Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 314

Low-pass to high-pass transformations

x LN ¼ x c = x; s LN ¼ x c = s :

Low-pass to band-stop transformations

; ;

x LN ¼ xx b = x 2 x g s LN ¼ sx b = s 2 þ x g where:

x g ¼

p

x b ¼ x u x l ; x u x l :

Circuit denormalization

R ! RZ; L ! L W ; C ! ZW , where:
Z = Impedance scaling factor (ISF)
W = Frequency scaling factor (FSF)

Hardware analog filter transformations

L
C

C 2 = 1 / ( LC )

i.e., L = 1 / (

ω g

LP BP 2 C )

ω g

& HP BS

L C

L

2 L )

and C = 1 / ( ω g

L 1 = 1 / C

C

e.g., if C = 2 F,
then L 1 = 1/2 H
Hardware filter design rules (using normalized LPF standard circuits)
HPF design:

LP HP

C 1 = 1 / L

L

1. Transform LP N ? HP N circuit components
2. Denormalize HP N ? HP
BPF design:

1. Denormalize LP N ? LP
2. Transform LP ? BP circuit components

BSF design:

1. Transform LP N ? HP N circuit components
2. Denormalize HP N ? HP
3. Transform HP ? BS circuit components

Magnitude response of Butterworth LPF:

q

1 þð x = x c Þ 2n

j H ð x Þj¼ G m =

;

where G m = maximum gain.