PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 567

R.6.27 Once a normalized LPF prototype is obtained (wnp = 1 rad/s and load R 0 = 1 Ω),

the denormalized LPF process starts by adjusting the wp = k 0 rad/s, where k 0 is the

desired BW by scaling the values of the C’s, L’s, and R’s. This adjustment is referred

to as an LP-to-LP transformation, and it is accomplished by what is known as a

frequency and impedance scaling.

R.6.28 The frequency scaling transformation, LPn (low-pass normalized) to LP (denormal-

ized) is accomplished by using the following relation:

H(jw) = Hn(jw/ k 0 )

R.6.29 In terms of circuit elements, frequency scaling is accomplished by dividing each

normalized capacitor and inductor labeled Cn and Ln by k 0 , and by leaving the resis-

tances unchanged, as follows:

C

C

f k

n

s


0

L

L

f k

n

s


0

RRfns
 ()unchanged

where the index fs denotes a frequency scale transformation.

R.6.30 The relation indicated in R.6.28 translates into the following:

if k 0 > 1 the BW of H(jw) spreads

if k 0 < 1 the BW of H(jw) contracts

R.6.31 By using frequency scaling, the transformation of a normalized LPF to an arbi-

trary LPF with an arbitrary BW is easily accomplished by simple divisions. Filter

handbooks, design tables, and graphs were used in the past to accomplish these

transformations. Tables are still used, but they are gradually being replaced by

sofi sticated computer software packages such as MATLAB.

R.6.32 Once frequency scaling was attained, the next objective in the synthesis process

is the impedance scaling, that is, the load connected to the fi lter output can be an

arbitrary load R (not necessarily Rn = 1 Ω).

R.6.33 Impedance scaling is attained by multiplying all inductors and resistors by R and

dividing each capacitor by R as follows:

R ← Rn R

LRL← fs

C

C

R

← fs

R.6.34 The transformation of a normalized LPF prototype into another denormalized fi lter

type can be accomplished by the variable substitution in H(s), shown in Table 6.1

and the corresponding circuit structures shown in Table 6.2. The variable substi-

tution translates into a mathematical mapping or a segment (elements) or block

substitution.

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