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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the area of
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
, the fractional Fourier transform (FRFT) is a family of
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s generalizing the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
— thus, it can transform a function to any ''intermediate'' domain between time and
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. Its applications range from filter design and signal analysis to phase retrieval and
pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics ...
. The FRFT can be used to define fractional
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
,
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
for phase-space rotations, and also by Namias, generalizing work of Wiener on
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well ...
. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain. A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a
z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
, and in particular for the case that corresponds to a
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
shifted by a fractional amount in frequency space (multiplying the input by a linear
chirp A chirp is a signal in which the frequency increases (''up-chirp'') or decreases (''down-chirp'') with time. In some sources, the term ''chirp'' is used interchangeably with sweep signal. It is commonly applied to sonar, radar, and laser syste ...
) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by
Bluestein's FFT algorithm The chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points along the unit circle, the chirp Z-transform samples along spiral arcs in the Z-plane, correspon ...
.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.


Introduction

The continuous
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
\mathcal of a function f: \mathbb \mapsto \mathbb is a
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
of xv 4 L^2 that maps the function f to its frequential version \hat (all expressions are taken in the L^2 sense, rather than pointwise): \hat(\xi) = \int_^ f(x)\ e^\,\mathrmx and f is determined by \hat via the inverse transform \mathcal^\, , f(x) = \int_^ \hat(\xi)\ e^\,\mathrm\xi\, . Let us study its ''n''-th iterated \mathcal^ defined by \mathcal^ = \mathcal mathcal^[f and \mathcal^ = (\mathcal^)^n when ''n'' is a non-negative integer, and \mathcal^ = f. Their sequence is finite since \mathcal is a 4-periodic automorphism: for every function ƒ, \mathcal^4 = f. More precisely, let us introduce the parity operator \mathcal that inverts x, \mathcal colon x \mapsto f(-x). Then the following properties hold: \mathcal^0 = \mathrm, \qquad \mathcal^1 = \mathcal, \qquad \mathcal^2 = \mathcal, \qquad \mathcal^4 = \mathrm \mathcal^3 = \mathcal^ = \mathcal \circ \mathcal = \mathcal \circ \mathcal. The FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers n = 2\alpha/\pi of the FT.


Definition

Note: some authors write the transform in terms of the "order " instead of the "angle ", in which case the is usually times . Although these two forms are equivalent, one must be careful about which definition the author uses. For any real , the -angle fractional Fourier transform of a function ƒ is denoted by \mathcal_\alpha (u) and defined by Formally, this formula is only valid when the input function is in a sufficiently nice space (such as L1 or Schwartz space), and is defined via a density argument, in a way similar to that of the ordinary
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
(see article), in the general case. If is an integer multiple of π, then the
cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
and
cosecant In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
functions above diverge. However, this can be handled by taking the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, and leads to a
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
in the integrand. More directly, since \mathcal^2(f)=f(-t)~, ~~\mathcal_ ~ (f) must be simply or for an even or odd multiple of respectively. For , this becomes precisely the definition of the continuous Fourier transform, and for it is the definition of the inverse continuous Fourier transform. The FRFT argument is neither a spatial one nor a frequency . We will see why it can be interpreted as linear combination of both coordinates . When we want to distinguish the -angular fractional domain, we will let x_a denote the argument of \mathcal_\alpha. Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is the Mehler kernel, \mathcal_\alpha(f)(\omega) = \sqrt e^ \int_^\infty e^ f(t)\, dt~.


Properties

The -th order fractional Fourier transform operator, \mathcal_\alpha, has the properties:


Additivity

For any real angles , \mathcal_ = \mathcal_\alpha \circ \mathcal_\beta = \mathcal_\beta \circ \mathcal_\alpha.


Linearity

\mathcal_\alpha \left sum\nolimits_k b_kf_k(u) \right \sum\nolimits_k b_k\mathcal_\alpha \left _k(u) \right /math>


Integer Orders

If is an integer multiple of \pi / 2, then: \mathcal_\alpha = \mathcal_ = \mathcal^k = (\mathcal)^k Moreover, it has following relation \begin \mathcal^2 &= \mathcal && \mathcal (u)f(-u)\\ \mathcal^3 &= \mathcal^ = (\mathcal)^ \\ \mathcal^4 &= \mathcal^0 = \mathcal \\ \mathcal^i &= \mathcal^j && i \equiv j \mod 4 \end


