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 theDFT 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- ofharmonic analysis whichunderlie 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 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 theFFT were observed,however, and the eigenvectors obtained are not hk- -dk=lz 0 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

Let e bean 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 eigenvectorsof 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 theonly 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), pcirculant 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. (These conditions are related to controllability con- than 2, and 2) any multiple eigenvalue of S is also an eigen- ditions for a particular linear system [ 121.) From (17) this value of TN-~. condition becomes The eigenvalues of Ti are the roots of the polynomial (- 1y-2 rN-2 (pi) = - 1, (23) ri(p) = det (Ti - PI), i 1. (15) > Thus the eigenvalues inherited are the ones corresponding to By expanding the determinant these polynomials are found to eigenvectors, e(pj),of TN-~satisfying obey the recursions [9, p.4231 e(pd = -Je(pi>. (24) ri(p) = cos(2 iu - p)rj-l (p) - ri-2 &) (16) From (18) and (20) the number of inheritedeigenvalues, and where ro(p)= 1 and r1(p) = 2 cos o - p. A set of unnormal- hence potential double eigenvalues of S is ized eigenvectors for TN-~is obtained when these polynomials (N- 1)/2, Nodd are evaluated at the eigenvalues of TN-1 [9, p. 3 161 (N- 2)/2, Neven. (25)

((1 9 -r1 (pi>>r2 (pi), . . . 9 (- 11N-’rN-2 (pi))’; We now use thecondition for eigenvalue inheritance, to- 1 , we find the values This will follow fromcertain even/odd symmetries of the of rN-.,(p) at {pi} satisfy eigenvectors of S. Let pi be an inherited eigenvalue of S. One eigenvector is

rN-Z(P1) > 0, rN-Z(p2) < 0,. . . > ‘N-Z(pN-l)> 0, then given by u(&) = (UO,u1, * ,UN-1)’ where N even u(pi)= (0,1, (pi), * * * 3 (- (pi))’. (26)

‘N-2 (111) < 0, rN-2 (p2)> 0, . Y rN-2 (PN-1) > 0, Thisfollows from the partitioned form of S [(21)]and the Nodd. (18) inheritancecondition [(23)]. From (24) it follows thatthis eigenvector has odd symmetry, in the usual (circular) sense Actually,more can be said. By the even symmetryof the u. = -u -imod~ O i

