Digital Signal Processing Reference
In-Depth Information
C
n
is called the Chebyschev polynomial of order
n
:
C
n
= cos(n cos
-1
(n)).
(6.8)
From (6.8), it can easily be determined that
C
0
= 1
and
C
1
= n.
(6.9)
To develop an algorithm for finding higher-order Chebyschev polynomials, we de-
fine a new variable,
f = cos
-1
(n),
so that we can now write
C
n
= cos(nf).
(6.10)
Now from Appendix A, we can use the trigonometric identities
cos(a ; b) = cos(a) cos(b) < sin(a) sin(b)
to determine that
cos [(n + 1)f] + cos [(n - 1)f] = 2 cos(nf) cos(f).
Using (6.10), we rewrite this as
C
n+ 1
(n) = 2 cos(nf) cos(f) - C
n- 1
(n).
Making use of (6.9), we write this result in a form that is more useful for determin-
ing Chebyschev polynomials:
C
n+ 1
(n) = 2nC
n
(n) - C
n- 1
(n).
(6.11)
Using (6.11) and (6.9), we can generate a list of Chebyschev polynomials as shown
in Table 6.1.
Figure 6.18(a) shows the magnitude frequency responses of four orders of nor-
malized Chebyschev filters. It is seen that the magnitude response decreases more
rapidly near the cutoff frequency as the order is increased. Figure 6.18(b) provides a
comparison of the frequency responses of fourth-order Butterworth and
Chebyschev filters.
MATLAB program to show the frequency response of Chebyschev filters
EXAMPLE 6.8
The following MATLAB program can be used to generate the Chebyschev filters of various
orders:
