Digital Signal Processing Reference
In-Depth Information
that the solution requires the minimization of an error function. To that end, we
first define the frequency response of an odd-order FIR filter with symmetrical
coefficients
h
(
n
), as shown in (7.33):
N
−
1
M
∑
∑
j
ω
−
j
ω
n
−
j
ω
M
H
(
e
)
=
h
(
n
)
⋅
e
=
e
c
(
m
)
⋅
cos(
m
ω
)
n
=
0
m
=
0
(7.33)
where
M
= (
N
− 1) / 2 and
⎧
h
(
M
)
for
m
=
0
c
(
m
)
=
(7.34)
2
⋅
h
(
M
-
m
),
for
m
=
1
2
…
,
M
⎩
We can then define the summation component of (7.33) as
M
∑
=
C
(
ω
)
=
c
(
m
)
⋅
cos(
m
ω
)
m
0
(7.35)
which is used to formulate the error function that will be the object of the
minimization. As shown in (7.36), the error function can be described in terms of
the desired frequency response
e
−
j
ω
M
D
(ω), the actual frequency response
e
−
j
ω
M
C
(ω), and a weighting function
W
(ω) that can be used to adjust the amount
of error in each filter band:
]
[
E
(
ω
)
=
W
(
ω
)
⋅
D
(
ω
)
−
C
(
ω
)
(7.36)
The desired frequency response function
D
(ω) is usually defined as being 1
within the passband of the filter and 0 within the stopband, although other values
can be assigned. The weighting function
W
(ω) can be defined equivalently
throughout the filter band, or it can be assigned a value of 1 within the passband
and 10 within the stopband if a smaller error value δ is desired in the stopband.
This result occurs because the minimization algorithm will produce equal amounts
of error throughout the defined frequency range, and since the stopband error has
been artificially increased by 10, the actual error will be 10 times smaller.
The optimum error function will produce variations within the passband and
stopband similar to those shown in Figure 7.2 (except that all ripple will be of the
same magnitude). The actual error function will alternate between positive and
negative δ values because of the summation of cosine functions. If we pick a set of
frequencies (
x
=
M
+ 1) at which the extremes of the error occur, (7.36) can be
written as
