Statistical analysis of pulse frequency modulation systems with white noise inpu

Chon, Y. T.

Hutchinson, C. E.

English

NASA Technical Reports Server

Presented at the 
1968 IEEE National Telemetering Conference 
April 11, 1968, Houston, Texas 
An Analysis of the Effect 
of a Particular Class of 
PFM on Noise Inputs* 
C. E. Hutchinson 
Y. T. Chon 
Electrical Engineering Department 
University of Massachusetts 
- 
I 
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*The research reported herein was partially supported by NASA 
under Grant NGR 22-010-018. 
. - -  
https://ntrs.nasa.gov/search.jsp?R=19680025093 2020-03-12T07:01:24+00:00Z
2 
PFM with Random Inputs - Hutchinson, et. al. 
w 
LIST OF SYMBOLS 
Expected value of [e] 
Effective cutoff frequency of band-limited white noise 
Bandwidth of CPFM 
Defined as EITnxo] 
Output of PFM integrator 
Probability of [ = ]  
Transition probability density 
Autocorrelation function of the impulse process and defined 
as E [S (t+i)S (t) 3 
PFM reset threshold level 
Impulse process 
Spectral density of a white noise 
Spectral density of the impulse process 
Sign of ( 0 )  
Instantaneous impulse period 
Impulse occurrence times 
Input of a PFM system 
Variance of [ - 3  
Random process 
Output of a PFM system 
Random intensity of impulse 
Average number of random impulses per unit time interval 
Dirac delta function 
Infinitesimal mean 
Defined as E[a.a ] 
Infinitesimal variance 
3 j+n 
Angular frequency, i.e., w = 2nf  
Number of pages - 13 
Number of Tables - 0 
Number of Figures - 5 
3 
ABSTRACT 
- 
degree of immunity to noise, it is presently of interest in 
many modern communication and control systems. 
presents a quantitative statistical analysis of PFM systems 
with a noise input (namely, white noise). Some concepts of 
the Impulse process and the first passage time problems of 
diffusian process are introduced, which are analogous to the 
noise transmission of PFM systems. 
spectral density functions are determined for the output o f  the 
PFM system with a white noise input. 
analytical and experimental results is presented. 
Because Pulse Frequency Modulation (PFM) provides a high 
This paper 
The autocorrelation and 
A comparison between the 
4 
I. INTRODUCTION 
During recent years communication and control engineering have 
merged in many related research areas, one of which is the application 
of Pulse Frequency Modulation (PFM). 
carrier is the time between the emission of two rectangular pulses 
with identical width and amplitude. 
to be impulses for analytical purposes. 
herein. 
In a PFM system the information 
The pulses are generally assumed 
This practice will be followed 
It has been demonstrated that PFM presents many advantages in 
engineering applications [Farrenkopf, Sabroff, and Wheeler, 19631. 
Most important among these advantages is a high degree of noise immunity 
However, little is known about the actual degree to which PFM systems 
are immune to noise and/or the responses to random inputs in a statistical 
sense. 
In this paper, a statistical analysis of the responses to a PFM 
system is developed, with particular attention given to the cases of 
Integral Pulse Frequency Modulation, IPFM [Li, 19611 and Sigma Pulse 
Frequency Modulation, CPFM [Pavlidis and Jury, 19651. 
Integral Pulse Frequency Modulation is defined as one that emits 
an impulse whenever the magnitude of the integral of the input signal 
reaches a given threshold value. 
pulse so that the successive integration always starts from zero. 
output is then a train of impulses whose instantaneous frequency in- 
creases linearly with the input signal magnitude. 
one type of IPFM, namely double-signed IPFM which can generate impulses 
OE both signs depending upon the sign of the time integral of the 
The integrator is reset after each 
The 
In this work, only 
activating signal is considered. 
Fig. 1. 
A block diagram of IPFM is shown in 
