Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 339

Table 7.3. Chebyshev polynomial T N (ω) for different
values of N

N T N ( ω )

1 ω

2 ω 2 − 1

4 ω 3 − 3 ω

8 ω 4 − 8 ω 2 + 1

1 6 ω 5 − 20 ω 3 + 5 ω

3 2 ω 6 − 48 ω 4 + 18 ω 2 − 1

6 4 ω 7 − 112 ω 5 + 56 ω 3 − 7 ω

8 128 ω 8 − 256 ω 6 + 160 ω 4 − 32 ω 2 + 1

256 ω 9 − 576 ω 7 + 432 ω 5 − 120 ω 3 + 9 ω

9

512 ω 10 − 1280 ω 8 + 1120 ω 6 − 400 ω 4 + 50 ω 2 − 1

10

7.3.2.2 Type I Chebyshev filter

The frequency characteristics of the Type I Chebyshev filter of order N are
defined as follows:

1

H ( ω ) = , (7.37)

1 + ε 2 T N ( ω/ω p )

where ω p is the pass-band corner frequency and ε is the ripple control parameter
that adjusts the magnitude of the ripple within the pass band. Substituting
ω p

= 1, the frequency characteristics of the normalized Type I Chebyshev filter
of order N are expressed in terms of the Chebyshev polynomial as follows:

1

H ( ω ) = . (7.38)

1 + ε 2 T N ( ω )

Based on Eqs. (7.35) and (7.38), we make the following observations for the
frequency characteristics of the normalized Type I Chebyshev filter.

(1) For ω = 0, the Chebyshev polynomial T N ( ω ) has a value of 1 or 0. This
can be shown by substituting ω = 0 in Eq. (7.33), which yields

N (2 n + 1) π
2

1 N is even

− 1 (0)) = cos

T N (0) = cos( N cos =

0 N is odd.

(7.39)