Pulse duration and pulse code modulations compared for input power and output signal to noise ratio at high and low signal strengt

Lorens, C. S.

English

NASA Technical Reports Server

N A T I O N A L  A E R O N A U T I C S  A N D  S P A C E  A D M I N I S T R A T I O N  
Technical Report 32-7069 
Nonlinear Characteristics of Pulse- Duration Modulation 
C. S. Lorens 
c 
GPO PRICE $ 
CFSTI PRICE(S) $ 
Hard copy (HC) 1 Tu-  
Microfiche (MF) ~ #45- 
ff 653 July 65 
r , 
(PAOES) 
L fJ/2y/ a*? i ii 
(NASA CR OR TMX OR AD NUMBER) CCMEOORY) 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  I N S T I T U T E  OF T E C H N O L O G Y  
P A  S A D  E N A ,  C A  L I F O  R N I A 
February 15, 1967 
https://ntrs.nasa.gov/search.jsp?R=19670007231 2020-03-16T18:44:15+00:00Z
N A T I O N A L  AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  
Technical Report 32-7069 
Nonlinear Characteristics of Pulse- Duration Modulation 
C, S. Lorens 
Approved by: 
y24.q &hU 
John J. Paulsoaanager 
Space Instrument Systems Section 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  I N S T I T U T E  O F  T E C H N O L O G Y  
P A S A D E N A ,  C A L I  F O  R N  I A 
February 15, 1967 
c 
Technical Report 32-1069 
Copyright @ 1967 
Jet Propulsion Laboratory 
California Institute of Technology 
Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration 
Contents 
1 . Introduction . . . . . . . . . . .  
II . Pulse-Duration Modulation . . . . . . .  
111 . Noise Reduction . . . . . . . . . .  
IV . Sampling . . . . . . . . . . . .  
V . limiter Noise . . . . . . . . . . .  
VI . Summation Noise . . . . . . . . . .  
VI1 . Bias Errors . . . . . . . . . . . .  
VIII . Error Noise . . . . . . . . . . .  
IX . Thresholds . . . . . . . . . . . .  
X . Pulse-Code Modulation Comparison . . . . .  
XI . Optimum Demodulation of Pulse-Code Modulation . 
Nomenclature . . . . . . . . . . . .  
References . . . . . . . . . . . . .  
. . . . .  1 
. . . . .  1 
. . . . .  2 
. . . . .  3 
. . . . .  3 
. . . . .  4 
. . . . .  4 
. . . . .  5 
. . . . .  5 
. . . . .  6 
. . . . .  7 
. . . . .  9 
. . . . .  9 
Figures 
1 . Pulse-duration modulation representing the sample a by the 
reversalof polarityatthetime ( 1  + a/A)(J/2) . . . . . .  1 
limiter to reduce the effects of noise . . . . . . . . . .  2 
bandwidth B = 1 / T  . . . . . . . . . . . . . .  2 
modulation . . . . . . . . . . . . . . . .  6 
2 . The demodulation of a PDM waveform. using a filter and a 
3 . Zero crossing of a PDM waveform that has been filtered with a 
4 . A comparison of pulse-duration modulation and pulse-code 
.. vcvnpr  - .. 32- ! 069 iii 
Abstract 
A comparison of pulse-duration demodulation with pulse-code demodulation 
indicates that there is a pulse-code modulation (PCM) system that will outper- 
form any pulse-duration modulation (PDM) system by about 2% db input power 
at 20 db output signal-to-noise ratio, 4% db at 30 db, and 7 db at 40 db. How- 
ever, the PDM systems outperform the PCM systems at a sufficiently high input 
signal-to-noise density ratio. Pulse-duration modulation thus tends to make better 
use of high signal strengths, while pulse-code modulation makes better use of low 
signal strengths. 
