Mode Extraction Techniques

The goal of resonator factoring is to identify and remove the least-damped resonant modes of the impulse response. In principle, this means ascertaining the precise resonance frequencies and bandwidths associated with each of the narrowest ``peaks'' in the resonator frequency response, and dividing them out via inverse filtering, so they can be implemented separately as resonators in series. If, in addition, the amplitude and phase of a resonance peak are accurately measurable in the complex frequency response, the mode can be removed by complex spectral subtraction (equivalent to subtracting the impulse-response of the resonant mode from the total impulse response); in this case, the parametric modes are implemented in a parallel bank as in [66]. However, in the parallel case, the residual impulse response is not readily commuted with the string.

In the inverse-filtering case, the factored resonator components are in cascade (series) so that the damped modes left behind may be commuted with the string and incorporated in the excitation table by convolving the residual impulse response with the desired string excitation signal. In the parallel case, the damped modes do not commute with the string since doing so would require somehow canceling them in the parallel filter sections. In principle, the string would have to be duplicated so that one instance can be driven by the residual signal with no body resonances at all, while the other is connected to the parallel resonator bank and driven only by the natural string excitation without any commuting of string and resonator. Since duplicating the string is unlikely to be cost-effective, the impulse response of the high-frequency modes can be commuted and convolved with the string excitation as in the series case to obtain qualitative results. The error in doing this is that the high-frequency modes are being multiplied by the parallel resonators rather than being added to them.

Various methods are available for estimating the mode parameters for inverse filtering:

• Amplitude response peak measurement
  
• Weighted digital filter design
  
• Linear prediction
  
• Sinusoidal modeling
  
• Late impulse-response analysis