Journal ArticleDOI
The fractional Fourier transform and time-frequency representations
Reads0
Chats0
TLDR
The authors briefly introduce the functional Fourier transform and a number of its properties and present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time- frequencies such as the Wigner distribution, the ambiguity function, the short-time Fouriertransform and the spectrogram.Abstract:
The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >read more
Citations
More filters
Journal ArticleDOI
Digital computation of the fractional Fourier transform
TL;DR: An algorithm for efficient and accurate computation of the fractional Fourier transform for signals with time-bandwidth product N, which computes the fractionsal transform in O(NlogN) time.
Journal ArticleDOI
Joint Radar and Communication Design: Applications, State-of-the-Art, and the Road Ahead
TL;DR: A novel scheme for joint target search and communication channel estimation, which relies on omni-directional pilot signals generated by the HAD structure, is proposed, which is possible to recover the target echoes and mitigate the resulting interference to the UE signals, even when the radar and communication signals share the same signal-to-noise ratio (SNR).
Journal ArticleDOI
Time--frequency feature representation using energy concentration: An overview of recent advances
TL;DR: Time-frequency domain signal processing using energy concentration as a feature is a very powerful tool and has been utilized in numerous applications and the expectation is that further research and applications of these algorithms will flourish in the near future.
Journal ArticleDOI
The discrete fractional Fourier transform
TL;DR: This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions, and is exactly unitary, index additive, and reduces to the D FT for unit order.
Proceedings ArticleDOI
The fractional fourier transform
Haldun M. Ozaktas,M. Alper Kutay +1 more
TL;DR: An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
References
More filters
Journal ArticleDOI
On Namias's fractional Fourier transforms
A. C. McBRIDE,F. H. Kerr +1 more
TL;DR: In this paper, it is stated que "for lever toute ambiguite, on montre qu'il est necessaire de modifier les operateurs fractionnaires de Namia On demontre des theoremes pour les operateur modifies and on developpe un calcul operationnel".
Journal ArticleDOI
Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms
TL;DR: Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains.
Journal ArticleDOI
The Fractional Fourier Transform in Optical Propagation Problems
TL;DR: In this article, the application of the fractional Fourier transform to optical propagation problems is discussed, and the conceptual and practical advantages of this new formulation are noted, as well as the theoretical advantages of the new formulation.
Proceedings ArticleDOI
An introduction to the angular Fourier transform
TL;DR: The author introduces the angular Fouriertransform (AFT), a generalization of the classical Fourier transform that can be interpreted as a rotation on the time-frequency plane and has a very simple and natural form.