THE SAMPLING PROCESS 15 
consist of a summation of weighted spectra, each of which is the same 
as the central term except for the weighting constant C;, and the shifted 
argument j(w — 2rk/T). 
The sampled spectrum F'*(jw) is sketched in Fig. 2.4. The original 
spectrum F(jw), from which the sampled signal spectrum F*(jw) is 
derived, is shown in this figure. The effect of the sampling process is to 
introduce a succession of spurious spectra which are proportional to the 
signal spectrum and which are shifted periodically by a frequency separa- 
[FUw)l 
0 
Frequency (w) 
(a) 
|F*(jw)| 

Frequency (w) 
(b) 
Fia. 2.4. Spectrum of finite-width sampled function. 
tion 2rk/T. It is evident that this spectrum bears little resemblance 
to that of the original signal and that means must be found to recover 
the original information in any useful application. 
The central signal spectrum can be extracted by rejecting the spurious 
higher-frequency spectra by means of a low-pass wave filter. It is evi- 
dent that even if this filter were “‘ perfect,’’ in the sense of passing perfectly 
a given spectrum and rejecting perfectly an unwanted spectrum, the 
signal spectrum could never be fully extracted unless it were band- 
limited, that is, unless it contained no components above a given fre- 
quency. The “spillover”’ effect produced by an infinite spectrum such 
as that shown in Fig. 2.4 always produces distortion in any recovered 
signal. Sampled-data systems are subject to a signal-recovery problem 
of this type, and imperfect signal recovery always produces ripple effects 
on the output of such a system. Basic limitations on the information 
capacity of sampled-data systems result from the information loss 
