Bilinear transform

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The bilinear transform (also known as Tustin's method) 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, the 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,  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 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  \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,  H_a(j \omega_a) \ to a corresponding point in the frequency response of the discrete-time filter,  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.

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Discrete-time approximation

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