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IIR Digital Filter Design via the Bilinear Transform

Module by: Douglas L. Jones

A bilinear transform maps an analog filter HasHa s to a discrete-time filter HzHz of the same order.
If only we could somehow map these optimal analog filter designs to the digital world while preserving the magnitude response characteristics, we could make use of the already-existing body of knowledge concerning optimal analog filter design.

Bilinear Transformation

The Bilinear Transform is a nonlinear ℂ→ℂ → mapping that maps a function of the complex variable s s to a function of a complex variable z z. This map has the property that the LHP in s s ( ℜs<0 s 0 ) maps to the interior of the unit circle in z z, and the ⅈλ=s λ s axis maps to the unit circle ⅇⅈω ω in z z.
Bilinear transform: s=αz-1z+1 s α z 1 z 1 Hz=H a s=αz-1z+1 H z H a s α z 1 z 1
Note: ⅈλ=αⅇⅈω-1ⅇⅈω+1=αⅇⅈω-1ⅇ-ⅈω+1ⅇⅈω+1ⅇ-ⅈω+1=2ⅈsinω2+2cosω=ⅈαtanω2 λ α ω 1 ω 1 α ω 1 ω 1 ω 1 ω 1 2 ω 2 2 ω α ω 2 , so λ≡αtanω2 λ α ω 2 , ω≡2arctanλα ω 2 λ α . Figure 1.
fig1.png
Figure 1
The magnitude response doesn't change in the mapping from λ λ to ω ω, it is simply warped nonlinearly according to Hω=H a αtanω2 H ω H a α ω 2 , Figure 2.
fig2a.png
Subfigure 2.1
fig2b.png
Subfigure 2.2
Figure 2: The first image implies the second one.
Note: This mapping preserves ∥L∥∞ L errors in (warped) frequency bands. Thus optimal Cauer ( ∥L∥∞ L ) filters in the analog realm can be mapped to ∥L∥∞ L optimal discrete-time IIR filters using the bilinear transform! This is how IIR filters with ∥L∥∞ L optimal magnitude responses are designed.
Note: The parameter α α provides one degree of freedom which can be used to map a single λ 0 λ 0 to any desired ω 0 ω 0 : λ 0 =αtan ω 0 2 λ 0 α ω 0 2 or α= λ 0 tan ω 0 2 α λ 0 ω 0 2 This can be used, for example, to map the pass-band edge of a lowpass analog prototype filter to any desired pass-band edge in ω ω. Often, analog prototype filters will be designed with λ=1 λ 1 as a band edge, and α α will be used to locate the band edge in ω ω. Thus an Mth Mth order optimal lowpass analog filter prototype can be used to design any Mth Mth order discrete-time lowpass IIR filter with the same ripple specifications.

Prewarping

Given specifications on the frequency response of an IIR filter to be designed, map these to specifications in the analog frequency domain which are equivalent. Then a satisfactory analog prototype can be designed which, when transformed to discrete-time using the bilinear transformation, will meet the specifications.
Example 1 
The goal is to design a high-pass filter, ω s = ω s ω s ω s , ω p = ω p ω p ω p , δ s = δ s δ s δ s , δ p = δ p δ p δ p ; pick up some α= α 0 α α 0 . In Figure 3 the δi δi remain the same and the band edges are mapped by λ i = α 0 tan ω i 2 λ i α 0 ω i 2 .
fig3a.png
Subfigure 3.1
fig3b.png
Subfigure 3.2
Figure 3: Where λ s = α 0 tan ω s 2 λ s α 0 ω s 2 and λ p = α 0 tan ω p 2 λ p α 0 ω p 2 .

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