Introduction
The continuousDefinition
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 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 ordinaryProperties
The -th order fractional Fourier transform operator, , has the properties:Additivity
For any real angles ,Linearity
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:__Related_transforms
There_also_exist_related_fractional_generalizations_of_similar_transforms_such_as_the__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._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___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__See_also
*_ Least-squares_spectral_analysis *__References
_Bibliography
*_ *_ *_ *_ *_ *__External_links
Fractional kernel
The FRFT is an integral transformRelated transforms
There also exist related fractional generalizations of similar transforms such as theGeneralizations
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 supersymmetricInterpretation
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 aApplication
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 equivalentlySee also
* Least-squares spectral analysis *References
Bibliography
* * * * * *External links
Fractional kernel
The FRFT is an integral transformRelated transforms
There also exist related fractional generalizations of similar transforms such as theGeneralizations
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 supersymmetricInterpretation
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 aApplication
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 equivalentlySee also
* Least-squares spectral analysis *References
Bibliography
* * * * * *External links