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

[Jens Simon Strom]

Hey Tim,
valuable advice from your side!  I haven't the files yet. And I fear 
that fiddling with the data will take more than a few days.  In any case 
I will give feedback when there is something to say.
------------------------------------------------------------------------------------------------------------------------------------

Am 08.09.2015 19:02, schrieb 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.
>