The Z transform is a generalization of the
Discrete-Time Fourier
Transform . It is used because the DTFT does not converge/exist for many important signals, and yet does for the z-transform. It is also used because it is notationally cleaner than the DTFT. In contrast to the DTFT, instead of using
complex exponentials of the form
$e^{i\omega n}$ ,
with purely imaginary parameters, the Z transform uses the more general,
$z^{n}$ ,
where
$z()$ is complex. The Z-transform thus allows one to bring in the power of complex variable theory into Digital Signal Processing.
The z-transform
Bilateral z-transform pair
Although Z transforms are rarely solved in practice using integration
(
tables and
computers (
e.g. Matlab) are much more
common), we will provide the
bilateral Z transform
pair here for purposes of discussion and derivation. These define the forward and inverse Z
transformations. Notice the similarities between the forwardand inverse transforms. This will give rise to many of the same
symmetries found in
Fourier
analysis .
Z transform
$X(z)=\sum_{n=()} $∞∞xnzn
Inverse z transform
$x(n)=\frac{1}{2\pi i}(z, , X(z)z^{(n-1)})$
We have defined the bilateral z-transform. There is also a
unilateral z-transform ,
$X(z)=\sum_{n=0} $∞xnzn
which is useful for solving the difference equations with nonzero initial conditions. This is similar to the
unilateral Laplace Transform in continuous time.
Relation between z-transform and dtft
Taking a look at the equations describing the Z-Transform and the Discrete-Time Fourier Transform:
Discrete-time fourier transform
$X(e^{i\omega})=\sum_{n=()} $∞∞xnωn
Z-transform
$X(z)=\sum_{n=()} $∞∞xnzn
We can see many similarities; first, that :
$X(e^{i\omega})=X(z)$
for all
$z=e^{i\omega}$
Visualizing the z-transform
With the DTFT, we have a complex-valued function of a real-valued variable
$\omega $ (and 2
$\pi $ periodic).
The Z-transform is a complex-valued function of a complex valued variable z.
With the Fourier transform, we had a
complex-valued
function of a
purely imaginary
variable ,
$F(i\omega )$ . This was something we could envision with two
2-dimensional plots (real and imaginary parts or magnitude andphase). However, with Z, we have a
complex-valued
function of a
complex variable .
In order to examine the magnitude and phase or real andimaginary parts of this function, we must examine
3-dimensional surface plots of each component.
Consider the z-transform given by
$H\left(z\right)=z$ , as illustrated below.
The corresponding DTFT has magnitude and phase given below.
While these are legitimate ways of looking at a signal in the
Z domain, it is quite difficult to draw and/or analyze.For this reason, a simpler method has been developed.
Although it will not be discussed in detail here, the methodof
Poles and Zeros is much easier to understand and is the way both the Z
transform and its continuous-time counterpart the
Laplace-transform are
represented graphically.
What could the system H be doing? It is a perfect all-pass, linear-phase
system. But what does this mean?
Suppose
$h\left[n\right]=\delta [n-{n}_{0}]$ . Then
Thus,
$H\left(z\right)={z}^{-{n}_{0}}$ is the
$z$ -transform of a system that simply delays the input by
${n}_{0}$ .
$H\left(z\right)$ is the
$z$ -transform of a unit-delay.
Now consider
$x\left[n\right]={\alpha}^{n}u\left[n\right]$
What if
$|\frac{\alpha}{z}|\ge 1$ ? Then
${\sum}_{n=0}^{\infty}\phantom{\rule{-0.166667em}{0ex}}{\left(\frac{\alpha}{z}\right)}^{n}$ does not converge! Therefore, whenever we compute a
$z$ -tranform, we must also specify the set of
$z$ 's for which the
$z$ -transform exists. This is called the
$regionofconvergence$ (ROC).
Using a computer to find the z-transform
Matlab has two functions,
ztrans and
iztrans , that are both part of the
symbolic toolbox, and will find the Z and inverseZ transforms respectively. This method is generally
preferred for more complicated functions. Simpler and morecontrived functions are usually found easily enough by using
tables .
Application to discrete time filters
The
$z$ -transform might seem slightly ugly. We have to worry about the region of convergence, and stability issues, and so forth. However, in the end it is worthwhile because it proves extremely useful in analyzing digital filters with feedback. For example, consider the system illustrated below
Hence, given a system the one above, we can easily determine the system's transfer function, and end up with a ratio of two polynomials in
$z$ : a rational function. Similarly, given a rational function, it is easy to realize this function in a simple hardware architecture.
Interactive z-transform demonstration
Conclusion
The z-transform proves a useful, more general form of the Discrete Time Fourier Transform. It applies equally well to describing systems as well as signals using the eigenfunction method, and proves extremely useful in digital filter design.
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