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This paper studies a spectrum estimation method for the case that the samplesare obtained at a rate lower than the Nyquist rate. The method is referred toas the correlogram for undersampled data. The algorithm partitions the spectruminto a number of segments and estimates the average power within each spectralsegment. This method is able to estimate the power spectrum density of a signalfrom undersampled data without essentially requiring the signal to be sparse.We derive the bias and the variance of the spectrum estimator, and show thatthere is a tradeoff between the accuracy of the estimation, the frequencyresolution, and the complexity of the estimator. A closed-form approximation ofthe estimation variance is also derived, which clearly shows how the varianceis related to different parameters. The asymptotic behavior of the estimator isalso investigated, and it is proved that this spectrum estimator is consistent.Moreover, the estimation made for different spectral segments becomesuncorrelated as the signal length tends to infinity. Finally, numericalexamples and simulation results are provided, which approve the theoreticalconclusions.

Finite-Length and Asymptotic Analysis of Correlogram for Undersampled Data