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evaluate the set of sufficient con dit ions (1 1 for uncoupled est imation:11

,=1

= R112 11

, = \ l + l

2 T / 1)1=1

= R23f

,=1 L I2 \ I 2a r - 11 - 1)

Jf ] = oRZc .0 5 [, = \ J + I. 1 2 T i 1

The condit ions given in I I are therefore satisfied, and weconclude that the est imates of azimuth and elevation angles madewith this two-ring array are uncoupled. It follows that an arrayconsist ing of several ring pairs, where each pair in (12) can havedistinct values of Z and R, also has the property that the angleestimates are uncoupled. Cylindrical arrays and spherical arrays areexamples of such arrays.

REFERENCESR. J. Kenefic, Maximum likelihood estimation of the parameters ofa plane wave tone at an equispaced linear array, IEEE Trans. SignulProcessing, vol. 36, no. I pp. 128-130, Jan. 1988.R. 0. Nielsen, Azimuth and elevation angle estimation with a three-dimensional array, IEEE I ceanic Eng., vol. 19, no. I pp , 8486 ,Jan. 1994.H . L. van Trees, Detection. Estimation and Modulation Theor?, Part I .New York: Wiley, 1968, pp. 66-85.

A Simple Algorithm for GeneratingDiscrete Prolate Spheroidal SequencesDon M . Gruenbacher and Donald R. Hum m e l s

Absfract- The discrete prolate spheroidal sequences are optimumwaveforms in many communication and signal processing applicationsbecause they comprise the most spectral efficient set of orthogonalsequences possible. Generation of the sequences has proven to be difficultin the past due to the absence of aclosed form solution. A new methodof easily generating any single discrete prolate spheroidal sequence,including sequences of very long length, is presented. Also shown aresome example sequences generated using the algorithm presented.

I . INTRODUCTIONThe set of discrete prolate spheroidal sequences DPSSs) haslong been known to have characterist ics which would make them

ideal candidates for use in communication and signal processingapplications. Possible applications of DPSSs include pulse shaping,secure communications [I], PAM [2], and signal extrapolation [ 3 ] .Initially, DPSS generation was difficult due to the lack of a closedform solution. In [4] a method was given for generating the singlemost frequency-compact sequence out of a set of DPSSs, however ,the method did not al low for the generation of ei ther DPSSs besidethe most compact sequence or for DPSSs of extremely long length.The recursive procedure in [4] does not work well for certainDPSSs with a length beyond appro ximately 32 samples. Applicationsrequiring sequences with long lengths would not be able to useDPSS pulses in these cases. This correspondence will give a fairlystraightforward, simple, and rel iable m ethod for the gen eration of anyDPSS of arbitrary length. DPSSs with length in excess of 10000samples have been generated with this method. First , a descriptionof the DPSSs and some of their properties will be given.

DPSSs are the discrete-time relatives of the prolate spheroidalwave functions [ 5 ] and discrete prolate spheroidal wave functions.Al though DPSSs have been studied by various authors, the mostextensive work has been done by Slepian [6]. The notation used bySlepian in deno ting a single DPSS is P ~ ) - \ - . UT) , here S ndicatesboth the number of DPSSs in the set and the number of samples ineach DPSS, and 11- is a bandwidth and shaping factor under theconstraint 0 < 11- < .5. The value of k indicates the particular DPSSout of the set k = 0 . 1 . . 1.The most well known property of the orthogonal DPSSs is thata set of DPSSs comprise the index l imited sequences with greatestamoun t of energy contained within a frequ ency band. It is also knownthat DPSS o) A\ - . l l - )as the most compact spectrum out of theset , while each succeeding DPSS, k = 1. . . ._ 1, has a largerbandwidth than the DPSS preceding i t [6].

11. DPSS GENERATIONIn 161 Slepian presented properties of DPSSs which can be used

to develop different techniques for generating any part icular set ofDPSSs. The two primary techniques find the DPSSs as eigenvectorsof one or the other of two matrices. Matrix H ; Y , T- is Hermitian,

Manuscript received September 29, 1993; revised May 16, 1994. This workwas supported by a grant from the Motorola Government Electronics Group.The associate editor coordinating the review of this paper and approving itfor publication was Prof. Henrik V . Sorensen.The authors are with Department of Electrical and Computer Engineering,Kansas State University, Manhattan, KS 66506 USA.IEEE Log Number 9404798.