If u(t) denotes the input, y(t) the output, p(t) the output of the 
5 
integrator, and r the threshold value, the following two equations 
describe the behavior: 
where sgn(p) = + - 1 depending on the sign of p and 6 is a unit impulse 
(a Dirac delta function). 
(1) represents a resetting of p to the zero value immediately after an 
output pulse occurs. 
The second term of the right hand side of 
A more general scheme of PFM is to feed the signal to a low-pass 
filter and emit an impulse when the output reaches a certain level. 
One of the cases where the filter is of first order and linear has been 
presented as CPFM, the block diagram of which is shown in Fig. 2.  
If u(t) denotes the input, y(t) the output, r the threshold value 
and K the bandwidth of CPFM, the following two equations describe the 
behavior: 
where p(t) is the integral of the sum of the input plus a function, 
Kp, of the integrator output. 
The CPFM systems present many advantages over the IPFM system 
such as improved stability and ease of physical implementation. 
obvious that IPFM is a special case of CPFM which occurs when K = 0. 
It is 
6 
11. PROBLEMS AND ASSUMPTIONS -
When the input u(t) to a PFM unit is a random function of time, the 
output of the modulator will be an impulse train whose instantaneous 
impulse period Tk is a random function of different distribution. If 
the statistdcal properties of Tk can be found through the statistical 
properties of the input u(t), the autocorrelation function of the 
modulator output y(t) and hence its power spectral density can be 
determined. 
available. 
Thus a quantitative measure of the effect of PFM will be 
The approach taken is to first determine properties of Tk for both 
IPFM and CPFM driven by random processes. 
functions and the power spectral densities will be determined. 
initial assumptions are in order, however, and these are given below. 
Then the autocorrelation 
Some 
Assumption I: 
process) which has: 
The input of the PFM unit is white noise (pure random 
a) 
b) stationary, normally distributed magnitude, and, 
c) zero mean. 
constant spectral density, So,  for all frequencies, 
Assumption 11: 
is effected instantaneously. 
The resetting of the integrators after each impulse 
Assumption 111: The emitted impulses at the output have unit area. 
111. IMPULSE PROCESS 
When attempting to discuss the statistical performance of PFM, the 
problem of determining the spectrum, and the autocorrelation of a random 
impulse train arises. 
consisting of an infinite train of impulses occurring at random times 
with random intensities. 
is defined as: 
The problem involves the study of a random process 
As in Lenemen[l966] the impulse process S ( t )  
OD 
a 6(t - tn) - 1  n n -- - -  
7 
where the firing times tn constitute a stationary point process [Beutler 
and leneman, 19661, and the intensity modulating coefficients an a 
stationary random process which is independent of the point process. 
That is, the impulse process consists of an infinite train of delta- 
functions occurring randomly in time and having random intensities (areas). 
In the same reference it was shown that the autocorrelation for 
S(t) is 
OD 
Rss(r) = BP(o)  TI + 8 1 P C ~ )  fn ( 7 )  ~ 2 0  (6) 
n= 1 
where 
B = average number of pulses per unit time 
P ( C 1  = E[e,21 
fnlT) = 6 f1lT-U) fn-l(u) du 
.p(n) = E[a.a ] J j+n 
T 
and fl(T) is the probability density function for the interval between 
two consecutive firing times. 
density function associated with n consecutive intervals. 
fn(r) therefore, denotes the probability 
The intensity of output impulses of PFM, an, constitute a stationary 
random process which is independent of the stationary point process {tnl 
and has outcomes of +1 or -1. In addition, it can be noted that in case 
of PFM with a white noise input, 
and 
p(n) = E[a.a ] = 0 n3l 
J j+n 
Consequently, the autocorrelation function and the power spectral density 
