Prolate spheroidal wave functions for solution of Fredholm equation for bandlimited signal detectio

Rino, C. L.

English

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The Application of Prolate Spheroidal 
Wave Functions to  the Detection 
and Estimation of Bandlimited Signals 
Charles L. Rino* 
Department of Applied Physics and Information Science 
University o f  California, San Diego 
La Jol la ,  California 92037 
ABSTRACT 
A solution t o  the detection Fredholm equation fo r  bandlimited 
noise and signal is presented that makes use of the dual orthogon- 
a l i t y  and completeness of the prolate spheroidal wave functions. 
Although data are available only on a f i n i t e  interval, the detector 
in principle achieves the same optimum performance as i f  the inter-  
val  were inf ini te .  This apparent anomaly is resolved in  an example. 
d&L- 
* Prepared under NASA grant NoJ 05:009-079. 
1 
https://ntrs.nasa.gov/search.jsp?R=19700003592 2020-03-12T01:42:39+00:00Z
The Fredholm integral equation 
and certain of its variants occur frequently in signal detection and estimation. 
The kernel R(t - s) is symmetric positive definite, and the solution can be 
formally represented as an infinite summation of the eigenfunctions of R(t - s) . 1 
These eigenfunctions have been computed only for a few simple kernels, and we 
normally assme R(t) has a rational spectrum. Then f(t) can be determined as 
the solution to a certain linear differential equation with constant coefficients. 2 ”  
We will show that for g(t) and R(t) bandlimited to IuI I W, the solution 
to (1) can be expressed as an infinite summation of prolate spheroidal wave 
functions (PSWF’s) . 
I. The Solution of (1) for Bandlimited g(t) and R(t) 
With the definition 
1 
the integral equation (1) can be rewritten 
Let ZT(u), i ( u )  and q ( w )  be the respective 
and R(t). The left hand side of (3) is an 
6 T/2 
> o  
-T/2 < t < T/2. 
Fourier transforms of fT(t), g(t), 
entire function of t since it is a 
linear superposition of bandlimited, and therefore entire, functions -R(t - s) . 
Now g(t) for It1 < T/2 is a piece of an entire function. Hence the left and 
2 
. .. . . . . . . . ., .-.- 
.. 
right hand sides of (3) are ent i re  functions that coincide over a f i n i t e  inter-  
val. We conclude that they are equal for  a l l  t, and we can Fourier transform 
both sides of (3) t o  obtain q ( w )  $,(w) = i ( w ) .  Now both q ( w )  and i ( w )  are zero 
for  I w I  > W, so that this  yields 
_. --_
only f o r  < W. But if we assume that ZT(w) is square integrable, we can then 
3 use the PSWF method t o  find a time limited function that is defined for  a l l  w 
. -  
and equal to  gT(w) almost everywhere in  (-W, w).  
The definitions of Qn(t), c, and x,(c) are those of Slepian and Pollack. 
The PSWF expansion 
n=O 
is defined for  a l l  w ,  where 
J 
Moreover, by taking the inverse Fourier transform of (5) we have the desired 
result  
An application of the general form of Parseval’s theorem t o  ( 6 )  shows that 
rc0 
. where 
an = in &)-’ [An(c)]-” I h(t)  $,(t) d t  
J-cc, 
(8 
Hence (7) can be rewritten 
3 
where hn is the n-th coefficient in the PSWF expansion of h(t). 
and ( l o ) ,  our main result, give the solution to (1). 
Equations (9) 
11. An Example from Detection Theory 
A sufficient statistic G for the detection of a known bandlimited signal 
s(t) in additive zero-mean, non-white gaussian noise n(t) is 
T 
G = 1 q(t) v(t) dt 
0 
where 
---. -- n(t) under hypothesis H, 
s(t) + n(t) under hypothesis H,. 
q(t)is the solution to the integral equation 
.~ - -  
T 
s(t> = J R(t.- s )  q(s) ds , o < t 7 T 
0 
J 
where R(t) is the covariance of n(t).4 With an appropriate change of variables 
(12) can be put in the form of (l), and we have the solution (10) with q(t + T/2) 
To evaluate the performance of this detection scheme we need only compute 
the mean and variance of G under the null hypothesis H, and the alternative 
hypothesis H,.' We have from (11) and (10) that E[G[H,] = 0 and 
__  - - 
4 
(15) follows from (9) and an application of Parseval's theorem. A similar com- 
putation w i l l  show that  
under both hypotheses. r( 
Note that  (15) is independent of T. Thus fo r  any T we achieve the same 
false-alarm and detection probabilities as f o r  an inf ini te  observation interval. 
In  reali ty,  because of computational diff icul t ies ,  we could approach this  
theoretical l imit  only i f  nearly a l l  the signal were contained in  our observation 
interval T.  This w i l l  be verified by an example. 
Now consider the signal s f t )  = sinc Q ( t  - t o )  with Z.rr/Q << T. 
We w i l l  take the noise spectral density t o  be the constant No over the interval 
1.1 I W. We have 
1 sinc Q[ (-T 2 - to) - t] - $,(t) d t .  
-T/ 2 
1 1 If we take the PSWF's with c = RT/2, then hn = ko in (7T - t o )  . If 17' - to I 
<< T/2, i .e .  s ( t )  is  contained in  T,  hn is nearly zero for  n > ncrit - [2 
If lTT - t o /  2 T/2, i . e .  to l i e s  outside the interval (0, T) ,  terms of higher 
- R 
T I .  
1 
order w i l l  be necessary to  compute a detection s t a t i s t i c .  However, for  
n > n  the terms cri t  
5 
are extremely sensitive to error since rl, (t] NN 0 in the interval It [ 5 T/2, and n 
we cannot hope to improve the detection. Now the statistic 
m 
G =c hn vn (19) 
n=O 
computed with c = nT/2 is not optimum, since from (9) we should use c = W/2. 
However, 
2c 1 7 1  
fT(w) -c an 1cl,(Tw/2n) 
n=O 
is nearly zero for 1.1 > Q , and we expect the improvement with c = WT/2 to be 
negligible, at least for to << T. 
We conclude that the PSwF's provide a convenient analytical tool for 
solving (1). 
only a finite T reflects the fact that all the information in a bandlimited 
That we attain theoretically maximum detectability (T = m) for 
signal is contained in a finite interval. 
results, however, we see that we cannot in reality attain this limiting detect- 
ability. 
approach has been presented. 
With careful interpretation of the 
An example from estimation theory which makes use of an alternative 
6 
ACKNOWLEDGMENT 
The author gratefully acknowledges a referee's suggestion for clarifying 
his original argument leading to (9) and (10). 
6 
, i 
FOCZNOTES 
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, 
Interscience Publishers, Inc., New-York, N. Y., 1966, p. 122. 
1. 
2. C. W. Helstrom, "Solution of the Detection Integral Equation for Stationary 
Filtered White Noise", . IEEE Trans. on Information Theory, IT-11, July 
1965, pp. 335-339. 
3. D. Slepian and H. 0. Pollak, "Prolate Spheroidal Wave Functions, Fourier 
Analysis and Uncertainty--I", BSTJ, - 40, January 1961, pp. 43-63. 
4. C. W. Helstrom, Statistical Theory of Signal Detection, Pergamon Press, 
New York, N. Y., 1968, p. 116. 
5. Ibid, p, 118. 
6 .. C. L. Rino , "Bandlimited Image Restoration by Linear Mean-Square Estimation", 
J. Opt. SOC. Am., 59, May 1969, pp. 547-553. -
7 


The application of prolate spheroidal wave functions to the detection and estimation of bandlimited signals

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