TABLE OF CONTENTS The Z-transform plays an important role in the fields of communication engineering and control engineering at the stage of analysis and representation of discrete-time linear shift invariance systems. When continuous signals are sampled, discrete-time functions arise. The application of the Z-transform in discrete time systems is similar to that of the Laplace transform in continuous time systems.
Let { f( n)} be a sequence defined for all positive integers n. Then the Z-transform of f( n) is defined as
where z is an arbitrary complex number.
This is a one sided Z-transform.
Notes :
If { f( n)} is defined for n = 0, 1, 2, ...... then 
and is called a two sided Z-transform.
If f( n) = 0 for n < 0 then { f( n)} is called a casual sequence.
The bracket { } represents a sequence.
The infinite series on the R.H.S. of (1) will be convergent only for certain values of z depending on the sequence f( n). The inverse Z-transform of Z{ f( n)} = F( z) is defined as Z 1{F( z)} = f( n).
It is defined as 
It is defined as 
We have
Also
