[sdiy] Prony's method [was: Re: Fourier Analysis Question]

[lanterma at ece.gatech.edu]

On Dec 18, 2010, at 12:39 PM, Magnus Danielson wrote:

> The Z-transform is essentially the same as the discrete Fourier Transoform (for which FFT is just a computational trick) in which rather than using the complex e^(iw) oscillator as basis it uses the superset function of e^(o+iw) which can form both accelerating and decelerating oscillations. The Fourier transform is a special case where o=0.
> 
> Practically you can pre-scale the sample-series with e^(-o*t) for analysis of the o offset. Positive o relates to accelerating amplitude and negative o relates to decelerating amplitudes.
> 
> It is best viewed as a 2D plot with the x-axis as frequency, y-axis for amplitude acceleration o values. Having both being log10 should help. Indication is then in log10 of absolute amplitude. You will see the zeros (no-responce) and poles (where responce blows up to infinity) quite clearly that way.
> 
> The Z-transform is relevant for transient responses (such as you hitting a key on your piano) where as the Fourier transform is actually only useful for continous amplitude signals (such as a perfect aligned sample of you holding a key on an organ and sample a part of that tone).

I've usually seen Z-transforms as more of a "pencil and paper" analytical tool - people generally don't numerically compute Z-transforms.

I think Prony's method (or one of the related damped exponential extraction techniques) might be the way to go:

http://en.wikipedia.org/wiki/Prony's_method

"Prony's method extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or sinusoids. This allows for the estimation of frequency, amplitude, phase and damping components of a signal."

Emphasis on the damping part; this is what you won't get (easily) from an FFT.

Google "prony method musical tones" (without the quotes) and some useful stuff comes up.

p. 598 of the "Computer Music Tutorial" by Curtis Roads has some useful comparison between Prony's method and the FFT.

- Aaron