The Z Transform: Definitionhttp://cnx.org/contenthttp://cnx.org/content/m10549/latest/m10549The Z Transform: Definition2.102002/03/292010/07/29 14:47:22.600 GMT-5MariyahPoonawalaMariyah Poonawalamariyah@rice.eduBenjaminFiteBenjamin Fitebfite@rice.eduPrashantSinghPrashant Singhprash@ece.rice.edubfitemariyah prashbfiteFourier transformROCz-transformMathematics and Statistics Science and TechnologyA brief definition of the z-transform, explaining its relationship with the Fourier transform and its region of convergence, ROC.enBasic Definition of the Z-Transform
The z-transform of a sequence is defined as
Xznxnzn
Sometimes this equation is referred to as the bilateral z-transform. At times the z-transform is defined as
Xzn0xnzn
which is known as the unilateral z-transform.
There is a close relationship between the z-transform and the
Fourier transform of a discrete time signal,
which is defined as
Xωnxnωn
Notice that that when the
zn
is replaced with
ωn
the z-transform reduces to the Fourier Transform. When the
Fourier Transform exists,
zω
, which is to have the magnitude of
z equal to unity.
The Complex Plane
In order to get further insight into the relationship between
the Fourier Transform and the Z-Transform it is useful to look
at the complex plane or z-plane. Take a look at
the complex plane:
The Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable
z. The position on the complex
plane is given by
rω
, and the angle from the positive, real axis around the plane is
denoted by ω.
Xz is defined
everywhere on this plane. Xω
on the other
hand is defined only where
z1
,
which is referred to as the unit circle. So for example,
ω1
at
z1
and
ω
at
z-1.
This is useful because, by representing the Fourier transform
as the z-transform on the unit circle, the periodicity of
Fourier transform is easily seen.
Region of Convergence
The region of convergence, known as the ROC, is
important to understand because it defines the region where
the z-transform exists. The ROC for a given
xn
, is defined as the range of
z
for which the z-transform converges. Since the z-transform is
a power series, it converges when
xnzn
is absolutely summable. Stated differently,
nxnzn
must be satisfied for convergence. This is best illustrated
by looking at the different ROC's of the z-transforms of
αnun
and
αnun1.
For
xnαnunXznxnznnαnunznn0αnznn0αz1n
This sequence is an example of a right-sided exponential
sequence because it is nonzero for
n0.
It only converges when
αz1.
When it converges,
Xz11αzzzα
If
αz1,
then the series,
n0αzn
does not converge. Thus the ROC is the range of values where
αz1
or, equivalently,
zα
For
xnαnun1Xznxnznnαnu-n1znn-1αnznn-1α-1znn1α-1zn1n0α-1zn
The ROC in this case is the range of values where
α-1z1
or, equivalently,
zα
If the ROC is satisfied, then
Xz111α-1zzzα