Recovering second-order statistics from compressive measurements

This paper focuses on the reconstruction of second order statistics of signals under a compressive sensing framework, which can be useful in many detection problems. More specifically, the focus is on general cyclostationary signals that are compressed using random linear projections, and using those compressive measurements, the cyclic power spectrum is retrieved. Subsequently, this can for instance be used to detect the occupation of specific frequency bands, which has applications in cognitive radio. Surprisingly, if the span of the random linear projections is larger than the period of the cyclostationary signals, the cyclic power spectrum can be recovered without putting any sparsity constraints on it, which allows for simple least squares reconstruction methods. This result indicates that significant compression can be realized by directly reconstructing the second-order statistics rather than the random signals themselves.