'^
NEW YORK UNIVCRSITY
AFCRC-TN-60-396
INSTITUTE 0;M\THEM^T:CAL SCIENCES
l;:
^E ET P R ^ ^ " ^'^^'^'y ^' - '' - Y-'< 3, N. Y.
NEW YORK UNIVr
Institute of Mathematical Sciences
/-"^w ^x^ J^ NEW YORK UNIVERSITY
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^ — ^ .s Division of Electromagnetic Research
^â– C>cccxX +
RESEARCH REPORT NO. EM-160
A Note on the Local Structure of Shot Noise
R. A. SILVERMAN
Contract No. AF 19(604)5238
MAY, 1 960
A FCRC-TN -60-396
NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research
Research Report No. EM-I60
A NOTE ON THE LOCAL STRUCTURE OF SHOT NOISE
R. A. Silverman
R. A. Silvennan
Morris Kline
Director
..^.â– yyW/V/
Dr. Werner Gerbes
Contract Monitor
fA
The research reported in this doc\ament has been sponsored by the
Electronics Research Directorate of the Air Force Cambridge Research
Center^ Air Research and Development Command, imder Contract No.
AE 19(601^)5258.
- 1
It is well known that the random velocity field produced by
wind tunnel turbulence at high Reynolds numbers has the following
statistical structure: Let the x axis be directed downstream beyond
the grid generating the turbulence and let u (x,y,z) be the downstream
component of the random velocity field, measured at any point of
the flow. Then, to a good approximation, u (x,y, z) is a normal
random variable, i.e., the velocity field is quite accurately
univariate normal. If the field were bivariate normal as well, then
writing 6u(r) = u (x+r,y,z) - u (x,y,z), where again (x,y,z) is any
point of the flow, we would expect the random variable 6u(r) to be
normal, in particular to have a skewness 7(r) = E 5u(r) /|e l6u(r)l >
equal to zero and a flatness 0(r) = E rSu(r) 1 / JE |5u(r) 1 i equal
to three, the appropriate values for a normal random variable.
What is actually found to be the case instead is that 7(r) and 0(r)
are appreciably different from zero and three, respectively, for
values of r such that the correlation function f(r)= Eu (x+r,y,z)
• u (x,y,z)/E u (x,y,z) is appreciably different from zero. For
experimental values of /(r) and 0(r) and a detailed discussion of the
statistical structiire of turbiilence, we refer to the last chapter of
Batchelor's monograph. <- J
The state of affairs just described raises the following
The symbol E denotes the expectation value or ensemble average.
- 2 -
question: Is it possible by suitably choosing the shot rate and the
pulse shape to construct shot noise which mimics the statistical
structure of turbulence^ i.e., which is quite accurately univariate
normal but exhibits marked departures from bivariate normality at
close ranges? The answer is in the affirmative, as we shall now
show. For simplicity, we consider first the case of a one -dimensional
random process
(1) u(t) =[/| f^ P.s(t-t.) - c l/^
stationary Poisson process with an average rate of p events per
second. The p. are a family of independent random variables, all
uniformly distributed over the unit interval (0,1). The centering
00
constant c is chosen to be -^ / s(t) dt, which assures that Eu(t) = 0,
2
and the nomialization is such that Eu (t) is independent of p and is
oo
in fact equal to / s (t) dt. (The author has given a similar construc-
-00
tion elsewhere./- -I
We must now suitably choose the p\ilse shape s(t) and the rate
p so that the random variable u(t) is quite accurately normal, while
for small t the difference random variable u(t-i-T) - u(t), in particular
u (t) = du(t)/dt corresponding to the limit t-*0, is quite markedly
non-normal. To achieve this, we choose s(t) to be the function
The parameter t is fixed but arbitrary.
- 5 -
(2)
Kt)
t/2€ , < t < 26
1 2e < t < a-e
(a-t)/e, a-e < t < a
, a < t
with derivative
s'(t)
' l/2e
-l/e
< t < 2e
2e < t < a-e
a-e < t < a
a < t
where e « a is a small parameter to "be adjusted later. Our choice
of s(t) is motivated by the fact that its support (a) is much larger
than the support (je) of its derivative; the significance of this will
emerge presently. We have also arranged to give u'(t) a negative
skewness, in order to resemble the turbulent velocity field.
