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| In [[Hamiltonian mechanics]], the '''linear canonical transformation''' ('''LCT''') is a family of [[integral transform]]s 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)|SL<sub>2</sub>('''R''')]] on the [[Time–frequency representation|time–frequency plane]] (domain).
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| The LCT generalizes the [[Fourier transform|Fourier]], [[fractional Fourier transform|fractional Fourier]], [[Laplace transform|Laplace]], [[Weierstrass transform|Gauss–Weierstrass]], Bargmann and the [[Fresnel transform|Fresnel]] transforms as particular cases. The name "linear canonical transformation" is from [[canonical transformation]], a map that preserves the symplectic structure, as SL<sub>2</sub>('''R''') can also be interpreted as the [[symplectic group]] Sp<sub>2</sub>, and thus LCTs are the linear maps of the time–frequency domain which preserve the [[symplectic form]].
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| ==Definition==
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| The LCT can be represented in several ways; most easily,<ref>de Bruijn, N. G. (1973). "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence", ''Nieuw Arch. Wiskd.'', '''III. Ser., 21''' 205-280.</ref> it can be viewed as a 2×2 matrix with determinant 1, i.e., an element of the [[special linear group]] SL<sub>2</sub>('''R'''). Taking a matrix <math>\left(\begin{smallmatrix}a & b\\c & d\end{smallmatrix}\right),</math> with ''ad'' − ''bc'' = 1, the corresponding [[integral transform]] is:
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| :{| border="0"
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| |-
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| | <math>X_{(a,b,c,d)}(u) = \sqrt{-i} \cdot e^{i \pi \frac{d}{b} u^{2}} \int_{-\infty}^\infty e^{-i 2 \pi \frac{1}{b} ut}e^{i \pi \frac{a}{b} t^2} x(t) \; dt \, , </math> {{pad|3em}} || when ''b'' ≠ 0,
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| |-
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| | <math>X_{(a,0,c,d)}(u) = \sqrt{d} \cdot e^{i \pi cdu^{2}} x(du) \, ,</math> || when ''b'' = 0.
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| |}
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| ==Special cases==
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| Many classical transforms are special cases of the linear canonical transform:
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| * The [[Fourier transform]] corresponds to rotation by 90°, represented by the matrix:
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| ::<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 0 & 1 \\
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| -1 & 0
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| \end{bmatrix}.
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| </math>
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| * The [[fractional Fourier transform]] corresponds to rotation by an arbitrary angle; they are the [[SL2(R)#Elliptic_elements|elliptic elements]] of SL<sub>2</sub>('''R'''), represented by the matrices:
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| ::<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| \cos \theta & \sin \theta \\
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| -\sin \theta & \cos \theta
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| \end{bmatrix}.
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| </math>
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| * The [[Fresnel transform]] corresponds to shearing, and are a family of [[SL2(R)#Parabolic_elements|parabolic elements]], represented by the matrices:
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| ::<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & \lambda z \\
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| 0 & 1
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| \end{bmatrix}.</math>
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| :where ''z'' is distance and ''λ'' is wave length.
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| * The [[Laplace transform]] corresponds to rotation by 90° into the complex domain, and can be represented by the matrix:
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| ::<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 0 & i \\
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| i & 0
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| \end{bmatrix}.
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| </math>
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| * The [[Fractional Laplace transform]] corresponds to rotation by an arbitrary angle into the complex domain, and can be represented by the matrix:<ref>P.R. Deshmukh & A.S. Gudadhe (2011) Convolution structure for two version of fractional Laplace transform. Journal of Science and Arts, 2(15):143-150. [http://core.kmi.open.ac.uk/display/1140726]</ref>
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| ::<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| i \cos \theta & i \sin \theta \\
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| i \sin \theta & -i \cos \theta
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| \end{bmatrix}.
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| </math>
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| ==Composition==
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| Composition of LCTs corresponds to multiplication of the corresponding matrices; this is also known as the "additivity property of the [[Wigner distribution function|WDF]]".
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| In detail, if the LCT is denoted by ''O<sub>F</sub><sup>(a,b,c,d)</sup>'', i.e.
