Digital Signal Processing Reference
In-Depth Information
discrete frequencies
{
Ω
1
<
Ω
2
< <
Ω
L
+
2
∈
S
, and solve Eq. (15.50) at the
selected frequencies. Assuming that the selected frequencies are the extremal
frequencies at which the maximum error changes between its peak value of
ε
max
, Eq. (15.50) reduces to
1
W
(
Ω
m
)
(
−
1)
m
ε
max
=
H
d
(
Ω
m
)
.
G
(
Ω
m
)
+
(15.51)
for 1
≤
m
≤
(
L
+
2). The resulting set of (
L
+
2) simultaneous equations is as
follows:
"
$
=
"
$
"
$
N
−
1
2
h
1
cos
Ω
1
cos
L
Ω
1
−
1
/
W
(
Ω
0
)
H
d
Ω
1
1
cos
Ω
2
cos
L
Ω
2
−
1
/
W
(
Ω
2
)
N
−
1
2
H
d
Ω
2
2
h
−
1
.
.
.
.
.
.
.
.
.
2
h
[0]
ε
max
.
(
−
1)
L
+
1
/
W
(
Ω
L
+
1
)
1 cos
Ω
L
+
1
cos
L
Ω
L
+
1
H
d
Ω
L
+
1
(
−
1)
L
+
2
/
W
Ω
L
+
2
1 cos
Ω
L
+
2
cos
L
Ω
L
+
2
Ω
L
+
2
H
d
(cos(
k
Ω
))
(15.52)
Once the extremal frequencies
Ω
1
<
Ω
2
< <
Ω
L
+
2
are known, Eq.
(15.52) can be used to solve for the coefficients of the FIR filter. The extremal
frequencies are computed using the Remez algorithm, which is based on Eq.
(15.52) (though it does not solve the simultaneous equations explicitly) and
consists of the following steps.
Initialization: pick
Ω
1
<
Ω
2
< <
Ω
L
+
2
∈
S
evenly over the pass and
stop bands.
Given: transfer function
H
d
(
Ω
) of the ideal filter and the weighting function
W
(
Ω
).
Step 1
Solve Eq. (15.52) to calculate
ε
max
. To compute
ε
max
, we do not need to
solve the complete set of simultaneous equations given in Eq. (15.52). Instead
the following expression, obtained from Eq. (15.52) is solved:
,
(15.53)
1
cos (
Ω
1
)
cos (
L
Ω
1
)
H
d
(
Ω
1
)
1
cos (
Ω
2
)
cos (
L
Ω
2
)
H
d
(
Ω
2
)
(
−
1)
L
+
3
(cos (
k
Ω
))
.
1 cos (
Ω
L
+
1
)
cos (
L
Ω
L
+
1
)
H
d
(
Ω
L
+
1
)
1 cos (
Ω
L
+
2
)
cos (
L
Ω
L
+
2
)
H
d
(
Ω
L
+
2
)
.
.
.
.
.
.
ε
max
=
where
(
)
denotes the determinant of the matrix
(
).
Step 2
Substituting the value of
ε
max
determined in step 1, compute the values
of
G
(
Ω
m
) at discrete frequencies
{
Ω
1
<
Ω
2
< <
Ω
L
+
2
}
using Eq. (15.51).
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