Improved discrete fractional Fourier transform

Soo-Chang Pei; Min-Hung Yeh

doi:10.1364/OL.22.001047

Improved discrete fractional Fourier transform

Soo-Chang Pei and Min-Hung Yeh

Optics Letters
Vol. 22,
Issue 14,
pp. 1047-1049
(1997)
•https://doi.org/10.1364/OL.22.001047

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Soo-Chang Pei and Min-Hung Yeh, "Improved discrete fractional Fourier transform," Opt. Lett. 22, 1047-1049 (1997)

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History
- Original Manuscript: March 11, 1997
- Published: July 15, 1997

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Abstract

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

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$N$	1	$- j$	$- 1$	$j$
$4 m$	$m + 1$	$m$	$m$	$m - 1$
$4 m + 1$	$m + 1$	$m$	$m$	$m$
$4 m + 2$	$m + 1$	$m$	$m + 1$	$m$
$4 m + 3$	$m + 1$	$m + 1$	$m + 1$	$m$

$N$	DFRFT Eigenvalues
$4 m$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m - 2, 4 m$
$4 m + 1$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m - 1, 4 m$
$4 m + 2$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m, 4 m + 2$
$4 m + 3$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m + 1, 4 m + 2$

$N$	1	$- j$	$- 1$	$j$
$4 m$	$m + 1$	$m$	$m$	$m - 1$
$4 m + 1$	$m + 1$	$m$	$m$	$m$
$4 m + 2$	$m + 1$	$m$	$m + 1$	$m$
$4 m + 3$	$m + 1$	$m + 1$	$m + 1$	$m$

$N$	DFRFT Eigenvalues
$4 m$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m - 2, 4 m$
$4 m + 1$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m - 1, 4 m$
$4 m + 2$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m, 4 m + 2$
$4 m + 3$	$exp (- jk α), k = 0, 1, 2, \dots, 4 m + 1, 4 m + 2$

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