Digital Signal Processing Reference
In-Depth Information
2.2
SOF TWARE FOR USE WITH THIS Topic
The software files needed for use with this topic (consisting of m-code (.m) files, VI files (.vi), and related
support files) are available for download from the following website:
http://www.morganclaypool.com/page/isen
The entire software package should be stored in a single folder on the user's computer, and the full
file name of the folder must be placed on the MATLAB or LabVIEW search path in accordance with the
instructions provided by the respective software vendor (in case you have encountered this notice before,
which is repeated for convenience in each chapter of the topic, the software download only needs to be
done once, as files for the entire series of four volumes are all contained in the one downloadable folder).
See Appendix A for more information.
2.3 DEFINITION & PROPERTIES
2.3.1 THE Z-TRANSFORM
The
z
-transform of a sequence
x
[
]
n
is:
∞
z
−
n
X(z)
=
x
[
n
]
n
=−∞
where
z
represents a complex number. The transform does not converge for all values of
z
; the region of
the complex plane in which the transform converges is called the
Region of Convergence (ROC)
, and
is discussed below in detail. The sequence
z
−
n
is a complex correlator generated as a power sequence of
the complex number
z
and thus
X(z)
is the correlation (CZL) between the signal
x
and a complex
exponential the normalized frequency and magnitude variation over time of which are determined by the
angle and magnitude of
z
.
[
n
]
2.3.2 THE INVERSE Z-TRANSFORM
The formal definition of the inverse
z
-transform is
1
2
πj
X(z)z
n
−
1
dz
x
[
n
]=
(2.1)
where the contour of integration is a closed counterclockwise path in the complex plane that surrounds
the origin (
z
= 0) and lies in the ROC.
There are actually many methods of converting a
z
-transform expression into a time domain
expression or sequence. These methods, including the use of Eq. (2.1), will be explored later in the
chapter.
2.3.3 CONVERGENCE CRITERIA
Infinite Length Causal (Positive-time) Sequence
When
x
to
z
n
(or in other words,
x
[
n
]
z
−
n
) must generally decrease in magnitude geometrically as
n
increases for convergence to a finite
sum to occur.
[
n
]
is infinite in length, and identically 0 for n
<
0, the ratio of
x
[
n
]