We now use the fact that the semi -invariants \x^ of u(t) are
given by the formula
f5^2
^^n = py
'"/
s"(t) dt
where p is any of the (3. .'- -' In our case Ep"^ = -^^ , since p is
uniformly distributed over the interval (0,l); the Inclusion of the
random variables p. in the expression (l) Is to assure that the
distributions of u(t) and u'(t) contain no delta function terms. In
2
view of the relations l^g " "^2' ^5 " "^5 ^^ ^]^" ^h, ~ ^2 ^^'^^^^'^ ^^^
- k
semi -invariants la^ |i and |i, and the central moments va. , m and
we find that the skewness of u(t) is given hy
1^:
'=7^72
and its flatness by
For the case of the pulse shape (2)^ we find that
(a-2e)^/^ V^
5 + ^ 2_^ rsjj, + ±1^, if e «a
(a-2e)'
so that for small e/a, the condition -/pa » 1 ass\ires that u(t) is
quite accurately normal. '- -I On the other hand^ differentiating (l)^
we see that the univariate distribution of the process u'(t) is
governed by the semi -invariants
i^f Ep- J [s'(t)]-dt.
with corresponding skewness
^ /2p£^ /^
- 5
and flatness
Examining the expressions for 7, 0, 7' and 0', we see that by
satisfying the conditions v^ » 1 and \/pe r~^l, we can arrange to have
simultaneously a quite accurately normal distribution of u(t) and a
markedly non-normal distribution of u'(t). Moreover, the two condi-
tions \/pa » 1 and /pi r-^ 1 are compatible, provided only that |/a/e » 1.
For example, if a = 1, e = lO" and p = 10 , we have 7^0, ^r^^,
7' ~-0.5 and 0'rs^ 3.9-
The skewness 7(1:) and the flatness 0(t) of the difference
u(t+T) - u(t) can be calculated in Just the same way by observing
(following a suggestion of E. N. Gilbert) that the process u(t+T) - u(t)
is given by (l) if we replace s(t) by s(t+T) - s(t) . It is found
that as T-0, 7(1:) and 0(t) reduce continuously to the limiting values
7' and 0', and that as T approaches the correlation distance a of the
process u(t), 7(t) and 0(1:) approach the normal values of zero
and three. Thus u(t) resembles the turbulent velocity field by
The correlation function of u(t) is
CD
00 / ^"
/ s(t+T)s(t)dt/ /
o / -co
5^(t)dt
-00
which vanishes for T > a • The function 7 (t) undergoes a sign
change in the interval (0,a).
- 6 -
having a markedly non-normal blvariate distribution at close ranges.
Precisely the same kind of construction can be carried out
in three dimensions by replacing the random times t. by a spatial
Poisson distribution and replacing the pulses s(t) by three-dimensional
"blobs". For example, suppose that
u(x,y,z)
J^Y. Pi s(x-x^) s(y-y^) s(z-z^) - c /^
where the function s is the same as in (2), while this time p is the
volume density of the points (x., y , z ) of a homogeneous spatial
Poisson process and the summation is over all the "centers" (x. , y., z.)
For each point (x. , y., z.), p is an independent random variable,
uniformly distributed over the interval (O, l). The centering constant
c is now -^ \ / s(x) dx > ^and Eu(x,y,z) = 0, Eu (x,y,z)
r 2 r
/ s (x)dx / . For the skewness y and the flatness of the
random variable u(x,y, z), we now have
1.3 A . 1.8
7 ^^
Pytt V
while for the skewness 7' and flatness 0' of the random variable
^â– : r. 7
^ u(x,y,z), we find
3 1
7' ~ - 17 ^^ - 0.
\
»^p^ >v^
^ 3 . ^^
P^a e
If yp^ » 1 and K P-yO. e ~1, the random field u(x,y,z) is quite
acciirately univariate normal but exhibits marked departures from
bivariate normality at close ranges. These two conditions are
compatible, provided that j/a/e » 1, as before. For example, if
a = 1, e = 10 and p = 10 , we have 7 ro 0,
In conclusion, we see that when a large number of elementary
waveforms are superimposed, even with high density azid considerable
overlap, there is no reason to expect a priori, in the absence of
detailed information about the shape of the waveforms, that the resulting
process is accurately normal.
We use the obvious modifications of the formulas for the semi-invariants
u. and u ' .
^n ^n
REZERENCES
[l] Batc±ielor, G. K.
I2J SilveimaxL, R. A.
[3]
Rice, S. 0.
!*â– Mlddleton, D.
The Theory of Homogeneous Turbulence j
Cambridge U. Press, 1955-
An 1 so spectral family of random
processes; N.Y.U., Inst. Math. Sci.,
Div. EM Res., Research Report No.
EM-I5I; to appear in IRE Trans, on
Information Theory.
Mathematical Analysis of Random
Noise; reprinted in the collection
'Selected Papers on Noise and
Stochastic Processes', (N. Wax, Ed. ) j
Dover Publications, Inc., New York,
19:>k, pp. 150-157.
An Introduction to Statistical
Communication Theory; McGraw-Hill,
New York, I960, pp. 505-506.
mJT
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Downey, California
Attn: J. C. Joerger
Trans-Tech, Inc.
P. 0. Box 3U6