  In this paper, two problems that show great similarities are examined. The
first problem is the reconstruction of the angular-domain periodogram from
spatial-domain signals received at different time indices. The second one is
the reconstruction of the frequency-domain periodogram from time-domain signals
received at different wireless sensors. We split the entire angular or
frequency band into uniform bins. The bin size is set such that the received
spectra at two frequencies or angles, whose distance is equal to or larger than
the size of a bin, are uncorrelated. These problems in the two different
domains lead to a similar circulant structure in the so-called coset
correlation matrix. This circulant structure allows for a strong compression
and a simple least-squares reconstruction method. The latter is possible under
the full column rank condition of the system matrix, which can be achieved by
designing the spatial or temporal sampling patterns based on a circular sparse
ruler. We analyze the statistical performance of the compressively
reconstructed periodogram including bias and variance. We further consider the
case when the bins are so small that the received spectra at two frequencies or
angles, with a spacing between them larger than the size of the bin, can still
be correlated. In this case, the resulting coset correlation matrix is
generally not circulant and thus a special approach is required.
