84 SAMPLED-DATA CONTROL SYSTEMS 
sampled-data systems. A type of transform calculus which bears the 
same relation to difference equations as does the Laplace transformation 
to differential equations can be adapted to the description of sampled- 
data systems. Known as the z transformation, a set of relations between 
variables and system constants which are analogous to those of ordinary 
Laplace transformations can be found. One approach to the z trans- 
formation is to consider it as a Laplace transform of impulse sequences, 
where the value of the function at a particular instant is the area of the 
impulse. An alternate viewpoint is to consider the exponent of the 
ordering variable z as the position of the number in a sequence. The 
resulting expressions are identical if it is assumed that e7* in the Laplace- 
transform sequences is replaced by z. The Laplace-transform approach 
is useful in obtaining the response of pulsed linear systems or filters. In 
this case, the pulse transfer function relating the input and output 
impulse sequences is the z transform of the impulsive response of the 
system. For digitai systems, the approach in which the z transform is 
regarded in the light of generating functions and weighting sequences is 
more meaningful. A pulse transfer function for a digital system is readily 
derived, and because of the similarity between this pulse transfer function 
and that of pulsed linear systems, a unified method of analysis can be 
applied. 
Theorems which are analogous to those of the continuous Laplace trans- 
form can be derived. The inversion theorem, the initial- and final-value 
theorems, and the pulse transfer function are readily derived and applied. 
The equivalent to the transfer function of continuous systems is the pulse 
transfer function of sampled systems. The pulse transfer function 
relates the z transforms of the input and output pulse sequences. It 
is emphasized that the pulse transfer function relates only the pulse 
sequences and does not give direct information of the value of the output 
function between sampling instants. It will be shown later how delayed 
z transforms can be used to obtain such information indirectly, but this 
requires special manipulation. For those systems where the linear ele- 
ment is a pulsed filter, the pulse transfer function is the z transform of the 
pulse sequence resulting from sampling the impulsive response of the 
filter. In the case of a digital device which implements linear recursion 
formulas between input and output number sequences, the pulse transfer 
function is regarded as a weighting sequence. The input and output 
number sequences are produced by generating functions in which the 
exponent of z in the transform represents the position of the number in 
the sequence. 
There is a one-to-one similarity between the z transform relations, 
whether they describe a pulsed linear filter or a digital system. This 
makes possible a unified analysis and synthesis procedure for mixed sys- 
