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Decaying sinusoid dictionaries

Notes on some calculations with decaying sinusoid atoms.

Consider an signal We will 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 [@PronyEssai1795;@BarkhuijsenRetrieval1985;@SerraSpectral1990], OMP is convenient in the current application [@GoodwinMatching1997] as we may re-use it in the next stage. Autocorrelograms of musical audio are typically highly sparse, achieving negligible residual error with (@Fig:sinusoidatomconverge).

We will apply the OMP with product weighted by returning parameters and code weights . We first find the normalized code product ([@eq:mpproduct]) in closed form. Substituting in @eq:atomdict gives {#eq:atomproduct}

Using Euler identities we find the following useful integrals:

and thus {#eq:sinusoidint}

Inner products of decaying sinusoidal atoms

With the use of @eq:sinusoidint we find analytic normalising factors for the atoms.

{#eq:sinusoidproduct}

If we choose a “top hat” weight it follows that we may expand this

{#eq:atomproduct} where we have defined etc.

Normalizing decaying sinusoidal atoms

To use matching pursuit we would need to normalize the atoms in our inner product formula @eq:mpproduct.

If we choose a top hat weight we find, as a special case of @eq:atomproduct, {#eq:atomsquarednorm}

Normalizing decaying sinusoidal molecules

We 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.

{#eq:moleculesquarednorm}

Once again, choosing we can apply @eq:atomsquarednorm and to find a (lengthy) closed-form expression for this normalising term.

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