ORTHONORMAL EXPANSION OF A CORRELATION FUNCTION 
In Fig. 1 a stationary random process gz (+) 
is applied to the inputs of a set of time-in- 
variant linear filters the impulse responses of 
which form a set of orthonormal functions 
{mn(t)} over the interval (0,00), i.e., 
1, n=n 
Ae n(t) Bg (t) at = (2) 
0o,n7m 
The response of the nth filter to g(t) applied 
at t = 0 is the random process v,(t), given by 
ies) 
v(t) |E gy(t-7) h(t) at (2) 
Multiplying the response Vn(t) by a second sta- 
tionary random process Bo(t) and time-averaging 
the product leads to 
lee) 
re go(t) va(t) v,(+) =f 3(0) eho) 6, (t-%) h,(Z) at (3) 
0 
Since 
g,(t) B4(t-Z) = goy(-T) = pra(7) (4) 
the cross-correlation function of £3 (+) and 
go(t), (3) may be written: 
fee) 
Ay = f dyy(t) n,(Z) at (5) 
an integral equation of the first kind, a solu- 
tion of which is 
[eje) 
$49(7) = a AD h, (©) (6) 
n= 
That (6) is a solution can be verified by sub- 
stituting (6) in (5), interchanging the order 
of summation and integration, and invoking (1). 
Hence averaging the products of g>(t) and the 
responses of the filters to g3(t)* leads to the 
coefficients, A,, of the orthonormal expansion 
of the Correlation function for T20, 
A second set of identical filters are ex- 
cited by a unit step, to give the indicial 
responses a(t): 
t 
a(t) =a h(E) at (7) 
Each a,(t) is then miltiplied by the correspond- 
ing coefficient A,, all such products are summed, 
and the resulting sum is differentiated, yielding 
348 
CO 
t 
and h,(@) atv = 2 A be) 4) 
which is the correlation function displayed in 
real time, 
A CLASS OF ORTHONORMAL FUNCTIONS AND FILTERS 
A class of orthonormal functions suitable for 
the expansion of correlation functions are defined 
by the following theorem: 
Theorem, The set of square-integrable time 
functions {h,(t)} form an ortho- 
normal set over the interval (0,0) 
if each member possesses a Laplace 
transform H,(s) with 
B (s) 
Hy(s) = Ky | 2 (9) 
and 
K Cc  (-s) 
H(s) = —2 el H,-1(s) n22 
(10) 
in which B 1(s) and the terms C,(s) 
are polynoitinals in s, and if 
the constants K, are such that 
al 
Hi H,(-s) H,(s) ds = 1 
r 
nosy 2 ene KI) 
in which Br denotes an appropriate 
Bromwich contour in the s-plane 
enclosing the poles of H(s). 
(A proof of this theorem appears in the 
Appendix, ) 
Examples of members of this cjass of ortho- 
nermal functions are the Laguerre~ functions, 
for which 
n-1 
(s -s) 
H(s) =¥25) +n = 1, 2,00, (12) 
(s)+s 
with multiple poles at s = -s, and multiple 
zeroes at s = s,; and the Kautz’ functions for 
which 
2s 
at 
S+S7 
Hy(s) = (13) 
