268 SAMPLED-DATA CONTROL SYSTEMS 
linear operation on the input message, such as ideal prediction, differentia- 
tion, or, in general, 
ca(t) = [", halr)m(t — 2) dr 
where ha(r) is the impulse response of the filter (which is not necessarily 
physically realizable) which will perform the desired operation. As this 
problem is formulated, the only information concerning the input which 
is required in order to perform the minimization is the autocorrelation 
functions of noise and message and their cross-correlation. 
The problem diagramed in Fig. 10.6 is particularly simple to solve when 
the successive samples of the input are statistically independent. In that 
case the autocorrelation of the input is zero for |r| equal to or greater than 
the sampling period 7’ and the input sampled power spectrum is a con- 
stant for all z. Such an input spectrum is said to be “almost white.” 
The mean-square value of the error defined in Fig. 10.6 is 
(e*(t)) = ([e(t) — ca(t)]?) 
(c2(t)) — 2e(t)ea(t)) + ca®(t)) 
$.-(0) — 26..,(0) + Beacg(O) (10.60) 
Each of the terms in (10.60) will be evaluated separately in terms of the 
relations defined by Fig. 10.6. 
The mean-square value of the output of the filter, ,.(0), can be 
obtained from the inversion of (10.39) as 
&,.(0) = = i h(x) dx | ; h(y) dy ®,.( — y) ia — jae 
k=—-0 
(10.61) 
On the assumption that the input is white, the autocorrelation function 
®,,(kT) is zero for k # 0 and 
(0) =f Wa) de [Wy dy Pale — ale - 9) 
ut fel i [h(x)]? &,,(0) dx (10.62) 
Tp 
The second term in (10.60) is given by 
To 
©7,,(0)'= io oF 7, cleat) dt 
1 re Tp 
tu slate |, h(x)r(t — x) 
eo 
° 6¢—a — kT) ae| ca(t) dt (10.63) 
k=—@ 
