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Eigenvectors and Functions of the Discrete Fourier Transform

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Eigenvectors and Functions of the Discrete Fourier Transform IEEETRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNALPROCESSING, VOL. ASSP-30, NO. 1, FEBRUARY 1982 25 Eigenvectors and Functions of the Discrete Fourier Transform BRADLEY W. DICKINSON, MEMBER, IEEE, AND KENNETH STEIGLITZ, FELLOW, IEEE Abstract-A method is presented for computing an orthonormal set Gauss [2]. This was pointedout by Good [3] and,later, of eigenvectors for the discrete Fouriertransform (DFT). The tech- McClellan 241. Recently,Auslander and Tolimieri [5] have nique is based on a detailed analysis of the eigenstructure of a special elaborated on the relationship of the multiplicity problem to matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applica- certain constructions of abstractharmonic analysis, even tions to multiplexing and transform coding are suggested. showing how the FFTarises from abstract principles! The structure of theeigenspaces of F is less well understood. I. INTRODUCTION It appears difficult to obtain an analytical form for an orthog- HIS paper deals with some mathematical aspects of the onal set of eigenvectors. This suggests that a computational discrete Fouriertransform (DFT), studied with linear approach may be necessary, and Yarlagadda [6] proposed one T such method. In Section I1 of this paper, we propose a dif- algebra andmatrix theory methods. The vast majorityof ferent approach based on the theory of commuting matrices. papers on the DFT have concerned computational issues, most Some interesting analytical results are obtained along the way. notably the extensive literature on the fast Fourier transform Functionsof the DFT are 'discussedin Section 111. The (FFT)algorithms and more recently the Winograd-Fouirier imbedding of the DFT into a family of transforms has been transformalgorithm. Yet the importance of the DFT stems used by Jain [7] to study the variation of general properties from more fundamental physical and mathematical principles of the transform family, especially for certain data compres- of harmonic analysis which underlie linear signal processing sionpurposes. We suggest anotherimbedding based on the technology. Thus a major goal of this paper is to present some idea of functions of a matrix. We point out that functions of novel ways of thinking about properties of the DFT in order the DFT matrixF are simple to compute and touse in compu- to stimulatefurther research along moretechnique-oriented tation. The family of (fractional) powers Ft is described in lines. detail.Finally, a novel matrixtransform pair relationship, To be more concrete, let F denote the DFT, regarded as a which generalizes theDFT relationship between vectors, is unitary mapping of complex N-space onto itself. With respect constructed from fractional powers of F, and some possible to the standard basis of N-space, F is simply an N X N matrix whose elements are normalized complex exponentials applications are mentioned. (F)ik = (e-j2nik/N)/fi, 0 < i, k <N - 1. (1) 11. A METHOD FOR OBTAININGORTHOGONAL EIGENVECTORSOF THE DFT Thus the methods of linear algebra and matrix theory may be applied to study the DFT. A. The General Approach McClellan and Parks [ 11 suggested that the eigenstructure of First, we recall some facts about the DFT matrix F in (1). F, being anatural linear algebraic wayof decomposing the The minimal polynomial of F is h4 - 1 [l], [3] -[5] ; thus matrix,might be ofcomputational interest as well. It was F obeys the equation hoped that the diagonal form of F with respect to an eigen- F4 =I (2) vector basis of N-space might be exploited. Through the use of highly original methods in [ 11, themultiplicities of the where I is the (N X N) identity matrix. The roots of h4 - I are distinct, so F is simple or diagonaZizabZe [8] with eigen- eigenvalues of F were determined and a set of eigenvectorswas values constructed. No computationalbenefits over the FFT were observed,however, and the eigenvectors obtained are not hk- -dk=lz 0<k<3 (3) orthogonal. thefourth roots of unity. The multiplicities of the eigen- It turns out that the eigenvalue multiplicity problem for F values are known, as discussed in Section I, and are given in has a long history, being equivalent to a problem solved by Table I. Since they are not distinct for N> 4, there are many possible sets of eigenvectors. Manuscript receivedAugust 4, 1980; revisedJuly 29,1981. This work was supported in part by the National Science Foundation under As motivationfor our approach, suppose that we have a Grants ENG77-28523 and ENG 79-16292, and also by the U.S. Army matrix S, with distinct 'eigenvalues, which commutes with F, Office of Research under