For musical instrument sounds containing partials, which are referred to as modes, the decaying processes of the modes significantly affect the timbre of musical instruments and characterize the sounds. However, their accurate decomposition around the onset is not an easy task, especially when the sounds have sharp onsets and contain the non-modal percussive components such as the attack. This is because the sharp onsets of modes comprise peaky but broad spectra, which makes it difficult to get rid of the attack component. In this paper, an optimization-based method of modal decomposition is proposed to overcome it. The proposed method is formulated as a constrained optimization problem to enforce the perfect reconstruction property which is important for accurate decomposition and causality of modes. Three numerical simulations and application to the real piano sounds confirm the performance of the proposed method.
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