 is normal in

.

.-.;*.-*=. . -

44

which case:

Var(s 2 )= _^li

# Pr'Lls2-<f2 1 ]^. kr^ MillJ

<**j

ie 2

*If n^t *

f N^r

where k is the value of the standard normal r. v. which will have a proba-

bility of being exceeded of # .

The confidence interval here was chosen to be 99% or .99 which implies

k^ = 2.57

ie. Pr ? N(Ojl) C 2. 57 7 = 0.99

Thus the confidence range C was constructed:

C = S 2(l -i -)k / —2—\e

fN-i

C indicates the region about s2 in which one has a probabilily greater

than . 99 of finding the true ^"2.

If we form C/s2 we have

-£5- X 100% =S^— o-"A X 100%sZ

1 + k *r~~24 r n-i

In the data analysed here k ^ 2. 57, N = 3900 and hence C/s 2 =

5.42%.

This means that we are confident that the true^" will lie within 5. 4%

of the estimated6"2 = s2 with probability .99.

Note that lim CR

45

In each of the runs it was desired to plot both the signal and its' auto-

correlation function. The units of the signal are convertedto volts by divid-

ing all data samples by 409.6 or .2048 levels

^

5 volts

Sine, square, and triangular waves were analysed to check the scal-

ing accuracy. It was found that the accuracy of measurements exceeded

that of the oscilloscope used to measure the input waves.

The D. C. signal was analysed to observe the amount of digitizing

noise and hum introduced by the process.

The maximum change in the digitized level was about 4 numbers or

fi^l. X 100% = 0.2% of the full scale. Over the, period of 0.8 seconds

the mean square power was 4. 74 x 10 D watts and the rms. voltage was

2. 17 x 10"3 volts. (A. change of one number -2. 44 m volts)

Thus for a 3. 025 volt signal, the noise and hum is 62. 8 db below the

signal.

The presence of a strong secondary peak in R(T* ) at f = 1/6 O**1 sec.

,

indicates the presence of a strong 60 cps component in the noise. Assuming

1 — kthat at T = —: sec. the noise is uncorrected, then R(l/60) = 2 x 10~° or

60

66. 6 db. with respect to a 3. 025 volt signal. Thus the digitizing noise is

3o 3 db stronger than the hum. If hum could be completely eliminated, the

noise power would then be -63. 3 db below a 3. 025 signal or -53. 7 db below

1 watt.

The "white" noise and the band-limited white noise samples were

taken primarily to check the whiteness of the noise generator. Measurement

46

of the — point of the R(T) curve indicates the generator output would be flat6

within 3 db out to 200 cps.

A considerable discrepancy between the rms voltage as measured with

a VTVM, and the calculated value of R(0) and s 2, was observed. Some of

the error is due to the motion of the needle, and some due to the fact that

the averaging time of the meter was about twice that of the correlater. How-

ever the most signifigant factor was a faulty measurement probe.

6. RECOMMENDATIONS

No attempt was made to "fix" the installation/ and hence some prob-

lems arose that would not have occurred in a permanent installation.

Noise and hum reduction was only achieved by careful grounding and

the use of BNC type coaxial signal leads. The use of coaxial cable is re-

commended for all digitizing work, for the above reason.

A clock pulse was used to synchronize left and right channel data.

Unfortunately, the act of switching Ampex tape recorders on or off in the

record mode causes transients to be generated on all tracks. This is a

characteristic observed on both the Ampex CP100 unit of E. E. department

and the Ampex FR-100 unit of the Aeronautical Engineering Department.

The transient pulses may be treated as clock pulses by the program,

causing loss of channel time synchronization.

The problem was partially overcome by removing the output of the

pulse generator until the transients were past. This is satisfactory unless

the first clock pulse is close to the transients. A standard interval of 20

47

seconds is recommended between the start of data and the first clock pulse,

as a temporary restriction pending hardware modifications.

A more permanent solution would be to imcorporate a time delay in all

record channels of the Ampex CP100 to allow transients to die out, when

starting and stopping the unit, before energizing the amplifiers. This should

prevent spurious clock pulses from being formed.

Despite great care, occasional spurious pulses appear on the clock

channel, and may cause false sampling to take place. The use of the short

optimal length pseudo-random code as a clock pulse, together with a match-

ed receiving filter would correct this problem.