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| :<math>X_{(a,b,c,d)}(u) = O_F^{(a,b,c,d)}[x(t)] \, </math>
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| then
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| :<math>O_F^{(a2,b2,c2,d2)} \left \{ O_F^{(a1,b1,c1,d1)}[x(t)] \right \} = O_F^{(a3,b3,c3,d3)}[x(t)] \, ,</math>
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| where
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| :<math>
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| \begin{bmatrix}
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| a3 & b3 \\
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| c3 & d3
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| a2 & b2 \\
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| c2 & d2
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| \end{bmatrix}
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| \begin{bmatrix}
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| a1 & b1 \\
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| c1 & d1
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| \end{bmatrix}.
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| </math>
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| ==In optics and quantum mechanics==
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| [[Paraxial optical system]]s implemented entirely with [[thin lens]]es and propagation through free space and/or graded index (GRIN) media, are quadratic phase systems (QPS); these were known before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform, a particular case of which was developed by Segal (1963) and Bargmann (1961) in order to formalize Fok's (1928) boson calculus.<ref>K.B. Wolf (1979) [http://www.fis.unam.mx/~bwolf/Book%27s/Integral%20Transforms/IX.pdf Ch. 9:Canonical transforms].</ref>
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| ==Applications==
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| Canonical transforms are used to analyze differential equations. These include [[diffusion]], the [[free particle|Schrödinger free particle]], the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the [[Fokker–Planck equation]]. Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these.<ref>K.B. Wolf (1979) Ch. [http://www.fis.unam.mx/~bwolf/Book%27s/Integral%20Transforms/IX.pdf 9] & [http://www.fis.unam.mx/~bwolf/Book%27s/Integral%20Transforms/X.pdf 10].</ref>
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| Wave propagation through air, a lens, and between satellite dishes are discussed here. All of the computations can be reduced to 2×2 matrix algebra. This is the spirit of LCT.
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| ===Electromagnetic wave propagation===
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| [[Image:TFA LCT fresnel.jpg|thumb|right|250px]]
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| <!-- ARRRGH so many different i's and j's , so confusing -->
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| Assuming the system looks like as depicted in the figure, the wave travels from plane ''x<sub>i</sub>'', ''y<sub>i</sub>'' to the plane of ''x'' and ''y''.
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| The [[Fresnel transform]] is used to describe electromagnetic wave propagation in air:
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| :<math> U_0(x,y) = - \frac{j}{\lambda} \frac{e^{jkz}}{z} \int_{-\infty}^\infty \int_{-\infty}^{\infty} e^{j \frac{k}{2z} [ (x-x_i)^2 +(y-y_i)^2 ] } U_i(x_i,y_i) \; dx_i\; dy_i, </math>
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| with
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| :{|
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| |-
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| | ''k'' = 2 ''π'' / ''λ'' {{pad|1em}} || : [[wave number]];
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| |-
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| | ''λ'' || : [[wavelength]];
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| |-
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| | ''z'' || : distance of propagation;
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| |-
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| | ''j'' || : imaginary unit.
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| |}
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| This is equivalent to LCT (shearing), when
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| :<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & \lambda z \\
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| 0 & 1
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| \end{bmatrix}.
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| </math>
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| When the travel distance (''z'') is larger, the shearing effect is larger.
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| ===Spherical lens===
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| [[Image:TFA LCT lens.jpg|thumb|right|250px]]
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| With the lens as depicted in the figure, and the refractive index denoted as ''n'', the result is:<ref>{{citation | title=Introduction to Fourier optics | first=Joseph W. | last=Goodman | edition=3rd | publisher=Roberts and Company Publishers | year=2005 | isbn=0-9747077-2-4 }}, §5.1.3, pp. 100–102.</ref>
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| :<math> U_0(x,y) = e^{jkn \Delta} e^{-j \frac{k}{2f} [x^2 + y ^2]} U_i(x,y) </math>
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| with ''f'' the focal length and ''Δ'' the thickness of the lens.
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| The distortion passing through the lens is similar to LCT, when
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| :<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda f} & 1
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| \end{bmatrix}.
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| </math>
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| This is also a shearing effect: when the focal length is smaller, the shearing effect is larger.
