Digital Signal Processing Reference
In-Depth Information
z
-transform manages to “bring down” the unit step because of the vanish-
ing power of
z
−n
for
1and
n
large and this is the case for all one-sided
sequences which grow nomore than exponentially. However, if
|
z
|
>
z
−n
|
|→
0for
l
g
r
,
y
i
d
.
,
©
,
L
s
n→∞
then necessarily
|z
−n
|→∞
for
n→−∞
and this may pose a problem
for the convergence of the
z
-transform in the case of two-sided sequences.
In particular, the
z
-transform does not converge in the case of periodic sig-
nals since only one side of the repeating pattern is “brought down” while the
other is amplified limitlessly. We can circumvent this impasse by “killing”
half of the periodic signal with a unit step. Take for instance the one-sided
cosine:
x
[
n
]=
cos
(
ω
0
n
)
u
[
n
]
its
z
-transform can be derived as
∞
z
−n
cos
X
(
z
)=
(
ω
0
n
)
u
[
n
]
n
=
−∞
∞
z
−n
cos
=
(
ω
0
n
)
n
=
0
∞
∞
1
2
1
2
e
j
ω
0
n
z
−n
e
−j
ω
0
n
z
−n
=
+
n
=
0
n
=
0
e
−j
ω
0
z
−
1
1
2
1
1
=
e
j
ω
0
z
−
1
+
1
−
1
−
1
−
cos
(
ω
0
)
z
−
1
1
−
2cos
(
ω
0
)
z
−
1
=
+
z
−
2
Similar results can be obtained for signals such as
x
[
n
]=
sin
(
ω
0
n
)
u
[
n
]
or
x
[
n
]=
α
n
cos
(
ω
0
n
)
u
[
n
]
.
Example 6.2: The impossibility of ideal filters
The
z
-transform of an FIR impulse response can be expressed as a simple
polynomial
P
1where
L
is the number of nonzero taps of
the filter (we can neglect leading factors of the form
z
−N
). The fundamental
theorem of algebra states that such a polynomial has at most
L
(
z
)
of degree
L
−
1 roots; as
a consequence, the frequency response of an FIR filter can never be iden-
tically zero over a frequency interval since, if it were, its
z
-transform would
have an infinite number of roots. Similarly, by considering the polynomial
P
−
C
,wecanprovethatthefrequencyresponsecanneverbeconstant
C
over an interval which proves the impossibility of achieving ideal (i.e. “brick-
wall”) responses with an FIR filter. The argument can be easily extended to
rational transfer functions, confirming the impossibility of a realizable filter
whose characteristic is piecewise perfectly flat.
(
z
)
−