The system as implemented has a 2. 5 Kc upper frequency and 4000

samples continous sampling capability. Since at best, samples may be taken

every 190 ixsecs. , of which only about 50 Lisecs. are chargeable to the A/D

converter, the most likely place to begin the search for higher sampling rates

is in the computer itself, rather than in a more expensive A/D converter. By

deleting certain functions from program DIGITIZE, the sampling may be made

faster but in no case is it possible to sample at intervals smaller than about

100 Usees.

48

Table 1(a). Program Digitize.

Cell Contents Code Explanations

0000 0101 PTA

1 0603 ADN032 7064 JPI64 Jump to INITIAL.,

3 7500 EXF.00 „

4 2410 2410 Set Enable

5 7500 EXF00 i

6 1401 1401 Call A/D Channel. 1

7 7600 INA INPUT0010 4176 STI7 6 Store sample in- (007 6)

11 2076 LDD7612 3464 5*&r SBN65 Enough Samples, yet?

13 6134 NZF34 If not, go to cell 0047

14 7500 EXF00 #,

15 2400 2400 Clear Enable

16 2074 LDD74 Load Current Run No.

17 4160 STI60 Store in cell 01330020 2067 LDD67 Load Spare ID. .

21 4161 STI61 Store in cell 0134

22 2075 LDD75 Load Current Block No.

23 4162 STI62 Store in cell 013524 2073 LDD73 Load No. of Blocks/Run25 4163 STI63 Store in cell 0136

26 7500 EXF00

11 2111 2111 Call 163 M.T.U.0030 7303 OUTOa Output from

31 0000 C computed L.W.A.32 6102 NZFC2 to

33 0133 0133 013334 2246 LDF46 SetA-013735 4076 STD7 6 Reset running storage address

36 2075 LDD75 Enough Blocks yet?

37 3473 SBD730040 6155 NZF55 If not, go to cell 0115

41 0401 LDN01$.-..•

42 4Q75 STD75 Reset block no.'<

43 5455 AOD55 Update total block count

44 3454 SBD54 Check if capacity exceeded45 6153 NZF53 If not, go to cell 012046 6061 ZJF61 If not, go to cell 012747 5476 AOD7 6 Beqin intersample delay loop

49

Table 1(b). Program Digitize.


0050 2066 LDD66 Load intersample delay word51 0701 SBN01 If not zero, go back 1

52 6501 NZB01 If zero, go to c&y. 000553 7056 JPI56 0760-M.T. U. block capacity54 0760 0760 Total No. Blocks written

55 0000 C address56 0005 0005 address

57 0003 0003

0060 0133 0133 Address of Current Run No. ID61 0134 0134 Address of Spare ID of ID62 0135 0135 Address of Cur. .Block No. ID63 0136 0135 Address of Tot. .Blocks ID

64 0100 0100 Address of INITIAL

65 C C Address of last word of data

66 M M Intersample delay word67 M M Spare ID anythinq

0070 M M Initial Run No. Set 1

71 M M No. runs desired

72 M M No. samples/block 76408

73 M M No. blocks/run ,

74 C C Current run no. .

75 C C Current Block Number76 C C Running Storage

rAddress

77 C C JUMP CONTROL.0100 4077 STD77 BEGIN INITIAL

101 2200 LDC00 ."

102 0137 0137 Set "A"-0137

103 4076 STD76 Initialize running storage address

104 3072 ADD72 Compute last address

105 4031 STD31 Store in 0031

106 0701 SBN01107 4065 STD65 Store in 0065

0110 2070 LDD70 f

11 4074 STD74 Initialize Run No.

12 0401 LDN0113 4075 STD75 Initalize Block No.

14 7077 JPI77 END INITIAL

15 5475 AOD75 Update Current Block No.

16 5455 AOD55 Update Current Block Count17 7057 TPI57 Go to 0003

50

Table 1(c). Program Digitize


0120 2074 LDD7 4 Have enough runs

21 3471 SBD71 been done yet?

22 6103 NZF03 If no, go to 012523 2074 LDD74 If yes, display last run no.

24 7701 HLT01 Halt 01

25 5474 AOD7 4 Update current run no.

