Digital Signal Processing Reference
In-Depth Information
250
10
(
a
)
(
b
)
0
150
-10
50
-20
-50
-30
-150
-40
-250
-50
-350
-60
0
50
100
0
0.2
0.4
0.6
0.8
1
Index,
k
Normalised Frequency,
F
Normalized Frequency,
F
Figure 2.17
Design of a notch filter from an all-pass and a single low-pass filter.
(a) Impulse
response,
h
N
and, (b) frequency response.
2.6.4.1 The Notch Filter
The notch filter can be realized exactly at frequency
F
notch
, as a special case of the
band-stop filter. This exhibits the narrowest rejection bandwidth for the given
order
L
, with maximum attenuation exactly at
F
notch
. In this case, the filter
coefficients are given by
=
2
(
)
h
h
h
F
(2.21)
notch
N
AP
LP
Note that the attenuation in the stop band is usually limited to about -35 dB.
However, if the cut-off frequency is 1/2
p
, where
p
is an integer, then attenuation of
-100 dB is readily realized. Figure 2.17 shows a 99-tap impulse response function
and its frequency response characteristics for
F
notch
= 0.35. If more attenuation is
needed one should try a higher filter order, or cascade two notch filters. If two
identical filters are cascaded, the result is a new filter whose decibel attenuation
and length are doubled. Cascaded filters will be discussed in Section 2.7.
2.6.5 A Low-Pass Filter with Gain Near Transition Edge
In some instances, it is required to design a filter with uncommon shape in the
pass band. Armed with an appropriate set of filter sections, the required filter
could be assembled. As an example, suppose a low pass filter is desired with cut-
off at say,
F
c
, and with peak gain of
close to the transition edge. This can be
made up from a low-pass filter with cut-off
F
c
and a band-pass section tailored so
that its peak magnitude is
(linear scale), and one of its edges at
F
c-off
is identical
to
F
c
. This picture is now readily transferred into the time domain by a linear
