Notes on some calculations with decaying sinusoid atoms as a sparse dictionary basis.
Consider an signal We overload notation and write it with free argument , so that for example, refers to the signal
We decompose each in the decaying sinusoid dictionary {#eq:atomdict} Note that although the original signal is discrete, our decomposition is a continuous near-interpolant for it. There are many methods of fitting decaying sinusoids to series (Prony 1795; Barkhuijsen et al. 1985; Serra and Smith 1990), OMP is convenient in the current application (Michael Goodwin 1997) as we may re-use it in the next stage. Autocorrelograms of musical audio are typically highly sparse, achieving negligible residual error with
We apply the OMP with product weighted by returning parameters and code weights . We first find the normalized code product (@ref(eq:eq:mpproduct)) in closed form. Substituting in @ref(eq:eq:atomdict) gives
Using Euler identities we find the following useful integrals:
and thus
Inner products of decaying sinusoidal atoms
With the use of @ref(eq:eq:sinusoidint) we find analytic normalizing factors for the atoms.
If we choose a “top hat” weight it follows that we may expand this
where we have defined etc. This top hat weight is clearly fairly choppy, although it is simple enough to mechanically calculate which is nice.
Normalizing decaying sinusoidal atoms
To use matching pursuit we would need to normalize the atoms in our inner product formula @ref(eq:eq:mpproduct).
If we choose a top hat weight we find, as a special case of @ref(eq:eq:atomproduct),
Normalizing decaying sinusoidal molecules
Now consider a signal which is a molecule of decaying sinusoid atoms, in the sense that Here we use as a free argument, as these identities will be applied in the autocorrelation domain.
Once again, choosing we can apply @ref(eq:eq:atomsquarednorm) and to find a (lengthy) closed-form expression for this normalizing term.
References
Cantor, and Evans. 1970.
“On Approximation by Positive Sums of Powers.” SIAM Journal on Applied Mathematics.
Goodwin, Michael. 1997.
“Matching Pursuit with Damped Sinusoids.” In
1997 IEEE International Conference on Acoustics, Speech, and Signal Processing.
Goodwin, M., and Vetterli. 1997.
“Atomic Decompositions of Audio Signals.” In
1997 IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, 1997.
Goodwin, M M, and Vetterli. 1999.
“Matching Pursuit and Atomic Signal Models Based on Recursive Filter Banks.” IEEE Transactions on Signal Processing.
Wiscombe, and Evans. 1977.
“Exponential-Sum Fitting of Radiative Transmission Functions.” Journal of Computational Physics.