26 7057 JPI57 Go to 0003

27 7500 EXF00 CallE.O.F.0130 1111 1111 Write an EOF Mark

31 2055 LDD55 Load block count

32 7702 HLT02 Halt 02

33 C C RUN #ID

34 C C SPARE #ID

35 C c BLOCK #ID36 C c BLOCKS/RUN ID.

0137 D D Data Storage

~) D D*r

7776 D D Data Storage

Notes:

1) C is ENTRY PROGRAM COMPUTEDM is MANUAL ENTRYD is DATA STORAGE

2) CDC 1604 Identifier will appear as follows:$

Word 1 RUN # Spare Block # . Blocks/RUN Format 016

Word 2 Sample 1 Sample 2 Sample 3 Sample 4

etc.

51

Table II. Modifications to Digitize

Mod Cell # From To Effect of Modification

0126 7057 7703 Individual blocks clocked at endof a run. To restart run at 0003

0014 7500 03000015 2400 0300

0117 7057 70560126 7057 7703

0014 7500 03000015 7500 0300

0040 6155 6113

0126 7057 7703

Initial block clocked. Subsequentblocks taken as fast as possible.

HLT in 0126. When run completereset FFI manually

Initial block clocked. Subsequentblocks taken immediately. All

blocks have same ID. HLT whenrun complete in 0126. FFI manual-ly reset.

0013 6134 6506 Eliminates variable sampling delay.

Samples possible in this modeevery 120 Usee.

52

Table III. Program TEST 160

Note: M means Manual Entry


0000 7500 EXF00 Begin Bias -set

1 2410 2410 Set Enable

2 7500 EXF00

3 1401 H*6^/ Call A/D Channel 1

4 7600 INA Input to "A" in Reg.

5 4070 STD7 Store "A" in 00706 7500 EXF00

r-

7 2401 2401 Call D/A Channel 1

0010 7303 OUT03 Output Cell 007 to

11 0072 0072 Cell 0071 INC.

12 6102 NZF0213 0070 007014 0400 LDN00 Load zero's.

15 4071 STD71 Store in 0071

16 6414 ZJB14 Jump back to 0000

17 7700 HLT00 HLT. End Bias -set

0020 7500 EXFOO Begin Timer21 2410 2410 Set Enable

22 7500 EXFOO Call A/D Ch. 1 .

23 1401 1401 v

24 7600 INA e

25 4143 STI43 Store Sample in 0043

26 0300 NOP No Operation-NOP27 0300 NOP These are Time

,

0030 0300 NOP Dummies to

31 0300 NOP Match Program DIGITIZE

32 0300 NOP Timing

33 0300 NOP34 0300 NOP35 0300 NOP36 2042 LDD42 Load Timing Control Word37 0701 SBN01 Subtract 1

0040 6501 NXB01 If not zero, go back 1

41 7044 JPI44 If is zero, jump rto 002242 M M Timing Control Word43 0043 004344 0022 002245 7701 HLT01 End Timer

53

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76

APPENDIX B

NOISE REMOVAL IN CORRELATION,

USING A PRIORI KNOWLEDGE OF NOISE

Assume that as apriori information we are given the expected form of

the interfering noise, represented by a normalized correlation function

Rjjjj(IcaT) formed in the absence of signal. We assume that when the re-

ceived signal is corrupted by noise, the statistics of the received noise are

described by this priori knowledge. Specification of the statistics of the

noise by the correlation function implies that the noise distribution is ad-

equately described by only it's first two moments. We shall assume that

the observed date is of mean zero.

Knowing the noise, it was shown that under certain assumptions we

could write

RgsOcAT) - Rg+N(kAT) - ^%N (kAT)' (l)

where %+N^ A ^ *s ^e °t>served normalized correlator output

/*

Roc(k^T) is to be determined.

R?TM (k a T) is known apriori

^SS'

lNN'

Obviously we must determine °^ before Rgg(k A'f) may be determined.

Since the correlator produces a correlation function the author's first thoughts

were to find aK using the observed and known correlation functions.

In the general case, the form of the correlation function of the signal

is not know. For the class of input signals having a non-periodic correlation

77

function, any choice of m, which forces the output to be periodic, ie max

R = R(0), would be fallacious.

Therefore we shall only consider the class of signals having periodic

correlation functions.r

Let us consider the function

F(<X) = ](Rs+n(0) -^N (°)) " MaxC T*o

which may be written

fRs+N (T)- <*RftN (T) (2)

1

A(3)FH ) = Max J(i -k )

-

Since the functional forms of Rs+n(7~) and R&N (T),are not known explicitly

no further analytic evaluation can be made of F(°* ).

