A brief definition of the z-transform, explaining its relationship with the Fourier transform and its region of convergence, ROC.
Basic definition of the z-transform
The
z-transform of a sequence is defined as
$X(z)=\sum_{n=()} $∞∞xnzn
Sometimes this equation is referred to as the
bilateral z-transform . At times the z-transform is defined as
$X(z)=\sum_{n=0} $∞xnzn
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
$X(e^{i\omega})=\sum_{n=()} $∞∞xnωn
Notice that that when the
$z^{-n}$ is replaced with
$e^{-(i\omega n)}$ the z-transform reduces to the Fourier Transform. When the
Fourier Transform exists,
$z()=e^{i\omega}$ , 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 lookat 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
$re^{(i\omega )}$ , and the angle from the positive, real axis around the plane is
denoted by
$\omega $ .
$X(z)$ is defined
everywhere on this plane.
$X(e^{i\omega})$ on the other
hand is defined only where
$\left|z\right|=1$ ,
which is referred to as the unit circle. So for example,
$\omega =1$ at
$z=1$ and
$\omega =\pi ()$ at
$z=-1$ .
This is useful because, by representing the Fourier transformas 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 wherethe z-transform exists. The ROC for a given
$x(n)$ , is defined as the range of
$z$ for which the z-transform converges. Since the z-transform is
a
power series , it converges when
$x(n)z^{-n}$ is absolutely summable. Stated differently,
$\sum_{n=()} $∞∞xnzn∞
must be satisfied for convergence. This is best illustratedby looking at the different ROC's of the z-transforms of
$\alpha ^{n}u(n)$ and
$\alpha ^{n}u(n-1)$ .
For
$x(n)=\alpha ^{n}u(n)$
$X(z)=\sum_{n=()} $∞∞xnznn∞∞αnunznn0∞αnznn0∞αz1n
This sequence is an example of a right-sided exponential
sequence because it is nonzero for
$n\ge 0$ .
It only converges when
$\left|\alpha z^{(-1)}\right|< 1$ .
When it converges,
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