The Experts below are selected from a list of 690 Experts worldwide ranked by ideXlab platform
Magdy Tawfik Hanna - One of the best experts on this subject based on the ideXlab platform.
-
1 Hermite-Gaussian-Like Eigenvectors of the Discrete Fourier Transform Matrix Based on the Direct Utilization of the Orthogonal Projection Matrices on its Eigenspaces1
2016Co-Authors: Magdy Tawfik Hanna, Nabila Philip, Attalla Seif, Waleed Abd, El Maguid AhmedAbstract:A new version is proposed for the Gram-Schmidt algorithm, the orthogonal procrustes algorithm and the sequential orthogonal procrustes algorithm for generating Hermite-Gaussian-like Orthonormal Eigenvectors for the discrete Fourier transform matrix F. This version is based on the direct utilization of the orthogonal projection matrices on the eigenspaces of matrix F rather than the singular value decomposition of those matrices for the purpose of generating initial Orthonormal Eigenvectors. The proposed version of the algorithms has the merit of achieving a significant reduction in the computation time. Index Terms: Discrete fractional Fourier transform, Hermite-Gaussian-like orthonorma
-
fractional discrete fourier transform of type iv based on the eigenanalysis of a nearly tridiagonal matrix
Digital Signal Processing, 2012Co-Authors: Magdy Tawfik HannaAbstract:A fully-fledged definition for the fractional discrete Fourier transform of type IV (FDFT-IV) is presented and shown to outperform the simple definition of the FDFT-IV which is proved to be just a linear combination of the signal, its DFT-IV and their flipped versions. This definition heavily depends on the availability of Orthonormal Eigenvectors of the DFT-IV matrix G. An eigenanalysis is performed of a nearly tridiagonal matrix S which commutes with matrix G. An involutary unitary matrix P is defined and used for performing a similarity transformation that reduces S to a block diagonal form where the two diagonal blocks are exactly tridiagonal matrices. Moreover the elements of those two diagonal blocks are derived in order to circumvent the need for performing the two matrix multiplications involved in the similarity transformation. Orthonormal even and odd symmetric Eigenvectors for S are generated - in terms of the Eigenvectors of the two diagonal blocks - and proved to always be Eigenvectors of G irrespective of the multiplicities of the eigenvalues of S. The relevance of the method contributed here is manifested in the case of a repeated eigenvalue of S with multiplicity 2 where a direct application of a general eigenanalysis procedure in any software package will not produce a pair of even and odd symmetric Eigenvectors corresponding to this repeated eigenvalue. It should be mentioned that the almost tridiagonal matrix S which commutes with the DFT-IV matrix G being dealt with here is distinct from matrix S which commutes with the DFT matrix F dealt with in a previous paper Hanna et al. (2008) [7].
-
Orthonormal Eigenvectors of the DFT-IV matrix by the eigenanalysis of a nearly tridiagonal matrix
2011 IEEE International Symposium of Circuits and Systems (ISCAS), 2011Co-Authors: Magdy Tawfik HannaAbstract:Orthonormal Eigenvectors are efficiently generated for the DFT-IV matrix G by a detailed eigenanalysis of a nearly tridiagonal matrix S which commutes with matrix G. Matrix S is reduced to a block diagonal form by means of a similarity transformation and the two diagonal blocks are proved to be tridiagonal matrices. Orthonormal Eigenvectors of S are generated by utilizing those of the two diagonal blocks. They are rigorously proved to always be Eigenvectors of matrix G irrespective of the multiplicities of the eigenvalues of S.
-
Discrete fractional Fourier transform based on the Eigenvectors of Grünbaum tridiagonal matrix
2008 IEEE International Symposium on Circuits and Systems, 2008Co-Authors: Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Abd El Maguid AhmedAbstract:The development of the discrete fractional Fourier transform (DFRFT) necessitates the availability of a complete set of Orthonormal Eigenvectors of the DFT matrix F. An eigenanalysis is performed for the original Grunbaum tridiagonal matrix T - which commutes with matrix F - having only one eigenvalue of multiplicity two and simple remaining eigenvalues. The two easily obtainable Eigenvectors of T corresponding to its repeated eigenvalue - which are not Eigenvectors of F - are exploited for analytically generating two Orthonormal Eigenvectors common to both T and F.
-
Direct Batch Evaluation of Optimal Orthonormal Eigenvectors of the DFT Matrix
IEEE Transactions on Signal Processing, 2008Co-Authors: Magdy Tawfik HannaAbstract:The generation of Hermite-Gaussian-like Orthonormal Eigenvectors of the discrete Fourier transform (DFT) matrix F is an essential step in the development of the discrete fractional Fourier transform (DFRFT). Most existing techniques depend on the generation of Orthonormal Eigenvectors of a nearly tridiagonal matrix S which commutes with matrix F. More sophisticated methods view the Eigenvectors of S as only initial ones and use them for generating final ones which better approximate the Hermite-Gaussian functions employing a technique like the orthogonal Procrustes algorithm (OPA). Here, a direct technique for the collective (batch) evaluation of optimal Hermite-Gaussian-like Eigenvectors of matrix F is contributed. It is a direct technique in the sense that it does not require the generation of initial Eigenvectors to be used for computing the final superior ones. It is a batch method in the sense that it solves for the entire target modal matrix of F instead of the sequential generation of the Eigenvectors. The simulation results show that the proposed method is faster than the OPA.
