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- In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered. Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates. (en)
- 在數學的文獻中，線性標準轉換(linear canonical transform, LCT)也稱作线性正则变换、ABCD轉換、广义Fresnel变换等。在漢米爾頓力學中，線性標準轉換是積分變換的一個代表家族，並且能夠將許多經典的轉換進行廣義化，例如傅立葉變換、分數傅立葉變換、拉普拉斯變換、菲涅爾轉換(電磁波在空氣中傳播)、、等等。此轉換提供了這些最常使用的線性轉換一個統一框架，並且在光學、信號轉換以及系統響應領域中都提供一般化的概念。尤其從系統工程的角度看來，線性標準轉換提供一個強大的光學系統設計和分析的工具。此轉換有四維變數的線性積分轉換和一個限制條件，因此實際上是一個三維自由度的積分變換的家族。在群論中，線性標準轉換屬於特殊線性群(SR(2))在時頻域上的一個作用群。 (zh)
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