Inverse

(\mathcal_\alpha)^=\mathcal_


Commutativity

\mathcal_\mathcal_=\mathcal_\mathcal_


Associativity

\left (\mathcal_\mathcal_ \right )\mathcal_ = \mathcal_ \left (\mathcal_\mathcal_ \right )


Unitarity

\int f^*(u)g(u)du=\int f_\alpha^*(u)g_\alpha(u)du


Time Reversal

\mathcal_\alpha\mathcal=\mathcal\mathcal_\alpha \mathcal_\alpha (-u)f_\alpha(-u)


Transform of a shifted function

Define the shift and the phase shift operators as follows: \begin \mathcal(u_0) (u)&= f(u+u_0) \\ \mathcal(v_0) (u)&= e^f(u) \end Then \begin \mathcal_\alpha \mathcal(u_0) &= e^ \mathcal(u_0\sin\alpha) \mathcal(u_0\cos\alpha) \mathcal_\alpha, \end that is, \begin \mathcal_\alpha (u+u_0)&=e^ e^ f_\alpha (u+u_0 \cos\alpha) \end


Transform of a scaled function

Define the scaling and chirp multiplication operators as follows: \begin M(M) (u)&= , M, ^ f \left (\tfrac \right) \\ Q(q) (u)&= e^ f(u) \end Then, \begin \mathcal_\alpha M(M) &= Q \left (-\cot \left (\frac\alpha \right ) \right)\times M \left (\frac \right )\mathcal_ \\ pt\mathcal_\alpha \left integral_transform \mathcal_\alpha_f_(u)_=_\int_K_\alpha_(u,_x)_f(x)\,_\mathrmx where_the_α-angle_kernel_is K_\alpha_(u,_x)_=_\begin\sqrt_\exp_\left(i_\pi_(\cot(\alpha)(x^2+_u^2)_-2_\csc(\alpha)_u_x)_\right)_&_\mbox_\alpha_\mbox\pi,_\\ \delta_(u_-_x)_&_\mbox_\alpha_\mbox_2\pi,_\\ \delta_(u_+_x)_&_\mbox_\alpha+\pi_\mbox_2\pi,_\\ \end Here_again_the_special_cases_are_consistent_with_the_limit_behavior_when____approaches_a_multiple_of_. The_FRFT_has_the_same_properties_as_its_kernels_: *_symmetry:_K_\alpha~(u,_u')=K_\alpha_~(u',_u) *_inverse:_K_\alpha^_(u,_u')_=_K_\alpha^*_(u,_u')_=_K__(u',_u)_ *_additivity:_K__(u,u')_=_\int_K_\alpha_(u,_u'')_K_\beta_(u'',_u')\,\mathrmu''.


_Related_transforms

There_also_exist_related_fractional_generalizations_of_similar_transforms_such_as_the_discrete_Fourier_transform_ In_mathematics,_the_discrete_Fourier_transform_(DFT)_converts_a_finite_sequence_of_equally-spaced__samples_of_a__function_into_a_same-length_sequence_of_equally-spaced_samples_of_the__discrete-time_Fourier_transform_(DTFT),_which_is_a_comple_...
. *_The_discrete_fractional_Fourier_transform_is_defined_by_ Zeev_Zalevsky._A_quantum_algorithm_to_implement_a_version_of_the_discrete_fractional_Fourier_transform_in_sub-polynomial_time_is_described_by_Somma. *_The_
Fractional_wavelet_transform Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform (FRFT). The FRWT inherits the advantages o ...
_(FRWT)_is_a_generalization_of_the_classical_
wavelet_transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
_in_the_fractional_Fourier_transform_domains. *_The_ chirplet_transform_for_a_related_generalization_of_the_
wavelet_transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
.


_Generalizations

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._These_have_been_generalized_into_a_ supersymmetric_FRFT,_and_a_supersymmetric_
Radon_transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...
.__There_is_also_a_fractional_Radon_transform,_a_ symplectic_FRFT,_and_a_symplectic_
wavelet_transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
.__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 In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
.__A_quantum_circuit_has_been_designed_which_implements_the_FRFT.