M T Hanna - One of the best experts on this subject based on the ideXlab platform.
-
revised direct batch evaluation of optimal Orthonormal Eigenvectors of the dft matrix
International Midwest Symposium on Circuits and Systems, 2012Co-Authors: M T HannaAbstract:A Revised and more numerically accurate version of the Direct Batch Evaluation by constrained Optimization Algorithm (RDBEOA) of Orthonormal Eigenvectors of the DFT matrix F is (RDBEOA) of Orthonormal Eigenvectors of the DFT matrix F is from the fact that it performs the singular value decomposition (SVD) of a matrix whose elements have almost half the range of values of the elements of the matrix to which the SVD is applied in the previous Direct Batch Evaluation by constrained Optimization Algorithm (DBEOA). Having more accurate Hermite-Gaussian-like (HGL) Orthonormal Eigenvectors of matrix F is a main requirement in the development of the discrete fractional Fourier transform (DFRFT).
-
direct sequential evaluation of optimal Orthonormal Eigenvectors of the discrete fourier transform matrix by constrained optimization
Digital Signal Processing, 2012Co-Authors: M T HannaAbstract:The recent emergence of the discrete fractional Fourier transform has spurred research activity aiming at generating Hermite-Gaussian-like (HGL) Orthonormal Eigenvectors of the discrete Fourier transform (DFT) matrix F. By exploiting the unitarity of matrix F - resulting in the orthogonality of its eigenspaces pertaining to the distinct eigenvalues - the problem decouples into finding Orthonormal Eigenvectors for each eigenspace separately. A Direct Sequential Evaluation by constrained Optimization Algorithm (DSEOA) is contributed for the generation of optimal Orthonormal Eigenvectors for each eigenspace separately. This technique is direct in the sense that it does not require the generation of initial Orthonormal Eigenvectors as a prerequisite for obtaining the final optimal ones. The resulting Eigenvectors are optimal in the sense of being as close as possible to samples of the Hermite-Gaussian functions. The technique is found to be numerically robust because total pivoting is allowed in performing the QR matrix decomposition step. The DSEOA is proved to be theoretically equivalent to each of the Gram-Schmidt algorithm (GSA) and the sequential orthogonal Procrustes algorithm (SOPA). However the three techniques are algorithmically quite distinct. An extensive comparative simulation study shows that the DSEOA is by far the most numerically robust technique among all sequential algorithms thus paying off for its relatively long computation time.
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces
IEEE Transactions on Signal Processing, 2006Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A new version is proposed for the Gram-Schmidt algorithm (GSA), the orthogonal procrustes algorithm (OPA) and the sequential orthogonal procrustes algorithm (SOPA) for generating Hermite-Gaussian-like Orthonormal Eigenvectors for the discrete Fourier transform matrix F. This version is based on the direct utilization of the orthogonal projection matrices on the eigenspaces of matrix F rather than the singular value decomposition of those matrices for the purpose of generating initial Orthonormal Eigenvectors. The proposed version of the algorithms has the merit of achieving a significant reduction in the computation time
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the singular value decomposition of its orthogonal projection matrices
IEEE Transactions on Circuits and Systems, 2004Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A technique is proposed for generating initial Orthonormal Eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like Orthonormal Eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram-Schmidt algorithm (GSA) the output Hermite-Gaussian-like Orthonormal Eigenvectors are invariant under the change of the input initial Orthonormal Eigenvectors.
Nabila Philip Attalla Seif - One of the best experts on this subject based on the ideXlab platform.
-
Discrete fractional Fourier transform based on the Eigenvectors of Grünbaum tridiagonal matrix
2008 IEEE International Symposium on Circuits and Systems, 2008Co-Authors: Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Abd El Maguid AhmedAbstract:The development of the discrete fractional Fourier transform (DFRFT) necessitates the availability of a complete set of Orthonormal Eigenvectors of the DFT matrix F. An eigenanalysis is performed for the original Grunbaum tridiagonal matrix T - which commutes with matrix F - having only one eigenvalue of multiplicity two and simple remaining eigenvalues. The two easily obtainable Eigenvectors of T corresponding to its repeated eigenvalue - which are not Eigenvectors of F - are exploited for analytically generating two Orthonormal Eigenvectors common to both T and F.
