18 SAMPLED-DATA CONTROL SYSTEMS 
sampling process. While the amplitude of the various repeated spectra 
depends on the switching function p(t), their respective bandwidths are 
unaffected by this function. The figure shows the condition for a 
sampling frequency which is twice the maximum frequency content W 
of the signal. Itis seen that there is no spectrum overlap, as had been the 
case for infinite spectra illustrated in Fig. 2.4. 
If it is desired to recover the original signal from the sampled sequence, 
it is necessary to separate the signal frequency spectrum from the infinite 
Fy iw) 

Frequency (w) 
Fic. 2.6. Sampled signal spectrum. 
repeated spectra by means of a low-pass wave filter. If this filter were 
ideal, that is, if its passband produced zero attenuation while perfect 
cutoff was produced outside this band, it would be possible to extract 
perfectly the original signal spectrum Fy(w). This shows that, for signals 
having finite spectra, it is possible to extract without distortion the 
original signal from a pulse sequence provided that the sampling fre- 
quency is twice the maximum frequency contained in that signal. It 
should be recognized that this result is based on an idealization of both 
the signal spectrum and the filter response. 
Practical considerations will show that signals generally do not have 
finite spectra nor do filters have perfect response characteristics. For 
that reason, the signal which is recovered from a sample sequence always 
has a certain amount of distortion, even though the sampling frequency 
is high relative to the signal frequencies. This distortion is roughly at 
sampling frequencies and their harmonics and is referred to as ‘‘ripple”’ 
in the recovered signal. In sampled-data feedback control systems, 
