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The bilinear transform 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 , often used to convert a transfer function H a {\displaystyle H_{a}} of a linear, time-invariant filter in the continuous-time domain to a transfer function H d {\displaystyle H_{d}} of a linear, shift-invariant filter in the discrete-time domain. It maps positions on the j ω {\displaystyle j\omega } axis, R e = 0 {\displaystyle \mathrm {Re} =0} , in the s-plane to the unit circle, | z | = 1 {\displaystyle |z|=1} , in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system and are implementable in the discrete domain by replacing a system's unit delays {\displaystyle \left} 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 {\displaystyle H_{a}} to a corresponding point in the frequency response of the discrete-time filter, H d {\displaystyle H_{d}} 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.