Harmonic distortion in frequency demodulator using feedback, delta modulation, recursive signal processing techniques, and multipath fadin

Clarke, K. K.

Pickholtz, R. L.

Schilling, D. L.

English

NASA Technical Reports Server

A Space Communications Study 
Status Report 
December 15, 1968- March 15, 1969 
Prepared for 
National Aeronautics and Space Administration 
Electronic Research Center 
under 
NASA Grant NGR - 33 - 006 - 020 
Principal Investigator Co- Principal Investigators 
Donald L. Schilling 
Associate 
Kenneth K. Clarke 
Professor 
Raythond L. Pickholtz 
Associate Professor 
Department of Electrical Engineering 
Polytechnic Institute of Brooklyn 
https://ntrs.nasa.gov/search.jsp?R=19690016977 2020-03-12T04:03:48+00:00Z
1 
Introduction 
This status report summarizes three areas  of research being supported 
under NASA Grant NGR - 3 3  - 066 - 020 during the period December 15, 1968- 
March 15, 1969. The topics covered in the letter are "Harmonic Distortion 
in the Frequency Demodulator Using Feedback", If Delta Modulationt1, 
Recursive Techniques in Signal Processingff and It Multipath Fadingf1. 
2 
I. HARMONIC DISTORTION I N  AN FMFB 
Harmonic distortion in an FMFB was considered in the first quarterly 
status report (September 15, 1968-December 15, 1968). Figure I. 2 of 
that report showed the variation of distortion with the various FMFB para- 
meters. Since then, we have been able to obtain a closed form solution for 
the harmonic distortion and a r e  presently determining an expression for  the 
int e rmodulat ion distortion. 
The equations governing the operation of the first-order FMFB have 
been given in the 1968 annual report. They are: 
+m sin E = -i G t l  - t- a A Ly 
and . 
_. A t A =  cos E 
CY 
where 
%co (3) 
is vc 0' Thus, the input 'to ' the vco, which is  directly proportional to  + 
equal to the sum of 9, 
the error .  
. 
the modulating signal, and E which represents 
The distortion is inherent in i. 
It can be shown that in  the region of interest where the harmonic 
distortion is less than -40dB that Eq. 1 can be used to calculate distortion 
with A set equal to  unity. This re$ult is clear from Eq. 2, since for low 
distortion E <<1, cos E fS 1 and AfS 1. 
Solving Eq. 1 for  E and then calculating the distortion yields 
3 
where D is the harmonic distortion in dB, /3 is the modulation index 
assuming + (t) = p sin w t, aW = PLJ, G is the feedback-factor in the 
FMFB, and 2,a, is the bandwidth of the IF filter preceding the l imiter-  
m m 
. discriminator in the FMFB. 
For  example, i f  the FMFB is to be designed so that the harmonic 
distortion is less  than or equal to -60dB and p=5,  
11. DELTA MODULATION 
Delta Modulation (DM) is a technique of transmitting an analog signal 
by first encoding the signal into binary digits (bits). 
then be transmitted using PSK, FSK, DPSK, etc. 
of delta modulation is the relative simplicity of the encoder and decoder as 
compared to the encoder and decoder for PCM. However, the quantization 
(or granular) noise in DM is much greater than that found for PCM: 
The bit stream may 
The principle advantage 
1 
($)DM 20(fMT) 3 
(apcM= 2N 
where f 
duration of the encoded signal, and N is the number of binary digits required 
t o  identify a signal which has been quantized into M levels. 
is the bandwidth of the signal to be delta modulated, T is the bit 
M 
N Thus M = 2 
Equations 1 and 2 can be compared by relating fMTand N. The 
1 duration of a bit in PCM is T = i f  sampling is performed at the 
Nyquist rate. Thus, if  w e  require T =T w e  have: 
4 
Thus, the "cost" required for the simplicity achieved in DM i s  20dB for 
this case. 
In many cases the increase in quantization noise is found to  be tolerable. 
We have listened to taped music played through our DM and have not found 
it unpleasant. The transmitter is shown in Fig. II. 1, and the receiver in 
Fig., II. 2. 
TRANSMITTER 
The encoding of the input signal is performed using the circuit of 
Fig. 11. 1. The DM, as seen from this figure, is a feedback circuit. The 
input and feedback signal a r e  subtracted in K:l and the e r r o r  is amplified 
in K2. T and T4 provide additional amplification so that the output of T 3  3 
is OV when the e r r o r  is negative and the output of T5 i s  OV when the 
e r r o r  is positive. The e r r o r  voltage is sampled in T 
positive pulse is emitted from T when the e r r o r  "is not" negative when 
sampled, and a positive pulse is emitted from T7 when the e r r o r  ' I  is not" 
and T7 so  that a 6 
6 
positive when sampled. 
