Digital Signal Processing Reference
In-Depth Information
CHAPTER
2
The z-Transform
2.1
OVERVIEW
In the previous chapter, we took a brief look at the Fourier and Laplace families of transforms, and a
more detailed look at the DTFT, which is a member of the Fourier family which receives a discrete
time sequence as input and produces an expression for the continuous frequency response of the discrete
time sequence. With this chapter, we take up the
z
-transform, which uses correlators having magnitudes
which can grow, decay, or remain constant over time. It may be characterized as a discrete-time variant
of the Laplace Transform. The
z
-transform can not only be used to determine the frequency response
of an LTI system (i.e., the LTI system's response to unity-amplitude correlators), it reveals the locations
of poles and zeros of the system's transfer function, information which is essential to characterize and
understand such systems. The
z
-transform is an indispensable transform in the discrete signal processing
toolbox, and is virtually omnipresent in DSP literature. Thus, it is essential that the reader gain a good
understanding of it.
The
z
-transform mathematically characterizes the relationship between the input and output
sequences of an LTI system using the generalized complex variable
z
, which, as we have already seen, can
be used to represent signals in the form of complex exponentials. Many benefits accrue from this:
• An LTI system is conveniently and compactly represented by an algebraic expression in the vari-
able
z
; this expression, in general, takes the form of the ratio of two polynomials, the numerator
representing the FIR portion of the LTI system, and the denominator representing the IIR portion.
• Values of
z
having magnitude 1.0, which are said to “lie on the unit circle” can be used to evaluate
the
z
-transform and provide a frequency response equivalent to the DTFT.
• Useful information about a digital system can be deduced from its
z
-transform, such as location of
system poles and zeros.
• Difference equations representing the LTI system can be constructed directly from inspection of
the
z
-transform.
• An LTI system's impulse response can be obtained by use of the Inverse
z
-transform, or by con-
structing a digital filter or difference equation directly from the
z
-transform, and processing a unit
impulse.
• The
z
-transform of an LTI system has, in general, properties similar or analogous to various other
frequency domain transforms such the DFT, Laplace Transform, etc.
By the end of this chapter, the reader will have gained a practical knowledge of the
z
-transform, and
should be able to navigate among difference equations, direct-form, cascade, and parallel filter topologies,
and the
z
-transform in polynomial or factored form, converting any one representation to another. Addi-
tionally, an understanding will have been acquired of the inverse
z
-transform, and use of the
z
-transform
to evaluate frequency response of various LTI systems such as the FIR and the IIR.
