Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 54

The required transformations are:
LP ! HP : x LN ¼ x c s LN ¼ x c

s

x ;

½ x c is the HP cutoff

LP ! BP : x LN ¼ x 2 x g
xx b

; s LN ¼ s 2 þ x g
sx b

½ x b ¼ x u x l ; x g ¼ x u x l

LP ! BS : x LN ¼ xx b
x 2 x g

; s LN ¼ sx b
s 2 þ x g

ð 1 : 48 Þ
An example filter design is presented later in this subsection to illustrate the use of
these transformations in filter design.
Note: Although Butterworth and Chebychev-I LPF's have no zeros, the above
transformations indicate that BP, HP, and BS filters do have zeros.

1.5.5.1 Circuit Design

Standard Tables provide not only filter transfer functions, but also suitable circuits
for implementing those transfer functions. After obtaining the transfer functions
and circuit diagrams from the Tables, one needs to perform appropriate transfor-
mations to suit the original filter specifications. The standard transformations for
physical implementation are shown in Fig. 1.36 .

1.5.5.2 Impedance Matching

For maximum power transfer in the passband, the load impedance R L should equal
(be matched to) the source impedance. In LPF design Tables, the values of R, L,
and C are given for R L = 1, x = 1, hence these circuits are normalized. For
impedance matching, normalized filter circuits should be denormalized by scaling
all R, L, and C values as follows:

Z
W ;

C ! C R ! RZ; L ! L

ZW ;

L
C

C 2 = 1 / ( LC )

ω g

LP
& HP

2 C )

i.e., L = 1 / ( ω g

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

LP HP

C 1 = 1 / L

L

Fig. 1.36 Hardware analog filter transformations