Low-Order Filter Implementations

When latency and time-efficiency is a concern, the length of the body's impulse response becomes an issue. Therefore, in these systems, the impulse response of the body is approximated using lower-order filters and a modal synthesis model.

To effectively shorten the body's impulse response for systems requiring low-latency and low-memory needs, methods for removing the peaks are applied to leave a residual signal and a low-cost representation for the removed resonating peaks. This reduces the length of the original impulse response while representing the primary modes parametrically. The two general approaches for dealing with spectral peak removal include subtraction and inverse-filtering methods.

The two basic methods are as follows:

1. Complex Spectral Subtraction

   $$H_r(z) = H(z) - \frac{b_0 + b_1 z^{-1}}{1+a_1z^{-1}+a_2z^{-2}}$$ (15)

   
   

   where $H_r(z)$ corresponds to the shortened body impulse response while $H(z)$ corresponds to the measured body impulse response. The parameters to be estimated are the second-order filter coefficients $b_0$ ,$b_1$ ,$a_1$ and $a_2$ .

   Complex Spectral Subtraction requires careful estimation of the phase, amplitude, frequency and bandwidth for peak removal. Furthermore, the resonators must run in parallel with the residual. Therefore, advantages obtained from Commuted Synthesis are lost as the approximated body impulse response model is not readily commutable with the string component of our physical model [45].

2. Inverse-Filtering

   $$H_r(z) = H(z)(1+a_1z^{-1}+a_2z^{-2})$$ (16)

   
   

   where $H_r(z)$ again corresponds to the shortened body impulse response with $H(z)$ equal to the measured body impulse response. In this form, the residual signal is readily commutable with the string component of our physical model as resonators are factored instead of subtracted. Furthermore, estimating the coefficients of the filter for peak removal requires only the frequency and bandwidth of the peak and not the amplitude and phase as is required for Complex Spectral Subtraction [28].

Figure 43: Time-domain signal of body response with peak at $120$ Hz removed.

Figure 44: Spectral view, magnitude and phase, of body response with peak at $120$ Hz removed. Note that the response is now shorter compared with the original body response in Figure 39.

Applying inverse-filtering as described above, we take the original body response shown in Figures 39 and 40, and remove the peak centered around $120$ Hz with a bandwidth of $10$ Hz. The residual, shown in Figure 43, is significantly shorter than the original response. Whereas in Figure 39, the response lasts for well over $50$ ms, in the residual signal, its amplitude oscillates near the noise floor at around $30$ ms. Figure 44 shows the spectrum, both magnitude and phase, of the response after inverse-filtering. Compared with Figure 41, the peak at $120$ Hz is completely removed.