264 SAMPLED-DATA CONTROL SYSTEMS 
of sampling frequency, 1/7, to signal “bandwidth,” w:/2z, is 2x in this 
case, which is a moderate sampling ratio, neither fast nor slow. The 
mean-square value of the signal x(t) is given by the area under T?S*, (jw) 
between —t <w <7. From the plot of Fig. 10.5 it is not easy to obtain 
a quantitative idea of the closeness with which the output of the zero-order 
hold approximates the original signal. Later, a comparison based on the 
mean-square error will be made between this system and other somewhat 
more sophisticated schemes for the reconstruction of random signals, but 
before a discussion of the filtering problem can be begun, some more 
results must be derived from Fig. 10.4. In particular, expressions are 
needed for the autocorrelation function or sampled power spectrum of the 
sampled output of the system and for the cross-correlation between output 
and input. 
Since y*(é) is the impulse-modulated y(¢), one would expect from (10.24) 
and (10.25) that the sampled power spectrum of y*(t) would be the two- 
sided z transform of S,,(jw). In other words, intuitively, one would 
expect that 
S,,(2) = Z[S2,(s)H(s)H(—s)] 
Sx2(2) H(z) H (27?) (10.48) 
This result will be derived directly from the fundamental relations 
between the signals and the filter impulse response h(t). The first step 
toward finding S,,(z) is to find ,,(k7), the autocorrelation function of 
y(t) at integral values of 7/T. 
By definition of impulse sampling, 
y*) = Y  y(eT)a¢ — kT) (10.49) 
k=—© 
and, from the convolution integral, 
o 
y(kT) = > h(nT)a(kT — nT) (10.50) 
n=0 
Since the process y(é) is ergodic the autocorrelation function of this pro- 
cess may be obtained from the time average 
N 
P 1 
$,,(kT) = ae IN +1 »; y(IT)yT + kT) 
=—N 
(y(IT)yUT + kT) (10.51) 