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42. NO. 1 I. NOVEMBER 1994 3217

and its elements are defined by

? T I 7) = 0.1. s 1. 1 )The symmetnc tndiagonal matrix T S. i - , is defined by

. j = / - l

0.where1 j = 0 . 1 . . .A\T 1.

I j > I

Matrix T S. t 7 )s a normal symmetric Jacobian matrix. It can beshown that T S . -1-)commutes with H iI-. i - , .e.H ~ ~ ~ , I i ~ ) ~ - I - . I i , . )T S.I17)H S.IT-) 3)

so all the eigenvectors of T S, t 7 are also the eigenvectors ofH 1V.IY 7].

In previous work [SJ the QR algorithm was used to find theeigenvectors DPSSs) of both T LT t7 and H A-, 14. . As expected,the same results were obtained from each matrix for smallHowever, for > 32 , H N.W ecomes ill-conditioned and itscomputed eigenvectors are no longer the true DPSSs. Thus, onlymatrix T N ,W) will provide reliable DPSS generation for large .Y.The only disadvantage of using T N s that its eigenvaluesdo not indicate true bandwidth concentrations of the DPSSs asdo the eigenvalues of H AV,lj7) .he remainder of this paperwill concentrate on DPSS generation from the tridiagonal matrixT X ,IY .

Using the QR algorithm or a variation of it generally requires theentire set of eigenvectors to be found. In various possible applicationsof DPSSs, only a select number of DPSSs will be desired. In thesecases, it would be beneficial to have a different algorithm which findsonly a single DPSS. The following d escribes a procedure for finding asingle DPSS (eigenvector) associated with the kth largest eigenvalueof normal symm etric Jacobian m atrices such as T V. W ) . n general,the algorithm first finds the desired eigenvalue, and the correspondingeigenvector is found using iterative techniques.Because T V. ) s originally in symmetric tridiagonal form,the well known method of bisection may be used to find a singleeigenvalue with potentially smaller relative error than if multipleeigenvalues were found using QR or QL iterations (p. 439 of [ 9 ] .Let the elements of T be denoted by

h,-1l oThe characteristic polynomial of the leading r x T principal submatrix,T,, of T is known to be

4)If we set po x ) = 1 and p l s) Q 1 s, hen a simple determinantalexpansion (p. 437 of [9] reveals the recursive relationship

p p r )= a , ) p T - 1 . r ) b:-lpr--2 .r)r = 2 . . .n. 5 )The roots of pn x ) are the eigenvalues of T , and bisection may

now be used to find any or all of the roots. However, finding aspecified eigenvalue, say the kth largest, also requires the use ofa theorem called the Sturm sequence property (p. 438 of [9]). Theportion of the theorem to be used here states that the number of sign

p r r ) = det Tr r 1 ) r = 1 . 2 . . I ? .

Fig. 1. DPSS d O ) N= 128,W = . l o ) .

S.,lp*

Fig. 2.changes n A ) in the sequence

DPSS d3) IV= 128,W = .lo .

PO A),Pl A),. ,Pn A))equals the number of eigenvalues that are less than A.TOL using the following algori thm:

while ]A, A, > TOLNow the kth largest eigenvalue may be found with an accuracy of

A k = Amax Am,n)/2if a A k ) 2 kAmax A kelse

end Amin = A kend.Of course, initially the interval [A,,,, A must contain the

desired eigenvalue. The interval in which all eigenvalues werecontained for a matrix T of size N was found using the Gershgorincircle theorem (p. 341 of [9] to be A E [ - y ,y ] , where

+ 0.25.N 1 * N + 2 + N * N + 18=Because the notation used by Slepian associates the most compactDPSS and largest eigenvalue with the index value k = 0 insteadof k = N 1, his reverse ordering must be accounted for whensearching for a specific eigenvalue.

Once the desired eigenvalue is found, the corresponding eigen-vector or DPSS may be found using the method of inverse iteration[lo]. Inverse i terat ion works fine for finding the eigenvector whichcorresponds to the known eigenvalue except when the eigenvalueis close in value to another eigenvalue, i.e., the eigenvalues areclustered. The procedure for inverse i terat ion is shown below forfinding the eigenvector associated with eigenvalue Ak.

let t i 1 be a unit vector

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0 0 8 -006O M -

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. I I NOVEMBER 1994

I20 40 6a 80 IW IN 140-0 2 SampleFig. 3. DPSS dl) .V = 12S,T.17 = . l o ) .