Certain features of F(°^ ) are known however in the case of periodic cor-

relation function. If F(<* ) < 0, then <* is too large,, and the output is not

a valid correlation function since R(T * 0) > R( T = 0). If F(°<) > 0, not

enough of the noise correlation has been subtracted, and the output will bek

excessively noisy for small T«

The best choice of <* would then seem to be ^ c such that F(o( c) = 0.

The determination of c< must be done iterativ^ly since no explicit

analytic formulation for the function. 3 is known.

78

Consider Figure 1 below

> <*

Initially set o<p= 0, o<

n = 1, and <*! = —* —

Evaluate F(°< ) and decide

a) oep= 0^

1if F(°C.) >

b) o<n= c<

iif F(*

4)< JD

c) Terminate if |f(«^.)| < £

Then o(i°<p + <Xn

i+1

. i.

^

Having found occsuch that - € 5 F(°( ) £ + € the portion of Ryy( fj due

to signal may be estimated by substitution in £q. (l) 4 that is

RssW = %*i^) - °t^(t) (4)

At this point we have a combinational ratio which causes a periodic out-

put. However there is no guarantee that the new correlation function formed

in Eq. (4) is the result of a physically realizable linear system. This is be-

cause the condition that the correlation be no larger for any T other than

T = is a necessary condition and not a sufficient condition. A sufficient

condition for realizability is that its transform, the power spectral density,

have no negative portions.

79

Rather than attempting to determine ^ in the correlation domain, it is

more fruitful to do a similar search in the frequency domain. The obvious

advantage is that physical realizability is easily satisfied.

Transforming Eq. (l) we have

Pss (k *$ ) = P|+N(k^ ) - ^ PJN (kA* ) (5)

where the asterick indicates normalization.

An iterative search, similar to the previous search, results in determina-

tion of e*c subject to physical realizability constraints.

This is ensured by terminating the search when for any k, the value of

Eq. (5) is less than .00001.

If any value greater than this is used then the contribution will be neg-

ative and realizability is violated. If a smaller value is used, more noise

will be left than in the case ®< =<*c .

This value of o< = ^ c was then used in Eq. (5) and in Eq„ (l).

The results of Eq. (5) were renormalized to have, unity area by the trap-

ezoidal rule, that is

i

L_1

-±-(kPl + kpL) + 2 j5 kPi = 1 (6)

2

from which we haveL-l

k = 2/ (p x+ Pl + 4 £ p^ (7)

2

The results of Eq. (l) were renormalized by a factor 1/ Rss(0).

To evaluate the results assume that the correlation of the desired signal

formed in the absence of noise Rss(k^ t) is available for comparison.

80

An estimate of the noise power in the correlated' output may be found by

evaluating

No = T^f *< ( ^SS^^T) - (1 - o<c) 'R^

S(U T))2

(8)

The signal power in the correlator output may be evaluated several ways.

One method is to assume that the scaled version of the pure signal, (l - k ) •

Rgs( T - 0), represents the peak value of the periodic output. In the case

of a single sinusoid signal, the output power would be given by ((l - <* )•

Rss(o))2/2.

The input signal-to-noise power ratio may also be determined in several

ways. One method is to compare the variance of the noise with that of signal

plus noise to give

(S/N)INpuT= (^ - i* )/<?N

2•

(9)

An alternate method to determine S/N ratios and processing gains achieved

is via the normalized power spectral density, obtained from the transform of

the correlation function. This method is felt to be more accurate than deter-

minations made from the correlation function because of the difficulties

encountered in separating the signal components of a correlation function

from the noise components.

Determination of signal to noise ratio from the normalized power spect-

ral density proceeds as follows.

The energy under the normalized density curve, with respect to a unit

change of index i, is given by

81

M+l- J X di = 1 (10)

Using the trapezoidal rule we may evaluate this as

1 J?E = 4 (Xx+XM+1 ) + 2 S. Ki + 2XS = 1+ € (11)

i=*2

I*S

where Xg is the component due to the signal.

The desired output, Y, is given in terms of frequency rather than unit

change of index and is given by

Yi = kXj/E (12)

where k = 2 ^ tM

** = k- 1

A t = interval between samples in seconds

M = the number of frequency estimates

Substitution of Eq. (L2) in Eq. (il) yields& •

^E = N+ S JL\3)

Mwhere N = i- 0^ + XM+1 ) + 2 £L Xj

Zi=2

i*S

S = 2 E Yg a f

The ratio of energy due to signal S, to that of noise, N is

A

S = 2E ^s = 2Y_s (14)

N E - 2£Yg 1-2'lJ Af

or in decibels

(S/N)db = 10 log! (S/N) (15)

82

2. Comparison of Auto and Cross-Correlation

No method of processing can improve the processing gain given by cross-

correlation, which is taken to mean comparison of a signal plus noise with

the completely specified input signal. This fact arises because of the a-

priori | information assumed to be available.

