Perturbation in the Fractional Fourier Span due to Erroneous Transform Order and Window Function

Authors: Sukrit Shankar, Chetana Shanta Patsa, Jaydev Sharma

Abstract:

Fractional Fourier Transform is a generalization of the classical Fourier Transform. The Fractional Fourier span in general depends on the amplitude and phase functions of the signal and varies with the transform order. However, with the development of the Fractional Fourier filter banks, it is advantageous in some cases to have different transform orders for different filter banks to achieve better decorrelation of the windowed and overlapped time signal. We present an expression that is useful for finding the perturbation in the Fractional Fourier span due to the erroneous transform order and the possible variation in the window shape and length. The expression is based on the dependency of the time-Fractional Fourier span Uncertainty on the amplitude and phase function of the signal. We also show with the help of the developed expression that the perturbation of span has a varying degree of sensitivity for varying degree of transform order and the window coefficients.

Keywords: Fractional Fourier Transform, Perturbation, Fractional Fourier span, amplitude, phase, transform order, filterbanks.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1072609

References:

[1] S. C. Pei and M. H. Yeh, "Improved discrete fractional Fourier transform," Optics Letters, vol. 22, pp. 1047-1049, July 15 1997.
[2] Ahmed I. Zayed, "Relationship between the Fourier and Fractional Fourier Transforms", IEEE Signal Processing Letters, vol. 3, no. 12, December 1996.
[3] Tatiana Alieva and Martin J. Bastiaans, "On Fractional Fourier Transform Moments", IEEE Signal Processing Letters, vol. 7, no. 11, November 2000.