#### Sinusoidal modeling

Another way to find the least-damped mode parameters is by means of an intermediate*sinusoidal model*of the body impulse response, or, more appropriately, the

*energy decay relief*(EDR) computed from the body impulse response (see §3.2.2). Such sinusoidal models have been used to determine the string loop filter in digital waveguide strings models. In the case of string loop-filter estimation, the sinusoidal model is applied to the impulse response (or ``pluck'' response) of a vibrating string or acoustic tube. In the present application, it is ideally applied to the EDR of the body impulse response (or ``hammer-strike'' response).

Since sinusoidal modeling software (

*e.g.*[424]) typically quadratically interpolates the peak frequencies, the resonance frequencies are generally quite accurately estimated provided the frame size is chosen large enough to span many cycles of the underlying resonance. The sinusoidal amplitude envelopes yield a particularly robust measurement of resonance bandwidth. Theoretically, the modal decay should be exponential. Therefore, on a dB scale, the amplitude envelope should decay

*linearly*. Linear regression can be used to fit a straight line to the measured log-amplitude envelope of the impulse response of each long-ringing mode. Note that even when amplitude modulation is present due to modal couplings, the ripples tend to average out in the regression and have little effect on the slope measurement. This robustness can be enhanced by starting and ending the linear regression on local maxima in the amplitude envelope. A method for estimating modal decay parameters in the presence of noise is given in [125,234,235]. Below is a section of matlab code which performs linear regression:

function [slope,offset] = fitline(x,y); %FITLINE fit line 'y = slope * x + offset' % to column vectors x and y. phi = [x, ones(length(x),1)]; p = phi' * y; r = phi' * phi; t = r\p; slope = t(1); offset = t(2);

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Linear prediction