Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 345

The frequency characteristics of the Type II Chebyshev filter are given by

ε 2 T N ( ω s /ω )
1 + ε 2 T N ( ω s /ω ) ,

1

H ( ω ) = = (7.53)

− 1

ε 2 T N ( ω s /ω ) 1 +

where ω s is the lower corner frequency of the stop band. To derive the normalized
version of the Type II Chebyshev filter, we set ω s

= 1 in Eq. (7.53) leading to
the following expression for the frequency characteristics of the normalized
Type II Chebyshev filter:

ε 2 T N (1 /ω )
1 + ε 2 T N (1 /ω ) .

1

H ( ω ) = = (7.54)

− 1

ε 2 T N (1 /ω ) 1 +

In the following section, we list the steps involved in the design of the Type II
Chebyshev filter.

7.3.3.1 Design steps for the lowpass filter

The design of the lowpass Type II Chebyshev filter is based on the following
specifications:

pass band (0 ≤ω≤ω p radians/s) 1 − δ p ≤ H ( ω ) ≤ 1 + δ p ;

stop band ( ω >ω s radians/s) H ( ω ) ≤δ s .

Normalizing the specifications with the stop-band corner frequency ω s ,we
obtain

pass band (0 ≤ω≤ω p /ω s )1 − δ p ≤ H ( ω ) ≤ 1 + δ p ;

stop band ( ω > 1) H ( ω ) ≤δ s .

Step 1 Compute the value of the ripple factor by setting the normalized fre-
quency ω = 1 in Eq. (7.54). Since the Type II Chebyshev filter is normalized
with respect to ω s , the normalized frequency ω = 1 corresponds to ω s and the
filter gain H (1) = δ s . Substituting H (1) = δ s in Eq. (7.54), we obtain

ε 2
1 + ε 2

H (1) = = δ s ,

which simplifies to

√

ε = , (7.55)

G s

with the gain term specified in Eq. (7.51).