Digital Signal Processing Reference
In-Depth Information
k
1
k
2
¼
f
1
f
2
¼
4
5
;
representing ratio of the smallest possible integer factors k
1
;
k
2
:
Therefore, con-
sidering the first equation one gets (for the cases (a) and (b)) the following window
lengths:
N
a
¼
f
S
k
1
f
1
¼
1000
4
100
N
b
¼
f
S
k
2
f
2
¼
600
4
100
¼
40;
¼
24
The output signal of the FIR filter applying first order Walsh function is
obtained using the following (also very simple) difference equation:
y
ð
n
Þ¼
X
N
=
2
1
x
ð
n
k
Þ
X
N
1
x
ð
n
k
Þ:
ð
6
:
18
Þ
k
¼
0
k
¼
N
=
2
In this case one also calculates a sum (with proper sign) of actual and preceding
input samples. The filter algorithm is here again as easy as before and requires
addition of a half of input samples and subtraction of remaining half of the samples
observed within the filter window (all coefficients are equal to either one or minus
one). Transfer function of the filter is calculated applying Z transform to both sides
of Eq.
6.18
:
2
W
1
ð
z
Þ¼
X
N
=
2
1
z
k
X
N
1
z
k
¼
1
z
N
=
2
:
ð
6
:
19
Þ
1
z
1
k
¼
0
k
¼
N
=
2
Substituting exp
ð
jX
Þ
for operator z and rearranging one gets the frequency
response of the first order Walsh function:
sin
2
N
4
sin
W
1
ð
jX
Þ¼
2j exp
j
N
1
2
X
2
:
ð
6
:
20
Þ
X
The formula (
6.20
) represents again a linear phase filter, however, the resulting
phase shift is by p
=
2 greater than before, giving the filter orthogonal to that of zero
order. Both magnitude and phase of frequency responses depend on window length
N as before. Doing the same calculations as before one can present these responses
graphically (Fig.
6.3
). It is seen that this time a band-pass filter is obtained, not of
high quality, but having very simple algorithm and easy to be applied with low
processor and memory burden.
The FIR filter applying second order Walsh function produces output signal
according to equation:
y
ð
n
Þ¼
X
N
=
4
1
x
ð
n
k
Þþ
X
3N
=
4
1
x
ð
n
k
Þ
X
N
1
x
ð
n
k
Þ
ð
6
:
21
Þ
k
¼
0
k
¼
N
=
4
k
¼
3N
=
4
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