One can obtain either exact realizations or useful approximations of linear systems or matrix-vector products that arise in many different applications by implementing them in the form of multistage or multichannel fractional Fourier-domain filters, resulting in space-bandwidth-efficient systems with acceptable decreases in accuracy. Varying the number and the configuration of filters enables one to trade off between accuracy and efficiency in a flexible manner. The proposed scheme constitutes a systematic way of exploiting the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent. © 1998 Optical Society of America

Arikan, O.

Candan Ç.

Erden, M.F.

Güleryüz Ö.

Kutay, M.A.

Ozaktas, H., M.

Bilkent University Institutional Repository

July 15, 1998 / Vol. 23, No. 14 / OPTICS LETTERS 1069Space–bandwidth-efficient realizations of linear systemsM. Alper KutayDepartment of Electrical Engineering, Bilkent University, TR-06533 Bilkent, Ankara, TurkeyM. Fatih ErdenSignal Processing Group, Tampere University of Technology, P. O. Box 553, Tampere, FinlandHaldun M. Ozaktas, Orhan Ar ıkan, Özgür Güleryüz, and Çaǧatay CandanDepartment of Electrical Engineering, Bilkent University, TR-06533 Bilkent, Ankara, TurkeyReceived March 6, 1998One can obtain either exact realizations or useful approximations of linear systems or matrix–vector productsthat arise in many different applications by implementing them in the form of multistage or multichannelfractional Fourier-domain filters, resulting in space–bandwidth-efficient systems with acceptable decreases inaccuracy. Varying the number and the configuration of filters enables one to trade off between accuracy andefficiency in a f lexible manner. The proposed scheme constitutes a systematic way of exploiting the regularityor structure of a given linear system or matrix, even when that structure is not readily apparent.  1998Optical Society of AmericaOCIS codes: 120.2440, 070.6110.Let f srd denote the input and gsrd ­RH sr, r0df sr0ddr0the output of an arbitrary linear system characterizedby the kernel H sr, r0d, where r ­ sx, yd. The samesystem can be written in operator notation as g ­ H f .In some applications, such as image enhancement, wemay wish to implement a linear system that is deliber-ately designed to impart a certain effect to the input.In others, such as image restoration or reconstruction,linear systems are used to recover a desired imagefrom whatever data or measurements are available.If we are working with images whose space–bandwidth products are N ­pN 3pN , the abovesystem can be approximated with the discrete formgkl ­PpNj­1PpNi­1 Hklijfij . This expression can repre-sent either a tensor or a matrix algebra operation thatwe wish to implement or an approximation of its con-tinuous counterpart. Digital implementation of suchgeneral linear systems takes OsN2d time. Commonsingle-stage optical implementations, such as opticalmatrix–vector multiplier or multifacet architectures,1require an optical system whose space–bandwidthproduct is OsN2d.The output of a space-invariant system character-ized by the impulse response hsrd is related to the inputby the convolution relation gsrd ­Rhsr 2 r0df sr0ddr0,which can again be expressed in tensor or matrix nota-tion, but this time the tensor or matrix is of a specialform (Toeplitz). Digital implementation of such sys-tems take OsN log N d time (by use of a fast Fouriertransform). Optical implementation requires an opti-cal system whose space–bandwidth product is OsN d.Because of the intrinsic nature of some problems,convolution-type systems are fully adequate. How-ever, in other cases, the use of such systems is eithertotally inappropriate or at best yields a crude approxi-mation. We may think of space-invariant systems0146-9592/98/141069-03$15.00/0and general linear systems as representing two ex-tremes in a cost–accuracy trade-off. Sometimes theuse of space-invariant systems may be inadequate,but at the same time the use of general linear sys-tems may be overkill and prohibitively costly. In thisLetter we consider multistage and multichannel frac-tional Fourier-domain filtering configurations that in-terpolate between these two extremes, offering greaterf lexibility in trading off between cost and accuracy.Common single-stage frequency-domain multiplicativefiltering is shown in Fig. 1(a). The dual of this op-eration is single-stage space-domain multiplicative f il-tering and is shown in Fig. 1(b). Figure 1(c) depictssingle-stage multiplicative filtering in the ath-orderfractional Fourier domain. This filtering configura-tion and its applications are discussed in Refs. 2–5.The ath-order fractional Fourier transform6,7 of f sxdis denoted as F af sxd. Then, F 0f sxd ­ f sxd itself,F 1f sxd ­ F snd, the ordinary Fourier transform, andF a2F a1 ­ F a21a1 . The ath fractional Fourier domainmakes an angle a ­ apy2 with the space domain in thespace–frequency plane [Fig. 1(d)].6,8 Generalizationto two dimensions is straightforward.9In the multistage system [Fig. 1(e)] that was sug-gested