#### 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);

**Next Section:**

Late impulse-response analysis

**Previous Section:**

Linear prediction