Normalized Butterworth and Chebyshev Functions - Digital Signal Processing: Fundamentals and Applications - page 809

2 2 into Equation (C.12)

To develop the transfer function B n ðsÞ , we let s ¼ ju and substitute u ¼s

to obtain

1

B n ðsÞB n ðsÞ¼ (C.15)

2 2

1 þ ε C

n ðs=jÞ

The poles can be found from

2 2

1 þ ε C

n ðs=jÞ¼ 0

or

C n ðs=jÞ¼ cos ðn cos 1 ðs=jÞÞ¼j 1 = ε (C.16)

If we introduce a complex variable v ¼ a þ jb such that
v ¼ a þ jb ¼ cos 1 ðs=jÞ (C.17)

we can then write

s ¼ j cos ðvÞ (C.18)

Substituting Equation (C.17) into Equation (C.16) and using trigonometric identities, it follows that

C n ðs=jÞ¼ cos ðn cos 1 ðs=jÞÞ

¼ cos ðnvÞ¼ cos ðna þ jnbÞ
¼ cos ðnaÞ cos h ðnbÞj sin ðnaÞ sin h ðnbÞ¼j 1 = ε

(C.19)

To solve Equation (C.19) , the following conditions must be satisfied:

cos ðnaÞ cos h ðnbÞ¼ 0 (C.20)

sin ðnaÞ sin h ðnbÞ¼ 1 = ε (C.21)

Since cos h ðnbÞ 1 in Equation (C.20) , we must let
cos ðnaÞ¼ 0

(C.22)

which therefore leads to

a k ¼ð 2 k þ 1 Þp=ð 2 nÞ; k ¼ 0 ; 1 ; 2 ; / ; 2 n 1 (C.23)

With Equation (C.23) , we have sin ðna k Þ¼ 1. Then Equation (C.21) becomes
sin h ðnbÞ¼ 1 = ε

(C.24)

Solving Equation (C.24) gives

b ¼ sin h 1 ð 1 = ε Þ=n (C.25)

Again from Equation (C.18) ,

s ¼ j cos ðvÞ¼j½ cos ða k Þ cos h ðbÞj sin ða k Þ sin h ðbÞ for k ¼ 0 ; 1 ; / ; 2 n 1 (C.26)