The section "The Derivation of the Formula" from the module titled "Filter Sizing"
used some of the properties of the Chebyshev
polynomials to develop the key formulas used for FIR filter sizing.
This appendix provides a very brief review of these polynomials and the
equations used to generate them.
Figure 1 shows a set of polynomials which have the
property that, for values of xx between -1 and 1, the polynomial has
peak magnitude of unity. A footnote in The section "The Derivation of the Formula" from the module titled "Filter Sizing"
pointed out
that the Russian engineer Chebyshev developed these polynomials as part
of design effort which required minimizing the maximum lateral excursion of
a locomotive drive rod. For each polynomial order, say MM, the objective is
to choose the polynomial's coefficients so that that it “ripples" between
x=-1x=-1 and x=1x=1 and then proceeds off proportional to |x|M|x|M for values
of |x|>1.|x|>1. Not only did Chebyshev find such polynomials, he found that
one exists for each positive value of MM, and that they are related thorugh
a recursion equation, that is, the polynomial for MM can directly obtained
for the polynomial for MM-1.
Consider the following recursion expression:
P
M
(
x
)
=
2
x
·
P
M
-
1
(
x
)
-
P
M
-
2
(
x
)
,
P
M
(
x
)
=
2
x
·
P
M
-
1
(
x
)
-
P
M
-
2
(
x
)
,
(1)
with initial conditions of
P
0
=
1
P
0
=
1
(2)
and
P
1
=
x
P
1
=
x
(3)
Note that both of these initial conditions meet (if
trivially) the stated
criteria for being Chebyshev polynomials.
Using this recursion expression we find, for MM from 0 to 5, that:
P
0
(
x
)
=
1
P
1
(
x
)
=
x
P
2
(
x
)
=
2
x
2
-
1
P
3
(
x
)
=
4
x
3
-
3
x
P
4
(
x
)
=
8
x
4
-
8
x
2
+
1
P
5
(
x
)
=
16
x
5
-
20
x
3
+
5
x
P
0
(
x
)
=
1
P
1
(
x
)
=
x
P
2
(
x
)
=
2
x
2
-
1
P
3
(
x
)
=
4
x
3
-
3
x
P
4
(
x
)
=
8
x
4
-
8
x
2
+
1
P
5
(
x
)
=
16
x
5
-
20
x
3
+
5
x
(4)
These polynomials are plotted in Figure 1
and it may be confirmed by inspection that they meet the stated criteria.
A surprising result is that there is yet another way to present these
polynomials. This method is given by the following equations:
P
M
(
x
)
=
c
o
s
[
M
·
c
o
s
-
1
(
x
)
]
,
f
o
r
|
x
|
≤
1
,
a
n
d
P
M
(
x
)
=
c
o
s
[
M
·
c
o
s
-
1
(
x
)
]
,
f
o
r
|
x
|
≤
1
,
a
n
d
(5)
P
M
(
x
)
=
c
o
s
h
[
M
·
c
o
s
h
-
1
(
x
)
]
,
f
o
r
|
x
|
>
1
.
P
M
(
x
)
=
c
o
s
h
[
M
·
c
o
s
h
-
1
(
x
)
]
,
f
o
r
|
x
|
>
1
.
(6)
Analytically it can be confirmed that these equations satisfy the recursion
seen in equation Equation 1. To see that they describe the same polynomials
as seen in Figure 1, consider Equation 5
for values of |x||x| between -1 and 1. For such values cos-1xcos-1x ranges
between ππ and 0. Thus M·cos-1xM·cos-1x ranges between
MπMπ and 0, and cos[M·cos-1x]cos[M·cos-1x] cycles between -1 or 1 and
1, hiting M+1M+1 extrema on the way, counting the endpoints. Similar
analysis shows that equation Equation 6 grows monotonically in
magnitude as |x||x| does. In fact it is easy to show that
|cosh[M·cosh-1x]||cosh[M·cosh-1x]| assymptotically approaches |x|M|x|M as
|x||x| gets much greater than one.
This second form of the definition for Chebyshev polynomials is very useful
since it is a closed form and because it involves cosines, a functional
form appearing frequently in frequency-domain representations of filters.
In light of this a final twist might be noted. Equation 6
is in fact superfluous given Equation 5. To see this,
consider evaluating Equation 5 for |x|=2|x|=2. It initially
appears that this won't work, since arccosine cannot be evaluated for
arguments greater than unity. In fact it can, it's just that the result is
purely imaginary. It is easy, using Euler's definition of the cosine, to see
that the cosine of jxjx is the same as the hyberbolic cosine of xx. Thus
the arccosine of 2 is jj times the inverse hyperbolic cosine of 2, that
is, j·1.31j·1.31. Multiplying by M and taking the cosine of the
product yields the cosine of jMxjMx, which is the hyperbolic cosine of MxMx.
Thus, if imaginary arguments are permitted, then Equation 5
suffices to describe all of the Chebyshev polynomials.
