STUDY OF THE' USE OF AMPLITUDE MODULATED ORTHOGONAL POLYNOMIAL WAVEFORMS IN A MULTIPLEX■SYSTEM
EYBrian K. Conant
A Thesis Submitted to the Faculty of the DEPARTMENT OF ELECTRICAL ENGINEERING
In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE
In the Graduate College THE UNIVERSITY OF ARIZONA
1963
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGHED: /6-Ua.,,
APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:
a. d. rl.
I'll (o'I*Granino A. Korn Date•
Professor of Electrical Engineering
ii
ACKNOWLEDGMENTS
The author is•indebted to Dr. Granino A. Korn for the assistance and encouragement he gave during the preparation of this thesis.
The author performed much of the investigation involved in this thesis as a research assistant in the Electrical Engineering Department, University of Arizona. This financial support is gratefully acknowledged.
iii
\
Chapter I
Chapter II Chapter III
3.13.2
Chapter IV4.1 .
4.2 Chapter V
5-15.2
Chapter VI Chapter VII Appendix
TABLE OP CONTENTS
INTRODUCTIONMATHEMATICAL DESCRIPTION OF POLYNOMIAL WAVEFORM GENERATORMATHEMATICAL DESCRIPTION OF MATCHED FILTERSCOMPUTER SETUPSPolynomial Waveform GeneratorMatched FiltersGENERATOR AND FILTER ACCURACYAccuracy of WaveformsCrosstalkSTATISTICAL MEASUREMENTS NoiseDetection ProbabilityAPPLICATION TO MULTIPLEX TELEMETRYCONCLUSIONINTRODUCTION TO THE ASTRAC COMPUTER
iv
LIST OF ILLUSTRATIONS Figure Page1.1 Legendre Polynomials. . . . . . . . . o . 31.2 Basic Transmitter . . . . . . . . . . . 62.1 Basic Receiver . . . . . . . . . . . . 82.2 Lq T) . . . . . . . . . . . . . . . 112.3 "1 T O . . . . O ti . o ® . '. . ... 123.1 . Computer Set Up For Polynomial
Table Page4-1 Accuracy of Polynomials 214-2 Accuracy of Filter Output 224-3 Crosstalk Measurements . 245-1 Noise Output of Matched Filters 25.5-2 Detection Probabilities 28
ABSTRACTA new method of multiplex transmission has
recently been developed. This system uses amplitude modulated orthogonal polynomial waveforms as carrier signals. .Orthogonal signals may be transmitted over a single wire or radio link simultaneously and can be separated at the receiver by a correlation detector or matched filters. The system is suitable for analog or digital information.
This thesis describes how such a communications system may be simulated on a fast iterative analog computer. Accuracy and crosstalk are measured. The system is also tested with additive Gaussian noise using a hybrid analog-digital statistics computer (sample averaging unit and distribution analyzer).
Vii
INTRODUCTION
A new method of multiplex transmission, recently developed by Bernard Electronics Company of Washington,D. C., employs repetitively generated Legendre poly-
inomials as carrier waveforms for a set of signals.Since the polynomials are orthogonal over a specified interval, they can be added and transmitted together over a single transmission link and separated at the receiver by a correlation detector or by matched filters.
The amplitude-modulated polynomial waveforms can be used to carry analog information in.a sampled-data fashion; digital information may be conveyed by simple polarity reversal of the waveforms.
It is the objective of this study to investigate this system with the aid of the Arizona Statistical Repetitive Analog Computer (ASTRA.C l) and to demonstrate the use of the digitally controlled repetitive computer for simulating a sophisticated communications system.
1
CHAPTER 1 THE POLYNOMIAL WAVEFORM GENERATOR
Two functions Qjn(x), Qn(x) of a single variable x are said to be orthogonal in some interval I if they satisfy the relation:
= 0 (m 7 n)/ F(x) ^(x) 01!(x)dx (1-1)
^ 0 (m = n)
where F(x) is a fixed real non-negative function of x.The Legendre polynomials satisfy this orthogonality condition over the interval -1 5 x £ 1 for F (x) = 1. The first five polynomials, illustrated in Fig. 1.1, are
Qq M = iQ1(x) = x
Q g M = I (-x2 • 1) (1.2)Qj(x) = ? (5x5 - 3x)Q4(x) = g (35x^ - 30xP- + 3)
Higher order polynomials may be derived from the two preceding lower order polynomials by the recurrence relation
%(x) = (2n-l) / Qn_1(x)dx + Q^_g(x) n g 2 (1.3)- 1
2
3
+1
+l
Legendre Polynomials
Fig. 1.1
To generate analogous polynomials orthogonal over the interval 0 § t 5 T on an iterative analog computer, a shift to the right and a change of variable is necessary:
Figure 1.2 illustrates how this concept can be applied to a multiplex telemetry system. The Pk (t) waveforms are generated by means of a polynomial waveform generator to be described later. The coefficients a^ represent the modulating information for each channel. The a^ must remain constant throughout the interval T to preserve orthogonality of the waveforms. Hence the
5modulating waveform must be sampled at a rate such that
remains essentially constant in the interval T; or the modulating voltage may be sampled and held constant in the interval by a sample-hold circuit.
