The approximation tolerances for a filter are very often given
in terms of the maximum, or worst-case, deviation within
frequency bands. For example, we
might wish a lowpass filter in a (16-bit) CD player to have no
more than
12
1
2
-bit deviation in the pass and stop bands.
Hω=1-1217≤|Hω|≤1+1217if|ω|≤
ω
p
1217≥|Hω|if
ω
s
≤|ω|≤π
H
ω
1
1
2
17
H
ω
1
1
2
17
ω
ω
p
1
2
17
H
ω
ω
s
ω
The Parks-McClellan filter design method efficiently designs
linear-phase FIR filters that are optimal in terms of worst-case
(minimax) error.
Typically, we would like to have the shortest-length filter
achieving these specifications.
Figure Figure 1 illustrates the amplitude frequency
response of such a filter.
Must there be a transition band?
Yes, when the desired response is discontinuous.
Since the frequency response of a finite-length filter
must be continuous, without a transition band the worst-case
error could be no less than half the discontinuity.
For a given filter length (MM) and
type (odd length, symmetric, linear phase, for example), and a
relative error weighting function
Wω
W
ω
, find the filter coefficients minimizing the maximum
error
argminhargmaxω∈F|Eω|=argminh∥Eω∥∞
h
ω
F
E
ω
h
E
ω
where
Eω=Wω
H
d
ω-Hω
E
ω
W
ω
H
d
ω
H
ω
and FF is a compact
subset of
ω∈0π
ω
0
(i.e., all ωω in
the passbands and stop bands).
Typically, we would often rather specify
∥Eω∥∞≤δ
E
ω
δ
and minimize over MM
and hh; however,
the design techniques minimize
δδ for a given
MM. One then repeats the design
procedure for different MM until
the minimum MM satisfying the
requirements is found.
We will discuss in detail the design only of odd-length
symmetric linear-phase FIR filters. Even-length and
anti-symmetric linear phase FIR filters are essentially the
same except for a slightly different implicit weighting
function. For arbitrary phase, exactly optimal design
procedures have only recently been developed (1990).
The Parks-McClellan method adopts an indirect method for finding the
minimax-optimal filter coefficients.
- Using results from Approximation Theory, simple
conditions for determining whether a given filter is
L
∞
L
∞
(minimax) optimal are found.
- An iterative method for finding a filter which
satisfies these conditions (and which is thus optimal) is
developed.
That is, the
L
∞
L
∞
filter design problem is actually solved
indirectly.
All conditions are based on Chebyshev's "Alternation Theorem,"
a mathematical fact from polynomial approximation theory.
Let FF be a compact subset on the real axis
xx, and let
Px
P
x
be and LLth-order polynomial
Px=∑k=0L
a
k
xk
P
x
k
L
0
a
k
x
k
Also, let
Dx
D
x
be a desired function of xx
that is continuous on FF, and
Wx
W
x
a positive, continuous weighting function on
FF. Define the error
Ex
E
x
on FF as
Ex=WxDx-Px
E
x
W
x
D
x
P
x
and
∥Ex∥∞=argmaxx∈F|Ex|
E
x
x
F
E
x
A necessary and sufficient condition that
Px
P
x
is the unique LLth-order polynomial minimizing
∥Ex∥∞
E
x
is that
Ex
E
x
exhibits at least
L+2
L
2
"alternations;" that is, there must exist at least
L+2
L
2
values of xx,
x
k
∈F
x
k
F
,
k=01…L+1
k
0
1
…
L
1
, such that
x
0
<
x
1
<…<
x
L
+
2
x
0
x
1
…
x
L
+
2
and such that
E
x
k
=-E
x
k
+
1
=±∥E∥∞
E
x
k
E
x
k
+
1
±
E
What does this have to do with
linear-phase filter design?
It's the same problem! To show that,
consider an odd-length, symmetric linear phase filter.
Hω=∑n=0M-1hnⅇ-ⅈωn=ⅇ-ⅈωM-12hM-12+2∑n=1LhM-12-ncosωn
H
ω
n
M
1
0
h
n
ω
n
ω
M
1
2
h
M
1
2
2
n
L
1
h
M
1
2
n
ω
n
(1)
Aω=hL+2∑n=1LhL-ncosωn
A
ω
h
L
2
n
L
1
h
L
n
ω
n
(2)
Where
L≐M-12
≐
L
M
1
2
.
