Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 340

Equation (7.37) implies that the dc component H (0) of the Type I

Chebyshev filter is given by

1

√ N is even

H (0) = (7.40)

1 + ε 2

1 N is odd.

(2) For ω = 1 radian/s, the value of the Chebyshev polynomial T N ( ω )isgiven
by

− 1 (1)) = cos(2 nN π ) = 1 .

T N (1) = cos( N cos (7.41)

Therefore, the magnitude H ( ω ) of the normalized Type I Chebyshev filter
at ω = 1 radian/s is given by

1

H (1) = √

1 + ε 2 ,

(7.42)

irrespective of the order N of the normalized Chebyshev filter.
(3) For large values of ω within the stop band, the magnitude response of the
normalized Type I Chebyshev filter can be approximated by

1
ε T N ( ω ) ,

H ( ω ) ≈ (7.43)

since ε T N ( ω ) ≫ 1. If N ≫ 1, then a second approximation can be made
by ignoring the lower degree terms in T N ( ω ) and using the approximation
T N ( ω ) ≈ 2 N − 1 ω N . Equation (7.43) is therefore simplified as follows:
H ( ω ) ≈ 1
ε

1
2 N − 1 ω N .

(7.44)

(4) Since

= H ( ω ) 2 , H ( s ) H ( − s )

s = j ω

H ( s ) H ( − s ) can be derived from Eq. (7.38) as follows:

1
1 + ε 2 T N ( s / j) .

H ( s ) H ( − s ) = (7.45)

The 2 N poles of H ( s ) H ( − s ) are obtained by solving the characteristic
equation,

1 + ε 2 T N ( s / j) = 0 , (7.46)

and are given by

2 n − 1
2 N

1
N

1
ε

− 1

s n = sin π sinh sinh

2 n − 1
2 N

1
N

1
ε

− 1

+ j cos π cosh sinh (7.47)

for 1 ≤ n ≤ 2 N − 1. To derive a stable implementation of the normalized
Type I Chebyshev filter, the N poles in the left-hand s-plane are included