Sparse Signal Decomposition for Periodic Signal Mixtures

Abstract

Periodicities are found in speech signals, musical rhythms, biomedical signals and machine vibrations. In many signal processing applications, signals are assumed to be periodic or quasi-periodic. Especially in acoustic signal processing, signal models based on periodicities have been studied for speech and audio processing. The sinusoidal modelling has been proposed to transform an acoustic signal to a sum of sinusoids [1]. In this model, the frequencies of the sinusoids are often assumed to be harmonically related. The fundamental frequency of the set of sinusoids has to be specified for this model. In order to compose an accurate model of an acoustic signal, the noise-robust and accurate fundamental frequency estimation techniques are required. Many fundamental frequency estimation techniques are performed in the short-time Fourier transform (STFT) spectrum by peak-picking and clustering of harmonic components [2][3][4]. These approaches depend on the frequency spectrum of the signal. The signal modeling in the time-domain has been also proposed to extract a waveform of an acoustic signal and its parameters of the amplitude and frequency variations [5]. This approach aims to represent an acoustic signal that has single fundamental frequency. For detection and estimation of more than one periodic signal hidden in a signal mixture, several signal decomposition that are capable of decomposing a signal into a set of periodic subsignals have been proposed. In Ref. [7], an orthogonal decomposition method based on periodicity has been proposed. This technique achieves the decomposition of a signal into periodic subsignals that are orthogonal to each other. The periodicity transform [8] decomposes a signal by projecting it onto a set of periodic subspaces. In this method, seeking periodic subspaces and rejecting found periodic subsignals from the observed signal are performed iteratively. For reduction of the redundancy of the periodic representation, a penalty of sparsity has been introduced to the decomposition in Ref. [9]. In these periodic decomposition methods, the amplitude of each periodic signal in the mixture is assumed to be constant. Hence, it is difficult to obtain the significant decomposition results for the mixtures of quasi-periodic signals with time-varying amplitude. In this chapter, we introduce a model for periodic signals with time-varying amplitude into the periodic decomposition [10]. In order to reduce the number of resultant 8