Fast numerical calculation of the offset linear canonical transform

Jian-Yi Chen; Jian-Yi Chen; Bing-Zhao Li; Bing-Zhao Li

doi:10.1364/JOSAA.482872

Journal of the Optical Society of America A
Vol. 40,
Issue 3,
pp. 427-442
(2023)
•https://doi.org/10.1364/JOSAA.482872

Fast numerical calculation of the offset linear canonical transform

Jian-Yi Chen and Bing-Zhao Li

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Jian-Yi Chen and Bing-Zhao Li, "Fast numerical calculation of the offset linear canonical transform," J. Opt. Soc. Am. A 40, 427-442 (2023)

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Abstract

The offset linear canonical transform (OLCT) is an important research topic in many fields, and it has a more universal and elastic performance due to its extra parameters. However, although much work has been done concerning the OLCT, its fast algorithms are rarely addressed. In this paper, an $O(N\log N)$ fast OLCT (FOLCT) algorithm that can significantly reduce the amount of calculation and improve accuracy is proposed. First, the discrete form of the OLCT is provided, and several important properties of its kernel are advanced. Next, the FOLCT based on the fast Fourier transform (FT) is derived for its numerical implementation. Then, the numerical results indicate that the FOLCT is a serviceable tool for signal analysis; additionally, the FOLCT algorithm can be used for the FT, fractional FT, linear canonical transform, and other transforms. Finally, its application to the detection of linear frequency modulated signals and optical image encryption, which is a basic case in signal processing, is discussed. The FOLCT can be effectively applied for the fast numerical calculation of the OLCT with valid and accurate results.

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Parameters $A$	Transform
$A = (α, β, γ, ω_{0}, u_{0})$	Offset linear canonical transform (OLCT)
$A = (0, 1, 0, 0, 0)$	Fourier transform (FT)
$A = (α, β, γ, 0, 0)$	Linear canonical transform (LCT)
$A = (c o t θ, c s c θ, c o t θ, 0, 0)$	Fractional Fourier transform (FRFT)
$A = (c o t θ, c s c θ, c o t θ, ω_{0}, u_{0})$	Offset fractional Fourier transform (OFRFT)
$A = (β, β, β, 0, 0)$	Fresnel transform (FST)

Property	Function
Time shift	$\begin{aligned} O_{A} [f (t - ξ)] (u) = e x p [j π ξ^{2} (γ - α \frac{γ^{2}}{β^{2}})] e x p [j 2 π ξ (\frac{γ}{β} (α u - α u_{0} + ω_{0}) + β (u_{0} - u))] O_{A} [f (t)] (u - ξ \frac{γ}{β}) \end{aligned}$
Frequency shift	$\begin{aligned} O_{A} [f (t) e x p (j 2 π ξ t)] (u) = e x p [- j π α \frac{ξ^{2}}{β^{2}}] e x p [j 2 π ξ \frac{α u - α u_{0} + ω_{0}}{β}] O_{A} [f (t)] (u - \frac{ξ}{β}) \end{aligned}$
Convolution	$\begin{aligned} h (t) & = K_{A} \int_{- \infty}^{+ \infty} f (ξ) g (t - ξ) e x p [- j 2 π γ ξ (t - ξ)] d ξ \\ \Rightarrow O_{A} [h (t)] (u) = O_{A} [f (t)] (u) O_{A} [g (t)] (u) e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \end{aligned}$
Product	$\begin{aligned} h (t) & = f (t) g (t) \Rightarrow O_{A} [h (t)] (u) = \frac{1}{2 π / β} e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \\ \times [O_{A} [g (t)] (u) e x p (j π (- α u^{2} + 2 u (α u_{0} - ω_{0})))] O_{A} [f (t)] (β u) \end{aligned}$
Correlation	$\begin{aligned} h (t) & = K_{A} K_{A}^{} \int_{- \infty}^{+ \infty} f (ξ) g^{} (t - ξ) e x p [j 2 π γ t (ξ - t)] d ξ \\ \Rightarrow O_{A} [h (t)] (u) = K_{A} O_{A} [f (t)] (u) O_{A}^{*} [g (t)] (u) e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \end{aligned}$
Scale	$g (t) = 1 / \sqrt{ξ} f (t / ξ) \Rightarrow O_{A} [g (t) (u) = O_{B} [f (t)] (u), B = (α, ξ β, ξ^{2} γ, ω_{0}, u_{0})$
Linear	$h (t) = a_{1} f (t) + a_{2} g (t) \Rightarrow O_{A} [h (t) (u) = a_{1} O_{A} [f (t) (u) + a_{2} O_{A} [g (t) (u)$
Retrotemporality	$g (t) = f (- t) \Rightarrow O_{A} [g (t) (u) = O_{B} [f (t)] (- u), B = (α, β, γ, - ω_{0}, - u_{0})$
Parseval	$\int_{- \infty}^{+ \infty} \| f (t)^{2} d t = \int_{- \infty}^{+ \infty} \| O_{A} [f (t)] (u) \|^{2} d u$

