Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> Direct calculation of fractional Fourier transforms from the expressions derived for their optical implementation is laborious. An extension of the discrete Fourier transform would have only O(N2) computational complexity. We define such a system, offer a general way to compute the fractional discrete Fourier transform matrix, and numerically validate the algorithm. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> A new method for the discrete fractional Fourier transform (DFRFT) computation is given in this paper. With the help of this method, the DFRFT of any angle can be computed by a weighted summation of the DFRFTs with the special angles. <s> BIB007 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> Existing versions of the discrete fractional Fourier transform (DFRFT) are based on the discrete Fourier transform (DFT). These approaches need a full basis of DFT eigenvectors that serve as discrete versions of Hermite-Gauss functions. In this letter, we define a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions. We develop a fast and efficient way to compute the multiangle version of the CDFRFT for a discrete set of angles using the FFT algorithm. We then show that the associated chirp-frequency representation is a useful analysis tool for multicomponent chirp signals. <s> BIB008 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security. <s> BIB009 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> Based on discrete Hermite-Gaussian like functions, a discrete fractional Fourier transform (DFRFT) which provides sample approximations of the continuous fractional Fourier transform was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be better discrete Hermite-Gaussian like functions than those developed before. Furthermore, by appropriately combining two linearly independent matrices which both commute with the DFT matrix, we develop a method to obtain even better discrete Hermite-Gaussian like functions. Then, new versions of DFRFT produce their transform outputs more close to the samples of the continuous fractional Fourier transform, and their application is illustrated. <s> BIB010 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> DFRFT based on eigenvectors <s> Discrete equivalents of Hermite-Gaussian functions play a critical role in the definition of a discrete fractional Fourier transform. The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this letter, we first characterize the space of DFT-commuting matrices and then construct matrices approximating the Hermite-Gaussian generating differential equation and use the matrices to accurately generate the discrete equivalents of Hermite-Gaussians. <s> BIB011
A possible approach for the DFRFT is based on searching the eigenvectors and eigenvalues of the DFT matrix and then computing the fractional power of the DFT matrix (e.g., BIB003 BIB005 BIB004 BIB001 ). This type of the DFRFT was proposed to combat the issues associated with previous implementations such as a lack of unitarity and index additivity (e.g., BIB006 BIB002 ) and the fact that most of these provide a satisfactory approximation to the continuous transform. In particular, the DFRFT is based on the eigendecomposition of the DFT kernel matrix BIB005 . The transform kernel of the DFRFT can be defined as  for N even. In order to ensure orthogonality of the DFT Hermite eigenvectors (u k ), the Gram-Schmidt algorithm (GSA) or the orthogonal procrustes algorithm (OPA) can be used BIB005 . The GSA minimizes the errors between the samples of the Hermite functions and the orthogonal DFT Hermite eigenvectors. On the other hand, the OPA minimizes the total errors between those samples. It should be also pointed out that the main difference between the approach proposed in BIB005 and similar approaches proposed (e.g., BIB003 BIB004 BIB001 ) is found in the obtained eigenvectors in previous contributions, in that they were just discrete Mathieu functions BIB005 . Although the Mathieu functions can converge to Hermite functions, the convergence for the eigenvectors obtained in previous approaches are not so fast for the high-order Hermite functions by the linear mean square error criterion BIB005 . Additionally, the authors investigated the relationship between the FRFT and the DFRFT and found that for a sampling period equal to ffiffiffiffiffiffiffiffiffiffiffiffi 2p=N p , the DFRFT performs a circular rotation of the signal in the time-frequency plane BIB005 . However, the DFRFT becomes an elliptical rotation in the continuous time-frequency plane for sampling periods different from ffiffiffiffiffiffiffiffiffiffiffiffi 2p=N p BIB005 . Therefore, for these elliptical rotations an angle modification and a post-phase compensation in the DFRFT are required to obtain results similar to the continuous FRFT BIB005 . This approach has been extended to the so-called multiple-parameter discrete fractional Fourier transform (MPDFRFT) BIB009 BIB008 . In fact, the MPDFRFT maintains all of the desired properties and reduces to the DFRFT when all of its order parameters are the same. A similar approach to BIB005 has been proposed in BIB006 . However, authors in BIB006 believe that the discrete time counterparts of the continuous time Hermite-Gaussians maintained the same properties, since these discrete time counterparts exhibited better approximations than the other proposed approaches BIB011 . In order to resolve this issue about the approximation of Hermite-Gaussian functions, a nearly tri-diagonal commuting matrix of the DFT and a corresponding version of the DFRFT was proposed in BIB010 . Most of the eigenvectors of this proposed nearly tri-diagonal matrix result in a good approximation of the continuous Hermite-Gaussian functions by providing a smaller approximation error in comparison to the previous approaches. Using the findings presented in BIB005 BIB006 , methods for parallel and cascade computations of DFRFT are proposed in BIB007 . By this new method, the DFRFT of any angle can be computed by a linear combination of the DFRFTs with special angles BIB007 . The parallel method is suitable for the signal whose DFRFT in special angles are already known. The chirp signal detection is a common one, and the cascade method has a regular structure; therefore, it is very suitable for VLSI implementation. However, it should be noted that these types of DFRFTs lack the fast computation algorithm and the eigenvectors cannot be written in a closed form.