Closed-form expression of smith forms for pseudo-circulants

The pseudo-circulant matrices arise naturally in the block filtering application of multirate systems. More recently, there has been considerable interests in pseudo-circulants for the applications of precoding and discrete multitone communications systems. In these systems, a scalar FIR channel P(z) is cast as a pseudo-circulant channel matrix. Many important channel properties have been derived from the Smith form and the decomposition of pseudo-circulants. In this paper, we will show that the Smith form of an FIR pseudo-circulant matrix can be given in terms of the zeros of the underlying scalar filter P(z). Once the zeros of P(z) are known, the Smith form of the corresponding pseudo-circulant matrix can be obtained in closed form.