DIRECT BATCH EVALUATION OF DESIRABLE EIGENVECTORS OF THE DFT MATRIX BY CONSTRAINED OPTIMIZATION Magdy Tawfik Hanna Department of Engineering Mathematics and Physics Fayoum University Fayoum 63514, Egypt Email: hanna@ieee.org Abstract - The process of the batch generation of orthonormal eigenvectors of a unitary matrix – like the DFT matrix – that are as close as possible to approximate eigenvectors having a desired feature – such as being samples of the Hermite Gaussian functions – is formulated as a constrained optimization problem. The adopted rationale is the collective evaluation of a complete set of eigenvectors in each eigenspace by the minimization of the squared Frobenius norm of the difference between the matrix whose columns are the sought vectors and the matrix whose columns are the corresponding approximate eigenvectors subject to the constraints that the sought vectors are orthonormal eigenvectors. Index Terms: DFT matrix, discrete fractional Fourier transform (DFRFT), Hermite-Gaussian-like eigenvectors, orthogonal procrustes algorithm (OPA), constrained optimization. I. INTRODUCTION The development of the discrete fractional Fourier transform (DFRFT) depends heavily on having orthonormal eigenvectors of the DFT matrix F that approximate samples of the Hermite Gaussian functions [1-3]. Some researchers first generated initial exact eigenvectors of matrix F and next applied an advanced technique such as the orthogonal procrustes algorithm (OPA) for getting final superior eigenvectors that are closer to samples of the Hermite Gaussian functions than the initial ones [2]. The approach contributed here is the Direct Batch Evaluation of the desirable eigenvectors by a constrained Optimization Algorithm (DBEOA). This method does not require the generation of initial eigenvectors. It will be shown to be faster than the OPA and more numerically accurate. II. MATHEMATICAL FRAMEWORK Let F be any unitary matrix of order N having the distinct eigenvalues k λ with algebraic multiplicity k r , K k , , 1 A = . Let k E be the eigenspace corresponding to k λ . The unitarity of F implies the orthogonality of the eigenspaces k E , K k , , 1 A = . Consequently the problem of generating orthonormal eigenvectors for F can be decoupled into K separate problems where in the k th problem one seeks orthonormal basis for k E . The unitarity of F also implies that the geometric multiplicity of k λ (which is the number of linearly independent eigenvectors corresponding to it) is equal to its algebraic multiplicity and consequently the dimension of k E is k r . Let k U be a matrix whose columns are approximate eigenvectors of F (having a desired feature) corresponding to the exact eigenvalue k λ . The columns of k U are assumed to be linearly independent. Let k U & be a matrix whose columns are exact orthonormal eigenvectors of F corresponding to k λ . The objective is to derive the matrix k U & such that it is the closest matrix to k U . III. A DIRECT BATCH EVALUATION BY A CONSTRAINED OPTIMIZATION ALGORITHM A. Formulation and Solution Orthonormal eigenvectors that span the eigenspace k E corresponding to the eigenvalue k λ of algebraic multiplicity k r of a unitary matrix F will be batch evaluated such that they will be the closest in the sense of Frobenius norm to approximate eigenvectors having a desired feature. The problem can be mathematically formulated as a constrained optimization one. Given an k r N × matrix k U whose columns are desired approximate eigenvectors of F find an k r N × matrix k U & that minimizes the squared Frobenius norm 2 F a J k U k U & - = (1) subject to the constraints ( ) 0 k U I F A = - ≡ & k λ (2) and 0 I U U B k r k H k = - ≡ & & . (3) Constraint (2) guarantees that the columns of k U & are eigenvectors of F and constraint (3) guarantees their orthonormality. Criterion (1) can be expressed as: ( ) ( ) [ ] ( ) ( ) { } k U H k U k U H k U k U k U k U k U & & & tr k r tr H tr a J Real 2 - + = - - = (4) where () tr is the trace of a matrix and {} Real denotes the real part. The above equation implies that minimizing a J w.r.t. k U & is equivalent to maximizing the following criterion: ( ) ( ) k U H k U k U H k U & & tr tr b J + = . (5) Augmenting this real criterion by the two sets of constraints (2) and (3), one gets ( ) ( ) ( ) ( ) µ B H µ µ B H µ µ A G µ µ A G µ * * * * c c c c T T T T b J J - - - - = (6) 825 1-4244-0921-7/07 $25.00 © 2007 IEEE.