When in general the desired signal is not known, auto-correlation, that

is comparison of a waveshape with a time shifted replica of itself, may allow

detection of the signal. In no case does it do it as efficiently as cross

-

correlation with an uncorrupted signal.

If the noise can be assumed to be completely specified by it's correla-

tion function, then it seems reasonable to assume that results approaching

that of cross -correlation might be obtained by suitable removal of the known

noise effects.

Lee [_lj has developed models to estimate the processing gains possible

for both auto and cross correlation, for the case of a single sinusoid *

Emsin(wt+ 9) corrupted by noise of zero mean and variance^-^

For a single sinusoid input the desired output of the correlator ^SS^) =

E2 cos w T where E = Em/ fl^*

At the input to the auto -correlator, let N± be tryS input noise in rms

value, and let

Si - -g- - -f Mbe the input noise-to-signal ratio. This definition while not conventional

leads to compact formulation.

83

The output noise in rms value may be shownQl] to be

Noa „ ^44

+2^*.|^)]1/2

(17)

where N is the number of data samples considered. . For sinusoidal input, the

rms signal output ^u«.

Soa = E 2/ fT ) (18)

At the output of the correlator the signal to noise ratio R is given by

Roa = 20 1og1Q

|2i(19)

Noa1-

Substitution of £q. (13), (14) and (15) into Eq. (16) yields

Roa = 10 lo9l0 ; j 4J,j ; 2fi4

(20)

If in the case of cross -correlation we assume the signal plus noise is

correlated with a local signal Em sin w t, the output noise in rms value

may be shown to be \_1 J.

Noc = JJL f.E2 p]

1/2(21 )

and the output signal rms

soc =~Tj

==k^ ^22 ^

[iven by

Roc = 20 log 4*- db (23)

The gain in db, Roc , is given by

soc.0 Noc

This may be written using Eq. (16)

Roc = 10 log ^ (24)

1 + 2^i2

r"

Inspection of Eq. (20) and Eq. (24) clearly shows the superiority of

cross -correlation especially for large ft (ie weak signals).

84

The superiority is expressed by the difference

G " Roc - Roa = 10 lo9l0 i + iJl2 '

' (25)

i The concept of processing gain is perhaps more useful for comparison

of any proposed method with these two standard methods.

Let us define processing gain as the difference^ in signal to noise ratio

at the output and the input of the processor.

_E2 >

ie Gpa

= Roa " 10 1og10 ^2 db -

I

Nfi

2N

= 10 log 10 £* 10 log 10 (26)

1 + 4jV + 2 f i

4,; ,

4+ 2 ^i2

E2 I

and Gpc = RQC - 10 log 10 —j- db'

= 10 logN 3 i2 - C=? 10 log

10 f- (27)

1 + 2 fiZ

Finally the difference in processing gains D_ '

ac

Dac " Gpa " Gpa = 10 lo9l0 <2 + 1 i2) dl

?(28)

The above forms are quite accurate forj1

^ >3. However the quantity

Ni

i i- -q -raus is not the most commonly used description of signal to noise

ratio. Accordingly equations (26) and (27) have been recast in terms of in-

put signal to noise power ratios in db, and plotted in Figure 2. Eq. (28)

was similarly treated and is shown in Figure 3.

Figure 3 conveniently shows the extent to which auto-correlation proces-

sing is able to emulate cross -correlation processing.

85

5 -JO -IS

Xnput SnR xm Db.

Figure %.'< R*ocess»*G Gmt-i vs. SNR

-S" -lo -15

Tnpot 5Nft xh Db-zo

Figure 3: Processing Aov/wrtfcS vs. SMR

-zsr

86

Figures 2 and 3 are for 3750 samples. >

3. Experimental Results

The results obtained for the case of unity signal to noise ratio are ex-

cellent, as may be seen by comparing Figures 5 and 10. Figure 5 is the

power spectral density of a sine wave formed without noise. Figure 11 is

the power spectral density of the db signal to noise ratio input after the

noise has been removed. A processing gain of 26 db was attained, as com-

puted from the correlation function. This is mid-way between the performance

of an auto-correlator and a cross -correlator.

