Bilinear transform

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 for converting a transfer function ${\displaystyle H_{a}(s)}$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often named an analog filter) to a transfer function ${\displaystyle H_{d}(z)}$ of a linear, shift-invariant filter in the discrete-time domain (often named a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the ${\displaystyle j\omega }$ axis, ${\displaystyle \mathrm {Re} [s]=0}$, in the s-plane to the unit circle, ${\displaystyle |z|=1}$, in the 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 ${\displaystyle \left(z^{-1}\right)}$ with first order all-pass filters.

The transform preserves stability and maps every point of the frequency response of the continuous-time filter, ${\displaystyle H_{a}(j\omega _{a})}$ to a corresponding point in the frequency response of the discrete-time filter, ${\displaystyle H_{d}(e^{j\omega _{d}T})}$ although to a somewhat different frequency, as shown in the 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. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.