_Interpretation

The_usual_interpretation_of_the_Fourier_transform_is_as_a_transformation_of_a_time_domain_signal_into_a_frequency_domain_signal._On_the_other_hand,_the_interpretation_of_the_inverse_Fourier_transform_is_as_a_transformation_of_a_frequency_domain_signal_into_a_time_domain_signal._Fractional_Fourier_transforms__transform_a_signal_(either_in_the_time_domain_or_frequency_domain)_into_the_domain_between_time_and_frequency:_it_is_a_rotation_in_the_ time–frequency_domain._This_perspective_is_generalized_by_the__linear_canonical_transformation,_which_generalizes_the_fractional_Fourier_transform_and_allows_linear_transforms_of_the_time–frequency_domain_other_than_rotation. Take_the__figure_below_as_an_example._If_the_signal_in_the_time_domain_is_rectangular_(as_below),_it___becomes_a_
sinc_function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
_in_the_frequency_domain._But_if__one_applies_the_fractional_Fourier_transform_to_the_rectangular_signal,_the_transformation_output_will_be_in_the_domain_between_time_and_frequency. The_fractional_Fourier_transform_is_a_rotation_operation_on_a_ time–frequency_distribution._From_the_definition_above,_for_''α'' = 0,_there_will_be_no_change_after_applying_the_fractional_Fourier_transform,_while_for_''α'' = ''π''/2,_the_fractional_Fourier_transform_becomes_a_plain_Fourier_transform,_which_rotates_the_time–frequency_distribution_with ''π''/2._For_other_value_of ''α'',_the_fractional_Fourier_transform_rotates_the_time–frequency_distribution_according_to_α._The_following_figure_shows_the_results_of_the_fractional_Fourier_transform_with_different_values_of ''α''.


_Application

Fractional_Fourier_transform_can_be_used_in_time_frequency_analysis_and_ DSP._It_is_useful_to_filter_noise,_but_with_the_condition_that_it_does_not_overlap_with_the_desired_signal_in_the_time–frequency_domain._Consider_the_following_example._We_cannot_apply_a_filter_directly_to_eliminate_the_noise,_but_with_the_help_of_the_fractional_Fourier_transform,_we_can_rotate_the_signal_(including_the_desired_signal_and_noise)_first._We_then_apply_a_specific_filter,_which_will_allow_only_the_desired_signal_to_pass._Thus_the_noise_will_be_removed_completely._Then_we_use_the_fractional_Fourier_transform_again_to_rotate_the_signal_back_and_we_can_get_the_desired_signal. Thus,_using_just_truncation_in_the_time_domain,_or_equivalently_
low-pass_filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
s_in_the_frequency_domain,_one_can_cut_out_any_
convex_set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
_in_time–frequency_space._In_contrast,_using_time_domain_or_frequency_domain_tools_without_a_fractional_Fourier_transform_would_only_allow_cutting_out_rectangles_parallel_to_the_axes. Fractional_Fourier_transforms_also_have_applications_in_quantum_physics._For_example,_they_are_used_to_formulate_entropic_uncertainty_relations. They_are_also_useful_in_the_design_of_optical_systems_and_for_optimizing_holographic_storage_efficiency.


_See_also

*_ Least-squares_spectral_analysis *_
Fractional_calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
*__Mehler_kernel Other_time–frequency_transforms: *_ Linear_canonical_transformation *_ Short-time_Fourier_transform *_
Wavelet_transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
*_ Chirplet_transform *_
Cone-shape_distribution_function The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994) (acronymized as the ZAM distribution or ZAMD), is one of t ...
*_ Quadratic_Fourier_transform


_References


_Bibliography

*_ *_ *_ *_ *_ *_


_External_links


DiscreteTFDs_--_software_for_computing_the_fractional_Fourier_transform_and_time–frequency_distributions

Fractional_Fourier_Transform
_by_Enrique_Zeleny,_ The_Wolfram_Demonstrations_Project.
Dr_YangQuan_Chen's_FRFT_(Fractional_Fourier_Transform)_Webpages

LTFAT_-_A_free_(GPL)_Matlab_/_Octave_toolbox
Contains_several_version_of_th
fractional_Fourier_transform
{{DEFAULTSORT:Fractional_Fourier_Transform Fourier_analysis Time–frequency_analysis Integral_transforms Articles_containing_video_clipshtml" ;"title="M, ^ f \left (\tfrac \right) \right ] &= \sqrt e^ \times f_a \left (\frac \right ) \end Notice that the fractional Fourier transform of f(u/M) cannot be expressed as a scaled version of f_\alpha (u). Rather, the fractional Fourier transform of f(u/M) turns out to be a scaled and chirp modulated version of f_(u) where \alpha\neq\alpha' is a different order.