become simply 
and 
CPFM 
time 
S S S ( d  = B (10) 
In the following Sectiongwill be evaluated in both the IPFM and 
cases with a white noise input using the techniques of first passage 
problems of homogeneous normal diffusion processes, following the 
8 
approach of Darling and Siegert [1953] as presented in Cox and Miller 
[ 19651 . 
IV. THE FIRST PASSAGE TIMES OF -- --
DIFFUSION PROCESSES 
If the input t o  the P W  unit is a guassian, white noise process' 
the output of the integrator, p(t), is a normal diffusion process which 
is time homogeneous. 
the state at time to and x that at a later time t then the transition 
probability density p = 
(diffusion equations). 
backward one. 
fixed state r as a function of the initial position xo, is desired, then 
the backward equation provides the appropriate method. 
homogeneous process the Kolomogorov backward equation becomes 
In the theory of the diffusion process if xo denotes 
(x ,t ;x,t) satisfies the Kolomogorov equations 
0 0  
There are two of these equations, a forward and 
However, if the first passage time distribution to a 
For the time 
where 
p = p(xo,x;t) and 
lim Var[x(At)-x0] 
o(xo) = At-bo 
At 
The functions of 0(x ) and (p(x ) are sometimes called the infinitesimal 
mean and variance of the process. 
equations may be found in Gnedenko [1962]. 
0 0 
A precise explanation of these 
For a process x ( t )  starting at xo with mean zero and variance 
parameter So, the first passage time T of X(t) to the point r>xo is 
defined by 
X(0)  = xo 
X(T) = r 
9 
If there is an absorbing barrier* at r and p(xo,x;t) is the 
probability density that X(t) = x and that the process does not reach 
the barrier in time (0,t) then 
P{X(T)<r for o < v t ,  X(t)<rlX(o) - = x03 
= ir P(xo,Y;t)dY 
= P(xo,r;t) (14) 
P(xo,r;t) = P(tsT) (15) 
ml(xo) = EITlxol (16) 
P(xo,r;t) is the probability that absorption has not yet occurred by 
time t, i.e., 
The above comments concerning first passage times is significant 
since if 
then 
It is straight forward, using Laplace Transform techniques to show that 
The appropriate boundary conditions for the PFM problem are 
If u(t) is assumed to have zero mean and spectral density S 
IPFM 
then for 
0’ 
and for CPFM 
* A point r is called an absorbing barrier if when the process X(t) starting 
at xo reaches that point, the motion ceases. 
10 
= - E  
n=l (2n) (2n-1) (n) 2K CPFM 
Utilizing these values in Eq. (18) yields 
V. ANALYTICAL AND EXPERIMENTAL RESULTS -
An experimental study was conducted to verify the analytical 
The EPFK equipment used in the results of the previous sections. 
experimental study was built by R. A. Leuchs 119671. 
generator was used to provide the input*. 
not truly "white", rather it was t'band-limitedlt. 
A laboratory noise 
In reality this generator was 
True white noise has constant power spectral density, i.e., 
S(w) = so (26) 
R ( T )  = Sofi(.r) (27) 
In the experimental work that follows, the input is a band-limited 
white noise which has spectral density and autocorrelation such as, 
sin aC7 R ( t )  = So 
Tf 
where wc = 2mfc and fc is the frequency band. 
For the particular noise generator used 
= 0.33 (volts)2/cps 
w =2n(150) rad/sec. 
C 
* Manufactured by Elgenco Inc. (Santa Monica, California), Model 321 A .  
11 
It is possible, however, to change the $agn€tuife of the power spectral 
density by increasing or decreasing the input gain to the PFM system. 
the response of PFM to a true white noise input were determined in 
Section I11 as 
The autocorrelation function and the spectral density function of 
R S S ( T )  = B &(TI (30) 
s s p  = B 
The calculation of the number of pulses per unit time, B, is given in 
Section IV, for IPFM 
r 
and for CPFM 
1 
(4r2K/So) 
B =  
(33) 
From Eqs. (32) and (33), it can be seen that Bis a function of both 
threshold value, r, and the bandwidth of CPFM, K. For the IPFM case, 
