In linear time-invariant systems, the Laplace transform can be
utilized in system analysis and design.
The Z-transform which will be now defined can be utilized in the
analysis of discrete-time systems modeled by difference equations of
the form given by equation [3]:
which is the general form of an nth-order linear difference equation.
A transform is defined for number sequences as follows. the function
is defined as a power series in 
with coefficients
equal to the values of the number sequence 
. This
transform, called the z-transform, is then expressed by the transform
pair
where 
and 
indicates the z-transform
operation and 
indicates the inverse
z-transform. in equations [4], [5] can be written in
more compact notation as
The z-transform is defined for any number sequence , and
may be used in the analysis of any type of system described by linear
time-invariant difference equations.