	No Noise ( $f_{1} (n T)$ )		Noisy ( $f_{2} (n T)$ )
Method	Relative Error	Computing Time (s)	Relative Error	Computing Time (s)
DOLCT	0.0833%	3.7724	0.0833%	3.7488
FOLCT	0	0.3045	0	0.2592

Parameters $A$	Transform
$A = (α, β, γ, ω_{0}, u_{0})$	Offset linear canonical transform (OLCT)
$A = (0, 1, 0, 0, 0)$	Fourier transform (FT)
$A = (α, β, γ, 0, 0)$	Linear canonical transform (LCT)
$A = (c o t θ, c s c θ, c o t θ, 0, 0)$	Fractional Fourier transform (FRFT)
$A = (c o t θ, c s c θ, c o t θ, ω_{0}, u_{0})$	Offset fractional Fourier transform (OFRFT)
$A = (β, β, β, 0, 0)$	Fresnel transform (FST)

Property	Function
Time shift	$\begin{aligned} O_{A} [f (t - ξ)] (u) = e x p [j π ξ^{2} (γ - α \frac{γ^{2}}{β^{2}})] e x p [j 2 π ξ (\frac{γ}{β} (α u - α u_{0} + ω_{0}) + β (u_{0} - u))] O_{A} [f (t)] (u - ξ \frac{γ}{β}) \end{aligned}$
Frequency shift	$\begin{aligned} O_{A} [f (t) e x p (j 2 π ξ t)] (u) = e x p [- j π α \frac{ξ^{2}}{β^{2}}] e x p [j 2 π ξ \frac{α u - α u_{0} + ω_{0}}{β}] O_{A} [f (t)] (u - \frac{ξ}{β}) \end{aligned}$
Convolution	$\begin{aligned} h (t) & = K_{A} \int_{- \infty}^{+ \infty} f (ξ) g (t - ξ) e x p [- j 2 π γ ξ (t - ξ)] d ξ \\ \Rightarrow O_{A} [h (t)] (u) = O_{A} [f (t)] (u) O_{A} [g (t)] (u) e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \end{aligned}$
Product	$\begin{aligned} h (t) & = f (t) g (t) \Rightarrow O_{A} [h (t)] (u) = \frac{1}{2 π / β} e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \\ \times [O_{A} [g (t)] (u) e x p (j π (- α u^{2} + 2 u (α u_{0} - ω_{0})))] O_{A} [f (t)] (β u) \end{aligned}$
Correlation	$\begin{aligned} h (t) & = K_{A} K_{A}^{} \int_{- \infty}^{+ \infty} f (ξ) g^{} (t - ξ) e x p [j 2 π γ t (ξ - t)] d ξ \\ \Rightarrow O_{A} [h (t)] (u) = K_{A} O_{A} [f (t)] (u) O_{A}^{*} [g (t)] (u) e x p [j π (- α u^{2} + 2 u (α u_{0} - ω_{0}))] \end{aligned}$
Scale	$g (t) = 1 / \sqrt{ξ} f (t / ξ) \Rightarrow O_{A} [g (t) (u) = O_{B} [f (t)] (u), B = (α, ξ β, ξ^{2} γ, ω_{0}, u_{0})$
Linear	$h (t) = a_{1} f (t) + a_{2} g (t) \Rightarrow O_{A} [h (t) (u) = a_{1} O_{A} [f (t) (u) + a_{2} O_{A} [g (t) (u)$
Retrotemporality	$g (t) = f (- t) \Rightarrow O_{A} [g (t) (u) = O_{B} [f (t)] (- u), B = (α, β, γ, - ω_{0}, - u_{0})$
Parseval	$\int_{- \infty}^{+ \infty} \| f (t)^{2} d t = \int_{- \infty}^{+ \infty} \| O_{A} [f (t)] (u) \|^{2} d u$

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Equations (60)

Journal of the Optical Society of America A