The Z Transform
The bilateral z transform of the discrete-time signal is
defined to be
where
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
![$ n = 0$](http://www.dsprelated.com/josimages_new/filters/img133.png)
![$ -\infty$](http://www.dsprelated.com/josimages_new/filters/img582.png)
![]() |
(7.2) |
The unilateral z transform is most commonly used. For inverting z transforms, see §6.8.
Recall (§4.1) that the mathematical representation of a
discrete-time signal maps each integer
to a complex
number (
) or real number (
). The z transform
of
, on the other hand,
, maps every complex number
to a new complex number
. On a higher
level, the z transform, viewed as a linear operator, maps an entire
signal
to its z transform
. We think of this as a ``function to
function'' mapping. We may say
is the z transform of
by writing
![$\displaystyle \zbox {X \leftrightarrow x}
$](http://www.dsprelated.com/josimages_new/filters/img629.png)
![$\displaystyle X(z) = {\cal Z}_z\{x(\cdot)\}
$](http://www.dsprelated.com/josimages_new/filters/img630.png)
![$\displaystyle X = {\cal Z}\{x\}.
$](http://www.dsprelated.com/josimages_new/filters/img631.png)
![$ X(z)
\leftrightarrow x(n)$](http://www.dsprelated.com/josimages_new/filters/img632.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
![$ n\in{\bf Z}$](http://www.dsprelated.com/josimages_new/filters/img387.png)
![$ z\in{\bf C}$](http://www.dsprelated.com/josimages_new/filters/img626.png)
The z transform of a signal can be regarded as a polynomial in
, with coefficients given by the signal samples. For example,
the signal
![$\displaystyle x(n) = \left\{\begin{array}{ll}
n+1, & 0\leq n \leq 2 \\ [5pt]
0, & \mbox{otherwise} \\
\end{array}\right.
$](http://www.dsprelated.com/josimages_new/filters/img633.png)
![$ X(z) = 1 + 2z^{-1}+ 3z^{-2} = 1 + 2z^{-1}+ 3(z^{-1})^2$](http://www.dsprelated.com/josimages_new/filters/img634.png)
Next Section:
Existence of the Z Transform
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Time Domain Representation Problems