Infinite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 203

C 2 ( Ω ) C 3 ( Ω )

1 1

1

1

C 4 ( Ω ) C 5 ( Ω )

1 1

1

1

( a )

e 2 C 4 2 (

10log(l + Ω ))

A p

Ω p = 1

( b )

Figure 4.8 Chebyshev polynomials and Chebyshev filter: (a) magnitude of Chebyshev
polynomials; (b) attenuation of a Chebyshev I filter.

= cos (n cos − 1 ) is indeed a polynomial of order n , consider
it in the following form:

To see that C n ()

= Re e jnφ cos (nφ)

Re φ φ 2 n

j ( 1 Re cos (φ) j sin (φ n = + = + −

= Re φ − 1 n

φ 2

+ (4.53)

Expanding φ − 1 n + φ 2 by the binomial theorem and choosing the real part,

we get the polynomial for

n(n − 1 )

2!

φ n φ n − 2 (φ 2 cos (nφ) = + − 1 )

n(n − 1 )(n − 2 )(n − 3 )

φ n − 4 (φ 2 1 ) 2 + − +··· (4.54)

4!