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 [297].

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.

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.

Matlab Code for Inverse Filtering

Below is the matlab source code used to extract the main Helmholtz air mode from the guitar body impulse response in Figures 8.14 through 8.17:

        freq = 104.98;  % estimated peak frequency in Hz
        bw = 10;        % peak bandwidth estimate in Hz

        R = exp( - pi * bw / fs);            % pole radius
        z = R * exp(j * 2 * pi * freq / fs); % pole itself
        B = [1, -(z + conj(z)), z * conj(z)] % numerator
        r = 0.9;     % zero/pole factor (notch isolation)
        A = B .* (r .^ [0 : length(B)-1]);   % denominator

        residual = filter(B,A,bodyIR); % apply inverse filter

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