Signals and Systems - Electrical Engineering

658 C H A P T E R 11: Introduction to the Design of Discrete Filters

11.4.2 Design of Butterworth Low-Pass Discrete Filters

Our aim in this section is to show how to design discrete low-pass filters based on the analog

Butterworth low-pass filter design using the bilinear transformation as a frequency transformation.

Applying the warping relation between the continuous and the discrete frequencies

=Ktan(ω/ 2 ) (11.25)

to the magnitude-squared function of the Butterworth low-pass analog filter

∣

∣HN(′)

∣

∣^2 =^1

1 +(′)^2 N

′=



hp

gives the magnitude-squared function for the Butterworth low-pass discrete filter:

|HN(ejω)|^2 =

1

1 +

[

tan(0.5ω)

tan(0.5ωhp)

] 2 N (11.26)

As a frequency transformation (no change to the loss specifications) we directly obtain the minimal orderN

and the half-power frequency bounds by replacing

s

p

=

tan(ωs/ 2 )

tan(ωp/ 2 )

(11.27)

in the corresponding formulas forNandhpof the analog filter, giving

N≥

log 10 [( 10 0.1αmin− 1 )/( 10 0.1αmax− 1 )]

2 log 10

[

tan(ωs/ 2 )

tan(ωp/ 2 )

]

2 tan−^1

[ tan(ω

p/^2 )

( 10 0.1αmax− 1 )^1 /^2 N

]

≤ωhp≤2 tan−^1

[

tan(ωs/ 2 )

( 10 0.1αmin− 1 )^1 /^2 N

]

(11.28)

The normalized half-power frequency′hp= 1 in the continuous domain is mapped into the discrete half-

power frequencyωhp, giving the constant in the bilinear transformation

Kb=

′

tan(0.5ω)

∣

∣∣

′=1,ω=ωhp=

1

tan(0.5ωhp)

(11.29)

The bilinear transformations=Kb( 1 −z−^1 )/( 1 +z−^1 )is then used to convert the analog filterHN(s),

satisfying the transformed specifications, into the desired discrete filter,

HN(z)=HN(s)

∣∣

∣s=Kb( 1 −z− (^1) )/( 1 +z− (^1) )
The basic idea of this design is to convert an analog frequency-normalized Butterworth magnitude-
squared function into a discrete function using the relationship in Equation (11.25). To understand