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces
IEEE Transactions on Signal Processing, 2006Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A new version is proposed for the Gram-Schmidt algorithm (GSA), the orthogonal procrustes algorithm (OPA) and the sequential orthogonal procrustes algorithm (SOPA) for generating Hermite-Gaussian-like Orthonormal Eigenvectors for the discrete Fourier transform matrix F. This version is based on the direct utilization of the orthogonal projection matrices on the eigenspaces of matrix F rather than the singular value decomposition of those matrices for the purpose of generating initial Orthonormal Eigenvectors. The proposed version of the algorithms has the merit of achieving a significant reduction in the computation time
-
Hermite-Gaussian-like Eigenvectors of the DFT matrix generated by the eigenanalysis of an almost tridiagonal matrix
2005 IEEE International Symposium on Circuits and Systems, 2005Co-Authors: Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Abd El Maguid AhmedAbstract:The development of the discrete fractional Fourier transform (DFRFT) necessitates having Orthonormal Eigenvectors for the DFT matrix, F. The objective of having the DFRFT approximate its continuous counterpart can be met if the Eigenvectors of F approximate samples of the Hermite-Gaussian functions. Orthonormal Hermite-Gaussian-like Eigenvectors for F are rigorously derived by a detailed analysis of an almost tridiagonal matrix, S, which commutes with F. By an appropriate similarity transformation, S is reduced to a 2/spl times/2 block diagonal form and the elements of the two exactly tridiagonal matrices forming the two diagonal blocks are explicitly derived in terms of the elements of matrix S.
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the singular value decomposition of its orthogonal projection matrices
IEEE Transactions on Circuits and Systems, 2004Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A technique is proposed for generating initial Orthonormal Eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like Orthonormal Eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram-Schmidt algorithm (GSA) the output Hermite-Gaussian-like Orthonormal Eigenvectors are invariant under the change of the input initial Orthonormal Eigenvectors.
Waleed A Ahmed - One of the best experts on this subject based on the ideXlab platform.
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the direct utilization of the orthogonal projection matrices on its eigenspaces
IEEE Transactions on Signal Processing, 2006Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A new version is proposed for the Gram-Schmidt algorithm (GSA), the orthogonal procrustes algorithm (OPA) and the sequential orthogonal procrustes algorithm (SOPA) for generating Hermite-Gaussian-like Orthonormal Eigenvectors for the discrete Fourier transform matrix F. This version is based on the direct utilization of the orthogonal projection matrices on the eigenspaces of matrix F rather than the singular value decomposition of those matrices for the purpose of generating initial Orthonormal Eigenvectors. The proposed version of the algorithms has the merit of achieving a significant reduction in the computation time
-
hermite gaussian like Eigenvectors of the discrete fourier transform matrix based on the singular value decomposition of its orthogonal projection matrices
IEEE Transactions on Circuits and Systems, 2004Co-Authors: M T Hanna, Nabila Philip Attalla Seif, Waleed A AhmedAbstract:A technique is proposed for generating initial Orthonormal Eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like Orthonormal Eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram-Schmidt algorithm (GSA) the output Hermite-Gaussian-like Orthonormal Eigenvectors are invariant under the change of the input initial Orthonormal Eigenvectors.
Waleed Abd El Maguid Ahmed - One of the best experts on this subject based on the ideXlab platform.
-
Discrete fractional Fourier transform based on the Eigenvectors of Grünbaum tridiagonal matrix
2008 IEEE International Symposium on Circuits and Systems, 2008Co-Authors: Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Abd El Maguid AhmedAbstract:The development of the discrete fractional Fourier transform (DFRFT) necessitates the availability of a complete set of Orthonormal Eigenvectors of the DFT matrix F. An eigenanalysis is performed for the original Grunbaum tridiagonal matrix T - which commutes with matrix F - having only one eigenvalue of multiplicity two and simple remaining eigenvalues. The two easily obtainable Eigenvectors of T corresponding to its repeated eigenvalue - which are not Eigenvectors of F - are exploited for analytically generating two Orthonormal Eigenvectors common to both T and F.
-
Hermite-Gaussian-like Eigenvectors of the DFT matrix generated by the eigenanalysis of an almost tridiagonal matrix
2005 IEEE International Symposium on Circuits and Systems, 2005Co-Authors: Magdy Tawfik Hanna, Nabila Philip Attalla Seif, Waleed Abd El Maguid AhmedAbstract:The development of the discrete fractional Fourier transform (DFRFT) necessitates having Orthonormal Eigenvectors for the DFT matrix, F. The objective of having the DFRFT approximate its continuous counterpart can be met if the Eigenvectors of F approximate samples of the Hermite-Gaussian functions. Orthonormal Hermite-Gaussian-like Eigenvectors for F are rigorously derived by a detailed analysis of an almost tridiagonal matrix, S, which commutes with F. By an appropriate similarity transformation, S is reduced to a 2/spl times/2 block diagonal form and the elements of the two exactly tridiagonal matrices forming the two diagonal blocks are explicitly derived in terms of the elements of matrix S.
