Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. 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. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> A new optical architecture is developed, based on fractional Fourier transforms, that compromises between shift-invariant (frequency) and position-dependent filtering. The analogy of this architecture to wavelet transforms and adaptive neural networks is also presented. The ambiguity and Wigner distribution functions are obtainable from special cases of the filter. The filter design corresponds to the training of the neural networks, and an adaptive learning algorithm is developed based on gradient-descent error minimization and error back propagation. The extension to multilayer architecture is straightforward. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> In this paper, we present a discrete rotational Gabor transform and apply it to signal representation on a non-rectangular time-frequency plane tiling. We use a modified version of the recently proposed discrete rotational Fourier transform. The rotational Gabor transform decomposes a signal using as basis functions scaled, translated and modulated by linear chirps windows. As a result, the time-varying frequency content of a signal is represented better than with sinusoidal modulated expansions. For a multi-component signal, the Gabor coefficients are obtained by combining different tilings to maximize a local energy concentration measure. This permits us to achieve a highly localized time-frequency signal representation. Examples are given to illustrate the transformation. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space–FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> A fractional-Fourier-domain realization of the weighted Wigner distribution (or S-method), producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the short-time Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the analysis of the second-order fractional Fourier transform moments. Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle /spl alpha/ in the time-frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> The paper shows that the fractional Fourier transform (FRFT) of a signal is the Radon transform of the time-frequency distribution of the same signal. Therefore, a time-frequency distribution known as the tomography time-frequency transform (TTFT) is defined as the inverse Radon transform of the FRFT of the signal. Because the computation of the TTFT does not explicitly require any window or kernel function, high resolutions in both the frequency and time domains can be achieved. When the signal contains multiple components, the cross terms can be effectively removed by an adaptive filtering process that is applied on the FRFT rather than the final result. Therefore, distortions made by the filtering process on the desired signal components can be minimized. <s> BIB007 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal. <s> BIB008 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> A new signal-adaptive joint time-frequency distribution for the analysis of nonstationary signals is proposed. It is based on a fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms. Improvement over the standard time-frequency representations is achieved when the principal axes of a signal (defined as mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum) do not correspond to time and frequency. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the windowed Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the knowledge of three second-order fractional Fourier transform moments. Numerical simulations confirm a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain. The approach can be generalized to the time-frequency distributions from the Cohen class. <s> BIB009 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> We present a fractional Gabor expansion on a non-rectangular time–frequency lattice. Sinusoidal analysis used in the traditional Gabor expansion is not appropriate for a compact representation for chirp-type signals. Basis functions of the proposed expansion are obtained via fractional Fourier basis. Completeness and biorthogonality conditions of the new expansion are derived. <s> BIB010 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> Signal processing can be found in many applications and its primary goal is to provide underlying information on specific problems for the purpose of decision making. Traditional signal processing approaches assume the stationarity of signals, which in practice is not often satisfied. Hence, time or frequency descriptions alone are insufficient to provide comprehensive information about such signals. On the contrary, time-frequency analysis is more suitable for nonstationary signals. Therefore, this paper provides a status report of feature based signal processing in the time-frequency domain through an overview of recent contributions. The feature considered here is energy concentration. The paper provides an analysis of several classes of feature extractors, i.e., time-frequency representations, and feature classifiers. The results of the literature review indicate that time-frequency domain signal processing using energy concentration as a feature is a very powerful tool and has been utilized in numerous applications. The expectation is that further research and applications of these algorithms will flourish in the near future. <s> BIB011 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Other time-frequency representations and properties <s> Uncertainty principle plays an important role in signal processing, physics and mathematics, and it represents the relations between time spread and frequency spread (or position and velocity). Linear canonical transform (LCT) is one generalisation of Fresnel transform, fractional Fourier transform and others. The LCT has been used in physical optics and signal processing. Three novel results of uncertainty principle in the LCT domains are obtained here, in which one is connected with parameters a and b and the other one is connected with c and d; the last one is connected with the four transformation parameters a, b, c and d. Their physical meanings are given as well. These results disclose the inequalities' relations between two spreads, between two group delays and between one spread and one group delay in the LCT domains. It also shows that any one of the three cases can reduce to classical uncertainty principle in time/frequency domain. The effects of time scaling on these results' bounds are also involved. <s> BIB012
A formal relationship between the wavelet transform and the FRFT has also been established BIB001 BIB002 . Specifically, the FRFT kernels corresponding to different values of a are closely related to the wavelet transform, which can be obtained from the quadratic phase function in BIB011 by scaling the coordinates and the amplitude BIB001 BIB002 . Furthermore, fractional-Fourier-domain realizations of several other time-frequency representations were also introduced. These include realizations of the short-time Fourier transform BIB008 , the weighted WD (i.e., S-method) BIB009 BIB005 , the Gabor expansion BIB003  BIB010 BIB004 , and the tomography time-frequency transform defined as the inverse Radon transform of the FRFT BIB007 . Some of the properties associated with the timefrequency analysis have been extended to the FRFT. For example, marginals associated with time-frequency representations based on the fractional Fourier transform were examined  in analogy to marginals associated with TFRs based on the standard Fourier transform. However, it is not very clear what is a precise meaning of those generalized marginals in comparison to the traditional time and frequency marginals, which have a specific meaning relating those quantities to the signal under analysis. Several publications also investigated the uncertainty principle in the fractional domain BIB006  BIB012  .