272 SAMPLED-DATA CONTROL SYSTEMS 
The entire optimum filter transfer function can be written from (10.80), 
(10.72), and (10.69) in two parts: 
1 5 
H(s) = exiop y(t)e-* dt 
Je At 
where v(t) = Taj = | ee dd (10.81) 
It is frequently convenient to express the desired output as the result 
of passing the message through an ideal filter, as shown in Fig. 10.6. If 
the ideal filter is linear and has the impulse response ha(t), then cross- 
correlating the total input of message and noise [m(t) + n(¢t)] and the 
desired output ca(t) expressed in terms of its convolution integral, the cross- 
correlation function ®,,,(7) 1s 
To C) 
o.(7) = ae a Cs Limo + n(t)] A ha(x)m(t +7— 2) dz; dt 
Wy ha(2)®mm(r — x) dx + ih ha(x)Pnam(r — x) dx (10.82) 
The cross power spectrum between input and desired output on the 
assumption (10.59) is given by the transform of (10.82), which is 
Sra) = Ha) [SnmQ)' + San) (10.83) 
which may be substituted in (10.81) for calculation of the optimum trans- 
fer function, if desired. 
Before working out an example of optimum least-squares filtering, it is 
instructive to determine an expression for the mean-square value of the 
error between the desired output and the output of the optimum filter. 
The mean-square value of the error in terms of an internal signal p(¢) 
whose mean-square value as expressed by ®,,(0) is unity may be obtained 
by a substitution of p for r and the optimum impulsive response for A(z), 
which causes the first integral of (10.65) to be zero. Thus, 
(2(0) = Beal) = 7 i © [Byee) de (10.84) 
However, ®,.,(x) is simply the inverse transform of 10.78, which is the 
y(t) defined in (10.81), so that, 
(2() = Peued(0) = 7 i} “WOR at (10.85) 
The transfer function for the optimum filter given in (10.81) is identical 
in form to the corresponding Wiener filter, but in the case of the sampled- 
data filter the factor 1/[S*.(s)]* eonregnonel to the transfer function of a 
discrete filter or digital controller of the type described in previous chap- 