JPL TECHNICAL REPORT 32- I069 
~ 
~~ 
Non I i nea r C ha racte r istics of Pu Ise-Du ration Mod u la t ion 
1. Introduction 
The most widely known analyses of pulse-duration 
demodulation are the linear analysis of A. J. Viterbi 
(Ref. 1) and the nonlinear analysis of M. H. Nichols and 
A. T. Bublitz (Ref. 2). The present report expands on 
these studies by producing new results on the nonlinear 
behavior of pulse-duration demodulation both above and 
below threshold. Although the nonlinear demodulator 
of Nichols and Bublitz exhibits better performance than 
the linear demodulator of Viterbi, its threshold behavior is 
much poorer than that assumed by Nichols and Bublitz. 
A comparison of pulse-duration with pulse-code de- 
modulation indicates that for any given output signal- 
to-noise ratio there is a PCM system that will outperform 
any PDM system by about 2% db input power at 20 db 
output signal-to-noise ratio, 4% db at 30 db, and 7 db at 
at 40 db. Considerably more bandwidth is required for 
a PDM system than for an equivalent PCM system. On 
the other hand, every PDM system will outperform every 
PCM system at a sufficiently high ratio of input signal 
energy to noise density; the PDM system excels in per- 
formance over a comparable PCM system at almost all 
power levels except the lower ones. Pulse-duration mod- 
ulation thus tends to make better use of high signal 
strengths, while pulse-code modulation makes better use 
of low signal strengths. 
II. Pulse-Duration Modulation 
Figure 1 shows a method of representing the sample 
a by pulse-duration modulation. This type of modulation 
has the capability of continuously representing samples 
Fig. 1. Pulse-duration modulation representing the 
sample a by the reversal of polarity at the time 
(1 4- a / A )  (1/21 
JPL TECHNICAL REPORT 32-1 069 1 
FILTER m ( t )  * (BANDWIDTH 8) 
ranging in value from -A to +A. The duration of the 
positive pulse is directly proportional to the value of 
the sample a. Demodulation of pulse-duration modula- 
tion thus requires synchronization to establish the zero 
time reference and isolation to separate the waveforms 
representing different samples. The various methods 
available for providing the synchronization and isolation 
will not be discussed in this report. 
- 
INTEGRATE 
X ( l )  LlMl  TER Y ( l )  AND DUMP Z ( f )  
* (OUTPUT-*I) * q:w dt 
0 
T 
The demodulation of the waveform of Fig. 1 is per- 
formed by filtering with a bandwidth B (cycles per sec- 
ond) equal to twice the low-frequency bandwidth BL, 
symmetrically limiting, and integrating the resultant 
waveform as in Fig. 2. 
Under noiseless conditions, the output of the integrate- 
and-dump circuit would be the integral of the pulse- 
duration waveform scaled to produce the sample a:  
z (T)  = $lT y(t) dt 
= A{+ T [(l + i):] - [T  - (1 + :):I} 
- - a  
Where the sample values are uniformly distributed be- 
tween + A  and - A, the output power is 
111. Noise Reduction 
A limited amount of noise can be removed from the 
pulse-duration waveform by first filtering the waveform 
and then limiting it to recover the transition point. 
Viterbi's analysis did not use the limiter that had been 
previously proposed by Nichols and Bublitz. 
A spectral density of noise power N / B  for frequencies 
from - B / 2  to + B / 2  corresponds to a high-frequency 
noise power N in a bandwidth B that has been inco- 
herently demodulated. Coherent demodulation (as used 
by Viterbi) would lead to a spectral density of N / ( 2 B ) .  
The filtering of the pulse-duration waveform distorts 
the transition so that when noise n(t)  is added to the 
waveform at the transition time, the transition moves by 
a value of t ,  approximately equal to the value of the 
noise divided by the slope 2S%/r of the transition as in 
Fig. 3. 
The fall time 7 is related to the double-sided band- 
width B in cycles per second by the relation (Ref. 3) 
1 
which also corresponds to the sampling theorem (Ref. 4). 
