Compressive sensing based stochastic process power spectrum estimation subject to missing data

A compressive sensing (CS) based approach for stationary and non-stationary stochastic process power spectrum estimation subject to missing data is developed. Stochastic process records such as wind and sea wave excitations can often be represented with relative sparsity in the frequency domain. Relying on this attribute, a CS framework can be applied to reconstruct a signal that contains sampling gaps in the time domain, possibly occurring for reasons such as sensor failures, data corruption, limited bandwidth/storage capacity, and power outages. Specifically, first an appropriate basis is selected for expanding the signal recorded in the time domain. In this regard, Fourier and harmonic wavelet bases are utilized herein. Next, an L1 norm minimization procedure is performed for obtaining the sparsest representation of the signal in the selected basis. Finally, the signal can either be reconstructed in the time domain if required or, alternatively, the underlying stochastic process power spectrum can be estimated in a direct manner by utilizing the determined expansion coefficients; thus, circumventing the computational cost related to reconstructing the signal in the time domain. The technique is shown to estimate successfully the essential features of the stochastic process power spectrum, while it appears to be efficient even in cases with 65% missing data demonstrating superior performance in comparison with alternative existing techniques. A significant advantage of the approach is that it performs satisfactorily even in the presence of noise. Several numerical examples demonstrate the versatility and reliability of the approach both for stationary and non-stationary cases.