A Low-Complexity Approach to Computation of the Discrete Fractional Fourier Transform

Abstract

This paper proposes an effective approach to the computation of the discrete fractional Fourier transform for an input vector of any length N. This approach uses specific structural properties of the discrete fractional Fourier transformation matrix. Thanks to these properties, the fractional Fourier transformation matrix can be decomposed into a sum of three or two matrices, one of which is a dense matrix, and the rest of the matrix components are sparse matrices. The aforementioned dense matrix has unique structural properties that allow advantageous factorization. This factorization is the main contributor to the reduction in the overall computational complexity of the discrete fractional Fourier transform computation. The remaining calculations do not contribute significantly to the total amount of computation. Thus, the proposed approach allows to reduce the number of arithmetic operations when calculating the discrete fractional Fourier transform.