AA where a’ = (1 0 0 l), we examinethe equations required Ui=U-imod~, O

The DFT is known to map even (respectively, odd) signals Thus, nearly half of the eigenvalues and eigenvectors of S into even (respectively, odd) transform sequences. Therefore, can be obtained from those of A - B;these are the odd eigen- by considering even and oddsymmetry weare still able to vectors of S. The rest of the eigenvalues must be found by conclude, as in (4)-(6), that u and u^ are eigenvectorsof F, using methods such as those described in [ 141. However, the even though they come from a double eigenvalue of S. Fur- eigenvectorsmay be determined from half-size problems as thermore, u and are orthogonal since they areof opposite follows. Let u(Xi) = (uo, ul, . . . , UN-1)’ be an eigenvector of symmetry types. S with(noninherited) eigenvalue hi. Bymultiplication by This completes the theoretical derivation of our main result. (XiI- S), we obtain However, there are a few other properties to be pointed out. Using DFT real/imaginary symmetryproperties, we may conclude that the even eigenvectors correspond to the eigen- values k1 of F andthe odd eigenvectorscorrespond to the where ti = (ul,* . , u~-~)’,a = (1 0 * 0 1)’ and ti = Jti. Equa- eigenvalues fj of F. All of the odd eigenvectors are accounted - - for by eigenvalues of S inherited from TN-~.Note that none tion (33b) is easily reduced to half size by reversal symmetry. of the even eigenvectors can have zero as its first component; E. A Comment on Limiting Behavior sucha vector would satisfy the inheritance condition (22), In this section, we will show that the problem of determin- which is impossible. As a check, we can see that the number ing the eigenstructure of the matrix S is a discrete counterpart of fj eigenvaluesof F fromTable I equals thenumber of of the eigenproblem for the Mathieu equation [ 171, a well- inherited eigenvalues (25). known periodic second-order self-adjoint differential equation. D. Computational Considerations Let e = (e,,, el, , eN-l)’ be an eigenvector of the matrix S corresponding to eigenvalue X. Using the form of S, the de- We want to emphasize thatthe construction of theodd- fining equations (S - W)e = 0 may be rewritten in the form of symmetriceigenvectors of S, given by (26), should not be a linear difference equation for the coordinate sequence used as the basis for a computational procedure. The evalua- tion of the recurrence relations [ (1 6)] is highly susceptible to (2 COS kW - X) ek t ek+l t ek-1 = 0; w = 2a/N. (34) numericalinstability [9, p. 3161. However,there are many This equation will hold for all integers k if we define the ex- standard numerical methods available which are well suited to tended ek sequence to be periodic: ek = ek mod^. Using the obtaining the eigenvectors of S. For example, the eigenprob- central second difference operator6 , where lemp for a class of matrices (Hill’s matrices [ 131) arising from ’ the discretization of periodic boundary value problems is dis- 62ek = ek+l - 2ek + ek-1 (3 5) cussed by Bjork andGolub [14]. While this class contains (34) becomes matrix S [(12)], there are additional symmetries which may be exploited in the eigenanalysis of S arising from the centrosym- 62ek + (2 COS ko- (h- 2)) ek = 0. (36) metric nature of TN-~[( 19)]. Now suppose we view ek as the sampled value of a contin: For simplicity, we treat the case of N odd, since the general uous time signal e(t) at time tk = k/N;as N gets large, (36) is case is notationally more cumbersome [ 151, [ 161. The matrix thefinite-difference approximation to the followingsecond- TN-1 can be written in partitioned form as order differential equation fore(t)

where whose periodicsolutions are the Mathieufunctions [17]. r2cos 1. 0 1 Thus, the eigenvectors of S, and, hence, the eigenvectorsof the DFT, maybe thought of as discrete Mathieu functions.

111. FUNCTIONSOF THE DFT AND FRACTIONAL TRANSFORMS We are accustomed to the usual interpretations of complex N-spaceas the time domain and of the image ofthis space under the DFT as the frequency domain. By using the defini- and J is the appropriate-sized interchange matrix as before. As tion of functions of a matrix, we may define the fractional above,centrosymmetry gives the eigenvectorsof TN-~the DFT Ft for 0 < t < 1, and regard the image of Ft as a gen- forms [x’, -x’J] ’ and [ y‘, y’J]’ for vectors x and y of size eralization of the time and frequency domains. This will allow (N - 1)/2. It is an easy verification that the vector [x’, -x’J]‘ the consideration of generalizationsof such ideas as multi- (respectively, [y’,$4‘) is an eigenvector of TN-~with eigen- plexing and transform coding. value pi, if andonly if the vector x (respectively, y) is an The definition of the matrix functionFt follows the standard eigenvector of the matrix A - B (respectively, A + B) corre- approach [X] since F is diagonalizable. For any functiong(F) sponding to eigenvalue pi [ 151, [ 161. we have DICKINSON AND STEIGLITZ:EIGENVECTORS AND FUNCTIONS OF THEDISCRETE FOURIER TRANSFORM 29