T7 "resets" F 1. 
A pulse from T b  l tsetsf l  F1 while a pulse from 
is the encoded bit stream. The output of F 
The feedback signal is derived by "integrating" the output of F1. 
The integration is performed by an RC (47KD, 0. OlpF) low pass filter. 
T8, an emitter follower is used for isolation. 
The sampling pulses a r e  generated by a monostable rnultivibrator 
The clock used to drive Mlis the astable having a pulse width of 1~ sec. 
multivibrator A1. The frequency of A 1  is adjustable from 50 KHz to 
5 
approximately 200KHz. 
clock can be inserted. 
RECEIVER 
I€ other clock frequencies a r e  required an external 
The receiver, shown in Fig. II. 2 illustrates the principal advantage 
of DM. 
and several stages of low pass filtering. 
Adaptive Delta Modulation 
The receiver consists of a bistable multivibrator, a linear amplifier, 
One of the problems associated with DM is ''slope overload noise". 
This noise results when the feedback signal fails t o  change a s  rapidly as  the 
input signal. 
techniques which varies the maximum slope of the feedback signal. 
currently investigating several different adaptive approaches. 
Slope overload noise can be drastically reduced by using adaptive 
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8 
III. RECURSIVE TECHNIQUES IN SIGNAL PROCESSING 
Various algorithms and their general application to diverse communi- 
cations problems have been studied, The objectives vary from providing 
optimum digital processing of signals for detection and estimation to such 
diverse and important problems as data equalization, data compression and 
pattern recognition. 
the fact that the processing of the signals i s  recursive in form, i. e. previous 
computations a re  used to effect subsequent outputs. 
Underlying al l  of the practical and useful methods is 
Since the las t  report, two significant advances have been worked out 
at  Polytechnic Institute of Brooklyn in  the area of recursive signal processing 
techniques. 
detection of known signal in noise. . 
The first is an extension of previous work on the recursive 
1 
2 The present work generalizes the recursive detector by permitting 
complete freedom to the dynamics of the noise. 
to detect one of M signals in the presence of additive, Gaussian, nonstationary 
noise which is described by being represented as  the output of a t ime vary- 
ing linear dynamical system driven by white noise. 
observes 
Basically, the problem is 
Specifically, the receiver 
Y( t )  = Si(t) + X(t) i = 1..  . M (1) 
when X(t) is the first component of the state vector generated by: 
- X (t) = A - X (t) + - b (t) W(t) (2) 
A is the state matrix and W(t) is white noise with covariance qO( t  )6(t-tl). 
The solution to the problem of finding the optirrium recursive detector is 
1 
1. Pickholtz, R. L. and R. Boorstyn "A Recursive Approach to Signal De- 
2. Pickholtz, R. and R. Boorstyn "Recursive Detection with Numerator 
tection" PGIT (PEEE) May 1968. 
Dynamics" Presented at  International Symposium on Information Theory, 
Jan. 1969. 
9 
outlined in Fig. III. 1. 
published in full at a later date. 
(The detailed derivations a r e  omitted and w i l l  be 
The figure shows the signal processing operations which a r e  to be 
performed on the observed signal Y(t). 
equivalent state is formed by using the waveform data Y(t)  as  the input to 
a recursive Kahan-Bucy filter. 
First, a vector estimate of the 
The dynamics of this operation a re  a s  
A - Y (t) = A(t)  - ? (t) f Z(t) [ G(t) - CTA(t)?(t) - ] 
when A(t)  is the dynamics matrix of the noise producing system as in (2) 
and the gain factor is given by 
M(t) is found from the Ricotti equation from Kalman-Bucy theory. 
and, 
C =  - 
Once the state estimate is formed, we show that this estimate is  Markov 
and thus our previously derived techniques fo r  a recursive test statistic 
1 
(log of the likelihood ratio) can be used. The result is 
where the Bn and Kn can be found from the statistics of the noise and - 
the index n refers to the sample time of the processed estimates. 
Equations (3) and (5) can be easily implemented in real  time in 
either digital or analog form. The advantages are: 
(i) 
(ii) They a re  causal, real-time operations. 
They are  the only known solutions f o r  nonstationary noise. 
10 
(iii) Being recursive operations they require a minimum of compu- 
tational effort and/or memory capacity. 
Possibly of greater significance is the general insight that js obtained 
. for general processing of signais for detection when the sample spacing used in 