in Ref. 6, the input is f irst transformed into thea1th domain, where it is multiplied by a filter h1srd.The result is then transformed back into the originaldomain. This process is repeated M times. The mul-tichannel f ilter structure consists of M single-stageblocks in parallel [Fig. 1(f)]. For each channel k, theinput is transformed to the akth domain, multiplied bya filter hksud, and then transformed back.Here, we let Lj denote the operator correspondingto multiplication by the filter function hj srd. Thenthe outputs gs and gp of the serial and the parallelconfigurations, respectively, are related to the input f 1998 Optical Society of America1070 OPTICS LETTERS / Vol. 23, No. 14 / July 15, 1998Fig. 1. (a) Fourier-domain f iltering. (b) Space-domainfiltering. (c) ath-order fractional Fourier-domain fil-tering. (d) ath-order fractional Fourier domain. (e)Multistage (serial) filtering. (f) Multichannel (parallel)filtering.bygs ­ sF 2aM LM . . . F a22a1L1F a1df , (1)gp ­µ MXk­1F 2ak LkFak∂f , (2)where F aj represents the aj th-order fractional Fouriertransform operator. Equations (1) and (2) can alsobe expressed as g ­ T f , where T is the operatorrepresenting the overall f iltering configuration.Both filtering configurations have at most MN 1 Mdegrees of freedom. Their digital implementationwill take OsMN log N d time, since the fractionalFourier transform can be implemented in OsN log N dtime.10 Optical implementation will require anM-stage or an M-channel optical system, each withspace–bandwidth product N .11 These configurationsinterpolate between general linear systems and shift-invariant systems in terms of both cost and f lexibility.If we choose M to be small, cost and f lexibility areboth low. If we choose a larger M , cost and f lexibilityare both higher; as M approaches N , the number ofdegrees of freedom approaches that of a general linearsystem.Increasing M allows us to approximate a givenlinear system better. For a given value of M , wecan approximate this system with a certain degreeof accuracy (or error). For instance, a shift-invariantsystem can be realized with perfect accuracy withM ­ 1. In general there will be a f inite accuracy foreach value of M . As M is increased, the accuracy willusually increase but never decrease, because systemswith larger values of M include systems with smallervalues of M as their special cases. In dealing with aspecific application, we can seek the minimum value ofM that results in the desired accuracy or the highestaccuracy that can be achieved for a given M . Thusthese systems give us considerable freedom in tradingoff efficiency and accuracy, enabling us to seek the bestperformance for a given cost or the least cost for a givenlevel of performance.Naturally, the value of M required for attaining agiven accuracy will be smaller for system whose ker-nels exhibit greater regularity. In such cases directimplementation of the linear system is clearly inef-ficient. The inherent structure can be exploited ona case-by-case basis through ingenuity or invention;the most-efficient algorithms and optical transformsare obtained in this manner. In contrast, our methodprovides a systematic way of obtaining an eff icientimplementation that does not require ingenuity on acase-by-case basis. This approach would be especiallyuseful when the regularity or the structure of the ma-trix is not simple or immediately apparent. A distinctcircumstance in which the method can be beneficial iswhen it is suff icient to implement the linear systemwith limited accuracy, as may be the case when someother component of the overall system limits the accu-racy to a lower value anyway, or simply when the ap-plication itself demands limited accuracy.The scheme suggested here offers improvementsin both digital and optical systems. We expect it tobe especially beneficial in optical systems, since thecost of optical components is a strong function of theirspace–bandwidth product. Direct single-stage imple-mentation of linear systems that use matrix–vectorproducts or multifacet architectures may be totallyunfeasible (a 256 3 256 image would require a space–bandwidth product of 2564). Furthermore, the in-trinsic amplitude- or intensity-level accuracy of analogoptical systems is usually limited to quite-modestdynamic ranges. Given this limited accuracy, it ispointless to try to implement the desired linear systemby use of an expensive scheme that could in principleaccommodate much greater accuracies (such as a con-ventional matrix–vector multiplier architecture thatmaps an exact matrix–vector product). Our proposalallows one to approximate the desired linear system toan accuracy that just matches the intrinsic accuracy ofanalog optical systems in a manner that reduces thecost as much as possible. When we employ analogoptical systems to image digital optical signals (e.g.,free-space optical interconnections), even-lower accu-racy is tolerable. Here it is possible to make do witheven-smaller numbers of stages, since because of thedigital nature of the signals