Digital Signal Processing Reference
In-Depth Information
combination of the composite impulse responses, so that the discrete impulse
response for the new filter
h
is given by
=
(
)
+
α
(
,
)
(2.22)
h
h
F
h
F
F
LP
c
-
off
BP
c
-
on
c
-
off
where
is the linear gain factor, and
h
BP
(
F
c-on
,
F
c-off
) is the band-pass filter at the
indicated cut-on and cut-off frequencies. Figure 2.18 shows a low-pass filter
derived in this way, where
F
c-off
= 0.55 and the number of coefficients for the low-
pass section is
L
LP
= 99, whereas for the band-pass section its length
L
BP
= 55,
with cutoff edge at
F
c-off
= 0.55, bandwidth of 0.1, and
= 0.1. However, it is
advisable that the length of major filter section, which in this case is the low-pass
filter, should be longer than the other composite filter sections. Moreover, all filter
coefficients should be center aligned before summing. A quadratic characteristic
could be realized instead of the band-pass filter section by using a second-order
differentiating filter.
350
1.2
(
a
)
(
b
)
New LPF
300
1
250
LPF
200
0.8
150
0.6
BPF
X10 scale
Figure 2.18
(a) Impulse response function for new low-pass at
F
c
= 0.55 (New LPF). (b) Frequency
response of new low-pass filter showing low-pass (LPF) and band-pass (BPF, on
100
0.4
50
0
0.2
-50
-100
0
0
0.2
0.4
0.6
0.8
1
0
50
100
Normalised Frequency,
F
Normalized Frequency,
F
Index,
k
×
10 scale) filter
composites.
2.6.5.1 Alternative Method
An alternative method of producing a low-pass filter with gain near the transition
edge is by using three low-pass sections. This is readily seen from (2.22) where
the band-pass section already comprises two low-pass sections, so that by
straightforward substitution of (2.19) into (2.22), the latter is reexpressed as
=
(
+
α
)
(
)
α
(
)
(2.23)
h
h
F
h
F
LP
c,
off
LP
c,
on