(5) (represented by the index n) is allowed to go to zero so that the signal 
i s  essentially processed in a continuous fashion. The results of this analysis 
leads to the following: 
A 
6 (t) = k (S(t) - ;,(t) ) (Y(t) - Y,(t) ) 
A 
with corresponding initial conditions. The subscripted quantities S2(t) and 
Y (t) a r e  the second scalar components of the output of the recursive state 
estimator. 
A 
2 
The most interesting and important aspect of this result is that 
it can be shown that forming [ Y (t) - $,(t)] from the observation Y(t)  is 
equivalent to performing a causal, and causally invertable, prewhitening 
operation on the data Y(t) .  [ Y (t) - Y z ( t ) ]  is shown to be a white process. 
3 This operation, which was also recently obtained by Kailath and Geesey , 
has been called an innovations process since the operation of forming the 
white process can be viewed as one of extracting all  the new and nonre- 
dundant data f rom the observation. 
out and some computer runs will be performed. 
h 
Explicit examples have been worked 
The second area  of work in recursive detection is the development 
The algorithm and analysis of a new iterative algorithm for  optimization. 
is called second order gradient (SOG) iteration. The new algorithm is shown 
to have more rapid convergence and smaller limiting mean square e r r o r  than 
the standard gradient algorithm when operating with noisy data. The 
3. Kailath, T. "The Innovations Concept in Detection and Estimation . 
Theory", International Symposium on Information Theory, Ellenville, 
N.Y., Jan. 1969. 
L 
11 
technique has been applied, at present, to adaptive equalization with 
excellent simulation results. The method has immediate application also 
to adaptive array processing, adaptive detection of F M  and other analog 
signals and to other optimization problems. The details and results will 
be reported more fully in the next quarter. 
12 
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13 
IV. MULTIPATH FADING 
A program of investigation of the effects of multipath fading upon 
the operation of threshold extending FM receivers is underway. 
The PIB random channel simulator is being used in a'yrozen channel 
mode" to produce results of the form illustrated in Figures 1 and 2. 
these figures one sees the wideband spectral response of the channel and 
the detected FM response of the channel. 
output via both a wideband and an "optimum" band low pass filter is shown. 
Figure 1 indicates a case with only two dominant paths. 
tone, or  binary FM suffers almost no distortion. 
with many small  transmission paths, 
signals a r e  almost completely wiped out [ the sine wave superimposed over 
the actual lloptimuml' low pass filter output signal was received via a 80dB 
attenuator instead of via the simulator. It merely serves to indicate that 
the system is functioning correctly. ] 
hi 
In the last case the discriminator 
In this case single 
Figure 2 indicates a case 
In this case single tone or  binary FSK 
14 
PROBABILITY OENSIM@F THE OETECTED ENVELOPE OF A 
FM SlW4.L UOOULAlED TO tKQkHI WITH A W J H I  
TRIANGULAR SIGNAL SCATTERER 3C RXEO POSITION 
R E S P W K O F C ~ ~ N ~ ~ ~ I + ~  
WED msinoN m ~ e c  FROM STARI 
cc P ~ T  m L E ~ T  E ~ E  OF TRACE 
%CKET"Al 3 T W H z  SCATXRER L 
- BV 
74dB- 
,7v 
CHANNEL RESPONSE 6v 
' W  
~ 4v 
8368 RECEIVER RESPONSE 
0668 
9MB 
- ZV 8040 ATTENUATOR 
EECTED FM SIGNAL lm IOLH2 
b(.?144LHI UPPER ?RACE BROAD 
BANNO L O W  mACE WA 1 ML Uz. 
JSM) 3700 37 
PDUR POLE NP*IMKLY PUT n m n  
FREWENCY IN KILOHERTZ 
DETECTED ENVELOPE RESWNSE 11 FREQUENCY M&ill,d#r AND OSr/dw SCATTERER 3r 
Fig. 1 
15 
77dB 
7 4 d ~  
86dB 
89dB 
9 2 d 8  
' 98dB 
104dB 
,RECEIVER RESPONSE 
80dB ATTENUATOR 
1 
3650 3700 3750 3800 3 f k O  
FREQUENCY IN KILOHERTZ 
i 
DETECTED ENVELOPE RESPONSE VI FREQUENCY 
O.2V/div 
lOpsec/div - 
RESFONSE OF CHANNEL TO A 3 . 5 ~ ~  
VACKEYAT 3.75MHr. SCAlTERER 4. 
flXED POSITION 680~5% FROM START 
OF PACI(ET TO LEFT EDGE OF TRACE. 
P ( V I  FOR RECEIVER ALONE 
[Bod8 ATTENUATiON] 
P(V) FpR CHANNEL 
p (v )  
DETECTED FM SIGNAL. fm=lOkH~,  
Af=t144k HI. UPPER TRACE BROAD 
BAND LOWER TRACE VIA A 20k Hr, 
FOUR POLE MAXIMALLY FLAT FILm. 
2Oparc/div AND 0.5v/div. SCATTERER 4. 
I I I  I 
6V dV 4V 3V 2V k 6 
PROBABILITY DENSIM OF THE DETECTED ENVELOPE'OF A 
FM SIGNAL MODULATED TO flOOkHz WITH A IOOHr 
TRIANGULAR SIGNAL. SCATTERER 4. FIXED POSITION. 
Fig. 2 


A space communications study Status report, 15 Dec. 1968 - 15 Mar. 1969

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