36
D. Rocchesso: Sound Processing
filter still has a lowpass shape. As compared to the first-order FIR filter, the
one-pole filter gives a steeper magnitude response curve. The fact that, for a
given filter order, the IIR filters give a steeper (or, in general, a more complex)
frequency response is a general property that can be seen as an advantage in
preferring IIR over FIR filters. The other side of the coin is that IIR filters can
not have a perfectly-linear phase. Furthermore, IIR filters can produce numerical
artifacts, especially in fixed-point implementations.
The one-pole filter can also be analyzed by watching its pole-zero distribution
on the complex plane. To this end, we rewrite the transfer function as a ratio of
polynomials in z and give a name to the root of the denominator: p
0
=
1
2
. The
transfer function has the form
H(z) =
1
2
z
z -
1
2
=
1
2
z
z - p
0
.
(32)
We can apply the graphic method presented in sec. 2.1.1 to have a qualitative
idea of the magnitude and phase responses. In order to do that, we consider the
point e
j
on the unit circle as the head of the vectors that connect it to the pole
p
0
and to the zero in the origin. Fig. 15 is illustrative of the procedure. While
we move along the unit circumference from dc to the Nyquist frequency, we
go progressively away from the pole, and this is reflected by the monotonically
decreasing shape of the magnitude response.
-1.5
-1
-0.5
0
0.5
1
1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 15: Single pole (×) and zero in the origin ()
To have a complete picture of the filter behavior we need to analyze the
transient response to the causal complex exponential. The Z transform of the
input has the well-known form
X(z) =
1
1 - e
j
0
z
-1
.
(33)
A multiplication of X(z) by H(z) in the Z domain gives
Y (z) = H(z)X(z) =
1
2
1
1 -
1
2
z
-1
1
1 - e
j
0
z
-1
=
1/2
1 - 2e
j
0
1
1 -
1
2
z
-1
+
1/2
1 - 1/2e
-j
0
1
1 - e
j
0
z
-1
,
(34)
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