Using trigonometric identities (such as
cosnα=2cosn-1αcosα-cosn-2α
n
α
2
n
1
α
α
n
2
α
), we can rewrite
Aω
A
ω
as
Aω=hL+2∑n=1LhL-ncosωn=∑k=0L
α
k
coskω
A
ω
h
L
2
n
L
1
h
L
n
ω
n
k
L
0
α
k
ω
k
where the
α
k
α
k
are related to the
hn
h
n
by a linear transformation. Now, let
x=cosω
x
ω
. This is a one-to-one mapping from
x∈-11
x
-1
1
onto
ω∈0π
ω
0
.
Thus
Aω
A
ω
is an LLth-order polynomial in
x=cosω
x
ω
!
The alternation theorem holds for
the
L
∞
L
∞
filter design problem, too!
Therefore, to determine whether or not a
length-
MM,
odd-length, symmetric linear-phase filter is optimal in an
L
∞
L
∞
sense, simply count the alternations in
Eω=WωAdω-Aω
Eω
Wω
Ad
ω
Aω
in the
pass and stop bands.
If there are
L+2=M+32
L2
M3
2 or more alternations,
hnhn,
0≤n≤M-10n
M1
is the optimal filter!
For M
M even,
Aω=∑n=0LhL-ncosωn+12
A
ω
n
0
L
h
L
n
ω
n
1
2
where
L=M2-1
L
M
2
1
Using the trigonometric identity
cosα+β=cosα-β+2cosαcosβ
α
β
α
β
2
α
β
to pull out the
ω2
ω
2
term and then using the other
trig identities, it can be shown that
Aω
A
ω
can be written as
Aω=cosω2∑k=0L
α
k
coskω
A
ω
ω
2
k
0
L
α
k
ω
k
Again, this is a polynomial in
x=cosω
x
ω
, except for a weighting function out in front.
Eω=Wω
A
d
ω-Aω=Wω
A
d
ω-cosω2Pω=Wωcosω2
A
d
ωcosω2-Pω
E
ω
W
ω
A
d
ω
A
ω
W
ω
A
d
ω
ω
2
P
ω
W
ω
ω
2
A
d
ω
ω
2
P
ω
(3)
which implies
Ex=
W
'
x
A
d
'
x-Px
E
x
W
'
x
A
d
'
x
P
x
(4)
where
W
'
x=Wcosx-1cos12cosx-1
W
'
x
W
x
1
2
x
and
A
d
'
x=
A
d
cosx-1cos12cosx-1
A
d
'
x
A
d
x
1
2
x
Again, this is a polynomial approximation problem, so the
alternation theorem holds. If
Eω
E
ω
has at least
L+2=M2+1
L
2
M
2
1
alternations, the even-length symmetric filter is
optimal in an
L
∞
L
∞
sense.
The prototypical filter design problem:
W=1if|ω|≤
ω
p
δ
s
δ
p
if|
ω
s
|≤|ω|
W
1
ω
ω
p
δ
s
δ
p
ω
s
ω
See Figure 2.
- The maximum possible number of alternations for a
lowpass filter is
L+3
L
3
: The proof is that the extrema of a polynomial
occur only where the derivative is zero:
∂∂xPx=0
x
P
x
0
. Since
P′x
P
x
is an
(
L
-
1
)
(
L
-
1
)
th-order polynomial, it can have at
most
L
-
1
L
-
1
zeros. However, the mapping
x=cosω
x
ω
implies that
∂∂ωAω=0
ω
A
ω
0
at
ω=0
ω
0
and
ω=π
ω
, for two more possible alternation
points. Finally, the band edges can
also be alternations, for a total of
L-1+2+2=L+3
L
1
2
2
L
3
possible alternations.
- There must be an alternation at either
ω=0
ω
0
or
ω=π
ω
.
- Alternations must occur at
ω
p
ω
p
and
ω
s
ω
s
. See Figure 2.
