Digital Signal Processing Reference
In-Depth Information
1
1
n
o
= M − = 2
2 2
h ( n )
h ( n )
n
o
= M = 3
N = 7 (odd)
N = 6 (even)
n
n
0 1 2 3 4 5 6
0 1 2 3 4 5
(a)
(b)
Fig. 2.18
Symmetric impulse response for a FIR filter. a odd length. b even length
If N is an odd integer, i.e., N = 2M ? 1, the midpoint of symmetry is
n
o
= M.IfN is an even integer, i.e., N = 2M, the midpoint of symmetry is
n
o
¼
M
2
[see Fig. (
2.18
)].
The frequency response of a linear phase FIR filter (symmetric or anti-sym-
metric) is given by:
H
ð
e
jxT
s
Þ¼
H
ð
z
Þj
z
¼
exp
ð
jxT
s
Þ
¼
A
o
ð
x
Þ
e
jxT
s
M
þ
ja
;
ð
2
:
22
Þ
N
¼
2M
þ
1(odd)
;
a
¼
0orp
A
e
ð
x
Þ
e
jxT
s
ð
M
1
=
2
Þþ
jb
;
N
¼
2M(even)
;
b
¼
0orp
where A
o
(x) and A
e
(x) are arbitrary real-valued functions.
Example
If h(n) = d(n) ? d(n - 1), then:
N
¼
2
;
M
¼
1
;;
n
o
¼ð
N
1
Þ=
2
¼
M
=
2
¼
1
2
;
and
H
ð
e
jxT
s
Þ¼
1
þ
e
jxT
s
¼
2e
jxT
s
=
2
½ð
e
jxT
s
=
2
þ
e
jxT
s
=
2
Þ=
2
¼
2e
jxT
s
=
2
cos
ð
xT
s
=
2
Þ:
Hence, the magnitude response is given by A
e
ð
e
jxT
s
Þ¼
2
j
cos
ð
xT
s
=
2
Þj;
and the
phase response is / =-xT
s
/2.
In the z-domain, the system function is H(z) = 1 ? z
-1
= (z ? 1)/z, so that there
is a pole at z = 0 and a zero at z =-1 [see Fig. (
2.19
)]. It is essentially a LPF. [As
an exercise plot the frequency response vs. true frequency f (Hz) or x (rad/s)!]. The
pole-zero diagram can be found on MATLAB using
zplane(A,B)
, where
A=[1 1]
and
B=[1 0]
.
2.6.4.6 Efficient Hardware Implementation of Linear Phase FIR Filters
Recall that the transfer function of a general FIR filter of length N is:
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