SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 269 
If the order of integration with respect to x and ¢ is reversed, 
rc) To = 
®..,(0) = dx h(x) lim isk / r(t — 2x)ca(t) 2 6(¢ — a2 — kT) dt 
0 Tir © 2T 0 —To 1a 
N 
[ dx h(x) lim OVD? De r(kT)ca(kT + 2x) (10.64) 
Il 
N- © 
= i JA@RuONE 
With the substitution of (10.63) and (10.64) in (10.60) 
(e*(t)) = oe ie [h(x)]? Cit ~ i) h(x) @re4(x) Ghe <P ®.4c4(0) 
®,,(0) e Prcq(L) : 
T i, Ee = ae ie 
1? (Brea)? 
i if See) dt + Be(0) (10.65) 

Since the last two terms in (10.65) do not include the unknown h(x) at 
all, and since the first term is either positive or zero, it is obvious that the 
least squared error will result if, and only if, 
Preg (x) 

ge) = ®,.(0) ae > (10.66) 
which leads to the transfer function 
ee eee emt 
(Ss) — [ a0) dx (10.67) 
A filter with the transfer function given by (10.67) will perform least- 
squares smoothing on inputs which are almost white, in the sense dis- 
cussed previously. For arbitrary input spectra, an extension is possible. 
t 
r(t) YP D's) pit) | ai | elt) 
Fie. 10.7. Block diagram for spectral shaping to form almost white noise. 
The extension discussed here is based on the ability to shape the spectrum 
of a signal with a realizable filter so that the output of the filter is almost 
white and consequently has a constant sampled power spectrum. A 
block diagram of the shaping operation is shown in Fig. 10.7. The pulse 
transfer function D(z) is assumed to be chosen such that the sampled 
power spectrum of the signal p(¢) is unity. This shaping operation is 
possible with a realizable discrete filter if the original signal, r(¢), has a 
