Databases Reference
In-Depth Information
Example14.3.2:
Consider a filter with
a
0
=
1 and
b
1
=
2. Suppose the input sequence is a single 1 followed
by 0s. Then the output is
y
0
=
a
0
x
0
+
b
1
y
−
1
=
(
)
+
(
)
=
(19)
1
1
2
0
1
y
1
=
a
0
x
0
+
b
1
y
0
=
(
)
+
(
)
=
(20)
1
0
2
1
2
y
2
=
a
0
x
1
+
b
1
y
1
=
1
(
0
)
+
2
(
2
)
=
4
(21)
.
.
2
n
y
n
=
(22)
30 is 2
30
, or more than a
Even though the input contained a single 1, the output at time
n
=
billion!
Although IIR filters can become unstable, they can also provide better performance, in
terms of sharper cutoffs and less ripple in the passband and stopband for a fewer number of
coefficients.
The study of design and analysis of digital filters is a fascinating and important subject.
We provide some of the details in Sections
14.5
,
14.6
,
14.7
,
14.8
. If you are not interested in
these topics, you can take a more utilitarian approach and make use of the literature to select
the necessary filters rather than design them. In the following section we briefly describe some
of the families of filters used to generate the examples in this chapter. We also provide filter
coefficients that you can use for experiment.
14.3.1 Some Filters Used in Subband Coding
The most frequently used filter banks in subband coding consist of a cascade of stages, where
each stage consists of a low-pass filter and a high-pass filter, as shown in Figure
14.5
.The
most popular among these filters are the
quadrature mirror filters
(QMF), which were first
proposed by Crosier, Esteban, and Galand [
203
]. These filters have the property that if the
impulse response of the low-pass filter is given by
{
h
n
}
, then the high-pass impulse response is
n
h
N
−
1
−
n
}
given by
. The quadrature mirror filters designed by Johnston [
204
] are widely
used in a number of applications. The filter coefficients for 8-, 16-, and 32-tap filters are given
in Tables
14.1
,
14.2
,
14.3
. Notice that the filters are symmetric; that is,
{
(
−
1
)
N
2
−
h
N
−
1
−
n
=
h
n
n
=
0
,
1
,...,
1
(23)
As we shall see later, the filters with fewer taps are less efficient in their decomposition than
the filters with more taps. However, from Equation (
18
) we can see that the number of taps
dictates the number of multiply-add operations necessary to generate the filter outputs. Thus,
if we want to obtain more efficient decompositions, we do so by increasing the amount of
computation.