The modulation process consists of four quadrant multiplication which is equivalent to double-sideband suppressed-carrier amplitude modulation.
The output of the waveform generator has theform
nE(t) = Z a. P„(t) 0 s t s T (1.7)
k=0
6TO WIRE OR RADIO LINK
SAMPLEHOLD
SAMPLEHOLD
SAMPLEHOLD
SAMPLEHOLDOSCILLATOR
TIMING
CONVENTIONAL AM OR FM
TRANSMITTER
SUMMING AND SCALING AMPLIFIER
POLYNOMIALWAVEFORM
SYNTHESIZER
INFORMATION INPUT SIGNALS
Basic Transmitter
Fig. 1.2
CHAPTER 2 THE RECEIVER
A basic receiver is illustrated in Pig. 2.1. The modulation coefficients a^ for each channel are determined by correlating the composite signal with the corresponding carrier waveform for that channel, i.e.,
i Tak = 4 / E(t)P.(t)dt (2.1)0
This equation follows immediately from the orthogonality condition (l.l) together with the definition (1.7) of the composite signal.
At the start of a RESET period, at time T, the correlator output voltages are sampled, and the integrators arc reset in preparation for the next transmission interval. The sequence of sampled voltages taken from the sample-hold circuit for each channel represents the original modulation information in sampled form. The outputs of the sample-hold circuits are then smoothed by low-pass filters.
The polynomial waveforms may also be separated at the receiver by matched filters, which does away with the need for explicit multiplication. For a given signal
7
8FROM WIRE ORRADIO LINK
AGO
POLYNOMIALWAVEFORM
SYNTHESIZER
INFORMATIONOUTPUT
SIGNALS
(t)
SAMPLEHOLD
AUTOMATICSYNCHRONIZER
LOWPASS
FILTER
INPUTAMPLIFIER
INTEGRATOR
CONVENTIONAL AM OR FM RECEIVER
Basic Receiver
Fig. 2.1
P(t), a matched filter is a network whose impulse respons is2
h(A) = P(T - t) (2.2)
For the Legendre polynomials, the impulse respons of the matched filters are
h0 (X) = 1
M X ) = -2,2
h2(X) = 6T T + %
h5(X) = -20 4%. + 20T
f „
T'12 >L_
T2 + %
hj^X) = 70 Vi\ - 140 Tb
90 x! - 20
(2.;)
A? + %rjY -L
Note that each impulse response has been multiplied by l/T, so that the expression for the filter output at time T will be independent of T.
Taking the Laplace transform of the impulse response, one finds the transfer function of the filters.
If noise is added into the filters to simulate interference, the noise output of the filters may be expressed in the following manner.^ We define the "autocorrelation function of the filter" as
T-XX ' T hkT(T,T) hk(^T)hk( M r >T)d?i (2.?)0
Figure 2.2 and 2.3 show two examples.The noise power at the filter output for t = T
with no signal present is then
Rss(T) - / \ A T(T,T)Rnn(T)dT ( 2 - 8 )
where Rnn(T ) is the autocorrelation function of the input noise. For white Gaussian noise passed through a simple low-pass RC filter, the autocorrelation function is
11
Rx x ( T ) - Rx x ( ° ) eRC
(2-9)
where Rxx(°) the mean square value of the noise or noise power. Then we have for the noise power output of the filters at time T for the two examples above
- T_+ 77 - 3V / 1) = 2Rnn(°)
R2C2o .
