Array Design for Superresolution Diretion-Finding Algorithms Naushad Hussein Dowlut BEng, ACGI, AMIEE Athanassios Manikas PhD, DIC, AMIEE, MIEEE Department of Eletrial Eletroni Engineering Imperial College of Siene, Tehnology Mediine Exhibition Road, London SW7 2BT, U.K. Abstrat. It is well-known that the sensor harateristis their physial arrangement plae fundamental limitations on the ultimate apabilities of an array system. In this paper, the design of linear arrays, tailored to the so-alled superresolution diretion-finding (DF) algorithms, is onsidered. The proposed design approah is based on the differential geometry of the array manifold. The novel onept of a sensor loator polynomial is presented its properties are investigated. Then it is shown how this onept an be applied to the design of a linear array with prespeified Cramer-Rao bounds on the aimuth estimates, without reourse to optimisation. 1. Introdution Diretion-finding array systems are mainly onerned with the estimation of the diretions-of-arrival (DOAs) of multiple emitters, present in the array environment, based on the statistis of the reeived signals. Currently, the most powerful DF algorithms are the soalled signal subspae algorithms, whih aptotially (i.e. infinite data or infinite Signal-to- Noise Ratio) exhibit infinite resolution apabilities. However, under non-aptoti onditions, their performane is influened by both the behaviour of the partiular algorithm employed the array geometry. It is therefore lear that areful array design is fundamental for the proper operation of superresolution DF systems. Array design for onventional beamforming tehniques, based on ahieving a gain pattern with a narrow mainlobe low sidelobes, is a well-researhed topi (see for instane [1]). Sensor plaement strategies tailored to the Maximum Entropy Maximum Likelihood Methods have also been proposed [2]-[3]. Array design for the powerful signal subspae algorithms, on the other h, has not been thoroughly investigated. The only realled attempt is one due to Spielman et al. [4], who showed that in the single-target sene, the design proedure for the MuSIC algorithm is equivalent to onventional beampattern design. Note that all the design proedures reviewed involve some kind of nonlinear optimisation. Signal subspae estimation tehniques onsist essentially of finding the intersetions between the signal subspae the array manifold, whih haraterises the array response over the entire parameter spae, e.g. aimuths to for the linear array ase. The array manifold is atually a vetor ontinuum lying in the omplex -dimensional spae, is the number of sensors, is determined by the sensor positions harateristis. In this paper, a new array design methodology based on the differential geometry of the array manifold is proposed. In Setion 2, the notational onventions used throughout this paper are introdued the signal model for the narrowb far-field passive array signal proessing problem is formulated. A formal definition of the array manifold is also presented. Setion 3 ontains some results of the appliation of differential geometry to the array manifold of a linear array of sensors ([], [6]). The new onept of a sensor loator polynomial is presented in Setion 4 it is shown that if the length urvatures of a manifold are known, the orresponding array an be synthesised from the roots of this polynomial. The sensor loator polynomial is then examined for various array strutures in Setion, with speial attention paid to the fully metri fully ametri linear array ases. In Setion 6, the well-known Cartan Matrix in differential geometry is shown to have a speial meaning to the array synthesis problem. The pratial problem of designing an array when the number of sensors, the desired performane levels are prespeified is examined in Setion 7. Finally, the paper is onluded in Setion 8. Address for orrespondene : Dr A. Manikas, Department of Eletrial & Eletroni Engineering, Imperial College of Siene, Tehnology Mediine, London SW7 2BT, U.K. e-mail : a.manikas@i.a.uk International Symposium on Digital Signal Proessing, ISDSP-96, London. 38
2. Notation Problem Formulation The following notation is used throughout the paper : salar vetor [ matrix / hermitian transpose ; transpose exp elemental exponential fix integer part elemental power * * absolute value of elements + + Eulidian norm [ power of matrix is a funtion of 4 number of soures sensor loations number of sensors normalised 3 number of snapshots ar length -dimensional real spae urvature -omplex spae ] Cartan matrix oordinate vetor The d omplex signal vetor % observed at the output of an array of sensors operating in the presene of 4 far-field narrowb emitters impinging from the aimuth-diretions Ä4 # ; with respet to the array axis, additive noise an be modelled as : % b [ 1 4 is the vetor of omplex signal d4 envelopes [ is the d4 matrix defined as : exponential. # [ Ä 4 2 with denoting the omplex array response to a unit amplitude wavefront from diretion. For a linear array of omnidiretional sensors with positions N given by the real vetor (in units of halfwavelengths), the response vetor is given by : ²³ exp os 3 exp denotes the element by element Definition 1 The array manifold is defined as the lous of all response vetors $ % it desribes a urve in. The array manifold ompletely haraterises an array's spatial response therefore reflets the intrinsi apabilities of the array independent of the type of proessing employed. In this paper, the proposed design approah determines the sensor loations of a linear array given ertain performane requirements whih an be translated into desired harateristis of the array manifold. 