GrantDAAG29-79-C-0024. that is, The authors are with the Department of Electrical Engineering and Computer Science, Princeton University, Princeton, NJ 08544. FS = SF, (4) 0096-3518/82/0200-0025$00.75 0 1982 IEEE 26 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-30, NO. 1, FEBRUARY 1982 TABLE I Circular convolution of time signals corresponds to pointwise multiplication in the transformdomain. Theaction of the filter D(z) is thus represented by multiplication of complex 4m ml1 m m m- 1 N-vectors (transforms) by the diagonal matrix AD, which has 4m+ 1 mi 1 m m m elements 4m+2 rn+ 1 rn m+ 1 m (AD)kk = D(eiznkIN), O<k<N- 1. (9) 4m+ 3 rn+l mal mcl m The correspondence between circular convolution and point- wise multiplication gives the matrix equation Let e be an eigenvector of S corresponding to eigenvalue A. FI'D = ADF. (10) Then From (lo), using the symmetry of each matrix withrespect to S(Fe) = F(Se) = F@e) = We. (5) transposition, it follows that the matrix So Fe is also an eigenvector of S corresponding to the same eigenvalue. However, S has distinct eigenvalues and, +AD therefore, zD=I'D (1 1) unique (up to normalization by a constant) eigenvectors [8]. commutes withF. Thus, for some constant0 This leaves the problem of determining a set of filter coeffi- Fe = pe (6) cients which leads to distinct eigenvalues for ED. In Appen- so e is an eigenvector of F also. dix I, we show, using perturbation methods, that sucha choice If S is real and symmetric, its eigenvectors will be real and is always possible. We will bypass the existence question by.a orthogonal and will be the desired set of eigenvectors of F. careful analysis of the simplest choice of filter, D(z) = z + zY1, Thecomputation of the eigenvectorsof a real symmetric corresponding to the matrix 1 0 ... 0 1 r: 2coso 1 0 io 1 2cos2w :I 0 1 matrix has been thoroughly studied, and reliable methods are where o = 21r/N. While we have been unable to make a com- available [9]. plete analysis, we will show that there is no difficulty in find- From knowledge that F has a set of real eigenvectors [ 11, we ing a complete set of real orthogonal eigenvectors for S which are assuredof the existence of suchmatrices S. For every are also eigenvectors of F, even when S has multiple eigen- dimension, it would be desirable to have a canonical choice of values. Based on extensive numerical evidence, as well as the S whose eigenvectors could be obtained analytically. No solu- analytical results to be presented next, we make the following tion is known for this problem. However, we will give a simple conjecture: the eigenvalues of the N X N matrix S [(12)] are choiceof S whosestructure is sufficiently rich to allow a distinct except when N is divisible by 4. (When N is divisible complete analysis of the associated eigenproblem. by four it can be proved that S has two zero eigenvalues. We also conjecture that this is the only multiplicity which ever B. A Class of Matrices Commuting withF occurs.) Ourchoice for thematrix S will comefrom a class of matrices which all commute with F constructed as follows. C The Eigenstructure of S Let D(z) be any real even FIR digital filter written in the For every N > 2, the matrix S has all of its eigenvalues in form the interval [-4, 41 by Gershgorin'scriterion [8]. To get D(z)=ao +a1(z+z-')+..-+ap(zP+z-p), p<N/2 more refined results, we will recall some facts about the rela- (7) tionsbetween the eigenvaluesof amatrix and those of its where time shifts are taken in the circular or mod N sense. principal submatrices; see [9] and [lo], for example. Theaction of thisfilter oncomplex N-space(time signals) Let Tibe the following i by i principal submatrix of S: corresponds to circular convolution or, equivalently, to multi- plication by the circulant matrix (N X N) 2coso 1 o... aoala2 . * . ap 0. * 0 ap * . a4a3u2a1 1 2 cos2w 1 oi alaOal . * . .a4a3a2 as 1 aza1ao * . * . a6a5a4a3 . (8) I ... DICKINSON AND STEIGLITZ: EIGENVECTORS AND FUNCTIONS OF THE DISCRETE FOURIER TRANSFORM 21 Ti is a symmetric, tridiagonal Jacobi matrix and has distinct to hold if pi is to be an eigenvalue of S to obtain the condition real eigenvalues [9, p. 3001. Let p1 > p2 >. > p~-~be the PO, P. 901 eigenvalues of TN-~and let Al h2 2.* 2 AN be the eigen- 2 - rank [a,&r-‘T~-l]=N- 2. (22) values of S. The interlacing inequalities for these eigenvalues are the inequalities [9, p. 1031,[ 111, and [ 121 There are a number of equivalent ways to express this condi- tion [ 111 , but because TN-~has eigenvectors of known form A1 >h2 >p2 >***>pN-l >AN. (14) (17), the most useful is that the vector a = (1 0 * 0 1)’ be From the distinctness of the cui} two important conclusions orthogonal to‘the eigenvector of TN-~corresponding to eigen- may be drawn: 1) no eigenvalue of S has multiplicity greater value pi.
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