The accuracy of the Distribution Derivative Method (DDM) [1] is evaluated on mixtures of chirp signals. It is shown that accurate estimation can be obtained when the sets of atoms for which the inner product is large are disjoint. This amounts to designing atoms with windows whose Fourier transform exhibits low sidelobes but which are once-differentiable in the time-domain. A technique for designing once-differentiable approximations to windows is presented and the accuracy of these windows in estimating the parameters of sinusoidal chirps in mixture is evaluated.

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In this paper, we introduce a function designed specifically for sparse audio representations. A progression in the selection of dictionary elements (atoms) to sparsely represent audio has occurred: starting with symmetric atoms, then to damped sinusoid and hybrid atoms, and finally to the re-appropriation of the gammatone (GT) and formantwave-function (FOF) into atoms. These asymmetric atoms have already shown promise in sparse decomposition applications, where they prove to be highly correlated with natural sounds and musical audio, but since neither was originally designed for this application their utility remains limited. An in-depth comparison of each existing function was conducted based on application specific criteria. A directed design process was completed to create a new atom, the ramped exponentially damped sinusoid (REDS), that satisfies all desired properties: the REDS can adapt to a wide range of audio signal features and has good mathematical properties that enable efficient sparse decompositions and synthesis. Moreover, the REDS is proven to be approximately equal to the previous functions under some common conditions.

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