3.7 Conversion of analog to digital transfer functions  (Page 3/5)

The bilinear transformation

A second method for converting an analog prototype filter into a desired digital filter is the bilinear transformation.This method is entirely a frequency-domain method, and as a result, some of the optimal properties of the analog filter arepreserved. As was the case with the impulse-invariant method, the time interval is not normalized to one, but is explicitly denotedby the sampling interval $T$ with units of seconds. The bilinear transformation is a change of variables (a mapping) that islinear in both the numerator and denominator [link] , [link] , [link] , [link] . The usual form is

The $z$ -transform transfer function of the digital filter $H\left(z\right)$ is obtained from the Laplace transform transfer function $F\left(s\right)$ of the prototype filter by substituting for $s$ the bilinear form of [link] .

This operation can be reversed by solving [link] for z and substituting this into $H\left(z\right)$ to obtain $F\left(s\right)$ . This reverse operation is also bilinear of the form

To consider the frequency response, the Laplace variable s is evaluated on the imaginary axis and the z-transform variable z isevaluated on the unit circle. This is achieved by

which gives the relation of the analog frequency variable u to the digital frequency variable $\omega$ from [link] and [link] to be

The bilinear transform maps the infinite imaginary axis in the-s plane onto the unit circle in the $z$ -plane. It maps the infinite interval of $-\infty <u<\infty$ of the analog frequency axis on to the finite interval of $-\pi /2<\omega <\pi /2$ of the digital frequency axis. This is illustrated in [link] .

There is no folding or aliasing of the prototype frequency response, but there is a compression of the frequency axis, which becomes extreme athigh frequencies. This is shown in [link] from the relation of [link] .

Near zero frequency, the relation of $u$ and $\omega$ is essentially linear. The compression increases as the digital frequency w nears $\pi /2$ . This nonlinear compression is called frequency warping. The conversion of $F\left(s\right)$ to $H\left(z\right)$ with the bilinear transformation does not change the values of the frequency response, but it changes thefrequencies where the values occur.

In the design of a digital filter, the effects of the frequency warping must be taken into account. The prototypefilter frequency scale must be prewarped so that after the bilinear transform, the critical frequencies are in the correctplaces. This prewarping or scaling of the prototype frequency scale is done by replacing s with Ks. Because the bilinear transform is also a change of variables, both can be performed inone step if that is desirable.

If the critical frequency for the prototype filter is $u_o$ and the desired critical frequency for the digital filter is ${\omega }_o$ , the two frequency responses are related by

The prewarping scaling is given by

Combining the prewarping scale and the bilinear transformation give

Solving for $K$ and combining with [link] give

All of the optimal filters developed in Continuous Frequency Definition of Error and most other prototype filters are designed with a normalized critical frequencyof $u_0=1$ . Recall that ${\omega }_0$ is in radians per second. Most specifications are given in terms of frequency $f$ in Hertz (cycles per second) which is related to $\omega$ or $u$ by