Analog Lowpass, Highpass, Bandpass, and Bandstop Filters - Practical Analog and Digital Filter Design - page 61

in (3.12). This is the reciprocal of the lowpass case, but since ω pass > ω stop , the
result still produces a value greater than 1. The Ω r ratio always produces a value
greater than 1 (as we will see in later sections as well) and as the value gets larger
and larger, the order of the filter will reduce as long as other characteristics remain
the same.

ω f

pass pass

Ω = = (3.12)

rH

ω f

stop stop

Once the normalized lowpass approximation is determined based on the
order, we can unnormalize the lowpass transfer function using an appropriate
unnormalization substitution. In the case of the highpass filter, the
unnormalization substitution is given in (3.13). As in the lowpass case, ω o will
take on the value of ω pass except for the inverse Chebyshev approximation where
it will have the value of ω stop .

ω

o

S =

(3.13)

H

s

3.2.1 Handling a First-Order Factor

For the first-order case, we start with (3.14) and make the substitution of (3.13).
The final result is then shown in (3.15). In this unnormalization case, we see that
there is a gain adjustment ( A 2 / B 2 ) that must be considered.

A ⋅ S + A A ⋅ ( ω s ) + A

1 2 1 o 2

H ( s ) = = (3.14)

B ⋅ S + B B ⋅ ( ω s ) + B

1 2 1 o 2

S = ω s

o

A s + ( A / A ) ⋅ ω A a ⋅ s + a

2 1 2 o 2 1 2

H ( s ) = ⋅ = ⋅

(3.15)

B s + ( B / B ) ⋅ ω B b ⋅ s + b

2 1 2 o 2 1 2

From careful observation we can draw the following information from these
equations:

• The gain constant is multiplied by A 2 / B 2 .
• The s -term coefficients become
a 1 = 1, b 1 = 1
• The constant term coefficients become

a 2 = ( A 1 / A 2 ) ω o , b 2 = ( B 1 / B 2 ) ω o