3.7 Conversion of analog to digital transfer functions  (Page 4/5)

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 order of the digital filter is the same as the prototype filter.
• The left-half s-plane is mapped into the unit circle onthe z-plane. This means stability is preserved.
• Optimal approximations to piecewise constant prototype filters, such as the four cases in Continuous Frequency Definition of Error , transform into optimal digital filters.
• The cascade of sections designed by the bilinear transform is the same as obtained by transforming the total system.

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.

Frequency transformations

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:

1. First, the lower and upper digital bandedge frequencies are specified as ${\omega }_1$ and or ${\omega }_1$ , ${\omega }_2$ , ${\omega }_3$ , and ${\omega }_4$ if an elliptic-function approximation is used.
2. These frequencies are prewarped using [link] to give theband edges of the prototype bandpass analogfilter.
3. These frequencies are converted into a single band- edge ${\omega }_p$ or ${\omega }_s$ for the Butterworth and Chebyshev and into ${\omega }_p$ and ${\omega }_s$ for the elliptic-function approximation of the prototype lowpass filter by using Equation 2 from Frequency Transformations and Equation 4 from Frequency Transformations .
4. The lowpass filter is designed for this ${\omega }_p$ and/or ${\omega }_s$ by using one of the four approximations in the sections in Continuous Frequency Definition of Error or some other method.
5. This lowpass analog filter is converted into a bandpass analog filter with the frequency transformation Equation 6 from Frequency Transformations .
6. The bandpass analog filter is then transformed into the desired bandpass digital filter using the bilineartransformation [link] .