RCRC (2.10)
R. R^C^1°1(?) = 2Rnn(0) -T“
- T
- 2R nn(°) e
P P R C . Tm9 + 3RC
RC[' *
2
(2.11)
For h0(X, T)
h0(X,T)+1
h0(X,T)
12For h,(A,T)
+1
1
R, , (t .T) = ^ - +. 2 ' ^ 1 - 3 Jt I T:birphIT iT
CHAPTER 3 COMPUTER SIMULATION
This system especially lends itself well to study on a repetitive analog computer since, in fact, the actual system would be implemented using similar analog computer techniques.3.1 Polynomial Waveform Generator
The computer setup for generating the Legendre polynomials is illustrated in Fig. 3-1• This setup makes the four quadrant multipliers unnecessary. ASTRAC I (see Appendix) is operated at 10 runs per second.^ The period of the waveforms is equal to 80 milliseconds. The polynomial period is a little shorter than the COMPUTE period of the computer so that an accurate sampling time at T could be set into the computer. Gain "A" is adjusted so that the output voltage Of the polynomial waveform generator is a convenient amplitude.3.2 Matched Filter
The computer setup for the matched filters, illustrated in Fig. 3-2, closely resembles the setup for the waveform generator. The gain "B" is adjusted so that the output voltage of the filters is a convenient amplitude.
13
Computer Set Up For Polynomial Waveform Generator
Fig. 2.1
HI CO
Computer Set Up For Matched Filters
Fig. 3.2
ui
CHAPTER 4 GENERATOR AND FILTER ACCURACY
4.1 Accuracy of the Waveforms
The output of the polynomial waveform generator was observed on an oscilloscope and the waveforms appeared as shown in the following sketches.
20 v/cm
80ms
Fig. 4.1
20 v/cm
80ms
16
17
20 v/cm
80msP
Fig. 4.3
10 v/cm
80msP
Fig. 4.4
18
1 v/cm
80msP4
Pig. 4.5
a4p4 v/as generated here merely to demonstrate the inaccuracy of higher order waveforms. The output amplifier takes the difference of large voltages and because of this, error and noise are magnified.
The output magnitude of higher-order polynomials becomes very small because of the necessity of scaling the recurrence relation. To improve this scaling situation, each polynomial would have to be generated separately at much greater equipment expense.
Fig. 4.6 is an example of the output sum of Pq, P ,P2, Py and P .
19
80msSum of P PP P and P2* 4
Fig. 4.6
The output of the matched filter is illustrated in the following sketches. The input polynomial sum is shown in Fig. 4.7 with a^ equal to zero. The corresponding S3(t) is shown in Fig. 4.8, and it can be seen that S- (T) is approximately equal to zero in agreement with theory. Fig. 4.9 shows the output of the filter, S^(t), with a^ unequal to zero.
4 v/cm
80msSum of PO' rV Fig.
and P
4 v/cm
80msS3(t>;
Fig. 4.8
21
10 v/cm
80msS_(t); 0
Pig. 4.9
Accuracy measurements were made on the four polynomial waveforms. This was done by checking the starting voltage a^-P^O) and comparing it with the input a^ and the voltage at time T, a P .(T). The results are shown in Table 4.1.
matched filter and compared to its theoretical output calculated from the input a^ in Table 4.1. The results are shown in Table 4.2,
TABLE 4.2 ACCURACY OF FILTER OUTPUT
sk(T)meas. sk(T)theo. Gain "B
s0(T) 63.17 64.0 1.60
s1(T) 21.2 21.4 1.60
s2(T) 61.7 64.0 8.00
s5(T) 43.0 45.7 16.00
The effects of computer inaccuracy shows up in the higher order polynomials, which require computer setups with more integrators connected in series. Any error will be successively integrated. Error in the integrators is probably mainly due to the tolerances of the input resistors and the feedback capacitor. Error at the start of the polynomial is probably due to the tolerances of the summing amplifier resistors. Noise, particularly chopper noise, will also contribute to the error.
23For the entire system, from input to output, the
sk(T)theo. - s, (T)meas. greatest error — s~X^TtKeo'J------ X ls about
6 percent for the third-order polynomial. These measurements were taken one polynomial, at a time, so that crosstalk did not enter into the measurements.4.2 Crosstalk
Crosstalk measurements were made in the following manner. A slowly varying modulating signal was sampled by a sample-hold unit (Fig. 1-1) and fed into one polynomial channel, and the output of the other channels was observed. The ratio of the magnitude of undesirable output signal to the input signal was used as a measure of crosstalk.
It was interesting to note that the crosstalk did not change appreciably when a signal was present in the channel being observed.
Crosstalk is due to inaccurate polynomial waveforms and to inaccuracies in the matched filter.
The crosstalk results are shown in Table 4.3«The crosstalk data are normalized so that Gain "A" and Gain "B" both equal unity. In each case, the crosstalk is less than 1.1 percent.
