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Sat Jun 19 13:38:34 CEST 2010

## Exponential modeling

I worked on sinusoidal modeling for a while, and if there is one thing
that I can remember, it is that there is a lot of wiggle room in how
to tackle the problem on the technical side, and that what we hear
isn't so clearly related to what we can measure: our perception fills
in a lot of gaps.
Sinusoidal modeling highlights:
* Fourier transform and DFT (FFT). The FT is interesting for
reducing the complexity of convolutions, performing a whitening
transform for adaptive filters (as a filterbank), and as an ad-hoc
method for analysing harmonic sounds.
* Autoregressive models + ladder filters (orthogonal polynomials and
Shur's algorithm). The theory behind this is quite pretty.
For structured matrices, modified Shur and rank-revealing
decompositions can be used to yield fast algorithms.
* Non-linear phase signals. Moving from linear phase (exponential
sinusoidal) to chirp and higher order polynomial phase adds more
complexity. There doesn't seem to be much structure to explore
here.
* Nonlinear optimization. Writing sinusoidal modeling as a generic
optimization problem can use the fact that derivatives of
exponentials in general do not look horrible. However, these
functions are usually multi-modal, and give rise to ambiguities,
suggesting that there are "many ways to see a signal" as a sum of
sine waves.
* Subspace based techniques. I see two distinct classes: one that
operates directly on the signal/noise subspaces obtained through
the SVD decomposition of signal marices, and one that uses signal
spaces to further derive approximate linear system parameters
(sinusoidal parameters).

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