Compressive sensing based power spectrum estimation from incomplete records by utilizing an adaptive basis

A compressive sensing (CS) based approach is developed in conjunction with an adaptive basis reweighting procedure for stochastic process power spectrum estimation. In particular, the problem of sampling gaps in stochastic process records, occurring for reasons such as sensor failures, data corruption, and bandwidth limitations, is addressed. Specifically, due to the fact that many stochastic process records such as wind, sea wave and earthquake excitations can be represented with relative sparsity in the frequency domain, a CS framework can be applied for power spectrum estimation. To this aim, an ensemble of stochastic process realizations is often assumed to be available. Relying on this attribute an adaptive data mining procedure is introduced to modify harmonic basis coefficients, vastly improving on standard CS reconstructions. The procedure is shown to perform well with stationary and non-stationary processes even with up to 75% missing data. Several numerical examples demonstrate the effectiveness of the approach when applied to noisy, gappy signals.