The Bilinear Transform

• The bilinear transform is an algebraic transformation between the continuous-time and discrete-time frequency variables s and z, respectively. It is therefore appropriate only when a closed-form filter representation in s exists.

• The bilinear transform is defined by the substitution:
  $$s = c \frac{1 - z^{-1}}{1 + z^{-1}}, \hspace{0.1in} c > 0.$$

• The bilinear transform maps the entire continuous-time frequency space, $-\infty \leq \Omega \leq \infty,$ onto the discrete-time frequency space $-\pi \leq \omega \leq \pi.$ Continuous-time dc ($\Omega = 0$) maps to discrete-time dc ($\omega = 0$) and infinite continuous-time frequency ( $\Omega = \infty$) to the Nyquist frequency ($\omega = \pi$). Thus, a nonlinear warping of the frequency axes occurs.

• Since the $j \Omega$-axis in the s plane is mapped exactly once around the unit circle of the z plane, no aliasing occurs.

• The constant c is typically given by c = 2/T_s. This parameter provides one degree of freedom for the mapping of a particular finite continuous-time frequency to a particular location on the z-plane unit circle. The warping of frequencies is given for the discrete-time frequency in terms of c and $\Omega$ as
  $$\omega = 2 \arctan(\Omega / c).$$

• If one wishes to map a specific “analog” frequency ($\Omega$) to the equivalent “digital” frequency ($\omega$), the constant c is found as
  $$c = \frac{\Omega}{\tan(\omega/2)}.$$

• Because of the nonlinear compression of the frequency axis, use of the bilinear transformation should be limited to filters with a single transition band (lowpass and highpass) or resonance (narrow bandpass and bandstop).

• The Matlab example, msd_fdbt.m, compares the results of using both the finite difference and bilinear transform approaches to simulate the motion of a mass-spring-damper system.