Normalized Butterworth and Chebyshev Functions - Digital Signal Processing: Fundamentals and Applications - page 806

where k ¼ 0 ; 1 ; / ; 2 n 1. Similarly, the phasor form is given by

r ¼ ε 1 =n ; and q k ¼ð 2 pk þ pÞ=ð 2 nÞ for k ¼ 0 ; 1 ; / ; 2 n 1 (C.7)

When n is an odd number, we can identify the poles on the LHHP as

p k ¼r; k ¼ 0 and

p k ¼r cos ðq k Þþjr sin ðq k Þ; k ¼ 1 ; / ; ðn 1 Þ= 2 (C.8)

Using complex conjugate pairs, we have
p k ¼r cos ðq k Þjr sin ðq k Þ

Notice that

ðs p k Þðs p k Þ¼s

2 2

þð 2 r cos ðq k ÞÞs þ r

and from a factor from the real pole ðs þ rÞ , it follows that

K

P n ðsÞ¼ (C.9)

ðs þ rÞ Q ðn 1 Þ= 2
k¼ 1

ðs

2

þð 2 r cos ðq k ÞÞs þ r

2

Þ

and

q k ¼ 2 pk=ð 2 nÞ for k ¼ 1 ; / ; ðn 1 Þ= 2

Setting P n ð 0 Þ¼ 1 for the unit passband gain leads to

K ¼ r n ¼ 1 = ε

When n is an even number, we can identify the poles on the LHHP as

p k ¼r cos ðq k Þþjr sin ðq k Þ; k ¼ 0 ; 1 ; / ; n= 2 1 (C.10)

Using complex conjugate pairs, we have

p k ¼r cos ðq k Þjr sin ðq k Þ

The transfer function is given by

K

P n ðsÞ¼ (C.11)

Q n= 2
k¼ 1 ðs

2

þð 2 r cos ðq k ÞÞs þ r

2

Þ

q k ¼ð 2 pk þ pÞ=ð 2 nÞ for k ¼ 0 ; 1 ; / ; n= 2 1