280 ; SAMPLED-DATA CONTROL SYSTEMS 
The asymptotic expression given by 10.112 should be compared with 
(10.99), which gives the absolute minimum mean-square value of the 
error that can be obtained by the use of a linear filter to reconstruct the 
particular random signal under consideration. In the case being con- 
sidered, the zero-order hold has an error which is at least 50 per cent 
greater than the optimum. To reconstruct a signal with a bandwidth of 
1 cps with a zero-order hold one needs 30,000 samples/sec instead of the 
20,000 samples required by the optimum filter. In many cases this 
increase in sampling rate is not a severe price to pay for the great reduc- 
tion in complexity of the zero-order hold as compared with the optimum 
filter. Asis the case in continuous filter theory, the optimum filter repre- 
sents a standard for comparison and not a very practical filter design. 
10.6 Summary 
The analysis of random signals in linear sampled-data systems can be as 
easily performed as the analysis of random signals in linear continuous 
systems. The use of the sampled power spectrum to describe impulse- 
modulated random signals permits the immediate extension of ideas from 
continuous systems to sampled systems in much the same way that the 
z transform and the pulse transfer function carry over familiar relations 
from continuous to sampled-data systems. A method is given for the 
calculation of the sampled power spectrum which corresponds to a given 
continuous signal spectrum if the given spectrum is a rational function 
of frequency s. This method shows that the sampled power spectrum is 
a rational function of z if the power spectrum of the continuous signal is 
a rational function of s. The mean-square value of a continuous signal 
can be calculated from the sampled power spectrum, and formulas are 
given for this purpose. 
The relation between the spectra of the signals at the input and the 
output of a sampler-and-filter combination is not obviously related to the 
continuous case, and a table of formulas is given as (10.59) to facilitate 
the use of the sampled power spectrum. The technique for calculating 
the spectra is straightforward, and additional relations are readily 
determined. 
It is possible to apply the theory of linear time-varying networks to the 
analysis of random signals in sampled-data systems, but this technique 
does not add any particular insight into the problem. The special 
character of the time variation in sampled-data systems permits the use 
of specialized formulas and techniques which lead quickly to the answers 
most generally required. It is obvious, of course, that the special tech- 
niques used for sampled-data systems do not apply to as broad a class of 
problems as do the methods of variable network theory. 
