THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 79 
which will minimize the sum of the squared error pulses, (4.67) proves to 
be extremely useful. 
EXAMPLE 
The z transform of a pulse sequence F(z) is the following: 
1 
1 — e-4Fz71 
1) 
To find the sum of the squared pulses described by F(z), the expression 
is substituted in (4.67), resulting in 
1 1 1 
yy we ical = 
© 20] il We) Waes222 * de 
This integral can be rearranged into the following form: 

S2=— ! nay dz 
Qrj Jr (2 — € *)(z — ef?) 

Evaluating the integral by taking the residue at the pole z = e~*, 
which is located inside the unit circle I’, the resultant expression for S? is 
1 
2 — aE 
2 1 — e724F 
For example, if the constant a were unity and if the sampling interval T 
were | sec, the sum of the squares of all the samples ranging from zero to 
infinity would be 
1 
a Ge 
= 1.16 units square 
S? = 
Evaluation of the integral for more complex expressions is carried out 
in the same manner. 
4.9 The Two-sided Z Transform 
In most cases, the behavior of a sampled-data system is required for 
positive time only. When a systematic input such as a step or ramp 
function is applied, this restriction is completely satisfactory, just as it is 
in the case of continuous systems. The input is assumed in these cases 
to be zero for all negative time. For some situations, however, and par- 
ticularly those in which the input is a stationary random time function, 
consideration of the input for all time, positive and negative, is required. 
The single-sided z transform, as described in previous sections, is unsatis- 
factory for this class of inputs. To treat systems with inputs of this 
type, the analogue of the continuous two-sided Laplace transform is used. 
