Digital Signal Processing Reference
In-Depth Information
Fig. 6.3 Normalized
frequency response of the
first order Walsh filter
W
(
j
Ω
)
1
W
(
j
Ω
)
1
1
N
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Ω
Ω
N
Fig. 6.4 Normalized
frequency response of the
second order Walsh filter
W
(
j
Ω
)
2
W
(
j
Ω
)
2
N
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Ω
Ω
N
The resulting filter transfer function is:
2
W
2
ð
z
Þ¼
X
N
=
4
1
z
k
þ
X
3N
=
4
1
z
k
X
N
1
z
k
¼
1
z
N
=
4
1
z
N
=
2
ð
6
:
22
Þ
1
z
1
k
¼
0
k
¼
N
=
4
k
¼
3N
=
4
From (
6.25a
,
b
,
c
), after known substitution, one obtains the filter frequency
response:
sin N
4
1
cos
N
4
W
2
ð
jX
Þ¼
2 exp
j
N
1
2
2
X
ð
6
:
23
Þ
X
sin
From (
6.26
) results that the 2nd order Walsh filter has also linear phase, it is
furthermore orthogonal to the first order Walsh function. As before the frequency
responses depend on window length N only (for given sampling frequency). The
responses are presented graphically in Fig.
6.4
. It is seen that in this case one also
gets a band-pass filter of not very high quality, but very simple to be realized.
Frequency responses of the filters using last two Walsh functions from Fig.
6.1
are shown in Fig.
6.5
. Certain improvement of filter features can be seen, first of all
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