a family of periodic functions in a way parametrized by an angle which determines ageneralized frequency domain. Several problems of signal design and analysis can now be where posed. We can ask forthe optimum value of t, (angle) for separating two signals. By this we would mean the slick of the hk- -eikT/’, Ofunction, we take principal tth roots of the frequency domain suggests applications &in to multiplexing. eigenvalues hk (39) giving Another application for the fractional DFT is in transform g(l) = coding [18], [19] for data compression purposes. Transform g(eikn/2) = eiknt/2 codingamounts to quantization of thetransformed signal or data before storage or transmission. This results in lower The formula for the fractional transform is storage or bandwidth requirements. The data may be approx- imatelyreconstructed by applying the inverse transform 3 Fr = Fiai(t) operation. i= 0 Various fractional DFT’s could be used as part of a trans- form coding scheme. However, here we suggest a novel way where of exploiting the fractional-DFT to obtain a twq-dimensional transformfor this purpose. We considerthe following in- O 4, an acceptablechoice for W is is the reflection operation. Finally we note that Ff defined the fractional DFT Ff,where t = 4/34, since the eigenvalues by (42) and (43) is a periodic matrix function with period 4 of Ft are [from (41)] eikn*l2, 1 < k < 4. It is doubly in- which agrees with the property F4 = I. Some correhponding triguing to observe that the usual signal, flowgraph for the FFT identities forthe coefficients of thepolynomials defining algorithm holds for this transform with the appropriate sub- various powers ofF include stitution of matrix operations with the matrix W playing the az(t+ 1) = aT(-t), where I = -(i - 1) mod 4. (45) role of the complex multiplications in the butterflies. From above, we know that multiplication by W can, itself, be per- q(t)= aF(-t), where I = -i mod 4. (46) formed using FFTmethods. It seemsworthwhile to note Since F is unitary, Ft is also unitary for any t arid’ Parseval’s that since matrix multiplication is noncommutative, there is relationship remains valid between signals in complex N-space nosubstantial generalization of the convolution theorem of and their fractional DFT’s computed by multiplication byFf. theDFT to this matrixtransform. However, for transform A nice geometric picture of the generalization of time and codingapplications, only the transform pair relationship, frequency domains is obtained by considering a function de- (47), is required. fined on N circles of radii 0 < ri < ri+l,0 <,i < N - 1, cen- As a closing note, we might point out that other unitary tered at the origin of the complex plane. The function values transforms may be used in place of the DFT to obtain a matrix atthe points on the positive real axis are thetime domain W for use in the transform pair (47). The properties required (signal) values. The function values along a ray of angle tn/2 are that the minimal polynomial of the transform takes the with respect to the positive real axis correspond to the image form hr - 1 for small r and that a fast implementation of the ofthe time domain values under Fr, the fractionalDFT transform be available for computing the polynomial expres- (transform) values. Hence, every signal may be imbedded into sion for W, defined as a fractionalpower. For example, the 30 IEEE TRANSACTIONSON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-30, NO. 1, FEBRUARY 1982 unitary Walsh-Hadamard transform [ 181 has r = 2 and with Let P(p, e) be the eigenprojection ontothe space spanned W equal to the 2/Mth power of the transform, multiplication by theeigenvectors of I;D(E) corresponding to eigenvalues by W is performed by taking the appropriate linear combina. y1(e) and p2(e). Then [20]

tion of the signal vectorand its Walsh-Hadamard transform m vector. i= 1 IV. CONCLUSIONS and We have described twoapplications of linear algebraand matrixtheory to the DFT. Using commutingmatrices, the m eigenanalysisof theDFT canbe performed using standard P(y, E) =Po + PiEi numericalmethods. Fractional powers of theDFT may be i=1 easily computed; also they suggest new signal processing tech- where niques for multiplexing and transform coding. These problems will be investigated in future research. Po = UUI t tic’ (A61 is the eigenprojection onto the eigenspace of S corresponding APPENDIXI to eigenvalue p. The coefficients of (A4) and (A5) are related EXISTENCEOF A MATRIX I;D HAVING DISTINCT by PO, (2.1>1 EIGENVALUES jii =(S1Pi-])/2i, trace i > 0. (‘47) Using perturbation methods on the matrix S [(12)] , we will show that it is always possible to construct a matrix ED as From the odd symmetryof u, we have described in (7)-(I 1) whose eigenvalues are distinct. Let e be SlU = 0 (A81 a complex number and define the parametric family of filters so that from (A2) u is an eigenvector of 2~ (E) corresponding WZ,€11 by to eigenvalue p. Thus we take pl(E) = p, a constant function. M From (A4) z+z-l +E 2 Zk, k=-M N odd,M= (N- 1)/2 D(z, E) = i=1 M Using (A7) to evaluate we find z + z-1 + e Zk t (e/2)(zN/’ t z-w), 2 ,, k=-M 2; = trace (s, (uu’ + u^u^‘)) I Neven,M=N/2 - 1. (Al) = trace (S, u^u^), by (A9) Applying theconstruction of (7)-(1 I), we obtain the para- metric family of matrices commuting withF

2D(E) = S + es, (-42) A AI where u^ = (Go, u 1, . . . , UN-.~). In Section I1 we showed that where S is given by (1 2) and Go # 0, so fi # 0 and the eigenvalue y2(E) is not a constant /Nt1 I 1 \ function. So for sufficiently small e, the eigenvalues p, (E) and p2 (E) are distinct. This completes the argument.