The variation t,, of the transition time, owing to noise, 
produces an additive noise zn at the output of the inte- 
gration: 
tn  z n = 2 A -  T 
Fig. 3. Zero crossing of a PDM waveform that has been 
filtered with a bandwidth B = 1 / ~  
2 JPL TECHNICAL REPORT 32-1069 
The average noise power, because of variations in the 
transition time, is thus 
- A2N 
z:, = s(TB)z 
which produces an output signal-to-noise power ratio 
where k = TB = T / T  is the number of fall times 7 in the 
interval T .  This relation is valid only for increasingly 
higher input signal-to-noise ratios as the index k is made 
larger (7% db for k=4; 9% db for k=16; and 10% db for 
k=64). Two other noise components become important 
as the input signal-to-noise ratio deteriorates. The first 
noise component is due to the breaking through of the 
limiter at times other than the transition time. The sec- 
ond noise component, which is found at still lower signal- 
to-noise levels, is bias distortion in the integration of the 
pulse-duration waveform and is also caused by the break- 
through of the limiter. 
IV. Sampling 
The use of the sampling theorem (Ref. 4) provides a 
means of characterizing the waveform x( t )  at the output 
of the filter of bandwidth B in Fig. 2. The waveform 
x ( t )  can be sufficiently described by k = TB equally 
spaced samples in the time interval T .  Each sample is 
plus or minus the signal amplitude SIh plus noise of 
variance N resulting from a spectral density of N/B be- 
ing passed through the filter of bandwidth B.  
The output of the limiter is then a sequence of equally 
spaced samples of + 1 and - 1, depending on the polarity 
and magnitude of the signal and noise. 
In order to perform a number of summations, the 
sample values of x ( t )  are assumed to occur on the odd 
integers between +2m and -2m. The actual transition 
time is assumed to occur at a time h belonging to the 
even integers. Thus, in the noiseless case, the limited 
waveform y(t) would be + 1  at sample times t < h and 
-1 at sample times t > h: 
h t < h at m +- samples 2 
h t > h at m -- samples 2 
+ 1, 
Y(t) = 
- 1, 
Assuming a fixed transition time h, integration of the lim- 
ited waveform y ( t )  is then the summation 
= A 2m [ ( m  + :) - (m - :)I 
which is the transition time h scaled by A/(2m). 
The variance of the output of the integrator in the 
noiseless case is then 
even 
- A 2 m + 1 - A 2 k + 2  A2 
3 m  3 k - 3  
_--_---- 
where the transition times are assumed to be equally 
distributed over the 2m + 1 possible transition times, 
k = TB = 2m, and 
m(m + 1)  (2m + 1)  
h = -2m 3 
even 
V. limiter Noise 
For Gaussian noise, the limiter in Fig. 2 reverses its 
polarity because of noise with probability p , :  
The polarity remains unchanged with probability p o  = 
1 - p1. 
The sample output of the limiter y(t) for times before 
the transition time h is then $ 1  with probability po  and 
- 1  with probability p , .  For times after the transition 
time h, these probabilities are reversed: 
t < h  {Ly: t > h P { y ( t )  = +1} = 
t < h  
{ P I ,  t > h P {y(t) = - 1 )  = Po7 
3 
. 
VI. Summation Noise 
The noise at the output of the summation with zero modulation (h = 0) due to breaking through the limiter is then 
The first term is the sum of the diagonal terms; the second term is the sum of the two principal minors, less the 
diagonal; and the third term is the sum of the two off-diagonal minors. 