Fractional kernel

The FRFT is an integral transform \mathcal_\alpha f (u) = \int K_\alpha (u, x) f(x)\, \mathrmx where the α-angle kernel is K_\alpha (u, x) = \begin\sqrt \exp \left(i \pi (\cot(\alpha)(x^2+ u^2) -2 \csc(\alpha) u x) \right) & \mbox \alpha \mbox\pi, \\ \delta (u - x) & \mbox \alpha \mbox 2\pi, \\ \delta (u + x) & \mbox \alpha+\pi \mbox 2\pi, \\ \end Here again the special cases are consistent with the limit behavior when approaches a multiple of . The FRFT has the same properties as its kernels : * symmetry: K_\alpha~(u, u')=K_\alpha ~(u', u) * inverse: K_\alpha^ (u, u') = K_\alpha^* (u, u') = K_ (u', u) * additivity: K_ (u,u') = \int K_\alpha (u, u'') K_\beta (u'', u')\,\mathrmu''.


Related transforms

There also exist related fractional generalizations of similar transforms such as the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. * The discrete fractional Fourier transform is defined by Zeev Zalevsky. A quantum algorithm to implement a version of the discrete fractional Fourier transform in sub-polynomial time is described by Somma. * The
Fractional wavelet transform Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform (FRFT). The FRWT inherits the advantages o ...
(FRWT) is a generalization of the classical
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
in the fractional Fourier transform domains. * The chirplet transform for a related generalization of the
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
.


Generalizations

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. These have been generalized into a supersymmetric FRFT, and a supersymmetric
Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...
. There is also a fractional Radon transform, a symplectic FRFT, and a symplectic
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
. 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 In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
. A quantum circuit has been designed which implements the FRFT.


Interpretation

The usual interpretation of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. On the other hand, the interpretation of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal. Fractional Fourier transforms transform a signal (either in the time domain or frequency domain) into the domain between time and frequency: it is a rotation in the time–frequency domain. This perspective is generalized by the linear canonical transformation, which generalizes the fractional Fourier transform and allows linear transforms of the time–frequency domain other than rotation. Take the figure below as an example. If the signal in the time domain is rectangular (as below), it becomes a
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
in the frequency domain. But if one applies the fractional Fourier transform to the rectangular signal, the transformation output will be in the domain between time and frequency. The fractional Fourier transform is a rotation operation on a time–frequency distribution. From the definition above, for ''α'' = 0, there will be no change after applying the fractional Fourier transform, while for ''α'' = ''π''/2, the fractional Fourier transform becomes a plain Fourier transform, which rotates the time–frequency distribution with ''π''/2. For other value of ''α'', the fractional Fourier transform rotates the time–frequency distribution according to α. The following figure shows the results of the fractional Fourier transform with different values of ''α''.


Application

Fractional Fourier transform can be used in time frequency analysis and DSP. It is useful to filter noise, but with the condition that it does not overlap with the desired signal in the time–frequency domain. Consider the following example. We cannot apply a filter directly to eliminate the noise, but with the help of the fractional Fourier transform, we can rotate the signal (including the desired signal and noise) first. We then apply a specific filter, which will allow only the desired signal to pass. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal. Thus, using just truncation in the time domain, or equivalently
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
s in the frequency domain, one can cut out any
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
in time–frequency space. In contrast, using time domain or frequency domain tools without a fractional Fourier transform would only allow cutting out rectangles parallel to the axes. Fractional Fourier transforms also have applications in quantum physics. For example, they are used to formulate entropic uncertainty relations. They are also useful in the design of optical systems and for optimizing holographic storage efficiency.


See also

* Least-squares spectral analysis *
Fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
* Mehler kernel Other time–frequency transforms: * Linear canonical transformation * Short-time Fourier transform *
Wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
* Chirplet transform *
Cone-shape distribution function The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994) (acronymized as the ZAM distribution or ZAMD), is one of t ...
* Quadratic Fourier transform


References


Bibliography

* * * * * *


External links


DiscreteTFDs -- software for computing the fractional Fourier transform and time–frequency distributions

Fractional Fourier Transform
by Enrique Zeleny, The Wolfram Demonstrations Project.
Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages

LTFAT - A free (GPL) Matlab / Octave toolbox
Contains several version of th
fractional Fourier transform
{{DEFAULTSORT:Fractional Fourier Transform Fourier analysis Time–frequency analysis Integral transforms Articles containing video clips>M, ^ f \left (\tfrac \right) \right &= \sqrt e^ \times f_a \left (\frac \right ) \end Notice that the fractional Fourier transform of f(u/M) cannot be expressed as a scaled version of f_\alpha (u). Rather, the fractional Fourier transform of f(u/M) turns out to be a scaled and chirp modulated version of f_(u) where \alpha\neq\alpha' is a different order.