even greater deviation ofthe obtained matrix T from the specified matrix H canbe tolerated while an acceptable eye pattern is main-tained. For these reasons, the proposed scheme wouldbe especially useful in optical systems, including analogsignal-processing systems, matrix processors, numeri-cal processors, algebraic processors, and opticalinterconnections.The proposed system can be used in one of twodistinct ways.12,13 Here we have space for only one ofthem: We take Eq. (1) or (2) as a constraint on theJuly 15, 1998 / Vol. 23, No. 14 / OPTICS LETTERS 1071Fig. 2. (a) Original image. (b) Blurred image. (c) Imagerestored by multistage f iltering. (d) Image restored bymultichannel f iltering.form of the linear-estimation matrix. Given a specificoptimization criterion, such as minimum mean-squareerror, we find the optimal values of aj and hj such thatthe given criterion is optimized.The repeated f iltering configuration is discussed atlength in Refs. 12 and 13, in which many examples aregiven of instances in which it is possible to obtain usefulapproximations by use of only a moderate number ofstages. Here we must satisfy ourselves with threeexamples.Our first example12 is the restoration of signalsthat were degraded by space-variant atmospheric tur-bulence by use of repeated filtering. The resultingmean-square error between the actual signal and itsestimate is found to be 1% with M ­ 4.In our second example we consider restoration ofimages blurred by a space-variant point-spread func-tion (Fig. 2). The mean-square estimation error is10% in the multistage case and 13% in the multichan-nel case for M ­ 4 and 3.7% in the multistage case and5.5% in the multichannel case for M ­ 8. OrdinaryFourier-domain filtering gives poor results, resultingin an error of 34%.In this example the orders were chosen to be equallyspaced between 0 and 1. Optimizing over the orderswill always yield better results but can be diff icultwhen M is large. (How to choose a set of ordersa1, a2, . . . , aM that are suitable for given applicationclasses or have certain desirable properties is thesubject of current research.)Finally, we consider the problem of recovering a sig-nal consisting of multiple chirplike components buriedin white Gaussian noise with a signal/noise ratio of 0.1.We assume that the signal consists of six chirps withuniformly distributed random amplitudes and timeshifts and that the chirp rates are known with a 65%accuracy. Use of M ­ 6 filters results in an estima-tion error of 2.6%.We employed an iterative algorithm to determinethe optimal filters hj that minimize mean-squareerror. The repeated system is highly nonlinear inthe filter coeff icients and does not admit of an exactsolution. The multichannel system is linear in thefilter coefficients and has an exact solution, but inpractice an iterative solution is preferred.The multistage and multichannel configurationsmay be based on other transforms with eff icient im-plementations, such as linear canonical transforms.14Concentrating on Eq. (2), the essential idea is toapproximate a general linear operator as a linearcombination of operators with fast algorithms orspace–bandwidth-efficient implementations. If anacceptable approximation can be found with a value ofM that is not too large, the cost can be significantlyreduced.The serial and parallel filtering configurations canbe combined in an arbitrary manner to give what weterm generalized filtering configurations or circuits.What types of circuit may be beneficial in what circum-stances is an area for future research.The authors acknowledge the contributions of HakanOzaktas to the solution of the optimization problemsdiscussed in this Letter.References1. D. Mendlovic and H. M. Ozaktas, Appl. Opt. 32, 5119(1993).2. M. F. Erden, H. M. Ozaktas, and D. Mendlovic, Opt.Commun. 125, 288 (1996).3. Z. Zalevsky and D. Mendlovic, Appl. Opt. 35, 3930(1996).4. M. A. Kutay, H. M. Ozaktas, O. Ar ıkan, and L. Onural,IEEE Trans. Signal Process. 45, 1129 (1997).5. M. A. Kutay and H. M. Ozaktas, J. Opt. Soc. Am. A 15,825 (1998).6. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L.Onural, J. Opt. Soc. Am. A 11, 547 (1994).7. L. B. Almeida, IEEE Trans. Signal Process. 42, 3084(1994).8. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).9. A. Sahin, H. M. Ozaktas, and D. Mendlovic, Appl. Opt.37, 2130 (1998).10. H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G.Bozdagi, IEEE Trans. Signal Process. 44, 2141 (1996).11. H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A12, 743 (1995).12. M. F. Erden, ‘‘Repeated f iltering in consecutive frac-tional Fourier domains,’’ Ph.D. dissertation (BilkentUniversity, Ankara, Turkey, 1997).13. M. F. Erden and H. M. Ozaktas, J. Opt. Soc. Am. A 15,1647 (1998).14. B. Barshan, M. A. Kutay, and H. M. Ozaktas, Opt.Commun. 135, 32 (1997).

Space-bandwidth-efficient realizations of linear systems

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Space-bandwidth-efficient realizations of linear systems

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