This technique avoids the problem of aliasing by mapping axis in the s-plane to one revaluation of the unit circle in the z-plane.

If is the continues time transfer function the discrete time transfer function is detained by replacing s with

Rearranging terms in equation (9.3) we obtain.

Substituting , we get

If , it is then magnitude of the real part in denominator is more than that of the numerator and so. Similarly if , than for all. Thus poles in the left half of the s-plane will get mapped to the poles inside the unit circle in z-plane. If then

So, , writing we get

rearranging we get

or

or

The compression of frequency axis represented by (9.5) is nonlinear. This is illustrated in figure 9.4.

Fig 9.4

Because of the nonlinear compression of the frequency axis, there is considerable phase distortion in the bilinear transformation.

We use the specifications given in the previous example. Using equation (9.5) with we get