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| ===Spherical Mirror===
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| [[Image:TFA LCT disk.jpg|thumb|right|250px]]
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| The spherical mirror—e.g., a satellite dish—can be described as a LCT, with
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| :<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| = | |
| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda R} & 1
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| \end{bmatrix}.
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| </math>
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| This is very similar to lens, except focal length is replaced by the radius of the dish. Therefore, if the radius is smaller, the shearing effect is larger.
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| ==Example== | |
| [[Image:TFA LCT dish2.jpg|thumb|right|250px]]
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| The system considered is depicted in the figure to the right: two dishes – one being the emitter and the other one the receiver – and a signal travelling between them over a distance ''D''.
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| First, for dish A (emitter), the LCT matrix looks like this:
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| :<math>
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| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda R_A} & 1
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| \end{bmatrix}.
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| </math>
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| Then, for dish B (receiver), the LCT matrix similarly becomes:
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| :<math>
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| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda R_B} & 1
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| \end{bmatrix}.
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| </math>
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| Last, for the propagation of the signal in air, the LCT matrix is:
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| :<math>
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| \begin{bmatrix}
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| 1 & \lambda D \\
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| 0 & 1
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| \end{bmatrix}.
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| </math>
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| Putting all three components together, the LCT of the system is:
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| :<math>
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| \begin{bmatrix}
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| a & b \\
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| c & d
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda R_B} & 1
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & \lambda D \\
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| 0 & 1
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & 0 \\
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| \frac{-1}{\lambda R_A} & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1-\frac{D}{R_A} & - \lambda D \\
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| \frac{1}{\lambda} (R_A^{-1} + R_B^{-1} - R_A^{-1}R_B^{-1}D) & 1 - \frac{D}{R_B}
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| \end{bmatrix}
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| \, .
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| </math>
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| ==See also==
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| * [[Segal–Shale–Weil distribution]], a metaplectic group of operators related to the chirplet transform
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| ; Other time–frequency transforms:
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| * [[Fractional Fourier transform]]
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| * [[Continuous Fourier transform]]
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| * [[Chirplet transform]]
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| ; Applications:
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| * [[Focus recovery based on the linear canonical transform]]
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| * [[Ray transfer matrix analysis]]
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| ==Notes==
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| <references />
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| ==References==
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| * J.J. Ding, "''Time–frequency analysis and wavelet transform course note''", the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
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| * K.B. Wolf, "''[http://www.fis.unam.mx/~bwolf/ Integral Transforms in Science and Engineering]''", Ch. 9&10, New York, Plenum Press, 1979.
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| * S.A. Collins, "Lens-system diffraction integral written in terms of matrix optics," ''J. Opt. Soc. Amer.'' '''60''', 1168–1177 (1970).
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| * M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," ''J. Math. Phys.'' '''12''', 8, 1772–1783, (1971).
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| * B.M. Hennelly and J.T. Sheridan, "Fast Numerical Algorithm for the Linear Canonical Transform", ''J. Opt. Soc. Am. A'' '''22''', 5, 928–937 (2005).
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| * H.M. Ozaktas, A. Koç, I. Sari, and M.A. Kutay, "Efficient computation of quadratic-phase integrals in optics", ''Opt. Let.'' '''31''', 35–37, (2006).
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| * Bing-Zhao Li, Ran Tao, Yue Wang, "New sampling formulae related to the linear canonical transform", ''Signal Processing'' ''''87'''', 983–990, (2007).
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| * A. Koç, H.M. Ozaktas, C. Candan, and M.A. Kutay, "Digital computation of linear canonical transforms", ''IEEE Trans. Signal Process.'', vol. 56, no. 6, 2383–2394, (2008).
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| * Ran Tao, Bing-Zhao Li, Yue Wang, "On sampling of bandlimited signals associated with the linear canonical transform", ''IEEE Transactions on Signal Processing'', vol. 56, no. 11, 5454–5464, (2008).
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| [[Category:Time–frequency analysis]]
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| [[Category:Integral transforms]]
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| [[Category:Fourier analysis]]
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| [[Category:Signal processing]]
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| [[Category:Hamiltonian mechanics]]
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