Smith form of FIR pseudocirculants

The pseudocirculant matrices have been found to be a very useful tool in the analysis and design of communication systems, e.g., precoding systems and discrete multitone transceivers. In these systems, a scalar channel P(z) is recast into a pseudocirculant channel matrix. Many important channel properties have been derived from the Smith form and the decomposition of pseudocirculants. In this letter, we will show that the Smith form of a pseudocirculant matrix can be given in terms of the zeros of the underlying finite-impulse response (FIR) channel P(z). Once the zeros of P(z) are known, the Smith form of the corresponding pseudocirculant matrix can be obtained in closed form.