3. Differential Geometry of Array Manifold The array manifold is onventionally parameterised in terms of the aimuth angle ; however for the purposes of studying the geometry of the urve, parameterisation in terms of its ar length, whih is the atual physial length in multidimensional spae, is more appropriate. The ar length is formally defined as : j j 4 or alternatively, + +. denotes differentiation with respet to parameter. In the ase of a linear array of isotropi sensors with loations (in units of half-wavelengths), it an be shown from Equations 3, 4 that + + os 6 sin 7 ++ is ased. The the initial ondition rate of hange of ar length,, is a loal property of the urve plays a ruial role in ditating the detetion resolution apabilities of the array [6]. Another important parameter of the array manifold urve is its total length, ++ 8 whih inreases in proportion to the sensor spaings. This parameter is important in the identifiation of manifold ambiguities of a linear array; if the manifold length is greater than the length of one winding ( odd) or one half-winding ( even) [], then spurious intersetions between the array manifold the signal subspae is possible. Furthermore, at every point along the manifold urve, a set of unit oordinate vetors o Ä # d urvatures $ Ä % an be defined, is the dimensionality of the manifold. The oordinate vetors urvatures are related aording to o o ] 9 denotes differentiation with respet to parameter ] is the Cartan matrix whih is a real skew-metri matrix of the urvatures defined as follows : x s Ä { ] s Ä s Ä 10 } Å Å Å Å Å y s J + + with 11 International Symposium on Digital Signal Proessing, ISDSP-96, London. 39
From Equation 9 bearing in mind that the oordinate vetors are of unit length, the following expressions for the first three manifold urvatures, for example, may be derived for a linear array of isotropi sensors : ++ h h + + h h b 12 + b + i 4 b i while in general, it an be shown that the manifold urvature an be alulated aording to the following reursive equation [] : 13 %² ³b ²³ Ä b (normalised sensor positions) + + ² ³( i.e. phase referene array entroid) In Equation 13, the oeffiients are given by : b b Ä Ä, b b 14 or reursively, b, 1 h with the initial onditions : 16 Note that the urvatures of a linear array of isotropi sensors depend on the relative rather than the absolute sensor spaings are independent of the ar length parameter. This implies that the manifold urve has the shape of a irular hyperhelix lying on a omplex - dimensional sphere of radius l in. 4. The Sensor Loator Polynomial It is known that for a urve embedded in a - dimensional spae,, only manifold urvatures exist, i.e. the urvature vanishes higher order urvatures are not defined. In the ase of a linear array of sensors, with metrial sensors about the entroid, it an be shown that : if E a sensor at array entroid H 17 otherwise Note that a sensor at the entroid ounts as a metrial sensor, i.e., therefore is always even, hene the ero urvature is always of even order. h Before proeeding further, linear arrays will be divided into three ategories : 1) Fully metri All the sensors our in metrial pairs about the entroid (e.g. Figure 1). 2) Partially metri At least one sensor has a metrial ounterpart about the entroid (e.g. Figure 2). 3) Fully ametri No sensor has a metrial ounterpart about the entroid (e.g. Figure 3). In the fully metri ase, for example, in Equation 17 is equal to, the number of sensors. Based on Equation 13, in onjuntion with the fat that, it an be shown that the roots of the polynomial : b Äb b 18 are the normalised sensor positions. This polynomial named the Sensor Loator Polynomial (SLP) has oeffiients given by Equation 14, for example, b The roots of the sensor loator polynomial will always our in pairs of opposite signs (sine the SLP onsist of only the even powers of ). Then, in the ase of fully ametri partially metri arrays, the roots will atually represent two arrays whih are mirror images of eah other, but whose manifold urvatures are idential. It is obvious in the fully metri ase that the roots will form a single array sine the array its mirror image are idential. In the fully ametri ase, the set of roots will onsist of two disjoint subsets orresponding to the two mirror arrays. In the partially metri ase, however, the two subsets will overlap at the metri sensors. Examples 1, 2 3 at the end of this setion, help to larify these points. The framework results of the previous disussion an be expressed in the following theorem : $ % Theorem 1 Given all the urvatures Ä the length of a manifold, then the loations of the elements of the array ( its mirror image) an be estimated from the following expression array/mirror array/mirror array mirror are two subsets of the set of the roots of the following polymomial 19 International Symposium on Digital Signal Proessing, ISDSP-96, London. 40