TABLE 4.3 CROSSTALK
CHAPTER 5 STATISTICAL MEASUREMENTS
5.1 NoiseNoise measurements were made with the ASTRAC I
noise generator, which filters a random telegraph wave generated by a radioactive source. The output autocorrelation function of the noise generator is
- MRx x M = Rxx(°)e (‘>.1)
where RC is the filter time constant.The mean square output at time T of the matched
filter 1iq( X) and h- (X) was measured (estimated) with the ASTRAC I hybrid analog-digital statistics computer.^The results are compared with the theoretical results from Eqs. (2.12) and (2.1;'), as shown in Table 5* !•
TABLE r.1NOISE OUTPUT OF MATCHED FILTERS h^ AMD h
Input Noise=313v2 RC=1000 Gain mB"=9*60Filter Mean-square output at time T
measured theoreticalh_ 704v2 712v2
h, 239v2 231v2
25
26The experimental results are seen to agree well with the calculated results.
5.2 DetectionDetection probability measurements were made on
the outputs of hQ and h-. Detection theory indicates that a test statistic useful for detection of a known signal in white Gaussian noise is generated by passing the signal corrupted by the noise through a matched filter.7
Figure 5-1 illustrates the simple detection problem for the detection of a d-c signal which has been corrupted by additive independent Gaussian noise. S^(T) is the expected value of the output of the matched filter when a signal is present. h is the hypothesis that a signal is present; and h^ is the hypothesis that there is no signal. 0[ftk(T)lh1] is the probability density function of the noise and signal and 0[ k(T) I 11q] is the probability density function of the noise with no signal. The input noise has zero mean. The noise creates a region of uncertainty, as illustrated.
For the purposes of this study, a critical level y is determined by a Neyman-Pearson test, that is, the critical level is chosen so that the detection probability, indicated by the area 1-B, is maximized for a given false- alarm probability, indicated by the area A.
27
1-B
Detection of a Known Signal in Gaussian Noise
Fig, 5.1
28The detection probability was estimated directly
with the aid of the AS.TRAC amplitude-distribution analyzer. ® The noise-power output of the filter was measured simultaneously with the ASTRAC sample-averaging unit. The measured detection probability was compared with that obtained from a table of the normal distribution (Table 5°l)•
TABLE 5-1 DETECTION PROBABILITY
Gain "B11 8.00Input Noise Output Noise Det.
meas.Prob.calc. sk(T) yc
h0 300v2 477v2 0.889 0.888 31.7v 5v
h1 3l6v2 153v2 0.797 0.808 21v lOv
The results agree very well.
CHAPTER 6 APPLICATION TO MULTIPLEX TELEMETRY
In actual use, the reset period of the integrators will be made very short compared to the operating period in order to save time.
Each even-order polynomial ends at the same voltage at which it starts. The odd-order polynomials end at minus their starting voltage. To eliminate the latter discontinuity and to conserve bandwidth, the odd-order polynomials can be reversed every other period.
The spectra of the polynomial waveforms are found by evaluation of the Fourier coefficients of the Fourier series for each polynomial over the period interval of length 2T, i.e.
00
Pk(z) = |Fk(0) + X Fk(ra)cosm=l
mTrzT (-TSZST) (6.1)
whereT
Fk(m) = | / Pk(z)cos mTrz
(k—0 , 2; m=0,2,4 - - - )T (6.2)
Fk(m) = f / pk(z)cos mirz
(k=l,3; m=l,3,5 - - -)
29
30
The spectrum for each waveform is given by
F0(m) = 1
(in) — p o7r'"m (m=l,3,5---- )
, . OilF2(m) = —p—g(6.4)
(m=2,4,6 - - -)7T "m
7r"m ir"m210
where m labels successive harmonics of 1/2T. The bar
have been reversed every other period. The spectrum is illustrated in Fig. 6-1.
The even-order waveforms contain only even harmonics, and the odd-order waveforms contain only odd harmonics. Each polynomial has a spectrum peak at the k— harmonic and drops off at 12 decibels per octave at higher frequencies.
should be at least wide enough to pass the spectrum peak for each channel. If an insufficient number of harmonics is passed, the resulting signals will no longer be orthogonal and increased crosstalk between odd-order channels and increased crosstalk between even-order channels will result. Note that crosstalk
on F^ and F^ is understood to mean that the waveforms
It is apparent that the transmission bandwidth
31between odd and even-order channels is only due to Interactions within the polynomial generator and the filter and not due to bandwidth considerations.
In cases where bandwidth is not critical, the polarity alternation of odd-order waveforms may not be required. In this case, the odd-order spectra will drop off at only 6 decibels per octave, and a wider bandpass filter will be needed to preserve orthogonality. The power in each waveform is given by
wk = 2 k V r (6-s)
If each polynomial is to be transmitted with equal power, the k— input signal should be scaled up by the factorV2k + l‘ .
