Bilinear Transform - Discrete-time Approximation

Discrete-time Approximation

The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of

$\begin{align} z &= e^{sT} \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align}$

where is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation. The above bilinear approximation can be solved for or a similar approximation for can be performed.

The inverse of this mapping (and its first-order bilinear approximation) is

$\begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &= \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align}$

The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function,

That is

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