-0 1

0 20 40 6a 80 100 120 140Sample-041

Fig. 4. DPSS t 3 1 2 7 ) S = 128,i.I. = . l o )for i = 1 , 2 ,

Solve T k 1 1 1 2 = 11 for 1 - 2Normalize I ~o get a new 11 ,end.

The desired DPS S is the last normalized solution, P I . Even thoughT-XkI) is nearly singular for eigenvalues Xk-of very good accuracy,

inverse i terat ion is known to work well in these cases, many t imesrequiring only one or two i terations [ IO] . The norm of the residueT = T X k 1 v l may be used to determine the accuracy of thefinal result . When the known eigenvalue Xk was accurate to withinTOL = l e 6 using the bisection method explained previously,i t was found that three i terat ions of the above procedure producedaccurate DPSSs. Because matrix T 11. is t ridiagonal, solvingthe system equations during each i terat ion for the latest eigenvectorestimate was also simplified (p. 155 of [ 9 ] .

111. S O M EE XAMPL E SFigs. 1-5 show some examples of DPSSs generated using the

technique described above.IV. CONCLUSION

A method for accurate and computationally efficient generationof any single DPSS of large length was presented. The method iseasy to implement and should prove useful in si tuations where onlya few DP SSs out of a large set are desired. The availabil i ty of longlength DPSSs will open the oportunity for their use in applicationsrequiring sequences of long length.

0068

Sampk

Fig. 5. DPSS d ) N = 1000O.W = . l o ) .REFERENCES

[l ] A. D. Wyner, An analog scrambling scheme which does not expandbandwidth, part I: Discrete time, IEEE Trans. Inform. Theory, vol.IT-25, pp. 261-274, May 197 9.[2] - Signal design for PAM data transmission to minimize excessbandwidth, Bell Sysr. Tech. J . vol. 57, pp. 3277-3307, Nov. 197 8.[3] A. K. Jain and S. Ranganath, Extrapolation algorithms for discretesignals with application in spectral estimation, IEEE Trans. Acoust.,Speech, Signal Processing, vol. ASSP-29, pp. 830-845, Aug. 1981.[4] J. D. Mathews, J. K. Breakall, and G. K. Karawas, The discrete prolatespheroidal filter as a digital signal processing tool, IEEE Trans. Acousr.,Speech, Signal Processing, vol. 33, pp. 1471-1478, Dec. 1985.[5] D. Slepian and H. 0. Pollak, Prolate spheroidal wave functions, Fourieranalysis and uncertainty -1 Bell Sysr. Tech. J . vol. 40, pp. 43-64,Jan. 1961.[6] D. Slepian, Prolate spheroidal wave functions, Fourier analysis anduncertainty-V: The discrete case , Bell Sysr. Tech. J . vol. 57, pp.1317-1430, May 1978.[71 F. P. Gantmacher, Matrix Theory.[81 D. M. Gruenbacher, Generation of discrete prolate spheroidal se-quences for signaling waveforms, M.S. thesis, Kansas State Univ.,Manhattan, KS, Oct. 1991.191 G. H. Golub and C. F. Van Loan, Matrix Comoutations. Baltimore:

New York: Chelsea, 1959, vol. I .

. _Johns Hopkins Univ., 1989, 2nd ed.[101 G. Peters and J. H. Wilkinson, Inverse iteration, ill-conditioned equa-tions and Newtons method, SIAM Rev., vol. 21, pp. 339-360, July1979.

Noniterative and Fast Iterative Methodsfor Interpolation and ExtrapolationPaulo Jorge S . G. Ferreira

Abstract-In this correspondence we study the band-limited interpola-tion and extrapolation problems for finite-dimensional signals. We showthat these problems can be easily reduced to the solution of a set oflinear equations with a real symmetric positive-definite matrix S withspectral radius p S ) < 1. Thus, the equations can be solved directly orusing successive approximation methods. A number of other well knownmethods which may substantially increase the convergence rate may alsobe readily applied and are briefly discussed. We state conditions for theirconvergence, and illustrate their performance through an example.Manuscript received May IO 1994; revised June 6, 1994. The associateeditor coordinating the review of this paper and approving it for publication

was Prof. Stanley J. Reeves.The author is with the Departamento de Electrhica e TelecomunicaGoes,Universidade de Aveiro, Instituto de Engenharia d e Sistemas e Computadores,3800 Aveiro, Portugal.IEEE Log Number 9404788.

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