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In this paper, we present a duality between two problems: the reconstruction of the angular periodogram from spatial-domain signals received at different time indices and that of the frequency periodogram from time-domain signals received at different wireless sensors. We assume the existence of a multiband structure in either the angular or frequency domain representation of the received spatial or time-domain signal, respectively, where different bands are assumed to be uncorrelated. The two problems lead to a similar circulant structure in the so-called coset correlation matrix, which allows for a strong compression and a least-squares (LS) reconstruction approach. The LS reconstruction of the periodogram is possible under the full column rank condition of the system matrix, which is achievable by designing the spatial or temporal sampling patterns based on a circular sparse ruler.