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Care must be taken with the elliptic-function filter where there are two critical frequencies that determine the transitionregion. Both frequencies must be prewarped.
The characteristics of the bilinear transform are the following:
The bilinear transform is probably the most used method of converting a prototype Laplace transform transfer function into adigital transfer function. It is the one used in most popular filter design programs [link] , because of characteristic 3 above that states optimality is preserved. The maximally flat prototype istransformed into a maximally flat digital filter. This property only holds for approximations to piecewise constant ideal frequencyresponses, because the frequency warping does not change the shape of a constant. If the prototype is an optimal approximation to adifferentiator or to a linear-phase characteristic, the bilinear transform will destroy the optimality. Those approximations have tobe made directly in the digital frequency domain.
To illustrate the bilinear transformation, the third-order Butterworth lowpass filter designed in the Example is converted into adigital filter. The prototype filter transfer function is
The prototype analog filter has a passband edge at ${u}_{0}=1$ . A data rate of 1000 samples per second corresponding to $T=0.001$ seconds is assumed. If the desired digital passband edge is ${f}_{0}=200$ Hz, then ${\omega}_{0}=\left(2\pi \right)\left(200\right)$ radians per second, and the total prewarped bilinear transformation from [link] is
The digital transfer function in [link] becomes
Note the locations of the poles and zeros in the z-plane. Zeros at infinity in the s-plane always map into the z = -1 point. The exampleillustrate a third-order elliptic-function filter designed using the bilinear transform.
For the design of highpass, bandpass, and band reject filters, a particularly powerful combination consists of using thefrequency transformations described in Section elsewhere together with the bilinear transformation. When using this combination, some care must betaken in scaling the specifications properly. This is illustrated by considering the steps in the design of a bandpass filter:
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