Since results were so encouraging, the -10 db signal to noise case was

then considered. Several problems became evident that previously were mask-

ed by the strong signal component.

Determination of ©< becomes difficult at low signal-to-noise ratios due

to the small signal component in the auto-correlation function, and led to

the formation of physically unrealizeable correlation functions, as evidenced

in Figures 13, 14 and 15. Inspection of Figure 12, the power spectral den-

sity of the -10 db signal plus noise before noise removal, shows a strong

component near 50 cps. It's transform, the correlation function (not shown)

is difficult to interpret.

This suggests that the signal-to-noise ratio measure of quality may not

be the most meaningful for this type of signal detection.

A more meaningful measure is the ability to distinguish a signal com-

ponent in the power spectral density from a large noise spike. This ability

87

may be expressed in db as

A = 10 log, n£§- for i ^ s (2 g)iU max yi

If A is computed before and after noise removal, the increase in A is a

direct measure of increased gain.

In other words, at low signal-to-noise ratios the problem shifts from

one of interpreting a correlation function, to that of evaluation of the psd.

During the course of experiment it became evident that the spectrum of

the noise may vary quite considerably from block to block. Since it was

desired to remove the expected noise spectral density, more than a single

block of noise must be used. By averaging the correlation function of noise

over ten blocks much smoother spectra resulted (Figures 6, 7 and 8). The

average noise correlation over ten blocks was used for the -10 db case.

This of course implies stationarity over the period of time represented by 10

blocks of noise.

The claim that averaging separately formed correlation functions is

equivelent to an increase in integration time, when the process is ergodic,

should be verified. Examination of Eq. (26) and Eq. (27) leads to the con-

clusion that G^ should increase by 3db when the number of samples is

doubled, ie

A Ni4 Gpa=10log 10 IT (30)

where N\ and N are the number of samples used.

This goal was not reached, but was closely approached, falling short

by about 1 to 1 . 5 db.

The results obtained for the -10 db signal to noise ratio input are shown

88

in Figures 17 and 19, before noise removal, and in Figures 20 and 21 after

noise removal. Noise removal had the effect equivalent to increasing the

input signal to noise ratio by 14.46 db. Discrimination against roise peaks

was increased by 3. 42 db.

89

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129

APPENDIX C

DETECTION AND LOCATION OF SIGNALS IN NOISE

In Appendix A and B methods were developed for digital analysis of

analogue signals.

The data utilized in the above appendices was in a sense artificial, in

as much as it was generated in the laboratory.

To test the proceedures developed using a live signal, where signal

characteristics and noise statistics are generally not known, a 200 cps

sinusoidal signal source was placed in an unknown noise field produced by

patrons of a cafeteria.

The signal source was set up at one end of the .dining room, and its'

volumn raised until complaints were voiced. The source level was then

lowered until the complaints ceased. This level, which we shall refer to as

the db level was measured to be 38 milliwatts at. the voice coil of the loud-

speaker. At this level the sound was occasionally audible at various loca-

tions within the room, but was not detectable aurally at the microphones.

A stereo recorder was set up at the far end of the room, using two micro-

phones symmetrically placed about the longtitudinal center line of the room.

A number of three minute recordings were made of signal plus noise, re-

duced signal plus noise (-9. 3 db), and noise alone.

The noise level of the crowd varied very considerably over the hour and

was punctuated by an occasional dropping plate. The period from 12:30 to

12:40 was selected for further study as the number of diners was essentially

130

constant over this period. The details are listed in Table I„

By cross -correlating the left and right channel information it is possible

to locate the source and to estimate its frequency.

.

The bearing of a source, with respect to the center-line may be deter-

mined by measurement of the difference in arrival times of sound waves at

the receiving sensors.

Assume as in Figure 1, a distant sinusoidal source, at.amangle 8 mea-

sured from the center-line or dead-ahead position, and two sensors symmet-

rically placed about the center-line spaced d units apart.

Assuming plane wave propagation, with no mulUpath effects, and in the

absence of noise, the difference in travel time or phase of ray 2 is a direct

measure of the angle 0.

2-

Ret)

K T,

/IN\

\i r 1 1 >

X

\yFigure 1. (a) The basic DF Problem, (b) Cross -Correlation of Mi, Hn

i rT/2

.