K = 0. 
To compare the analytical and experimental results, several 
experiments were conducted. 
shown in Figs. 3, 4, and 5 .  In Fig. 3, the system is IPFM and the plot 
is B versus threshold, r. In Fig. 4, the system is CPFM with constant 
K but variable r while Fig. 5 is constant r with variable K. 
The results of three such experiments are 
For low threshold values the experimental results do not agree with 
the analytical results as shown in the figures because the white noise 
input was band-limited. 
VI. CONCLUSIONS 
The statistical properties of random impulse trains from pulse 
frequency modulators (IPFM and CPFM) have been studied for a random 
input (namely, white noise). It was found that the response of the PFM 
12 
system to a white noise input has certain characteristics such as: 
1) The autocorrelation, R s s ( t ) ,  is an impulse with intensity B 
(pulses per second). 
The spectral density, Sss(w), is constant with magnitude $ 
B depends upon the spectral density, So, of the white noise 
input, the threshold value, r, and the bandwidth of CPFM, K. 
$ decreases nonlinearly when r and/or K are increased. 
2) 
3) 
4) 
13 
BIBLIOGRAPHY 
E. , J .  Beut le r  and 0. A. Z. Leneman, l'Random Sampling Processes," 
Information and Control,  vol .  9, 1966, pp. 325-346. 
D. R. Cox and €1. D. Miller, The Theory o f  S tochas t i c  Processes,  (book) 
John Wiley 6 Sons, N e w  York, 1965, Ch. 5. 
D. A. Darling and A. J. F. S i ege r t ,  "The F i r s t  Passage Problem For 
A Continuous Markov Process," Ann. Math. S t a t i s t . ,  24,  624-39, 1953. 
R. L. Farrenkopf, A. E. Sabroff ,  and P. C. Wheeler, " In tegra l  Pulse 
Frequency ON-OFF Control", P rep r in t  63-328, presented at A I A A  Guidance 
and Control Conference, Cambridge, Massachusetts, August 12-14,  1P63. 
R .  V. Gnedenko, The Theory of Probabi l i ty ,  (book), Chelsea, New York, 
1962, p .  344. 
C .  C. L i ,  " In tegra l  Pulse Frequency Modulated Control Systems", Ph. D. 
d i s s e r t a t i o n ,  Northwestern Universi ty ,  Evanston, I l l i n o i s ,  1961. 
T. Pavl id is  and E.  I .  Jury,  "Analysis of  a New Class o f  Pulse- 
Frequency-Modulated Feedback Systemsf1, I .E.E.E. Transactions on 
Automatic Control,  vol .  AC-10, No. 1, January, 1965, pp. 35-43. 
0. A. Z. Leneman, " R a n d o m  Sampling of  Random Processes: Impulse 
Processes", Information and Control,  vol. 9, 1966, pp. 347-363. 
R. A. Leuchs, l'A C i r c u i t  Design f o r  a Pulse Frequency Modulator", 
M.S.E.E. Thesis,  Universi ty  of Massachusetts, June, 1967. 
Figure Legends 
Fig.  1. Block Diagram of IPFM 
Fig.  2. Block Diagram of CPFM 
Fig. 3 .  Frequency of Pulses vs Threshold Value Holding 
(volts) 2 
CPS 
K = 0 (IPFM); So = 15.7 
Fig.  4 .  Frequency of Pulses vs Threshold Value Holding 
(volts) 2 K = 50; so = 15.7  
CPS 
Fig.  5. Frequency of Pulses vs Bandwidth of CPFM K 
Holding r = 1.0 Vo l t s ;  So = 15.7 
(volts) 1 
CPS 
Information and Control 
Hutchinson, et. a l ,  
Figure (1) 
- 
f l;W FCt) Pulsing 
Circuit Integrator u(t) 
* 
Infopation and Control 
Hutchinson, et. al. 
Figure (2) 
Information and Control 
Hutchinson, et. al. 
Figure (3) 
200 
180 
160 
240 
120 
100 
80 
60 
40 
20 
0 
pulseshec.  
- - -.- - - Experimental 
Analytical 
I I I I I I I I I I 
0 0.2 0.4 0 . 6  0.8 1.0 1.2 1.4 1.6 1.8 2.0 
Threshold value, r (volts) 
Information and Control 
Hutchinson, et. al. 
Figure (4) 
pulses/sec. 
I 
\ 
\ 
I 
\ 
\ 
\ 
\ 
\ 
- - -. -- Experimental 
Analytical 
2 20 
200 
180 
160 
140 
120 
100 
80 
60 
40 
20 
0 
Threshold value, r (volts) 
20 
18 
16 
4 .  
2 -  
0 
Information and Control 
Hutchinson, et. al. 
I I I 
Figure (5) 
pulses/sec. 
I 6 ' \  \ \ \ 
\ 
- - - - - Experimental 
Analytical 
0 10 20 30 40 50 60 70 80 90 
Bandwidth of cPFM, K 


An analysis of the effect of a particular class of PFM on noise inputs

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