Bilinear Transformation

The bilinear transform is defined by the substitution

$$s = c\frac{1-z^{-1}}{1+z^{-1}}, \quad c>0, \; c=2/T\;$$ (J.5)

$$\,\,\implies\,\, z = \frac{1+s/c}{1-s/c} = 1 + 2(s/c) + 2(s/c)^2 + 2(s/c)^3 + \cdots$$ (J.6)

where $c$ is some positive constant [77,304]. That is, given a continuous-time transfer function $H_a(s)$, we apply the bilinear transform by defining

$$H_d(z) = H_a\left(c\frac{1-z^{-1}}{1+z^{-1}}\right)$$

where the ``$d$'' subscript denotes ``digital,'' and ``$a$'' denotes ``analog.''

It can be seen that analog dc ($s=0$) maps to digital dc ($z=1$) and the highest analog frequency ($s=\infty$) maps to the highest digital frequency ($z=-1$). It is easy to show that the entire $j\omega$ axis in the $s$ plane is mapped exactly once around the unit circle in the $z$ plane. Therefore, it does not alias. With $c$ real and positive, the left-half $s$ plane maps to the interior of the unit circle, and the right-half $s$ plane maps outside the unit circle.

The constant $c$ provides one remaining degree of freedom which can be used to map any particular finite frequency from the $j\omega_a$ axis in the $s$ plane to a particular desired location on the unit circle $e^{j\omega_d}$ in the $z$ plane. All other frequencies will be warped. In particular, approaching half the sampling rate, the frequency axis compresses more and more. Note that at most one resonant frequency can be preserved under the bilinear transformation of a mass-spring-dashpot system. On the other hand, filters having a single transition frequency, such as lowpass or highpass filters, map beautifully under the bilinear transform; one simply uses $c$ to map the cut-off frequency where it belongs, and the response looks great. In particular, ``equal ripple'' is preserved for optimal filters of the elliptic and Chebyshev types because the values taken on by the frequency response are identical in both cases; only the frequency axis is warped.

The bilinear transform is often used to design digital filters from analog prototype filters [323]. An on-line introduction is given in [426].