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Compressive sampling (CS) is famous for its ability to perfectly reconstruct a sparse signal based on a limited number of measurements. In some applications, such as in spectrum sensing for cognitive radio, perfect signal reconstruction is not really needed. Instead, only statistical measures such as the power spectrum or equivalently the auto-correlation sequence are required. In this paper, we introduce a new approach for reconstructing the power spectrum based on samples produced by sub-Nyquist rate sampling. Depending on the compression rate, the entire problem can be presented as either under-determined or over-determined. In this paper, we mainly focus on the over-determined case, which allows us to employ a simple least-squares (LS) reconstruction method. We show under which conditions this LS reconstruction method yields a unique solution, without including any sparsity constraints.