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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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