VII. Bias Errors 
The error in the summation 
is the difference 
which has the nonzero mean value 
Thus for any particular transition time h, the integration produces a bias error of ( -  Ahp,)/m. The average squared 
value of the bias error is then 
even 
4 J P L  TECHNICAL REPORT 32- I069 
VIII. Error Noise 
The average squared error is the sum of the summation noise N ,  and the bias error noise N b .  This can be obtained 
directly by finding the average squared error N h  for a fixed transition time h and then averaging over the possible 
transition times : 
- 
After the averages y(t)y(s) and y ( t )  are evaluated, the average squared error is 
where the first term is the summation noise N ,  and the second term is the square of the bias error. Further averaging 
of the transition times h produces the result 
N = N ,  + N b  
where 
IX. Thresholds 
Above threshold, the only noise in a PDM system is 
the noise (A2N)/(Sk2) due to a slight displacement in the 
transit time. However, at threshold, the noise begins to 
change the polarity of the output of the limiter and thus 
produces an increase in the output noise resulting from 
the noisy summation. This term varies essentially with 
the first power of the breakthrough probability p l  and 
has a maximum value of A2/k.  Further, below threshold, 
the bias error predominates by adding in the term 
(4/3)  AZp; varying with the square of the breakthrough 
probability p l  and having a maximum value of AZ(4/3).  
The total average squared noise is then essentially 
N = -  A2N 4A2plpo 4 Sk2 + --i g A'p; k 
as compared with the average squared signal 
A2 s = -  
3 
The output signal-to-noise ratio is then 
Figure 4 is a plot of the output signal-to-noise ratio as a 
function of 
- 
B 
for various values of k.  Viterbi's linear result (using non- 
coherent demodulation to baseband) 
is obtained for k = 1 and is considerably less than that 
obtained with nonlinear demodulation. 
JPi TECiiEiiCAi REPORT 31-! 069 5 
I 
6 -  
4 -  
2 -  
lo4 - 
6 -  
4 -  
0 
I- 
2 2 -  
(L E 103- 
w 
I- 
A 6 -  
Ll 4 -  a 
2 2 -  
z 
v) 
W 
3 
$5 102- 
6 -  
z a 4 -  
W 
5 
2 -  
put signal-to-noise ratio of 
s o  - L' 
N o  1 + 4L2p -- 
where L is the number of quantization levels and p is 
the probability of binary error: 
where the lower integration limit is 
ST 
(+log, L )  
This result is obtained by noting that the average squared 
signal uniformly distributed over + A  is 
A' s = -  
O 3  
The average squared quantization error N ,  resulting 
from L quantization intervals is similarly 
IO' 2 4 6 IO2 2 4 6 IO3 2 4 t 
u = s r/(N/B/ 
Fig. 4. A comparison of pulse-duration modulation 
and pulse-code modulation 
The 3-db deterioration points (sometimes defined as 
threshold) occur at higher and higher input signal-to- 
noise ratios for large time-bandwidth products (8% db 
for k=8, 9% db for k=16, 10 db for k=32, and 10% for 
k=64). For a fixed time interval T and a coefficient k, the 
bandwidth required is k/t: 
i = O  
k BPD,  = - T 
X. Pulse-Code Modulation Comparison 
As a means of evaluating the effectiveness o PD it 
is compared with PCM, which is known to have an out- 
A' N ,  = - 3L' 
False detection of the most significant binary digit 
would result in an error of magnitude A. False detection 
of the next most significant digit would result in an error 
of magnitude A/2, etc. The average squared false detec- 
tion error is then 
where p is the probability of false detection of a single 
binary digit. 
The output signal-to-noise ratio then follows imme- 
diately when the output noise is set equal to the sum of 
the quantization noise plus the false detection noise. 
6 J P l  TECHNICAL REPORT 32-1069 
The integrate-and-dump circuit used in the detection 
of the binary digits has an effective bandwidth Be of 
l/t, where t is the integration time: 
where the linear transfer function of the integrate-and- 
dump circuit is 
where the lower integration limit is 
(qh 
assuming that each binary digit uses an equal portion of 
the interval T for detection, 
The output signal-to-noise ratio for PCM is plotted in 
Fig. 4 for various quantization levels. The double-sided 
bandwidth in cycles per second required for a PCM 
system is 
The variance of the detection noise is then 
as compared with the bandwidth for the PDM system 
1 N  N e  = -- 
t B  
where the baseband spectral density is N / B .  
