### Sinusoidal Modeling of Mode Decays

When the amplitude envelope, frequency, phase, and onset time are all
accurately estimated (for all time), it is possible to *subtract*
the synthesized modal impulse response from the measured impulse
response. (This contrasts with purely spectral modal parameters which
are amplitude, frequency, bandwidth, and phase.) This method of
``sinusoidal track removal'' is used in sines-plus-noise spectral
modeling. (See [424] for further details and supporting C
software). In this approach, the resonant mode is subtracted out
rather than divided out of the frequency response.

There are some disadvantages of subtraction relative to inverse filtering. First, more parameters must be accurately measured; the precise gain and phase of the resonance are needed in addition to its frequency and bandwidth. Inverse filtering on the other hand requires only estimation of frequency and bandwidth (or frequency and time-constant of decay). In addition, the residual impulse response after subtraction cannot be precisely commuted with the string for commuted synthesis.

The advantages of subtraction over inverse filtering are that amplitude modulation due to mode coupling can be retained in the measured modal decay and subtracted out, whereas a second-order inverse filter cannot subtract out the modulation due to coupling. Also, if the system is time varying (as happens, for example, when the performer's hand is pressing against the resonating body in a time-varying way), the subtraction method can potentially track the changing modal parameters and still successfully remove the modal decay. To compete with this, the inverse filtering method would have to support a time-varying filter model. As another example, if the guitar body is moving, the measured response is a time-varying linear combination of fixed resonant modes (although some Doppler shift is possible). The subtraction method can follow a time-varying gain (and phase) so that the subtraction still will take out the mode.

The inverse filtering method seems most natural when the physical
resonator is time-invariant and is well modeled as a *series* of
resonant sections. It is also the only one strictly valid for use in
commuted synthesis. The subtraction method seems most natural when
the physical resonator is best modeled as a *sum* of resonating
modes. As a compromise between the two approaches, all parametric
modes may be separated from the nonparametric modes by means of
inverse filtering, and the parametric part can then be split into
parallel form.

**Next Section:**

Parallel Body Filterbank Design

**Previous Section:**

Inverse Filtering