Compressive Sampling for Power Spectrum Estimation

Compressive sampling is a well-known approach to reconstruct sparse signals based on a limited number of measurements. In spectrum sensing applications for cognitive radio though, only reconstruction of the power spectrum of the signal is required, instead of the signal itself. In this paper, we present a new method for power spectrum reconstruction based on samples produced by a sub-Nyquist rate sampling device. The stationary assumption on the received analog signal causes the measurements at the output of the compressive sampling block to be cyclo-stationary, or the measurement vectors to be stationary. We investigate the relationship between the autocorrelation matrix of the measurement vectors and that of the received analog signal, which we represent by its Nyquist rate sampled version. Based on this relationship, we are able to express the autocorrelation sequence of the received wide sense stationary signal as a linear function of the vectorized autocorrelation matrix of the measurement vectors. Depending on the compression rate, we can present the problem as either over-determined or under-determined. Our focus will be mainly on the over-determined case, in which the reconstruction does not require any additional constraints. Two types of sampling matrices are examined, namely complex Gaussian and multi-coset sampling matrices. For both of them, we can derive conditions under which the over-determined system will result in a unique solution for the power spectrum by adopting a simple least squares (LS) algorithm. In the case of multi-coset sampling, further improvement on the quality of the power spectrum estimates can be attained by optimizing the condition of the sampling matrix.