  This paper studies two spectrum estimation methods for the case that the
samples are obtained at a rate lower than the Nyquist rate. The first method is
the correlogram method for undersampled data. The algorithm partitions the
spectrum into a number of segments and estimates the average power within each
spectral segment. We derive the bias and the variance of the spectrum
estimator, and show that there is a tradeoff between the accuracy of the
estimation and the frequency resolution. The asymptotic behavior of the
estimator is also investigated, and it is proved that this spectrum estimator
is consistent.
  A new algorithm for reconstructing signals with sparse spectrum from noisy
compressive measurements is also introduced. Such model-based algorithm takes
the signal structure into account for estimating the unknown parameters which
are the frequencies and the amplitudes of linearly combined sinusoidal signals.
A high-resolution spectral estimation method is used to recover the frequencies
of the signal elements, while the amplitudes of the signal components are
estimated by minimizing the squared norm of the compressed estimation error
using the least squares technique. The Cramer-Rao bound for the given system
model is also derived. It is shown that the proposed algorithm approaches the
bound at high signal to noise ratios.
