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Empirical Notes on Inverse Filtering

In experiments factoring guitar body impulse responses, it was found that the largest benefit per section comes from pulling out the main Helmholtz air resonance. Doing just this shortens the impulse response (excitation table) by a very large factor, and because the remaining impulse response is noise-like, it can be truncated more aggressively without introducing artifacts.

It also appears that the bandwidth estimate is not very critical in this case. If it is too large, or if ``isolation zeros'' are not installed behind the poles, as shown in Figs. 8.18b and 8.21b, the inverse filtering serves partially as a preemphasis which tends to flatten the guitar body frequency response overall or cause it to rise with frequency. This has a good effect on the signal-to-quantization-noise ratio versus frequency. To maximize the worst-case signal-to-quantization-noise versus frequency, the residual spectrum should be flat since the quantization noise spectrum is normally close to flat. A preemphasis filter for flattening the overall spectrum is commonly used in speech analysis [363,297]. A better preemphasis in this context is an inverse equal-loudness preemphasis, taking the inverse of an equal-loudness contour near the threshold of hearing in the Fletcher-Munson curves [475]. This corresponds to psychoacoustic ``noise shaping'' so that the quantization noise floor is perceptually uniform, and decreasing playback volume until it falls below the threshold of hearing results in all of the noise disappearing across the entire spectrum at the same volume.9.19

Since in some fixed-point implementations, narrow bandwidths may be difficult to achieve, good results are obtained by simply setting the bandwidth of the single resonator to any minimum robust value. As a result, there may still be some main-air-mode response in the residual signal, but it is typically very small, and early termination of it using a half-window for table shortening is much less audible than if the original impulse response were similarly half-windowed. The net effect on the instrument is to introduce artificial damping the main air mode in the guitar body. However, since this mode rings so much longer than the rest of the modes in the guitar body, shortening it does not appear to be detrimental to the overall quality of the instrument. In general, it is not desirable for isolated modes to ring longer than all others. Why would a classical guitarist want an audible ``ringing'' of the guitar body near $ 100$ Hz?

In computing figures 8.16 and Fig. 8.16b, the estimated $ Q$ of the main Helmholtz air mode was only $ 10$. As a result, it is still weakly present in the inverse filter output (residual) spectrum Fig. 8.16b.

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Matlab Code for Inverse Filtering
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