Editing Fractional Fourier transform (section)

===Properties===
The {{math|''α''}}-th order fractional Fourier transform operator, <math>\mathcal{F}_\alpha</math>, has the properties:

====Additivity====
For any real angles {{math|''α, β''}},
<math display=block>\mathcal{F}_{\alpha+\beta} = \mathcal{F}_\alpha \circ \mathcal{F}_\beta = \mathcal{F}_\beta \circ \mathcal{F}_\alpha.</math>

====Linearity====
<math display=block>\mathcal{F}_\alpha \left [\sum\nolimits_k b_kf_k(u)  \right ]=\sum\nolimits_k b_k\mathcal{F}_\alpha \left [f_k(u) \right ]</math>

====Integer Orders====
If {{math|''α''}} is an integer multiple of <math>\pi / 2</math>, then:
<math display=block>\mathcal{F}_\alpha = \mathcal{F}_{k\pi/2} = \mathcal{F}^k = (\mathcal{F})^k</math>

Moreover, it has following relation

<math display=block>\begin{align}
\mathcal{F}^2 &= \mathcal{P} && \mathcal{P}[f(u)]=f(-u)\\
\mathcal{F}^3 &= \mathcal{F}^{-1} = (\mathcal{F})^{-1} \\ 
\mathcal{F}^4 &= \mathcal{F}^0 = \mathcal{I} \\
\mathcal{F}^i &= \mathcal{F}^j && i \equiv j \mod 4
\end{align}</math>

====Inverse====
<math display=block>(\mathcal{F}_\alpha)^{-1}=\mathcal{F}_{-\alpha}</math>

====Commutativity====
<math display=block>\mathcal{F}_{\alpha_1}\mathcal{F}_{\alpha_2}=\mathcal{F}_{\alpha_2}\mathcal{F}_{\alpha_1}</math>

====Associativity====
<math display=block> \left (\mathcal{F}_{\alpha_1}\mathcal{F}_{\alpha_2} \right )\mathcal{F}_{\alpha_3} = \mathcal{F}_{\alpha_1} \left (\mathcal{F}_{\alpha_2}\mathcal{F}_{\alpha_3} \right )</math>

====Unitarity====
<math display=block>\int f(t)g^*(t)dt=\int f_\alpha(u)g_\alpha^*(u)du</math>

====Time Reversal====
<math display=block>\mathcal{F}_\alpha\mathcal{P}=\mathcal{P}\mathcal{F}_\alpha</math>
<math display=block>\mathcal{F}_\alpha[f(-u)]=f_\alpha(-u)</math>

====Transform of a shifted function====
{{see also|Generalizations of Pauli matrices#Construction: The clock and shift matrices}}

Define the shift and the phase shift operators as follows:

<math display=block>\begin{align}
\mathcal{SH}(u_0)[f(u)] &= f(u+u_0) \\
\mathcal{PH}(v_0)[f(u)] &= e^{j2\pi v_0u}f(u)
\end{align}</math>

Then
<math display=block>\begin{align}
\mathcal{F}_\alpha \mathcal{SH}(u_0) &= e^{j\pi u_0^2 \sin\alpha \cos\alpha} \mathcal{PH}(u_0\sin\alpha) \mathcal{SH}(u_0\cos\alpha) \mathcal{F}_\alpha,
\end{align}</math>

that is,

<math display=block>\begin{align}
\mathcal{F}_\alpha [f(u+u_0)] &=e^{j\pi u_0^2 \sin\alpha \cos\alpha}  e^{j2\pi uu_0 \sin\alpha} f_\alpha (u+u_0 \cos\alpha)
\end{align}</math>

====Transform of a scaled function====
Define the scaling and chirp multiplication operators as follows:
<math display=block>\begin{align}
M(M)[f(u)] &= |M|^{-\frac{1}{2}} f \left (\tfrac{u}{M} \right) \\
Q(q)[f(u)] &= e^{-j\pi qu^2 } f(u)
\end{align}</math>

Then, 
<math display=block>\begin{align}
\mathcal{F}_\alpha  M(M) &= Q \left (-\cot \left (\frac{1-\cos^2 \alpha'}{\cos^2 \alpha}\alpha \right ) \right)\times M \left (\frac{\sin \alpha}{M\sin \alpha'} \right )\mathcal{F}_{\alpha'} \\ [6pt]
\mathcal{F}_\alpha \left [|M|^{-\frac{1}{2}} f \left (\tfrac{u}{M} \right)  \right ] &= \sqrt{\frac{1-j \cot\alpha}{1-jM^2  \cot\alpha}} e^{j\pi u^2\cot \left (\frac{1-\cos^2 \alpha'}{\cos^2 \alpha}\alpha \right )} \times f_a \left (\frac{Mu \sin\alpha'}{\sin\alpha} \right )
\end{align}</math>

Notice that the fractional Fourier transform of <math>f(u/M)</math> cannot be expressed as a scaled version of <math>f_\alpha (u)</math>. Rather, the fractional Fourier transform of <math>f(u/M)</math> turns out to be a scaled and chirp modulated version of <math>f_{\alpha'}(u)</math> where <math>\alpha\neq\alpha'</math> is a different order.<ref>An elementary recipe, using the contangent function, and its (multi-valued) inverse, for <math>\alpha'</math> in terms of <math>\alpha</math> and <math>M</math> exists.</ref>