Digital Signal Processing Reference
In-Depth Information
1
F
c
>
0.2
0.1
0.1
0.01
10
100
1000
Filter Length,
L
+1
Figure 3.3
Effect of filter length on 90%-10% transition width
∆
F
.
though less standard, is to use the width
F
, defined as the bandwidth between the
90% and 10% positions on the frequency response curve. This is smaller in value
than the standard transition width measure (shown in Figure 3.2) but is less
dependent on the attenuation levels in the rejection band. Moreover, this approach
works in our case because the ripples in the pass and rejection bands are free
parameters and are not actively used in the design of the filter. If the slope of the
transition region
∆
µ
T
is required, this is readily recovered from
0
. (3.1)
µ
=
T
f
∆
F
N
In the rest of the topic, transition width should be taken to mean
∆
F
, unless
otherwise stated.
The gross features of the 90%-10% transition width
∆
F
are shown in Figure
3.3. It is seen that to realize a
F
less than 0.01, more than 500 taps are needed.
Given there is an inverse proportionality between the transition width
∆
∆
F
and
filter length,
L
+1, this relationship takes on the form
a
∆
F
=
(3.2)
L
+
1
where for Gaussian-windowed filters (
= 3.5, and
L
D
~ 90% of
L
+1),
a
= 6.2368.
This empirical result (3.2) is valid for 0.15
α
0.9, and has been used to design
low-pass filters. Filter design rules are given in Chapter 2. This result however
suggests that the rate of increase in edge sharpness is fastest for small
L
, but gets
progressively slower for larger
L
. Thus, there is no optimum filter length for a
given set of design requirements.
≤
F
≤