REFERENCES J. H. McClellan and T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” ZEEE Trans. Audio Electl-oacoust., vol. AU-20, pp. 66-74, 1972. The elements of ED(E) are analytic (linear!) functions of e, E. Landau, Vorlesungeniiber Zahlentheorie, voi. 1. New York: and ED(€) is real and symmetric for real E. Thus the eigen- Chelsea, 1927/1950 (reprint), p. 164. I. J. Good, “Analogues of Poisson’s summation formula,” Amer. values and the projections onto the eigenspaces of 2 D (e) are Math. Monthly, vol. 69, pp. 259-266, 1962. analytic functions of E in a neighborhood of 0 in the complex J. H. McClellan, “Comments on ‘Eigenvalue and eigenvector E-plane [20]. Being continuousfunctions of E, twodistinct decomposition of the discrete Fourier transform’,’’ ZEEE Truns. Audio Electroacoust., vol. AU-21, p. 65, 1973. eigenvalues of S cannot coalesce for small E, so we must show L. Auslander and R. Tolimieri, “Is computingwith thefinite that the multiple (double) eigenvalues of ED (0) = S split apart Fourier txansform pure or appliedmathematics?” Bull. Amer. for small e to show that ZD(e) hasdistinct eigenvalues for Math. SOC.,vol. 1, pp. 847-891, 1979. R. Yarlagadda, “A note on the eigenvectors of DFT matrices,” sufficiently small nonzero E. ZEEETrans. Acoust., Speech, SignalProcessing, vol. ASSP-25, Let p be a double eigenvalue of S with odd normalized eigen- pp. 586-588, 1977. vector u and even normalizedeigenvector u^ asdescribed in A. K. Jain. “A sinusoidal family of unitary transforms,” ZEEE Trans. Pattern Anal. Machine Zntell., vol. PAMI-1, pp. 356-365, Section 11. Let pl (e) and p2(e) be theanalytic functions 1979. describing the eigenvalues of ED (E), where p1(0) = p2 (0) = p. P. Lancaster, Theory ofMatrices. New York: Academic, 1969. IEEETRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-30, NO. 1, FEBRUARY 1982 31

