Digital Signal Processing Reference
In-Depth Information
]
[
i
E
(
ω
)
=
W
(
ω
)
⋅
D
(
ω
)
−
C
(
ω
)
=
(
−
1
δ
,
i
i
i
i
for
i
=
0
1
…
,
x
(7.37)
Equation (7.37) can be expanded into a matrix equation by considering these
x
+1 frequencies, which are typically called extremals and play a crucial role in the
optimization process.
⎡
⎤
1
1
cos
ω
cos
M
ω
⎢
⎥
0
0
W
(
ω
)
⎢
⎥
c
(
0
)
0
⎡
⎤
⎡
D
(
ω
)
⎤
−
1
⎢
⎥
0
c
(
1
cos
ω
cos
M
ω
⎢
⎥
⎢
⎥
⎢
⎥
1
1
D
(
ω
)
W
(
ω
)
1
1
⋅
=
⎢
⎥
⎢
⎥
⎢
⎥
c
(
M
)
D
(
ω
)
⎢
⎥
⎣
⎦
⎢
⎥
x
δ
⎣
⎦
()
x
−
1
1
cos
ω
cos
M
ω
⎢
⎥
x
x
W
(
ω
)
⎣
⎦
x
(7.38)
In (7.38), ω
0
- ω
x
represents the extremal frequencies and δ is the error. With
this expression the filter design problem has been set into a form that can be
manipulated by the Remez exchange algorithm.
7.3.2 The Remez Exchange Algorithm
The Remez exchange algorithm is a powerful procedure that uses iteration
techniques to solve a variety of minimax problems. (A minimax problem is one in
which the best solution is the one that minimizes the maximum error that can
occur.) Before initiating the process, a set of discrete frequency points is defined
for the passband and stopband of the filter. (Transition bands are excluded.) This
dense grid of frequencies is used to represent the continuous frequency spectrum.
Extremal frequencies will then be located at particular grid frequencies as
determined by the algorithm. The basic steps of the method as it is applied to our
filter design problem are shown below.
Remez Exchange Algorithm
I.
Make an initial guess as to the location of
x
+ 1 extremal
frequencies, including an extremal at each band edge.
II.
Using the extremal frequencies, estimate the actual frequency
response by using the Lagrange interpolation formula.
III.
Locate the points in the frequency response where maximums
occur and determine the error at those points.
IV.
Ignore all new extremals beyond the number initially set in I.
