Editing Bilinear transform (section)

{{Short description|Signal processing operation}}
{{more citations needed|date=June 2009}}
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 map]]ping (namely, a [[Möbius transformation]]), often used for converting a [[transfer function]] <math> H_a(s) </math> of a [[linear]], [[time-invariant]] ([[LTI system theory|LTI]]) filter in the [[continuous function|continuous]]-time domain (often named an [[analog filter]]) to a transfer function <math>H_d(z)</math> of a linear, shift-invariant filter in the [[discrete signal|discrete]]-time domain (often named a [[digital filter]] although there are analog filters constructed with [[switched capacitor]]s that are discrete-time filters). It maps positions on the <math> j \omega </math> axis, <math> \mathrm{Re}[s]=0 </math>, in the [[s-plane]] to the [[unit circle]], <math> |z| = 1 </math>, in the [[complex plane|z-plane]].  Other bilinear transforms can be used for warping the [[frequency response]] of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays <math> \left( z^{-1} \right) </math> with first order [[all-pass filter]]s.

The transform preserves [[BIBO stability|stability]] and maps every point of the [[frequency response]] of the continuous-time filter, <math> H_a(j \omega_a) </math> to a corresponding point in the frequency response of the discrete-time filter, <math> H_d(e^{j \omega_d T}) </math> although to a somewhat different frequency, as shown in the [[#Frequency warping|Frequency warping]] section below.  This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency.  The change in frequency is barely noticeable at low frequencies but is quite evident at frequencies close to the [[Nyquist frequency]].