[9] J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford, ingand Computer Science, Princeton Univer- England: Clarendon, 1965. sity, Princeton, NJ, where he is now an Associ- [ 101 A. S. Householder, The Theory of Matrices in Numerical Anal- ate Professor. ysis. New York: Blaisdell, 1964. Dr. Dickinson is a member of TauBeta Pi, [ 111 D. Slepian and H. Landau, “A note on the eigenvalues of Her- S.A.B.R., and Sigma Xi. He has been Associate mitian matrices,” SIAM J. Math.Anal., vol. 9, pp. 291-297, Editorfor Identification for theIEEE TRANS- 1978. ACTIONS ON AUTOMATICCONTROL and is currently [12] B. W. Dickinson 2nd K. Steiglitz, “Eigenvalues of symmetric a member of the Board of Governors of the matrices:System theoryconditions for distinctness,” IEEE IEEE Information Theory Group. Trans. Automat. Contr., vol. AC-25, pp. 284-285,1980. [13] H. Hochstadt, “On the theory of Hill’s matrices and related in- verse spectral problems,” Lin.Alg. Appl., vol. 11, pp. 41-52, 1975. [ 141 A. Bjork and G. H. Golub, “Eigenproblems for matrices associ- ated with periodic boundary conditions,” SIAM Rev., vol. 19, pp. 5-16,1977. 1151 A. L. Andrew. “Eigenvectors of certain matrices,” Lin. A&. LA Kenneth Steiglitz (S’57-M’64-SM779-F’81) was ~ppl.,vol.7, pi. 151-162,1973. born in Weehawken, NJ, on January 30, 1939. [ 161 A. Cantoni and P. Butler, “Eigenvalues and eigenvectors of sym- He received the B.E.E. degree in 1959, the metric centrosymmetric matrices,” Lin. Alg. Appl., vol. 13, pp. M.E.E. degree in 1960, and theEng. Sc. D. degree 275-288,1976. in 1963, all from New York University, NY. [ 171 N. W. McLachlan, Theory and Application of Mathieu Functions. Since September 1963 he has been with the Oxford, England: Clarendon, 1947. Department of Electrical Engineering and Com- [181 N. Ahmed and K. R. Rao, OrthogonalTransforms for Digital puter Science, Princeton University, Princeton, Signal Processing. Berlin, Germany: Springer-Verlag, 1975. NJ, wherehe is now Professor,teaching and [19] P.A. Wintz, “Transform picture coding,” Proc. IEEE, vol. 60, conducting research in the computer and sys- pp. 809-820,1972. tems areas. He is Associate Editor of the [20] T. Kato, Perturbation Theory for Linear Operators, ch. 2. New Journal of the Association for Computing Machinery, and the journal York: Springer-Verlag, 1966. Networks. He is the author of Introduction to Discrete Systems (Wiley, 1974), and coauthor, with C. H. Papadimitriou, of Combinatorial Optimiza- Bradley W. Dickinson (S’7O-M’74) was born in St. Marys, PA, on April tion: Algorithms and Complexity (Englewood Cliffs, NJ: Prentice-Hall, 28, 1948. He received the B.S. degree in engineering from Case Insti- 1982). tute ofTechnology, of Case Western Reserve University, Cleveland, Dr. Steiglitz is a member of the Digital Signal Processing Committee OH, in 1970, and the M.S. and Ph.D. degrees in electrical engineering of the Acoustics, Speech, and Signal Processing Society, and has served from StanfordUniversity, Stanford,CA, in 1971 and 1974, respectively. as an Administrative Committee member and Awards Chairman of the Since 1974 he has been with the Department of Electrical Engineer- society. He is a member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi.

Realizable Wiener Filtering in Two Dimensions

MICHAEL P. EKSTROM, SENIOR MEMBER, IEEE

Abstruct-The extension of Wiener’s classical mean-square estimation to play a prominent role in modern time series analysis. In- theory to a two-dimensional setting is presented. In analogy with the deed, its extension to new applications is a topic of active re- one-dimensional problem,the optimal realizable filter is derived by search interest. solution of a two-dimensional, discrete Wiener-Hopf equation using a spectral factorization procedure. Filters are developed for the cases of The representative problem addressed by Wiener was that of prediction, filtering, and smoothing, and appropriate error expressions optimallyestimating an unobserved time signal s(t) given a are derived to characterize their performance. noise corrupted observation, s(t) t w(t), where w(t)is a noise process. Both signal and noise are modeled as wide-sense sta- I. INTRODUCTION tionary processes; the estimator is constrained to be linear, and HILE developed in antiquity, Wiener’s minkum mean- derived optimal in the sense of achieving the MMSE associated squareerror (MMSE) estimation theory [l] continues with the estimate. Two classes of estimators were described W by Wiener: the so-callednoncausal (unrealizable, bilateral) Manuscript received May 28, 1981. This work was supported in part filter which uses all past, present, and future observations in by a Senior Scientist Award of the Alexander von Humboldt Founda- forming the estimate, and the so-called causal (realizable, uni- tion, while the author was Visiting Scientist at the Lehrstuhl furNach- lateral) filter which uses only pastand present observations. richtentechnik, Universitat Erlangen-Niirnberg, Erlangen, West Germany. Theauthor is with Schlumberger-Doll Research, Ridgefield,CT With appropriate generalization, the fundamental estimation 06877. problem addressed by Wiener, that of estimating signals from

0096-3518/82/0200-003 1$00.75 0 1982 IEEE