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