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A comparative study of commuting matrix approaches for the discrete fractional fourier transform

As an extension of the conventional Fourier transform and as a time-frequency signal analysis tool, the fractional Fourier transforms (FRFT) are suitable for dealing with various types of non-stationary signals. Computation of the discrete fractional Fourier transform (DFRFT) and its chirp concentration properties are both dependent on the basis of DFT eigenvectors used in the computation. Several DFT-eigenvector bases have been proposed for the computation of transform, and there is no common framework for comparing them. In this paper, we compare several different approaches from a conceptual viewpoint and review the differences between them. We discuss five different approaches to find centered-DFT (CDFT) commuting matrices and the various properties of these commuting matrices. We study the properties of the eigenvalues and eigenvectors of these commuting matrices to determine whether they resemble those of corresponding continuous Gauss-Hermite operator. We also measure the performance of these five approaches in terms of: mailobe-to-sidelobe ratio, 10-dB bandwidth, quality factor, linearity of eigenvalues, chirp parameter estimation error, and, finally the peak-to-parameter mapping regions. We compare the five approaches using these performance metrics and point out that the modified QMFD approach produces the best results in terms of bandwidth of the spectral peak for a chirp, invertibility of the peak-parameter mapping, linearity of the eigenvalue spectrum and chirp parameter estimation errors.