  This paper studies a spectrum estimation method for the case that the samples
are obtained at a rate lower than the Nyquist rate. The method is referred to
as the correlogram for undersampled data. The algorithm partitions the spectrum
into a number of segments and estimates the average power within each spectral
segment. This method is able to estimate the power spectrum density of a signal
from undersampled data without essentially requiring the signal to be sparse.
We derive the bias and the variance of the spectrum estimator, and show that
there is a tradeoff between the accuracy of the estimation, the frequency
resolution, and the complexity of the estimator. A closed-form approximation of
the estimation variance is also derived, which clearly shows how the variance
is related to different parameters. The asymptotic behavior of the estimator is
also investigated, and it is proved that this spectrum estimator is consistent.
Moreover, the estimation made for different spectral segments becomes
uncorrelated as the signal length tends to infinity. Finally, numerical
examples and simulation results are provided, which approve the theoretical
conclusions.