False detection occurs where the noise changes the 
polarity of the output digit. For Gaussian noise, this 
occurs with probability 
1 exp (G) du 
k 
BPDM = - T 
The wider bandwidth of PDM results in an increasing 
probability of error, which is partly offset by the PDM 
system’s reducing the noise by averaging. However, the 
smaller noise bandwidth of the PCM demodulation pre- 
dominates in an exponential manner to produce a better 
performance at any fixed output signal-to-noise ratio. 
Over wide ranges of input signal to noise, the PDM 
system makes better use of the higher signal-to-noise 
ratios and thus can outperform any fixed PCM system. 
XI. Optimum Demodulation of Pulse-Duration Modulation 
A more sophisticated approach to the demodulation of pulse-duration modulation is the use of the optimum solu- 
tion for minimum mean-square error. This solution is highly nonlinear and at present is not in a form that can be 
constructed readily. 
The optimum mean-square filter requires that the output z ( T )  of the filter be the conditional average 
JPL TECHNICAL REPORT 32-1069 7 
4 
where m is the input waveform, a is the data sample, and p(a I m) is the conditional probability of the data sample for 
the given input waveform m. Where the data sample is uniformly distributed, this expression can be written in the 
more computable form 
where p(m I a)  is the conditional probability of input waveform m, given data sample a. The denominator of this ex- 
pression normalizes the averaging process. The averaging process of the numerator is based on the probability of the 
waveform m resulting from a particular data sample a. The data associated with small values of p(m I a) are ignored, 
while the other data are averaged in relation to their probability of occurrence. A close approximation to this type of 
demodulation is the use of a limiter followed by an integrate-and-dump circuit. 
Another approximation can be obtained by assuming a discrete model where the input waveform is sampled on an 
odd integer base between +2m and - 2m, and where the transition time h is assumed to occur at an even integer. 
The pulse-duration signal S ( t )  has the representation 
On the assumption of additive Gaussian noise with distnuution 
p(n) = ( h ) m  1I +, 1 %  e q  [& E 1 + Itu n(t)n(u)] 
t = - 2 m  u=-2m 
the conditional distributions become 
where + is the correlation matrix and 4 is the tu cofactor. I l tu 
The optimum processor is then the mechanization of the formula 
8 JPL TECHNICAL REPORT 32-1069 
V 
On the assumg ion that the correlation matrix 9 is diagonal (9 = NI), the formula reduces to 
which still represents a rather formidable computing task. The computation could, however, be feasibly performed on 
a high-speed computing system. 
Nomenclature 
sampling interval 
sample 
bandwidth 
effective bandwidth 
low-frequency bandwidth 
number of fall times in the interval T 
number of quantization levels 
input waveform 
high-frequency noise power 
bias error noise power 
average squared false detection error power 
variance of detection noise power 
Nh average squared error power for fixed transition 
time 
N, average squared quantization error power 
T 
9 
summation noise power 
spectral density of noise 
noise waveform 
pulse-duration signal 
variation of the transition time 
waveform at output of filter 
waveform at output of limiter 
output waveform 
fall time 
correlation matrix 
References 
1. Viterbi, A. J., Classification and Evaluation of Coherent Synchronous Sampled- 
Data Telemetry Systems, Technical Report No. 32-123, Jet Propulsion Labora- 
tory, Pasadena, Calif., June 15, 1961. 
2. Nichols, M. H., and Bublitz, A. T., “The Effect of Different Types of Video 
Filters on PDM-FM and PCM-FM Radio Telemetry,” IRE Tramactions on 
Space Electronics and Telemetry, Vol. SET-6, No. 2, June 1960, pp. 85-92. 
(The appendix is of particular interest.) 
3. Guillemin, E. A., Communication Networks, Vol. 11, John Wiley and Sons, 
Inc., 1951, p. 487. 
4. Shannon, C. E., “Communication in the Presence of Noise,” Proceedings of 
the IRE, Vol. 37, January 1949, pp. 10-21. 
JPl TECHNICAL REPORT 32-1069 9 


Nonlinear characteristics of pulse-duration modulation

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server