Orthogonal Projections and Discrete Fractional Fourier Transforms

A summary of results from linear algebra pertaining to orthogonal projections onto subspaces of an inner product space is presented. A formal definition and a sufficient condition for the existence of a fractional transform given a unitary periodic operator is given. Next, using an orthogonal projection formula the class of weighted discrete fractional Fourier transforms (WDFrFTs) is shown to be completely determined by four integer parameters. Particular choices of these parameters yield the Dickinson-Steiglitz and Santhanam-McClellan WDFrFTs. Another choice gives a WDFrFT which agrees with any eigenvector decomposition-based DFrFT for terms of degree less than four. Applications of the proposed algorithm to chirp filtering is discussed