78 SAMPLED-DATA CONTROL SYSTEMS 
4.8 Sum of Squares of Sample Sequence 
A useful relation in the application of optimizing design criteria for 
sampled-data systems is the expression for the sum of the squares of a 
pulse sequence. This is analogous to the integrated-square integral used 
in continuous systems. The definition of this sum is given by the 
following: 
= ) [faT)P (4.62) 
n=0 
where f(T) is the nth sample in the sequence whose z transform is F(z). 
Using the inversion theorem as given in (4.20), 
f(nT) = — al ered (4.63) 
and substituting the result in (4.62), there results 
= > f(nT) os} il z”—F (z) dz (4.64) 
n=0 
By interchanging the order of the summation and the integration, 
a ae = ee dz : iu ee (4.65) 
The summation in (4.65) is recognized to be the z transform of the pulse 
sequence, of which f(n7’) is a typical term except for the fact that the 
exponents of z are positive instead of negative. In this case, the following 
identity is recognized: 
F(e-?) = » f(nT)2" (4.66) 
n=0 
Substituting back into (4.65), the expression for S? becomes 
Sybase / F(2)F(e-2)z7 de (4.67) 
21) Tr 
where it is recalled that the contour I is the unit circle since all poles of 
F(z) are contained therein for stable systems. The advantage of being 
able to express the sum of the squared samples in the z domain is that 
evaluation of (4.67) can be readily carried out. In those cases where 
optimization of the system is sought by adjustment of system parameters 
