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Concept# Bilinear transform

Summary

The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used to convert a transfer function H_a(s) of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function H_d(z) of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the j \omega axis, \mathrm{Re}[s]=0 , in the s-plane to the unit circle, |z| = 1 , in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linea

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When it comes to designing active control disposals, sound pressure sensing is usually required. In the framework of active control of acoustic impedance, this work aims at conceiving a digital estimator that processes voltage and current measurement to compute an estimation of the pressure in the vicinity of the diaphragm of an electro-dynamic speaker. After dealing with modeling concerns of the actuator, the bilinear transform is used to obtain equations of two IIR filters. An adaptive identification using the LMS algorithm and a SVD-based identification are tested to compute filters coefficient, both leading to poor experimental results. Accordingly, the actual transfer functions of the speaker are measured and optimal IIR filters are designed using the simulated annealing algorithm. It is then shown that good results can be achieved experimentally.

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