Bilinear transformation:

One method to the numerical solution of an ordinary linear constant-coefficient differential equation is laid on the application of the trapezoidal principle to the first order approximation of integration. Consider the given equivalent pair of equations

where we have used the trapezoidal rule to calculate the area under a curve.

Taking the z transform of the above we get

Thus the continuous time system y(t) = ∫x(t)dt shown by the following block diagrams

is replaced in the discrete-time domain by the given block.

In other words, given the Laplace transfer function Ha(s), the corresponding digital filter is given by replacing s with

This is known as bilinear transformation (both numerator and denominator are first order polynomials), also called as bilinear z-transformation (BZT).

Here again, note that at that point we have not given how Ha(s) was calculated, but rather we are define how to calculate the digital filter H(z) from any given Ha(s) taking bilinear transformation.

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