[Scilab-users] Identifying parameters of a multy frequency damped oscillation

[Tim Wescott]

On Mon, 2015-09-07 at 14:26 +0200, Jens Simon Strom wrote:
> Hi Scilab users, I want to analyse a microphone recording of the sound
> of a bell or gong. Given ia the equidistantly sampled sound pressure
> y(t) after a stroke for 10 s.
> 
> The ansatz 
> 
> y=sum(  A_i*exp(-d_i*t))*cos(2*%pi*f_i+alpha_i)  )  for i=1,2,...5 or more
> 
> is assumed to approximate the signal where A_i, d_i, f_i, and alpha_i
> are the unknown amplitudes, danping factors, frequencies, and phase
> angles of y. The analysis may be restricted to the lowest 5 frequencies
> fi.

> Does Scilab offer a method for this? FFT seems not to be adequate
> because the signal is aperiodic (silence after 10 s).

> Is nonlinear regression (with which I'm fmiliar) a promising way. The
> lowest frequencies can probably be concluded from a short time DFT
> where damping is negligible.

Hey Jens:

I'm not sure that a gong or a bell that's not "tuneful" (e.g. a cowbell,
as opposed to a church bell) would have components that are well-enough
separated to be easily picked out by any numerical analysis -- but I'm
ready to be proven wrong.  Certainly it should be possible to pick out
the components of a bell that might be used for making music.

I would start with an FFT to get rough values for the f_i and alpha_i.
Then I would use those as starting points for nonlinear regression.

Alternately, if the components are well-enough separated in the FFT of
the captured audio you could identify each component, then for each
component you could zero out the FFT outside of that component's
contribution (remember to retain both positive and negative frequency
components symmetrically around 0Hz), then do an IFFT to get the
waveform for just that component.  This _might_ recover each individual
component well enough that you can eyeball it for A_i and d_i -- at
which point you could either call it good enough, or you could make an
even better starting point for your regression analysis.

-- 

Tim Wescott
www.wescottdesign.com
Control & Communications systems, circuit & software design.
Phone: 503.631.7815
Cell:  503.349.8432