Forming RM , M (T) = lim T -*•<*> ~ J (cos w t)(cos w (t-r))at1 4 i _T/2

(1)

131

= B |T/W

?RM1M2< T) = M. J i [cos

(Wo^ * Wo^ + CW dt

2TT _ ^/Wq * u

o

w f **°r 1

RM1M2 (T) " ° + cos (W T -a) (3)

The first peak in the correlation curve occurs when

Wo Tj = V or T,

= JL. = A sin>9 (4)

Subsequent peaks in RjvilM2 (1") occur when

W f* - y = N (2TT ) ,N = 0, 1, fe, 3 ...

Peaks obtained for n\0 represent ambiguities in the bearing angle. The

_ d_

P" cfirst ambiguity occurs when T D = — sin &8orA0 = 9a -6= sin

-1 E

= sin-l C^TT_

dW

The first peak at 1"! is of interest. In this case using room measure-

ments and an assumed velocity of propogation of 1129 ft/second, Ttwas

calculated to be 1. 5506 msec, for an angle of 10. 4°. For small angles we

may assume the time and the associated angle are linearly related. Under

this assumption the arrival time would change very nearly 150 U. sec. per

degree.

The available processing system however could- sample the record at

best every 200 M. sec. , resulting in a bearing accuracy of about 1. 33°. To

improve the bearing accuracy, the data was slowed by a factor of four,

grequency shifting all components down by a factor of four. This has the

132

effect of increasing the arrival time by a factor of four resulting in a bearing

accuracy of 1. 33/4° or . 333°.

The mechanics of the speed reduction are shown in Table 2.

This is more bearing accuracy than is justifiable but since a reduction

in data speed of at least two was necessary to digitize, a 5 Kc analogue re-

cord, the extra factor of two was used to increase bearing accuracy.

The data speed reduction meant that the original. 200 cps signal would

appear as 50 cps with the first peak of the cross -correlation curve at T =

6. 02 msec. The data was sampled at intervals of 205 JLLsec. , and hence

the signal, if present, should occur between 29 and 30 shifts of the correla-

tion function argument.

2. Data Analysis

Twenty blocks of data were digitized for each of three cases; signal to

noise ratio db; -10 db; and noise alone.

Due to difficulties experienced with the clocking system, only the first

block of a twenty block run was clocked, with clocking of subsequent blocks

internally. Due to the variations in the time necessary for the 163 tape unit

to recover from a writing operation, a cumulative error in clocking arose:

Hence only the first few blocks are time synchronized to an acceptable degree.

Fortunately this was long enough for sucessful signal detection.

The data was normalized in the computer by the computed estimator of the

standard deveation, and the cross -correlation of the normalized data formed.

The time of integration was fixed due to the finite block length. Assum-

133

ing a process is stationary and ergodic, (at least short-time stationary) the

effective integration time may be extended by averaging in time, ie

N

N=l* R

iab(r) " -£ % f a (t)fb(t-r) (5)

and R^bCf) = — i> R- ab (1 )

K i=l

- K NRiab(T) = ± £ 0- £ f a CT) f h(t-T))K ^ N N=l

NK= j^ £ f-.(-r) i b (t -T) (6)

The running average cross -correlation function of the db signal plus

noise was formed over the first five blocks and is shown in Figures 3 through

5 as a "hatched-in" curve.

3. Results

Experiments with averaging more than five curves were not successful,

due to the previously mentioned cumulative timing error.

The cross -correlation of db signal plus noise was abserved to be es-

sentially sinusoidal with the first peak occuring between 29 and 30 shifts

exactly as predicted.

The frequency of the sinusoid however was observed to be 150 cps, or

three times the expected signal frequency, indicating detection was being

made on the third harmonic of the source!

No firm explanation for this is offered. However it is noted that the

134

oscillator produced 0. 1% third hormonic, and the 20 W. push-pull amplifier

used would be expected to produce considerable cross -over distorion when

operated at such low power levels. In addition the tape re-recording process

tended to favour the upper registers.

A power spectral density was made of the db signal plus noise, and

of noise alone, using a 6. 125 cps filter bandwidth. The normalized results,

shown in Figure 2, reveal the pronounced 150 cps component in the signal

recording, with the general shape of the remaining portion being similar to

the noise power spectral density.

The programming mecessary is included as Program SIMSIG in Appendix

B.

