16 SAMPLED-DATA CONTROL SYSTEMS 
brought about by the sampling process. For this reason, a good under- 
standing of the phenomenon is required by a study of the sampling 
theorem. 
2.2 The Sampling Theorem 
Insight into the limitations imposed by the sampling operation in 
sampled-data systems can be obtained by consideration of the sampling 
theorem developed by Shannon.*” Essentially, this theorem states that, 
for signals having a finite bandwidth, including frequency components 

Frequency (w) 
Fig. 2.5. Finite bandwidth signal spectrum. 
up to but not beyond a frequency of W radians/sec, a complete descrip- 
tion of that signal is obtained by specification of the values of the signal 
at instants of time separated by 4(27/W) sec. The converse of this 
theorem is particularly useful in sampled-data systems, where a signal 
must be reconstructed from samples separated by a particular sampling 
interval. The converse can be stated that, for a band-limited signal 
which contains no frequency components beyond W radians/see, it is 
theoretically possible to recover completely the signal from a sequence of 
samples which are separated by time intervals of 4(27/W) sec. In 
view of the basic importance of the theorem, a proof will be given. 
The Fourier spectrum Fy(w) of the signal f(¢) is plotted as a solid line 
in Fig. 2.5. It is noted that the spectrum contains no frequencies higher 
than W radians/sec. A convenient representation of this spectrum is by 
means of a Fourier series whose fundamental period is 2W. This repre- 
sentation produces additional spectra shown by dashed lines in Fig. 2.5, 
but these are of no consequence if it is recalled that the representation is 
valid only over a frequency range of from —W to W radians/sec. The 
periodic spectrum can be represented by means of a Fourier series in w 
with a fundamental frequency of 2W as follows: 
ar ner 
Pylo)= ) Cie ae NN (2.6) 
n=—o 
4 
