# Decaying sinusoid dictionaries


Notes on some calculations with decaying sinusoid atoms.

Consider an $L_2$ signal $f: \bb{R}\to\bb{R}.$ We will overload notation and write it with free argument $\xi$, so that $f(r\xi-\phi),$ for example, refers to the signal $\xi\mapsto f(r\xi-\phi).$

We decompose each $\hat{G}=\omp_{\cc{S},C}(\cc{A}\{g\})$ 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 $C\leq 4$ (@Fig:sinusoidatomconverge).

We will apply the OMP with product $\inner{\cdot}{\cdot}_v$ weighted by $v(\xi):=\bb{I}\{[0,L)\}(\xi)/L,$ returning parameters $\{\tau_i, \omega_i, \phi_i\}$ and code weights $\mu_i$. 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 $v=\bb{I}[0,L],$ it follows that we may expand this

{#eq:atomproduct} where we have defined $\omega_{+}=\omega+\omega',$ $\omega_{-}=\omega-\omega'$ 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 $v=\bb{I}[0,L]$ we find, as a special case of @eq:atomproduct, {#eq:atomsquarednorm}

## Normalizing decaying sinusoidal molecules

We consider a signal $F$ which is a molecule of decaying sinusoid atoms, in the sense that $F:\xi\mapsto \sum_{k=1}^K \alpha_k \cos( \omega_k \xi +\phi_k)\exp \tau_k \xi.$ Here we use $\xi$ as a free argument, as these identities will be applied in the autocorrelation domain.

{#eq:moleculesquarednorm}

Once again, choosing $v=\bb{I}[0,L]$ we can apply @eq:atomsquarednorm and to find a (lengthy) closed-form expression for this normalising term.