THE SAMPLING PROCESS 17 
where the C,’s are the various Fourier coefficients of the exponential 
series are defined by 
1 +W jm 
Cn = 2W Ie Fy(w)e W' dw (2.7) 
The significance of this representation is that, if all the values of C, 
can be obtained, the Fourier spectrum Fy(w) is fully defined and the 
signal is thereby fully specified. Thus, if information is available which 
permits the evaluation of the coefficients as expressed in (2.7), the Fourier 
transform of the signal—and, consequently, the signal itself—is fully 
specified. ‘Turning one’s attention to the time function f(¢) and its 
Fourier transform Fyw(w), the following inverse transform relationship 
exists: 
+W 
Wo = = es Fur (w)et** de (2.8) 
where the finite limits are justified by the fact that the Fourier spectrum 
Fy(w) is band-limited at —W and +W radians/sec. Now, if the time 
in (2.8) is set to a specific value nz/W, then the integral (2.8) becomes 
+W nm 
f(ne/W) = = iy Fy(we” dw (2.9) 
It is seen that, except for a constant, the integral (2.9) is identical to 
that in (2.7), which is required to specify the Fourier coefficients in (2.6). 
The significance of this result is that a specification of the time function 
only at instants of time nz/W is required to specify the Fourier transform 
of that signal and, therefore, the signal itself. 
The implications of this result are that for the class of signals whose 
Fourier spectra are finite, that is, are zero for frequencies above a specified 
value W, the function is completely described by a set of samples whose 
interval is 7/W sec. This corresponds to one-half the highest frequency 
content of the signal. Thus, for band-limited signals whose highest 
frequency component is W radians/sec (or F cps), no loss of information 
is experienced if that signal is sampled at a frequency of W/z (or 2F) 
samples/sec. In sampling such signals, the sampling frequency must 
be two or more times the highest frequency contained in the signal to 
ensure no loss of information. For reasons that will become more evident 
later, practical considerations dictate a sampling frequency considerably 
higher than this theoretical minimum. 
The converse of this theorem can be readily understood by referring to 
(2.5), which gives the Fourier spectrum of a signal f(¢) which has been 
sampled by a periodic sampling function p(t). Figure 2.6 shows a finite 
spectrum Fy(w) and the repeated spectra F*,(w) which result from the 
