Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 338

Stepwise implementation of the above code returns the following values for
different variables:

Instruction II: N = 2; wc = 9.0360;
Instruction III: num = [0 0 81.6497]; den = [1.0000
12.7789 81.6497];
Instruction IV: Ht = 1/(sˆ2 + 12.78s + 81.65);

The magnitude spectrum is the same as that given in Fig. 7.7(b).

7.3.2 Type I Chebyshev filters

Butterworth filters have a relatively low roll off in the transitional band, which
leads to a large transitional bandwidth. Type I Chebyshev filters reduce the
bandwidth of the transitional band by using an approximating function, referred
to as the Type I Chebyshev polynomial, with a magnitude response that has
ripples within the pass band. We start with the definition of the Chebyshev
polynomial.

7.3.2.1 Type I Chebyshev polynomial

The N th-order Type I Chebyshev polynomial is defined as

− 1 ( ω ))

ω≤ 1

cos( N cos

T N ( ω ) = (7.33)

− 1 ( ω ))

ω > 1 ,

cosh( N cosh

where cosh( x ) denotes the hyperbolic cosine function, which is given by

cosh( x ) = cos( j x ) = e x

− x

+ e

. (7.34)

2

Starting from the initial values of T 0 ( ω ) = 1 and T 1 ( ω ) = ω , the higher orders
of the Type I Chebyshev polynomial can be recursively generated using the
following expression:

T n ( ω ) = 2 ω T n − 1 ( ω ) − T n − 2 ( ω ) . (7.35)

Table 7.3 lists the Chebyshev polynomial for different values of n within the
range 0 ≤ n ≤ 10.
Using Eq. (7.33), the roots of the Type I Chebyshev polynomial T N ( ω ) can
be derived as follows:

(2 n + 1) π
2 N

ω n = cos , (7.36)

for 0 ≤ n ≤ N − 1.