Discrete linear canonical transforms based on dilated Hermite functions

Soo-Chang Pei; Yun-Chiu Lai

doi:10.1364/JOSAA.28.001695

Journal of the Optical Society of America A
Vol. 28,
Issue 8,
pp. 1695-1708
(2011)
•https://doi.org/10.1364/JOSAA.28.001695

Discrete linear canonical transforms based on dilated Hermite functions

Soo-Chang Pei and Yun-Chiu Lai

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Soo-Chang Pei and Yun-Chiu Lai, "Discrete linear canonical transforms based on dilated Hermite functions," J. Opt. Soc. Am. A 28, 1695-1708 (2011)

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Related Topics
Table of Contents Category
- Fourier Optics and Signal Processing
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fourier optics and signal processing (070.0070)
- Paraxial wave optics (070.2580)
- ABCD transforms (070.2590)
- Fractional Fourier transforms (070.2575)

About this Article
History
- Original Manuscript: April 19, 2011
- Revised Manuscript: June 10, 2011
- Manuscript Accepted: June 11, 2011
- Published: July 27, 2011

Abstract

Linear canonical transform (LCT) is very useful and powerful in signal processing and optics. In this paper, discrete LCT (DLCT) is proposed to approximate LCT by utilizing the discrete dilated Hermite functions. The Wigner distribution function is also used to investigate DLCT performances in the time–frequency domain. Compared with the existing digital computation of LCT, our proposed DLCT possess additivity and reversibility properties with no oversampling involved. In addition, the length of input/output signals will not be changed before and after the DLCT transformations, which is consistent with the time–frequency area-preserving nature of LCT; meanwhile, the proposed DLCT has very good approximation of continuous LCT.

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	Function Descriptions^a	Length N
$F_{1}$	Chirped pulse function $\exp (- π t^{2} - i π t^{2})$ , $d t = 0.1$	101
$F_{2}$	Trapezoidal function $1.5 tri (t / 3) - 0.5 tri (t)$ , $tri (t) = rect(t) * rect (t)$ , $d t = 0.1$	101
$F_{3}$	Binary sequence 00001101010000	280
$F_{4}$	Rectangular function $rect (t / 2)$ , $d t = 0.2$	171
$F_{5}$	Complex sinusoidal function (time-limited) $[2 \cos (2 π t) + i \sin (π (t - 1))] \exp (- t^{2})$ , $d t = 0.1$	101
$F_{6}$	ECG signal downloaded from MIT/BIH database 100 [52]	250

	$M_{1}$	$M_{2}$	$M_{3}$	$M_{4}$
$[\begin{matrix} a & b \\ c & d \end{matrix}]$	$[\begin{matrix} 0.5 & - 0.5 \\ 0.5 & 1.5 \end{matrix}]$	$[\begin{matrix} 1 & - 2 / π \\ 0 & 1 \end{matrix}]$	$[\begin{matrix} - 1.2 & 0.8 \\ - 2 & 0.5 \end{matrix}]$	$[\begin{matrix} 0.4 & 1.2 \\ - 0.9 & - 0.2 \end{matrix}]$
$(ξ, σ, α)$	$(- 1, 1 / \sqrt{2}, - π / 4)$	$(- 0.45, 1.19, - 0.57)$	(1.35, 1.44, 2.55)	$(- 0.38, 1.26, 1.25)$

	Complexity	Oversampling Factor^a	MSE^b	Additivity	Reversibility
Method II [24]	$Ω (N \log N)$	k	$5.37 \times 10^{- 5}$	×	×
Proposed [Eq. (34)]	$Ω (N^{2} \log N)$	1	$1.61 \times 10^{- 25}$	O	O

	Function Descriptions^a	Length N
$F_{1}$	Chirped pulse function $\exp (- π t^{2} - i π t^{2})$ , $d t = 0.1$	101
$F_{2}$	Trapezoidal function $1.5 tri (t / 3) - 0.5 tri (t)$ , $tri (t) = rect(t) * rect (t)$ , $d t = 0.1$	101
$F_{3}$	Binary sequence 00001101010000	280
$F_{4}$	Rectangular function $rect (t / 2)$ , $d t = 0.2$	171
$F_{5}$	Complex sinusoidal function (time-limited) $[2 \cos (2 π t) + i \sin (π (t - 1))] \exp (- t^{2})$ , $d t = 0.1$	101
$F_{6}$	ECG signal downloaded from MIT/BIH database 100 [52]	250

	$M_{1}$	$M_{2}$	$M_{3}$	$M_{4}$
$[\begin{matrix} a & b \\ c & d \end{matrix}]$	$[\begin{matrix} 0.5 & - 0.5 \\ 0.5 & 1.5 \end{matrix}]$	$[\begin{matrix} 1 & - 2 / π \\ 0 & 1 \end{matrix}]$	$[\begin{matrix} - 1.2 & 0.8 \\ - 2 & 0.5 \end{matrix}]$	$[\begin{matrix} 0.4 & 1.2 \\ - 0.9 & - 0.2 \end{matrix}]$
$(ξ, σ, α)$	$(- 1, 1 / \sqrt{2}, - π / 4)$	$(- 0.45, 1.19, - 0.57)$	(1.35, 1.44, 2.55)	$(- 0.38, 1.26, 1.25)$

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Journal of the Optical Society of America A

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