Fractional kernel

The FRFT is an integral transform \mathcal_\alpha f (u) = \int K_\alpha (u, x) f(x)\, \mathrmx where the α-angle kernel is K_\alpha (u, x) = \begin\sqrt \exp \left(i \pi (\cot(\alpha)(x^2+ u^2) -2 \csc(\alpha) u x) \right) & \mbox \alpha \mbox\pi, \\ \delta (u - x) & \mbox \alpha \mbox 2\pi, \\ \delta (u + x) & \mbox \alpha+\pi \mbox 2\pi, \\ \end Here again the special cases are consistent with the limit behavior when approaches a multiple of . The FRFT has the same properties as its kernels : * symmetry: K_\alpha~(u, u')=K_\alpha ~(u', u) * inverse: K_\alpha^ (u, u') = K_\alpha^* (u, u') = K_ (u', u) * additivity: K_ (u,u') = \int K_\alpha (u, u'') K_\beta (u'', u')\,\mathrmu''.


Related transforms

There also exist related fractional generalizations of similar transforms such as the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. * The discrete fractional Fourier transform is defined by Zeev Zalevsky. A quantum algorithm to implement a version of the discrete fractional Fourier transform in sub-polynomial time is described by Somma. * The
Fractional wavelet transform Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform (FRFT). The FRWT inherits the advantages o ...
(FRWT) is a generalization of the classical
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
in the fractional Fourier transform domains. * The chirplet transform for a related generalization of the
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
.


Generalizations

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. These have been generalized into a supersymmetric FRFT, and a supersymmetric
Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...
. There is also a fractional Radon transform, a symplectic FRFT, and a symplectic
wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
. 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 In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
. A quantum circuit has been designed which implements the FRFT.


Interpretation

The usual interpretation of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. On the other hand, the interpretation of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal. Fractional Fourier transforms transform a signal (either in the time domain or frequency domain) into the domain between time and frequency: it is a rotation in the time–frequency domain. This perspective is generalized by the linear canonical transformation, which generalizes the fractional Fourier transform and allows linear transforms of the time–frequency domain other than rotation. Take the figure below as an example. If the signal in the time domain is rectangular (as below), it becomes a
sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
in the frequency domain. But if one applies the fractional Fourier transform to the rectangular signal, the transformation output will be in the domain between time and frequency. The fractional Fourier transform is a rotation operation on a time–frequency distribution. From the definition above, for ''α'' = 0, there will be no change after applying the fractional Fourier transform, while for ''α'' = ''π''/2, the fractional Fourier transform becomes a plain Fourier transform, which rotates the time–frequency distribution with ''π''/2. For other value of ''α'', the fractional Fourier transform rotates the time–frequency distribution according to α. The following figure shows the results of the fractional Fourier transform with different values of ''α''.


Application

Fractional Fourier transform can be used in time frequency analysis and DSP. It is useful to filter noise, but with the condition that it does not overlap with the desired signal in the time–frequency domain. Consider the following example. We cannot apply a filter directly to eliminate the noise, but with the help of the fractional Fourier transform, we can rotate the signal (including the desired signal and noise) first. We then apply a specific filter, which will allow only the desired signal to pass. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal. Thus, using just truncation in the time domain, or equivalently
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
s in the frequency domain, one can cut out any
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
in time–frequency space. In contrast, using time domain or frequency domain tools without a fractional Fourier transform would only allow cutting out rectangles parallel to the axes. Fractional Fourier transforms also have applications in quantum physics. For example, they are used to formulate entropic uncertainty relations. They are also useful in the design of optical systems and for optimizing holographic storage efficiency.


See also

* Least-squares spectral analysis *
Fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
* Mehler kernel Other time–frequency transforms: * Linear canonical transformation * Short-time Fourier transform *
Wavelet transform In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
* Chirplet transform *
Cone-shape distribution function The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994) (acronymized as the ZAM distribution or ZAMD), is one of t ...
* Quadratic Fourier transform


References


Bibliography

* * * * * *


External links


DiscreteTFDs -- software for computing the fractional Fourier transform and time–frequency distributions

Fractional Fourier Transform
by Enrique Zeleny, The Wolfram Demonstrations Project.
Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages

LTFAT - A free (GPL) Matlab / Octave toolbox
Contains several version of th
fractional Fourier transform
{{DEFAULTSORT:Fractional Fourier Transform Fourier analysis Time–frequency analysis Integral transforms Articles containing video clips