Introduction
z transforms are used to solve problems in
discrete systems in a manner similar to the use of Laplacetransforms for piecewise continuous systems. We take z transforms of sequences.
We shall assume that our sequences begin with the zeroth term and have terms
for positive n .f0, f1, f2,.
. ., fn, . . . is an input sequence to the system. However,
when considering the initial conditions for a difference equation it is
convenient to assign them to y?j , . . . , y?2,y?1, etc., where j is the order of the difference
equation. So, in that case, we shall allow some elements in the sequence with
negative subscript. Our output sequence will be of the form y?j, . . . , y?2, y?1, y0, y1,y2, . . . , yn, . . . where the difference equation
describing the system only holds for n ³ 0.
z-transform
definition
The z-transform of
a sequence f0, f1, f2, .
. . , fn, . . . is given by
As this is an infinite
summation it will not always converge. The set of values of z for which
it exists is called the region of convergence. The sequence, f0,f1, f2, . . . , fn,. . . is a function of an integer, however, its z-transform is a
function of a complex variable z. The operation of taking the z transform
of the sequence fn is represented by Z{fn} = F(z). As this is an infinite summation it will
not always converge. The set of values of z for which it exists is
called the region of convergence. The sequence, f0, f1,f2. . . fn, . . . is a function of
an integer, however, its z-transform is a function of a complex variablez. The operation of taking the z transform of the sequence fn is represented by Z{fn}=F(z).