An example of the usefulness of such a system can be illustrated by the following example. IRIG (Inter-Range Instrumentation Group) channel three which has an 11 cps response and occupies 110 cps of bandwidth can be subdivided into at least 5 channels of this system, running at 44 cps, .e.ac,h having 11 cps response.
32
\ 12 db/octaveX T"i— t. ,*,.1 3 5 7
-Z r.'l~:.~ ~ .x.,T T„ z- T%5„'2 4 6 8 10
1/I7 3 5 7 9 11
m
m
m
m
2/ T ~~I ~ ~ i r : —1/ 4 6 8 10 12 mfrequency
m " 1/2T
Frequency Spectrum
Fig. 6.1
CHAPTER 7 CONCLUSION
The fast, digitally controlled repetitive analog computer has proven particularly useful in studying this communications system. Such a machine makes it easy to study the effects of noise on signal-to-noise ratio and detection probability by actual statistical measurements on a large sample of computer runs.
The results of the computer study of this system agree closely with those predicted by the theory of matched filters and detection theory.
33
BIBLIOGRAPHY
1. Ballard, A. H., "A New Multiplex Technique for Telemetry," Proc. National Telemetering Conference, Vol. T, Session 6-2, Communication Theory and Techniques I, 1962
2. Helstrom, Carl, W., Statistical Theory of. Signal Detection, Pergamon Press, 19 0, pp. 91-95-
3- Middleton, David, An Introduction To StatisticalCommunication Theory, McGraw Hill, i960, pp. l6l-l64.
4. Brubaker, T. A., "The Design, Development, and Applications of the ASTRAC Computer," Doctoral Dissertation, Department of Electrical Engineering, University of Arizona, Tucson, Arizona, 1963.
5. Manelis, J. R., "Generating Random Noise With Radioactive Sources," Electronics, September 8,1961, pp. 66-69.
6. Conant, B. K., "A Hybrid Analog-Digital Statistics Computer," Analog-Hybrid Computer Laboratory Memorandum No. 45j Department of Electrical Engineering, University of Arizona, Tucson, Arizona, December,1962.
7- Helstrom, Carl W., Statistical Theory of SignalDetection, - Pergamon Press, 1966, pp. 8"4 "87.
8. Brubaker, T. A. and Korn, G. A., "Accurate AmplitudeDistribution Analyzer Combines Analog and .Digital Logic," Rev. of Scientific Instruments, Vol. 32,No. 3, March, 1961, pp. 317-322.
34
APPENDIXINTRODUCTION TO THE ASTRAC COMPUTER
The Arizona Statistical Repetitive Analog Computer (ASTRAC I) combines a fast memory equipped repetitive/ iterative analog computer with digital logic and control. The resulting synthesis of high speed analog computation with digital automatic programming is of particular interest in connection with Monte-Carlo-type studies of random processes.
Referring to Fig. A.1, an analog computer simulated control system, communications system, queuing problem, etc. is supplied random inputs, random initial conditions, and/or random parameters from noise generators whose statistics are known. Reset pulses from a simple digital control unit cause repetitive simulation of the process under study at 10, 25, 50 and 100 times per second at computing frequencies of 10 to 10,000 radians per second. Accurate sample-hold (analog memory) units read selected process variables at the push-button preset times t^ and tg seconds after the start of each computer run. A hybrid analog-digital statistics computer accepts these samples to computer statistical averages over 100 to 9900 computer runs, as determined
35
36by a preset run counter in the control unit. An amplitude distribution analyzer computes probabilities.
37
DIGITAL TO AMPLITUDE READOUT* DISTRIBUTION ANALYZER
RANDOMIN PU T S
SAMPLE- HOLD I
SAMPLE- H O L D 2
X(T()
Y(T,)
ENSEMBLE
S T A T .COMPUTER
C O R R E L A T IO NF U N C T I O N S ,E T C .
« RUN I — M — RUN 2 M— RUN 3 — M — RUN 4 — ►4 -I------ ►
R E P RATE
t, "I", t, t,START RUN AND t , P U L S E S
P U L S E SC O M PU T E R
X(T,)X(t.)X(T.)
X(t,)
C O M P U T E R O U T P U T
nSAMPLE-HOLD R E S E T J P U L S E S
'A------ 11X1 ^ ______ ___ —. ------- 1 II III 1 , 1 1 1 ii ii M ! !
S A M P L E -H O L D O U T P U T
A Typical Repetitive-Analog-Computer Set Up Together with Four Sample Computer Runs
