Digital Signal Processing Reference
In-Depth Information
for Type II filters,
M
−
1 is an odd number, which leads to the conclusion
that
H
.
Similar results can be demonstrated for the other filter types, and they are
summarized below:
(
−
1
)=
0; in other words, Type II filters
must
have a zero at
ω
=
π
l
g
r
,
y
i
d
.
,
©
,
L
s
Filter Type
Relation
Constraint on Zeros
Type I
H
(
z
−
1
)=
z
M−
1
H
(
z
)
No constraints
Type II
H
(
z
−
1
)=
z
M−
1
H
(
z
)
Zero at
z
=
−
1(i.e.
ω
=
π
)
z
−
1
z
M−
1
H
Type III
H
(
)=
−
(
z
)
Zeros at
z
=
±
1(i.e.at
ω
=
0,
ω
=
π
)
Type IV
H
(
z
−
1
)=
−
z
M−
1
H
(
z
)
Zero at
z
=
1(i.e.
ω
=
0)
These constraints are important in the choice of the filter type for a given
set of specifications. Type II and Type III filters are not suitable in the design
of highpass filters, for instance; similarly, Type III and Type IV filters are not
suitable in in the design of lowpass filters.
Chebyshev Polynomials.
Chebyshev polynomials are a family of or-
thogonal polynomials
T
k
(
)
k∈
x
which have, amongst others, the follow-
ing interesting property:
cos
n
ω
=
T
n
(
cos
ω
)
(7.13)
in other words, the cosine of an integer multiple of an angle
ω
can be ex-
pressed as a polynomial in the variable cos
ω
. The first few Chebyshev poly-
nomials are
T
0
(
x
)=
1
T
1
(
x
)=
x
2
x
2
T
2
(
x
)=
−
1
4
x
3
T
3
(
x
)=
−
3
x
8
x
4
8
x
2
T
4
(
x
)=
−
+
1
and, in general, they can be derived from the recursion formula:
T
k
+
1
(
x
)=
2
xT
k
(
x
)
−
T
k−
1
(
x
)
(7.14)
From the above table it is easy to see that we can write, for instance,
4cos
3
cos
(
3
ω
)=
ω
−
3cos
ω
The interest inChebyshev polynomials comes from the fact that the zero-
centered frequency response of a linear phase FIR can be expressed as a lin-
ear combination of cosine functions, as we have seen in detail for Type I