b b Äb with Ä Ä b b b b array r mirror + array + + mirror + array mirror array mirror Example 1 : Fully Symmetri Linear Array The normalised sensor positions of the fully metri linear array of Figure 1 are :,, # its manifold urvatures, using Equation 13, are : # The SLP alulated from the above urvatures is given by : with roots : array # whih are idential to the normalised sensor positions of the array. Note that the sensor at the entroid does not appear as a root sine it does not affet the urvatures. However, its absene might give rise to ambiguities in the array manifold. Example 2 : Partially Symmetri Linear Array Figure 2 depits a partially metri linear array, whose normalised sensor loations urvatures are respetively given by : # # The SLP is given by : with roots : f f f f # The subsets of the roots, whih satisfy the onditions given by Equation 21 are : # array mirror, # Clearly, the two subsets overlap. Note that the minor disrepanies between the roots of the SLP the atual normalised sensor loations are due to numerial error aused by rounding off the urvatures the polynomial oeffiients. Example 3 : Fully Ametri Linear Array The fully ametri linear array of Figure 3 has normalised sensor positions urvatures given by : # # respetively. The SLP orresponding to the above urvatures is : 21 with roots : f f f f # the subsets satisfying the onditions of Equation 21 are : array # mirror # Note that in this ase, the subsets are disjoint. The omputation of the oeffiients of the sensor loator polynomial using Equation 14 requires all the urvatures. However, in many situations only a limited set of urvatures may be known or an be estimated from the problem speifiations. For instane, we will see in Setion 7 that, sometimes, only the first urvature an be estimated from the problem speifiations. Therefore, to allow the use of the sensor loator polynomial to situations a subset of the urvatures is known, new expressions for the oeffiients of the sensor loator polynomial, aommodating the inomplete knowledge of all the urvatures, need to be derived. This is possible only if the designs are restrited to partiular onfigurations a priori knowledge about the sensor positions is then exploited to derive simpler expressions. In the next setion, it is shown that for the speial ases of fully metri fully ametri arrays, it is possible to ompute the sensor loator polynomial oeffiients with a small subset of the urvatures.. Fully Symmetri Fully Ametri Arrays Consider the SLP b b bäb 22 b Ä 23 is interpreted as is a funtion of. The problem is to evaluate the oeffiients when not all of the urvatures are known. A useful theorem by Newton [7], states that : The s of the similar powers of the roots of an equation an be expressed rationally in terms of the oeffiients. Using the onverse of the above theorem, it an be shown that the oeffiient of the SLP is given by : ²³ 24 2 International Symposium on Digital Signal Proessing, ISDSP-96, London. 41
i.e. is the of the power of all the positive or negative roots (sine is always even) of the SLP. Note that Ä 26 Interestingly, the s of similar powers of the normalised sensor loations,, an be expressed in terms of the manifold urvatures, as an be realised by exping the expliit expressions in Equation 12 : 4 4 4 b b 4 b it an be proved that, in general, b > D 4 Ä E? ] 28 This implies that the of the power of the normalised sensor loations an be alulated as the first term of the power of a matrix with idential struture to the Cartan matrix (Equation 10) but ontaining only the first manifold urvatures. It is worth noting that 27 4 Ä 29 The next step is to find out the relationship between. As previously mentioned, the roots of the SLP represent a normalised array together with its mirror image about the entroid, hene it an be dedued that for the fully metri fully ametri arrays : a therefore, from Equations 26 29, Ä I 30 31 i.e. the oeffiient of the SLP of a fully metri or fully ameti array is a funtion of only the first manifold urvatures. In partiular, Ä 4 hene : Theorem 2 The SLP of a fully metri or fully ametri array an be formed using only the first of the manifold urvatures. However, in the partially metri ase, there exists no suh relationship as Equation 30 between p, beause of the overlap of the roots, therefore, Equation 24 is not appliable. Considering the speifi example of, defined as : ++ 32 it an be easily verified that, a 33 p Note that the exat value of determined. p annot be Using Equations 24, 27 30, the first three oeffiients of the SLP for a fully metri array are given by : 4 4 4 b b b b b 34 6. Sensor Loations the Cartan Matrix The Cartan matrix is a real skew-metri matrix of the manifold urvatures, with a speial struture as illustrated in Equation 10. The Cartan matrix therefore ontains all the information about the loal behaviour of the manifold, in the ase of a linear array of isotropi sensors, it atually desribes the whole manifold-hyperhelix, sine the urvatures are onstant independent of the ar length parameter. Sine the array manifold is in turn determined by the sensor positions harateristis, it an be inferred that the Cartan matrix should also ontain this