Digital Signal Processing Reference
In-Depth Information
) in (4.52) is expressed in terms of
an infinite number of coefficients, and to obtain a realizable filter, we must truncate
(4.52), which yields the approximated transfer function
with
C
n
=
C
-
n
. The desired transfer function
H
d
(
Q
=
Â
()
=
H
C e
jn
p
(4.55)
a
n
nQ
where
Q
is positive and finite and determines the order of the filter. The larger the
value of
Q
, the higher the order of the FIR filter and the better the approximation
in (4.55) of the desired transfer function. The truncation of the infinite series with
a finite number of terms results in ignoring the contribution of the terms outside a
rectangular window function between
Q
. In Section 4.6 we see how the
characteristics of a filter can be improved by using window functions other than
rectangular.
Let
z
-
Q
and
+
e
j
p
; then (4.55) becomes
=
Q
=
Â
()
=
Hz
Cz
n
(4.56)
a
n
nQ
with the impulse response coefficients
C
-
Q
,
C
-
Q
+1
,...,
C
-1
,
C
0
,
C
1
,...,
C
Q
-1
,
C
Q
.The
approximated transfer function in (4.56), with positive powers of
z
, implies a non-
causal or not realizable filter that would produce an output before an input is
applied. To remedy this situation, we introduce a delay of
Q
samples in (4.56) to
yield
Q
=
Â
()
=
()
=
Hz
z H z
-
Q
Cz
nQ
-
(4.57)
a
n
nQ
Let
n
-
Q
=-
i
; then
H
(
z
) in (4.57) becomes
2
Q
=
Â
0
()
=
Hz
C
z
Qi
-
i
(4.58)
-
i
Let
h
i
=
C
Q
-
i
and
N
-
1
=
2
Q
; then
H
(
z
) becomes
N
-
Â
0
1
()
=
Hz
hz
i
-
i
(4.59)
i
=
where
H
(
z
) is expressed in terms of the impulse response coefficients
h
i
, and
h
0
=
C
Q
,
h
1
=
C
Q
-1
,...,
h
Q
=
C
0
,
h
Q
+1
=
C
-1
=
C
1
,...,
h
2Q
=
C
-
Q
. The impulse response
coefficients are symmetric about
h
Q
, with
C
n
=
C
-
n
.
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