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The bilinear transform (also known as Tustin's method) is used in digital signal processing and discretetime control theory to transform continuoustime system representations to discretetime 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 of a linear, timeinvariant (LTI) filter in the continuoustime domain (often called an analog filter) to a transfer function of a linear, shiftinvariant filter in the discretetime domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discretetime filters). It maps positions on the axis, , in the splane to the unit circle, , in the zplane. Other bilinear transforms can be used to warp the frequency response of any discretetime linear system (for example to approximate the nonlinear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays with first order allpass filters.
The transform preserves stability and maps every point of the frequency response of the continuoustime filter, to a corresponding point in the frequency response of the discretetime filter, 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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Discretetime approximation
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