- The filter must be equiripple except at possibly
ω=0
ω
0
or
ω=π
ω
. Again see Figure 2.
The alternation theorem doesn't directly suggest a
method for computing the optimal filter. It simply tells us
how to recognize that a filter is
optimal, or isn't optimal. What we need
is an intelligent way of guessing the optimal filter
coefficients.
In matrix form, these
L+2
L
2
simultaneous equations become
1cos
ω
0
cos2
ω
0
...cosL
ω
0
1W
ω
0
1cos
ω
1
cos2
ω
1
...cosL
ω
1
-1W
ω
1
⋮⋮⋱...⋮⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮...⋱⋮1cos
ω
L
+
1
cos2
ω
L
+
1
...cosL
ω
L
+
1
±1W
ω
L
+
1
hLhL-1⋮h1h0δ=
A
d
ω
0
A
d
ω
1
⋮⋮⋮
A
d
ω
L
+
1
1
ω
0
2
ω
0
...
L
ω
0
1
W
ω
0
1
ω
1
2
ω
1
...
L
ω
1
-1
W
ω
1
⋮
⋮
⋱
...
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
⋮
⋮
⋮
...
⋱
⋮
1
ω
L
+
1
2
ω
L
+
1
...
L
ω
L
+
1
±
1
W
ω
L
+
1
h
L
h
L
1
⋮
h
1
h
0
δ
A
d
ω
0
A
d
ω
1
⋮
⋮
⋮
A
d
ω
L
+
1
or
Whδ=Ad
W
h
δ
A
d
So, for the given set of
L+2
L
2
extremal frequencies, we can solve for
h
h and
δ
δ via
hδT=W-1Ad
h
δ
W
A
d
. Using the FFT, we can compute
Aω
A
ω
of
hn
h
n
, on a dense set of frequencies. If the old
ω
k
ω
k
are, in fact the extremal locations of
Aω
A
ω
, then the alternation theorem is satisfied and
hn
h
n
is
optimal. If not, repeat the process
with the new extremal locations.
OL3
O
L
3
for the matrix inverse and
Nlog2N
N
2
N
for the FFT (
N≥32L
N
32
L
, typically), per
iteration!
This method is expensive computationally due
to the matrix inverse.
A more efficient variation of this method was
developed by Parks and McClellan (1972), and is based on the
Remez exchange algorithm. To understand the Remez exchange
algorithm, we first need to understand Lagrange Interpoloation.
Now
Aω
A
ω
is an L
Lth-order polynomial in
x=cosω
x
ω
, so Lagrange interpolation can be used to
exactly compute
Aω
A
ω
from
L+1
L
1
samples of
A
ω
k
A
ω
k
,
k=012...L
k
0
1
2
...
L
.
Thus, given a set of extremal frequencies and
knowing δ
δ, samples of the amplitude response
Aω
A
ω
can be computed directly from the
A
ω
k
=-1k+1W
ω
k
δ+
A
d
ω
k
A
ω
k
1
k
1
W
ω
k
δ
A
d
ω
k
(5)
without solving for the filter
coefficients!
This leads to computational savings!
Note that Equation 5 is a set of
L+2
L
2
simultaneous equations, which can be solved for
δ
δ to obtain (Rabiner, 1975)
δ=∑k=0L+1
γ
k
A
d
ω
k
∑k=0L+1-1k+1
γ
k
W
ω
k
δ
k
0
L
1
γ
k
A
d
ω
k
k
0
L
1
1
k
1
γ
k
W
ω
k
(6)
where
γ
k
=∏i=0i≠kL+11cos
ω
k
-cos
ω
i
γ
k
i
i
k
0
L
1
1
ω
k
ω
i
The result is the Parks-McClellan FIR filter design method,
which is simply an application of the Remez exchange algorithm
to the filter design problem. See
Figure 3.
The cost per iteration is
O16L2
O
16
L
2
, as opposed to
OL3
O
L
3
; much more efficient for large L
L. Can also interpolate to DFT sample frequencies,
take inverse FFT to get corresponding filter coefficients, and
zeropad and take longer FFT to efficiently interpolate.
Frequency Sampling Design Method for FIR Filters
Overview of IIR Filter Design
Adaptive Filters
