Bilinear Transformation

The bilinear transform is defined by the substitution

$$s = c\frac{1-z^{-1}}{1+z^{-1}}, \quad c>0, \; c=2/T\;$$ (8.6)

$$\,\,\Rightarrow\,\, z = \frac{1+s/c}{1-s/c} \eqsp 1 + 2(s/c) + 2(s/c)^2 + 2(s/c)^3 + \cdots$$ (8.7)

where $c$ is some positive constant [83,329]. That is, given a continuous-time transfer function $H_a(s)$ , we apply the bilinear transform by defining

$$H_d(z) = H_a\left(c\frac{1-z^{-1}}{1+z^{-1}}\right) \protect$$ (8.8)

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 (where $s\isdeftext \sigma+j\omega$ ) is mapped exactly once around the unit circle in the $z$ plane (rather than summing around it infinitely many times, or ``aliasing'' as it does in ordinary sampling). 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. This means stability is preserved when mapping a continuous-time transfer function to discrete time.

Another valuable property of the bilinear transform is that order is preserved. That is, an $N$ th-order $s$ -plane transfer function carries over to an $N$ th-order $z$ -plane transfer function. (Order in both cases equals the maximum of the degrees of the numerator and denominator polynomials [452]).^8.6

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 frequency-warping of the bilinear transform is readily found by looking at the frequency-axis mapping in Eq.(7.7), i.e., by setting $s=j\omega_a$ and $z = e^{j\omega_d T}$ in the bilinear-transform definition:

$$j\,\omega_a \;=\; c \, \frac{1-e^{-j\omega_d T}}{1+e^{-j\omega_d T}} \;=\; j \, c \, \tan\left(\frac{\omega_dT}{2}\right).$$

Thus, we may interpret $c$ as a frequency-scaling constant. At low frequencies, $\tan(x)\approx x$ , so that $\omega_a\approx c \omega_d T / 2$ at low frequencies, leading to the typical choice of $c = 2/T = 2f_s$ , where $f_s$ denotes the sampling rate in Hz. However, $c$ can be chosen to map exactly any particular interior frequency $\omega_a\in(0,f_s/2)$ .

The bilinear transform is often used to design digital filters from analog prototype filters [346]. An on-line introduction is given in [452].