---
title: "Bilinear transformation method of IIR design"
subject: "Digital Signal Analysis and Processing"
chapter: 6
key: "semester_7/digital_signal_analysis_and_processing:bilinear-method-explain"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/bilinear-method-explain
site: BCT Engineering
---

# Bilinear transformation method of IIR design

Digital Signal Analysis and Processing · Chapter 6: IIR Filter Design

- Asked in: 2073 Shrawan, 3 marks
- Video: [Bilinear Transformation and Frequency Warping in IIR Filter Design Explained](https://www.youtube.com/watch?v=VVfvFyKSY-E) — EC Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/bilinear-method-explain
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-6

## Question

Explain briefly about bilinear transformation method of IIR filter design.

## Short answer

Bilinear transformation substitutes s=(2/T)(1-z^-1)/(1+z^-1) into the analog transfer function. It maps the jΩ axis to the unit circle with Ω=(2/T)tan(ω/2), eliminating aliasing and preserving stability; pre-warping the critical frequencies compensates its nonlinear frequency warping.

## Answer

The bilinear transformation designs a digital IIR filter from an analog prototype H_a(s) by the substitution

s = (2/T) · (1 − z⁻¹)/(1 + z⁻¹),  so H(z) = H_a(s) at s = (2/T)(1 − z⁻¹)/(1 + z⁻¹)

It comes from approximating an integral by the trapezoidal rule.

**Mapping:**
- With s = jΩ and z = e^jω, the relation becomes **Ω = (2/T) tan(ω/2)**, which maps Ω = −∞…∞ onto ω = −π…π once only.
- The left half of the s-plane maps inside the unit circle, so a stable analog filter gives a stable digital filter.
- The jΩ-axis maps onto the unit circle.

**Advantage:** no aliasing, and it works for low-pass, high-pass, band-pass and band-stop filters.

**Disadvantage:** the relation is nonlinear (frequency warping), so the phase response is distorted and the edges are shifted. This is compensated by **pre-warping**: the digital edges are converted to analog edges by Ω = (2/T) tan(ω/2) before designing H_a(s).

**Steps:** (1) pre-warp ωp, ωs; (2) design H_a(s) (Butterworth: find N and Ωc); (3) substitute the bilinear relation to get H(z).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the bilinear substitution?
   A. s = (1/T)(1 - z^-1)
   B. s = (2/T) ln z
   C. s = z^-1/T
   D. s = (2/T)(1 - z^-1)/(1 + z^-1)
   Answer: D. It comes from the trapezoidal rule approximation of an integral. (Key point: Substitution)

2. What is the relation between analog and digital frequencies?
   A. Omega = (2/T) tan(w/2)
   B. Omega = w/T
   C. Omega = (2/T) sin(w)
   D. Omega = (T/2) tan(w)
   Answer: A. This tan relation maps the infinite Omega range onto -pi to pi, causing warping. (Key point: Frequency relation)

3. What is pre-warping used for?
   A. To avoid instability
   B. To convert digital edges to analog edges so warping does not shift the edges
   C. To remove phase delay
   D. To reduce the filter order
   Answer: B. Designing with pre-warped edges compensates for the tan nonlinearity. (Key point: Pre-warping purpose)

## Related in Digital Signal Analysis and Processing

- [Advantages of bilinear transformation](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aadvantage-bilinear-transformation) — Digital Signal Analysis and Processing, chapter 6, asked 3×
- [Bilinear Butterworth, 0.27π and 0.58π](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Abilinear-butter-0-27-0-58) — Digital Signal Analysis and Processing, chapter 6
- [Bilinear Butterworth, 0.25π passband, −15 dB at 0.45π](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Abilinear-butter-0-25-1db-0-45) — Digital Signal Analysis and Processing, chapter 6, asked 3×
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- [Bilinear Butterworth, 0.2π passband, −15 dB at 0.4π](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Abilinear-butter-0-2-1db-0-4) — Digital Signal Analysis and Processing, chapter 6, asked 2×
- [Bilinear Butterworth, 0.89125 and 0.17783 limits](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Abilinear-butter-0-89125) — Digital Signal Analysis and Processing, chapter 6

## Related in other subjects

- [Find a bilinear transformation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afind-a-bilinear-transformation) — Mathematics III, chapter 2
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