Inverse Filtering

Whatever poles are chosen for the least-damped part, and however they are computed (provided they are stable), the damped part can be computed from the full impulse response and parametric part using inverse filtering, as illustrated in the computed examples above. The inverse filter is formed from zeros equal to the estimated resonant poles. When the inverse filter is applied to the full resonator impulse, a ``residual'' signal is formed which is defined as the impulse response of the leftover, more damped modes. The residual is in exactly the nonparametric form needed for commuting with the string and convolving with the string excitation signal, such as a ``pluck'' signal. Feeding the residual signal to the parametric resonator gives the original resonator impulse response to an extremely high degree of accuracy. The error is due only to numerical round-off error during the inverse and forward filtering computations. In particular, the least-damped resonances need not be accurately estimated for this to hold. When there is parametric estimation error, the least-damped components will fail to be completely removed from the residual signal; however, the residual signal through the parametric resonator will always give an exact reconstruction of the original body impulse response, to within roundoff error. This is similar to the well known feature of linear predictive coding that feeding the prediction error signal to the LP model always gives back the original signal [299].

The parametric resonator need not be restricted to all-pole filters, however, although all-pole filters (plus perhaps zeros set manually to the same angles but contracted radii) turn out to be very convenient and simple to work with. Many filter design techniques exist which can produce a parametric part having any prescribed number of poles and zeros, and weighting functions can be used to ``steer'' the methods toward the least-damped components of the impulse response. The equation-error method illustrated in Fig. 8.13 is an example of a method which can also compute zeros in the parametric part as well as poles. However, for inverse filtering to be an option, the zeros must be constrained to be minimum phase so that their inverses will be stable poles.