Features of Wave Digital Filters (and Bilinear Transforms)

• We obtain an exact, one-to-one mapping of the frequency response $S(j\omega)$ (a reflectance) to the unit circle in the z plane, even though each first-order element is reduced to a mere unit delay with a possible sign flip (or to nothing but 0 in the case of a resistor/dashpot).

• One can show that a unit-sample delay in the digital version corresponds to an exactly correct phase-shift at the mapped analog frequency. The frequency mapping is, for this bilinear transform,
  $$\omega_a = \tan\left(\frac{\omega_d T}{2}\right),$$
  where $\omega_a$ = analog radian frequency and $\omega_d$ = digital radian frequency.

• If it is possible to warp the frequencies of an input signal in the same way, that is, if we can perform the bilinear transform also on our analog input signal
  $$x(t) \leftrightarrow X(s)$$
  to produce a digital version
  $$X\left(\frac{z-1}{z+1}\right)\leftrightarrow x_d(n),$$
  then we can obtain exact LTI processing of an infinite-bandwidth signal inside a wave digital filter, no matter what sampling rate we choose!
  • Unfortunately, this is usually not possible.
  • Exercise: List the practical problems associated with trying to do something like this in practice, however approximate.

• Because the frequency axis is warped, we cannot expect nonlinear or time-varying systems to behave in discrete time as they did in continuous time:
  • Modulation sidebands land at ``wrong'' frequencies.
  • Harmonic relationships are destroyed $\Rightarrow$
    ``harmonic distortion'' becomes inharmonic.