Digital Signal Processing Reference
In-Depth Information
L
+
1
L
+
1
D
W
W
W
W
W
L
L
L
2
2
+
1
1
2
+
2
h
h
h
h
h
N
L
N
N
L
N
L
N
L
+
1
+
1
+
2
2
2
2
2
2
2
2
2
Filter coefficients
h
h
h
h
h
1
2
L
L
+
1
L
2
+
1
Array cells containing data
Discarded array cells
Figure 2.3
Window function
W
k
and discretized impulse response function
H
k
are multiplied and
truncated to yield the filter coefficients
h
k
, starting with arbitrarily large
N
.
k
=
1
2
L
+
1
(2.4)
h
=
W
H
k
k
k
where
L
is even and chosen to be a few taps longer than
L
D
. All
h
k
are discarded
for
k
outside the given range. The design filter order
L
D
is used as a guide to the
final filter length. Note that the total filter length is
L
+1, and moreover, since in
this case
L
is chosen to be even, then
h
k
is an odd-symmetric linear phase filter
(e.g.,
[5])
.
Figure 2.3 shows the design process and interrelationships between
parameters. It is the
L
+1 values of
h
k
that are compiled and presented in this topic.
The filter response
h
ω
is obtained by replacing the discarded values of
h
k
with
zeroes, then performing the FFT on the resulting sequence. Figure 2.4, curve (c),
shows the resulting FFT response when
H
k
is Gaussian windowed, but not
truncated. It is quite noticeable that the attenuation in the rejection band is close to
-180 dB (i.e. a signal power rejection by a factor of 10
−
9
). But how is the filter
