Digital Signal Processing Reference
In-Depth Information
l
g
r
,
y
i
d
.
,
©
,
L
s
Chapter 6
The Z-Transform
Mathematically, the
z
-transform is a mapping between complex sequences
and analytical functions on the complex plane. Given a discrete-time signal
x
[
n
]
,the
z
-transform of
x
[
n
]
is formally defined as the complex function of
acomplexvariable
z
∈
∞
)=
x
]
=
z
−n
X
(
z
[
n
x
[
n
]
(6.1)
n
=
−∞
Contrary to the Fourier transform (as well as to other well-known trans-
forms such as the Laplace transform or the wavelet transform), the
z
-trans-
form is not an analysis tool
per se
, in that it does not offer a new physical
insight on the nature of signals and systems. The
z
-transform, however, de-
rives its status as a fundamental tool in digital signal processing from two
key features:
•
Its mathematical formalism, which allows us to easily solve constant-
coefficient difference equations as algebraic equations (and this was
precisely the context in which the
z
-transform was originally inv-
ented).
•
Its close association to the DTFT, which provides us with easy stabil-
ity criteria for the design and the use of digital filters. (It is evident
that the
z
-transform computed on the unit circle, i.e. for
z
e
j
ω
,is
=
nothing but the DTFT of the sequence).
Probably the best approach to the
z
-transform is to consider it as a
clever mathematical
transformation
which facilitates the manipulation of
complex sequences; for discrete-time filters, the
z
-transformbridges the al-
gorithmic side (i.e. the CCDE) to the analytical side (i.e. the spectral proper-
ties) in an extremely elegant, convenient and ultimately beautiful way.
