We desire that the oscillatory portion of the polynomial shown in
Figure 1 in the module titled "Filter Sizing"
correspond to the
stopband region of the filter response and the xMxM portion to correspond
to the transition from the stopband to the passband. This is achieved by
employing a change of variables from frequency ff to the polynomial
argument xx:
x
=
(
x
0
+
1
2
)
c
o
s
(
2
π
f
f
s
)
+
(
x
0

1
2
)
.
x
=
(
x
0
+
1
2
)
c
o
s
(
2
π
f
f
s
)
+
(
x
0

1
2
)
.
(1)While many different types of variable changes could be employed, this one
matches the boundary conditions (an obvious requirement) but happens to
employ the cosine function, a member of the same family used to define the
Chebyshev polynomials.
With this change of variables we see that the transition band ΔfΔf is
defined by the difference between x=1x=1 and x=xpx=xp. Using the closed, but
nonintuitive form of the Kth order Chebyshev polynomial, valid for
x>1x>1, we have that
P
K
(
x
)
=
c
o
s
h
(
K
·
c
o
s
h

1
(
x
)
)
P
K
(
x
)
=
c
o
s
h
(
K
·
c
o
s
h

1
(
x
)
)
(2)To synthesize the desired impulse response using this windowing technique we
multiply the resulting window function by the sampled sinc function. In this
case, however, we desire that the cutoff frequency be as low as possible,
limiting at zero Hz. The associated sinc function equals unity for all
nonzero
coefficients of the impulse response. Since the final impulse response is
the pointbypoint product of the window and the sampled sinc function, in
this case the window itself is the resulting impulse response. It suffices then
to examine the properties of the Nth order
Chebyshev polynomial to see how the Npoint optimal filter will behave.
To find the relationship between the required filter order NN and the
attainable transition band ΔfΔf, we first determine the proper
value of KK and
then evaluate Equation 2 at the known combinations of
xx and PK(x)PK(x). To select KK we note that all but one of the ripples in
the polynomial's response are used in the stopband and these are split
evenly between the positive and negative frequencies. Thus a filter and
window of order NN implies a Chebyshev polynomial of order
K
=
N

1
2
K
=
N

1
2
(3)With this resolved we observe from Figure 1 in the module titled "Filter Sizing"
that
P
N

1
2
(
1
)
=
1
P
N

1
2
(
1
)
=
1
(4)
P
N

1
2
(
x
p
)
=
1

δ
1
δ
2
P
N

1
2
(
x
p
)
=
1

δ
1
δ
2
(5)
P
N

1
2
(
x
0
)
=
1
+
δ
1
δ
2
P
N

1
2
(
x
0
)
=
1
+
δ
1
δ
2
(6)These equations are manipulated to yield an expression for xpxp.
Equation 1 is then used to obtain values for fstfst,
corresponding to x=1x=1, and fcfc, corresponding to x=xpx=xp. Their
difference, defined earlier to be the transition band ΔfΔf, is then
given by
Δ
f
=
f
s
π
(
N

1
)
[
c
o
s
h

1
(
1
+
δ
1
δ
2
)

{
(
c
o
s
h

1
(
1
+
δ
1
δ
2
)
)
2

(
c
o
s
h

1
(
1

δ
1
δ
2
)
)
2
}
1
2
]
.
Δ
f
=
f
s
π
(
N

1
)
[
c
o
s
h

1
(
1
+
δ
1
δ
2
)

{
(
c
o
s
h

1
(
1
+
δ
1
δ
2
)
)
2

(
c
o
s
h

1
(
1

δ
1
δ
2
)
)
2
}
1
2
]
.
(7)Under suitable conditions this equation can be simplified considerably. For
example, in the limits of small δ1δ1 and large NN, Equation 7 reduces to
Equation 4 in the module titled "Filter Sizing".