Fractional Fourier transform

The α-th order fractional Fourier transform operator, ${\displaystyle {\mathcal {F}}_{\alpha }}$, has the properties:

• Additivity. For any real angles α, β,

${\displaystyle {\mathcal {F}}_{\alpha +\beta }={\mathcal {F}}_{\alpha }\circ {\mathcal {F}}_{\beta }={\mathcal {F}}_{\beta }\circ {\mathcal {F}}_{\alpha }.}$

• Linearity.

${\displaystyle {\mathcal {F}}_{\alpha }\left[\sum \nolimits _{k}b_{k}f_{k}(u)\right]=\sum \nolimits _{k}b_{k}{\mathcal {F}}_{\alpha }\left[f_{k}(u)\right]}$

• Integer Orders. If α is an integer multiple of ${\displaystyle \pi /2}$, then:

${\displaystyle {\mathcal {F}}_{\alpha }={\mathcal {F}}_{\frac {k\pi }{2}}={\mathcal {F}}^{k}=({\mathcal {F}})^{k}}$

Moreover, it has following relation

${\displaystyle {\begin{aligned}{\mathcal {F}}^{2}&={\mathcal {P}}&&{\mathcal {P}}[f(u)]=f(-u)\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}=({\mathcal {F}})^{-1}\\{\mathcal {F}}^{4}&={\mathcal {F}}^{0}={\mathcal {I}}\\{\mathcal {F}}^{i}&={\mathcal {F}}^{j}&&i\equiv j\mod 4\end{aligned}}}$

• Inverse.

${\displaystyle ({\mathcal {F}}_{\alpha })^{-1}={\mathcal {F}}_{-\alpha }}$

• Commutativity.

${\displaystyle {\mathcal {F}}_{\alpha _{1}}{\mathcal {F}}_{\alpha _{2}}={\mathcal {F}}_{\alpha _{2}}{\mathcal {F}}_{\alpha _{1}}}$

• Associativity

${\displaystyle \left({\mathcal {F}}_{\alpha _{1}}{\mathcal {F}}_{\alpha _{2}}\right){\mathcal {F}}_{\alpha _{3}}={\mathcal {F}}_{\alpha _{1}}\left({\mathcal {F}}_{\alpha _{2}}{\mathcal {F}}_{\alpha _{3}}\right)}$

• Unitarity

${\displaystyle \int f^{*}(u)g(u)du=\int f_{\alpha }^{*}(u)g_{\alpha }(u)du}$

• Time Reversal.

${\displaystyle {\mathcal {F}}_{\alpha }{\mathcal {P}}={\mathcal {P}}{\mathcal {F}}_{\alpha }}$

${\displaystyle {\mathcal {F}}_{\alpha }[f(-u)]=f_{\alpha }(-u)}$

• Transform of a shifted function

Define the shift and the phase shift operators as follows:

${\displaystyle {\mathcal {SH}}(u_{0})[f(u)]=f(u+u_{0})}$

${\displaystyle {\mathcal {PH}}(v_{0})[f(u)]=e^{j2\pi v_{0}u}f(u)}$

Then

${\displaystyle {\begin{aligned}{\mathcal {F}}_{\alpha }{\mathcal {SH}}(u_{0})&=e^{j\pi u_{0}^{2}\sin \alpha \cos \alpha }{\mathcal {PH}}(u_{0}\sin \alpha ){\mathcal {SH}}(u_{0}\cos \alpha )F_{\alpha }\\{\mathcal {F}}_{\alpha }[f(u+u_{0})]&=e^{j\pi u_{0}^{2}\sin \alpha \cos \alpha }e^{j2\pi uu_{0}\sin \alpha }f_{\alpha }(u+u_{0}\cos \alpha )\end{aligned}}}$

• Transform of a scaled function

Define the scaling and chirp multiplication operators as follows:

${\displaystyle M(M)[f(u)]=|M|^{-{\frac {1}{2}}}f\left({\tfrac {u}{M}}\right)}$

${\displaystyle Q(q)[f(u)]=e^{-j\pi qu^{2}}f(u)}$

Then,

${\displaystyle {\begin{aligned}{\mathcal {F}}_{\alpha }M(M)&=Q\left(-\cot \left({\frac {1-\cos ^{2}\alpha '}{\cos ^{2}\alpha }}\alpha \right)\right)\times M\left({\frac {\sin \alpha }{M\sin \alpha '}}\right){\mathcal {F}}_{\alpha '}\\[6pt]{\mathcal {F}}_{\alpha }\left[|M|^{-{\frac {1}{2}}}f\left({\tfrac {u}{M}}\right)\right]&={\sqrt {\frac {1-j\cot \alpha }{1-jM^{2}\cot \alpha }}}e^{j\pi u^{2}\cot \left({\frac {1-\cos ^{2}\alpha '}{\cos ^{2}\alpha }}\alpha \right)}\times f_{a}\left({\frac {Mu\sin \alpha '}{\sin \alpha }}\right)\end{aligned}}}$

Notice that the fractional Fourier transform of ${\displaystyle f(u/M)}$ cannot be expressed as a scaled version of ${\displaystyle f_{\alpha }(u)}$. Rather, the fractional Fourier transform of ${\displaystyle f(u/M)}$ turns out to be a scaled and chirp modulated version of ${\displaystyle f_{\alpha '}(u)}$ where ${\displaystyle \alpha \neq \alpha '}$ is a different order.

The FrFT is an integral transform

${\displaystyle {\mathcal {F}}_{\alpha }f(u)=\int K_{\alpha }(u,x)f(x)\,\mathrm {d} x}$

where the α-angle kernel is

${\displaystyle K_{\alpha }(u,x)={\begin{cases}{\sqrt {1-i\cot(\alpha )}}\exp \left(i\pi (\cot(\alpha )(x^{2}+u^{2})-2\csc(\alpha )ux)\right)&{\mbox{if }}\alpha {\mbox{ is not a multiple of }}\pi ,\\\delta (u-x)&{\mbox{if }}\alpha {\mbox{ is a multiple of }}2\pi ,\\\delta (u+x)&{\mbox{if }}\alpha +\pi {\mbox{ is a multiple of }}2\pi ,\\\end{cases}}}$

Here again the special cases are consistent with the limit behavior when α approaches a multiple of π.

The FrFT has the same properties as its kernels :

• symmetry: ${\displaystyle K_{\alpha }~(u,u')=K_{\alpha }~(u',u)}$
• inverse: ${\displaystyle K_{\alpha }^{-1}(u,u')=K_{\alpha }^{*}(u,u')=K_{-\alpha }(u',u)}$
• additivity: ${\displaystyle K_{\alpha +\beta }(u,u')=\int K_{\alpha }(u,u'')K_{\beta }(u'',u')\,\mathrm {d} u''.}$

There also exist related fractional generalizations of similar transforms such as the discrete Fourier transform. The discrete fractional Fourier transform is defined by Zeev Zalevsky in (Candan, Kutay & Ozaktas 2000) and (Ozaktas, Zalevsky & Kutay 2001, Chapter 6). A quantum algorithm to implement a version of the discrete fractional Fourier transform in subpolynomial time is described by Somma.[9]

Fractional wavelet transform (FRWT):[10] A generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FRFT) domains. The FRWT is proposed in order to rectify the limitations of the WT and the FRFT. This transform not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to the FRFT. Compared with the existing FRWT, the FRWT (defined by Shi, Zhang, and Liu 2012) can offer signal representations in the time-fractional-frequency plane.

See also the chirplet transform for a related generalization of the Fourier transform.

The Fourier transform is essentially bosonic; it works because it is consistent with the superposition principle and related interference patterns. There is also a fermionic Fourier transform.[11] These have been generalized into a supersymmetric FRFT, and a supersymmetric Radon transform.[11] There is also a fractional Radon transform, a symplectic FRFT, and a symplectic wavelet transform.[12] Because quantum circuits are based on unitary operations, they are useful for computing integral transforms as the latter are unitary operators on a function space. A quantum circuit has been designed which implements the FRFT.[13]