SAMPLED-DATA SYSTEMS 95 
It is readily apparent that the magnitude of a summation is always less 
than the summation of the magnitudes. Thus, 
le@mT)| SY |g(eT)| [rm — 1) TI (5.35) 
k=0 
The second set of terms in the summation is all bounded in view of the 
fact that it represents the input pulse sequence. Thus, |r[(m — k)T]|is 
always less than some finite positive number M, and therefore 
le(mT)| SM Y g(t) (5.36) 
k=0 
Hence, c*(t) is to be bounded if 
Y kT) < & (5.37) 
k=0 
Thus, it follows that a sufficient condition for the sampled-data system to 
be stable is that the summation of the magnitudes of the samples in the 
sampled impulsive response be bounded. 
That the condition given in (5.37) is necessary as well as sufficient can 
be ascertained by finding at least one bounded input which makes the 
condition necessary. Such an input is one in which the signs of the 
various input samples in (5.34) are the same as those of the samples 
g(kT). In this ease, all the terms of (5.34) are positive, and their sum is 
identically that given by (5.35). Thus, in order for c(mT) to remain 
bounded for a bounded input, it is necessary for the summation of the 
magnitudes of g(k7’) to be bounded since the signs have effectively been 
all made positive by the choice of signs for the input sequence. The 
necessary and sufficient condition for the stability of a system is given by 
(5.37). This summation is the analogue of the one applying to continu- 
ous systems which states that the time integral of the magnitude of the 
impulsive response of the system must be bounded in a stable system. 
While the criterion for stability given by (5.37) is rigorous, it is not 
readily applicable to the problems normally encountered in system syn- 
thesis and analysis. It is desirable to relate the condition to character- 
istics of the pulse transfer function in much the same way as is done in 
continuous systems. In the latter, satisfaction of the requirement that 
the integral of the impulsive response be bounded is tested by the pres- 
ence or absence of poles of the transfer function in the right half of the com- 
plex frequency plane. A similar condition must be sought for sampled- 
data systems whose characteristics are contained in the pulse transfer 
function. 
To relate the condition for stability to the singularities of the pulse 
