Inversion of block-circulant matrices and circular array approach

Closed-form expressions of the inverse of a block-circulant matrix (BCM) are well known. Such expressions are derived here following two alternative procedures, which are based on the discrete Fourier transform (DFT) and appear to be simple and brief when compared to previously published procedures. In the case of circulant matrices of scalars, the formulas of inversion are derived using the theory of circular arrays and this approach suggested an extension of the DFT to the BCM case.