INTRODUCTION 9 
The form and constants of the pulse transfer function G(z) are a property 
of the system and can be found in terms of the system constants. These 
relations will be rigorously derived in later chapters. 
The relation expressed in (1.2) is dependent on the fact that the 
input and output samples are taken with the same sampling instants. 
en ae LORE nee IEE 
\ 
\ 
\ Ge 
\ \ 
: ~~ _ Rez) S Cle] 
Input Sampled Sampled 
input Linear output 
system 
Fia. 1.7. The pulse transfer function G(z). 
It is possible to extend the concepts of the z transformation to include 
the case where the output and input samples are taken at some integral 
multiple of a basic sampling rate. For instance, if the basic rate is 
taken as unity, the input and output sampling operations can take place 
at two and three times the basic rate, respectively. Such systems are 
referred to as multirate sampled-data systems, and suitable modifications 
in z-transform theory can be made to cover these cases. A typical system 
is shown in Fig. 1.8, where the input sampler operates with a sampling 
Gi(z,) 
Th R(z) T/n Clz ) 
Input Sampled 
Multi-rate 
input Linear sampled 
system output 
Fic. 1.8. Multirate sampled-data system. 
interval 7, while the output sampler has a sampling interval T/n. The 
pulse transfer function relating these sequences at input and output is the 
multirate pulse transfer function G(z,). If R(z) is the z transform of the 
input sequence and C(z,) is the multirate z transform of the output, then 
they are related as follows: 
C(én) = Glen) RZ) (1.3) 
Inversion of (1.3) will yield the multirate output sequence. 
Sampled-data systems are often subjected to inputs or disturbances 
which are random. To handle this situation analytically, a number of 
concepts and definitions analogous to those for continuous systems must 
be devised. Such terms as auto- and cross-correlation function of sample 
sequences, sampled power spectra, and cross spectra are used. Relations 
which give the shaping effects of a linear sampled-data system can be 
