SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 267 
Syy(s) a Sz,(s) H (s)H(—s) 
Syy (2) ag Size () ell (eth (2am) 
Say(s) = 7 Seals) (s) 
(10.59) 
Sx*(s) = T Szr(s) H*(s) 
Szory(s) = Sz,(s)H(s) 
Sxty*(Z) = Szr(2)H (2) 
10.5 Linear Filtering of Sampled Random Data 
In this section attention is turned from the analysis problem to the 
synthesis problem. Sections 10.3 and 10.4 were concerned with analyz- 
ing the random signals which might appear in sampled-data systems and 
determining the effects of the system on the signal characteristics. The 

Fie. 10.6. Block diagram illustrating formulation of the linear filtering problem. 
present section is concerned with the design or synthesis of systems 
intended to operate in the presence of random sampled data. The design 
of any system depends on the characteristics of the signals on which the 
system operates, on the constraints placed on the system behavior, on 
the desired operation required, and on the criterion used to evaluate 
the actual performance. The value of the design to the ‘consumer”’ 
will depend upon how closely these four conditions match his situation. 
Although a variety of problems suitable for different situations in the 
filtering of discrete data have been treated in the literature,” 1! 12,3745 only 
one of these will be discussed here. That one is the linear least-squares 
filtering—smoothing and prediction—of sampled random data in the 
presence of additive random noise. The approach used is that of “‘spec- 
tral shaping,’”’ introduced by Zadeh and Ragazzini,®’ and is closely 
related to the method of Bode and Shannon.$ 
The filtering problem under consideration is most easily visualized by 
considering Fig. 10.6. The design objective is the selection of a filter 
with the transfer function H(s) which will operate on the sampled signal 
r*(t) in such a way as to minimize the mean-square value of the error 
e(t). The input signal is assumed to be the periodic impulse modulation 
of the sum of a random message, m(t), and a random noise, n(¢), and the 
error is defined as the difference between the filter output and the ideal or 
desired output, ca(t). Typically the desired output is the result of a 