information. The question is how it an be easily retrieved. It an be proved that there exists a diret relationship between the eigenvalues of the Cartan matrix the normalised sensor positions, as stated in the following theorem : $ % Theorem 3 Given all the urvatures Ä of a manifold, the roots of the orresponding sensor loator polynomial are simply given by : deig ] 3 eig ] represents the eigenvalues of the Cartan matrix. The roots an then be partitioned into the normalised array-mirror pair saled to the speified manifold length by using Equations 21 20 respetively. In mary, if all the manifold urvatures are known, then it is more onvenient to use Theorem 3, rather than the SLP, to design the array. However, if only a limited number of urvatures is available, as illustrated by the example in the next setion, then the design should be based on the SLP onept. International Symposium on Digital Signal Proessing, ISDSP-96, London. 42
7. Array Design Based on the CR Bound The most popular bound in array proessing is the Cramer-Rao Bound (CRB), whih represents the minimum estimation error variane ahievable by any unbiased estimator. In other words, the CRB is estimator-independent, it an be shown to depend on the number of snaphots 3, the Signal-to-Noise ratio, SNR, the sensor plaement harateristis. Its use for the design of arrays for superresolution algorithms is justified by the fat that it is aptotially attainable by the MuSIC algorithm [8]. The expliit expressions for the CRB in terms of the array manifold harateristis, hene, the array geometry, derived in [9], will be used Asing the phase referene to be taken at the array entroid, the differential geometri versions of the one-soure two-soure CRB's are respetively given by : 4 ++* os os* CRB 36 LSNR CRB 37 3d SNR 38 Note that Equation 37 is an exellent approximation of the exat CRB expression proposed in [8] for two lose emitters at b respetively. Next, onsider the following design example : Design a metrial -sensor linear array whih exhibits a one-soure CRB of on the stard deviation of the DOA estimate of an emitter at, a two-soure CRB of on the stard deviation of the DOA estimate of the same emitter, in the presene of a seond emitter at, using a number of snapshots SNR produt, 3 d SNR ² e.g. 3 SNR ³. Note that under the same onditions, a stard - sensor uniform linear array, would exhibit lcrb l CRB. From the design speifiations in onjuntion with Equations 36 37, the following properties of the array to be designed an be alulated : H ++ 39 it is lear that in this ase only the first urvature is available. The sensor loator polynomial, omputed using Equations 34, is given by the following expression : 40 with roots f f#, whih, together with a sensor at the origin, atually onstitute the normalised version of the desired array. After saling, the designed array is : # ; As a hek of the validity of the design proedure, the CRBs of the proposed array are omputed using the exat expressions derived in [8] : lcrb l CRB are found to math the speifiations very losely. 8. Conlusion The important problem of linear array design tailored to superresolution DF algorithms has been addressed. The design approah is based on the properties of the array manifold, whih reflets the intrinsi apabilities of the array. The proposed onepts were supported by a number of examples the onnetion between the sensor loator polynomial the Cartan matrix was established. Current work is foussed on extending the onepts presented in this paper deriving new superresolution design rules for general array geometries. Referenes [1] Y. T. Lo, Aperiodi Arrays, in Antenna Hbook, Theory, Appliations, Design, Y. T. Lo S. W. Lee, Editors, Van Nostr, New York, 1988. [2] S. W. Lang, G. L. Dukworth J. H. MClellan, Array Design for MEM MLM Array Proessing, IEEE ICASSP-81 Proeedings, Vol. 1, pp. 14-148, Marh 1981. [3] X. Huang, J. P. Reilly M. Wong, Optimal Design of Linear Array of Sensors, IEEE ICASSP-91 Proeedings, pp.140-1408, May 1991. [4] D. Spielman, A. Paulraj T. Kailath, Performane Analysis of the MUSIC Algorithm, IEEE ICASSP-86 Proeedings, pp. 1909-1912, April 1986. [] I. Daos A. Manikas, The Use of Differential Geometry in Estimating the Manifold Parameters of a One-Dimensional Array of Sensors, Journal of the Franklin Institute, Engineering Applied Mathematis, Vol. 332B, No. 3, pp. 307-332, 199. [6] A. Manikas, H. R. Karimi, I. Daos, Study of the Detetion Resolution Capabilities of a One-Dimensional Array of Sensors by Using Differential Geometry, IEE Proeedings on Radar, Sonar Navigation, Vol. 141, No. 2, pp. 83-92, Apr. 1994. [7] W. S. Burnside A. W. Panton, Theory of Equations, Vols I & II, Dover Publiations In., New York, 1960. [8] P. Stoia A. Nehorai, MUSIC, Maximum Likelihood Cramer-Rao Bound, IEEE Transations on Aoustis, Speeh Signal Proessing, Vol. 37, No., pp. 720-741, May 1989. [9] H. R. Karimi A. Manikas, The Manifold of a Planar Array its Effets on the Auray of Diretion-Finding Systems, aepted for publiation in IEE Proeedings on Radar, Sonar Navigation, Apr. 1996. Figure 1. Fully Symmetri Linear Array Figure 2. Partially Symmetri Linear Array Figure 3. Fully Ametri Linear Array International Symposium on Digital Signal Proessing, ISDSP-96, London. 43