Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Conclusion <s> We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Conclusion <s> The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It t is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Conclusion <s> The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Conclusion <s> Motor current signature analysis (MCSA) is a well-established method for the diagnosis of induction motor faults. It is based on the analysis of the spectral content of a motor current, which is sampled while a motor runs in steady state, to detect the harmonic components that characterize each type of fault. The Fourier transform (FT) plays a prominent role as a tool for identifying these spectral components. Recently, MCSA has also been applied during the transient regime (TMCSA) using the whole transient speed range to create a unique stamp of each harmonic as it evolves in the time-frequency plane. This method greatly enhances the reliability of the diagnostic process compared with the traditional method, which relies on spectral analysis at a single speed. However, the FT cannot be used in this case because the fault harmonics are not stationary signals. This paper proposes the use of the fractional FT (FrFT) instead of the FT to perform TMCSA. This paper also proposes the optimization of the FrFT to generate a spectrum where the frequency-varying fault harmonics appear as single spectral lines and, therefore, facilitate the diagnostic process. A discrete wavelet transform (DWT) is used as a conditioning tool to filter the motor current prior to its processing by the FrFT. Experimental results that are obtained with a 1.1-kW three-phase squirrel-cage induction motor with broken bars are presented to validate the proposed method. <s> BIB004
The FRFT is a powerful mathematical transform that generalizes the ordinary Fourier transform through the order parameter a. As such, it has been rediscovered in the literature several times. In this paper, we provided an overview of the FRFT from the signal processing point of view. In particular, our goal was to attract signal processing practitioners to this mathematically eloquent concept and depict its advantages. In order to do so, the paper is thematically divided into three topics. In the first part, we presented the definition of the FRFT and tied it to other mathematical representations. The second part of the paper covered various practical realizations of the FRFT, while the practical applications of the FRFT were discussed in the third part of the paper. The conclusions stemming from each part were as follows: Besides its ties to the ordinary Fourier transform, the FRFT can be further related to various time-frequency transforms. In particular, we showed the direct relationship between the AF, the WD and the FRFT. This relationship enables us to relate the FRFT to a wide class of time-frequency transforms. Furthermore, we showed that the FRFT can be considered as a special case of the second-order LPFT and LPP. Also, the FRFT can be considered as the special case of the LCT. Such a relationship provides a direct generalization link between the FRFT and many higher-order/affine mathematical transformations. In particular, we outlined the connection between the FRFT and L-WD for the first time in the literature. Future contributions in this theme will include a further understanding of the FRFT and its ties to other mathematical transforms. The digital realizations of the FRFT can be divided into three major streams. One stream is represented through direct sampling of the FRFT. It is the least complicated approach, but these discrete realizations could lose many important properties of the FRFT. Furthermore, the kernel associated with the FRFT can introduce very fast oscillations that require excessively large sampling rates. A second stream relies on a linear combination of ordinary Fourier operators raised to different powers. Nevertheless, these realizations often produce an output that does not match the output of the continuous FRFT. The third stream approaches the discrete realizations based on the idea of an eigenvalue decomposition. The realizations obtained through this stream tend to closely resemble the representations obtained by the continuous FRFT. However, we should point out that the major drawbacks of this approach are that they cannot be written in a closed form and might have computational costs. Lastly, we also outlined other approaches that appeared in the literature. We specifically suggested that potential realizations of the FRFT could be achieved with the so-called quadratic phase transform. Anticipated contributions in this field need to deal with issues associated with the current schemes. The number of publications discussing practical applications of the FRFT has been steadily rising. In particular, we have witnessed a strong expansion of the FRFT in several fields, including watermarking, radar applications, and wireless communications. Furthermore, we also have the FRFT present in other fields as well (e.g., fractal signal processing, pattern recognition, filtering). The main advantage is that the FRFT-based schemes increase processing accuracy. For example, a group of authors described an adaptive filtering scheme in a recent contribution. The presented results showed a superior performance in comparison to its time domain counterparts. Similar trends have been observed in multimedia signal processing, where the FRFT has been used for watermarking. In particular, the watermark could be detected only if the angle under which the watermark was embedded was known. This enabled us to increase the number of watermarks embedded within multimedia data. From these past contributions, we expect an expansion of FRFT-based methods in different applications to dominate the future contributions. In particular, we expect to see increased application of the FRFT based time-frequency representations in speech and music processing, biomedical signal processing and mechanical vibration analysis. Problems stemming from such applications require the employment of such advanced time-frequency transforms. Overall, the paper provided a compact summary of the recent contributions regarding the FRFT (for convenience, Table 5 contains reference sorted according to their topics). The FRFT is a very powerful tool and has been applied to many fields. Further research and applications of existing schemes will increase in the near future. BIB001  BIB002  BIB003 BIB004 