The results obtained with the signal at -9.3 db were inconclusive and

are not included.

4. Conclusion

Provided accurate time synchronization between right and left channels

can be maintained during the digitizing process, cross -correlation of the two

channels provides both bearing and frequency information.

The process of averaging cross -correlation curves must be done with

care, ensuring accurate time synchronization and continuing the averaging

process only as long as the signal characteristics remain stable.

135

ANALOGUETIME

DATA DIGITALSTATE RUN

12:30-12:33 "O" DB. 0001 0021 0001 0024 (R)

TO0001 0021 0024 0024

0001 0032 0001 0024 (L)

TO0001 0032 0024 0024

12:35-12:33 "-10"DB. J

12:41-12:44 NOISE C

Digitized on Reserve Tape 18

Sampling Interval 205 Ms.

0002 0021 0001

TO0024 (R)

0002 0021 0024 0024

0002 0032 0001

TO0024 (L)

0002 0032 0024 0024

0003 0021 0001TO

0024 (R)

0003 0021 0024 0024

0003 0032 0001TO

0024 (L)

0003 0032 0024 0024

TABLE I: Details of Data Digitized.

STEP PLAYBACK BAND-W. RECORD BAND-W. SIGNAL

Concertone 50 cps Ampex FR-100 - 1. 1 Kc. 200

7.5 IPS 10 Kc. cps

Ampex - 650 cps Concertone 20 - 650 100

7.5 IPS 7.5 IPS cps cps

Concertone 10 - 325 cps Ampex 10 - 325 50

3.75 IPS 7.5 IPS cps cps

TABLE II: Frequency Reduction Process.

136

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X Scale- 2.05 msec/inch. Y Scale- 0.5 units/inch,

Figure 3- ODB. signal cross-correlation over 3950samples.

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X' Scale- 10.25 msec/inch. Y Scale- 0.1 volts/inch,

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figure 6- Relative power spectral densities of noisealone, ,and signal .plus noise, Normalizedto 100/* .

145

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 20Cameron Station

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2. Library 2

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3 Commanding Officer 1

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2450 Mass. Ave. N.W.Washington, D.C. 20008

4. Dr. Glen A. Myers (Thesis Advisor) 1

Department of Electronics Engineering

U. S. Naval Postgraduate School

Monterey, California

5. Dr. L. V. Schmidt 1

Department of Aeronautical Engineering



6. LT Donald Leichtweiss, USN 1

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7. LTNeilA. Barrett, R.C.N. 1

1038 Broncho Road (

Pebble Beach, California

8. Mr. W. H. Littrell 1

Code 3150U. S. Naval Electronics Laboratory

San Diego, California

9. Dr. H. A. Titus 1

Department of Electrical Engineering



146

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2a. REPORT SECURITY CLASSIFICATION

UNCLASSIFIED26 GROUP

NOT applicable3. REPORT TITLE

Extracting Analogue Signals from Noise Using a Digital Computer

4. DESCRIPTIVE NOTES (Type ot report and Ineluaiva dates)

Master's Thesis in Engineering Electronics5 AUTHORfSJ (Lett name, tint name. Initial)

Barrett, Neil A.

6. REPORT DATE 7a. TOTAL NO. OF PACES

144

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11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Department of National Defence CanadianDefence Forces Headquarters, Ottawa,Ontario, Canada.

13. ABSTRACT

It is frequently convenient in data processing to convert analogue to digital

data for computer assimilation. .A convenient method of such conversion has beendeveloped and used in the study of correlation detection of the audio signals cor-

rupted by noise.

?•

A method to use apriori knowledge of the corrupting noise to increase proces-

sing gain has been studied. In the case of detection of a sinusoid in noise, anadditional gain over convertial auto-correlation of up to 14. 5 db has been achieved.

Finally, a signal source located in an unknown random noise field was detectedclassified and located in relative bearing by the cross -correlation of the signals

received from two spatially separated sensors.

DD .MS. 1473 UNCLASSIFIED

147Security Classification

UNCLASSIFIEDSecurity Classification

14.KEY WORDS

Analogue to Digital

Correlation DetectionNoise Removal

LINK A

ROLE WTLINK B

ROLELINK C

ROLE

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DPJ FORMVJ 1 JAN 94 1473 (BACK) mssSaWLusn

148 Security Classification

lhesB2376

Extracting analogue signals from noise u

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Extracting analogue signals from noise using a digital computer.

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