ANGLE OF OF ARRIVAL ESTIMATION WITH A POLARIZATION DIVERSE ARRAY

Transcription

1 ANGLE OF OF ARRIVAL ESTIMATION WIT A POLARIZATION DIVERSE ARRAY DR. DR. WILLIAM ILLIAM P. P. BALLANCE ** RALP ALP A. A. COAN COAN MARC * No it Te Boeing Company WPB-

2 Problem formulation OUTLINE Obseration model is linear in polarization Maximum Likeliood Estimator MLE) Te MLE is identical to te Beamform, Witen, and Energy Detect BWE) tecnique of Allan Steinardt Problem is a minor ariation on an already soled MLE problem, te single snapsot -source problem Example to proide intuitie feel for solution Cramer-Rao Bound CRB) on estimation performance Agreement beteen MLE and CRB asymptotic efficiency) Performance impact due to required estimation of polarization Performance implications of assumed polarization Angle of arrial AOA) estimate generally as reduced standard deiation but is potentially biased WPB-

3 PROBLEM FORMULATION: ARRAY OBSERVATION MODEL Standard array obseration model for an N-element 3-dimensional array is x d )a + n ere [φ;]. Te direction ector is gien by jωoτ n ) [ d )] f )e ; n,, N n n ere f n ) is te complex-alued element pattern and te pase term is due to element separation. Te unknon model parameters are and a, for a total of 4 real-alued unknons. Te generalization for a polarization dierse array is simply x d, )a + n ere te polarization state is gien by [ ; ]. Te direction ector is gien by jωoτ n d, ) f, )e z c [ ] ) n n x φ c y Azimut φ 9 o - φ c, Eleation 9 o - c Time sift τ n ) n n + n c x sinφ cos + y cosφ cos z sin ) ere x n,y n, z n ) is te location of te n t element. WPB-3

4 PROBLEM FORMULATION: POLARIZATION SPECIFICS Electric field of a polarized ae traces an ellipse as time progresses E φˆ ˆ Transerse Plane ˆ β φˆ ere + Te real-alued quantities γ and η determine te ellipse orientation β and axial ratio r minor axis)/major axis). E cosγ E sinγ e jη Examples orizontal Vertical Linear RCP LCP Elliptical WPB-4

5 PROBLEM FORMULATION: POLARIZATION SPECIFICS CONT.) Just as electric field can be represented in terms of a basis set of polarizations e.g., orizontal and ertical, RCP and LCP, an elliptical polarization and its ortogonal polarization pol and cross-pol )), so can te element pattern. f n, ) f ) +,n f, n ) So tat d, ) d ) + d ) ere jωoτ n ) jωoτ n ) [ d )] f )e, [ d )] f ) e n,n Te unknon model parameters are, γ, η, and a, for a total of 6 real-alued unknons. Te model is linear in polarization en ieed in terms of te parameters and. n,n WPB-5

6 Rater tan begin it te likeliood function, begin it te resultant standard ML performance index and ten proide te requisite incremental deriation Maximizing te likeliood function is equialent to maximizing te folloing performance index Te key is tat te direction ector is linear in te unknon polarization parameters d, ) d ) + d ) D ) So tat ere J, ), MLE DERIVATION: AN INCREMENTALIST APPROAC, ) x, ) R, ) D ) ere R d, ) [ d ) d )], W ) [ ) )] ), ) ) + ) W ) R d ), ) R d ) R D ), WPB-6

7 MLE DERIVATION: AN INCREMENTALIST APPROAC CONT.) Define te by ector output of a orizontal and ertical polarization beamformer as y, and te corresponding by coariance matrix by R yy. Making use of, ) W ), y W ) x, R yy ) W ) R W ) Te performance index may be ritten J, ), ) x, ) R, ) R y ) yy ) ic is a Rayleig quotient in te unknon polarization state. Te closed-form ML estimate of polarization is ten R yy y Substituting tis into J results in te simple expression J ) y ) R- yy y ) ence, te performance index need only be searced oer. Note tat J is identical to te BWE performance index and so te BWE is te Maximum Likeliood Estimator. Tis assertion as first made by Roland Stougton of SAIC.) WPB-7

8 MLE DERIVATION: SUCCINCT PROOF FROM FIRST PRINCIPLES Te model may be ritten as x d, )a + n a d ) + a d ) + n D )a + n ere a a a. a Tis model is analogous to te single snapsot -source model x a d ) + a d ) + n D, + )a n except tat D [d ) d )] and a [a ; a ] are replaced by D [d ) d )] and a [a ; a ], respectiely. Te deriation readily follos. Te likeliood function is gien by N L π det R ) exp{ x D )a) R x D )a) } Maximizing te likeliood function is equialent to minimizing te nonlinear least squares cost function C x D )a) R x D )a) Witen for notational conenience C x D )a WPB-8

9 MLE DERIVATION: SUCCINCT PROOF FROM FIRST PRINCIPLES CONT.) Te problem is a separable nonlinear least squares problem. It is nonlinear in and linear in a, alloing a closed form solution for a. Substitution yields C I P) x ere P D D D ) D is a projection matrix. Minimizing C is equialent to maximizing or C a D J x D a) a D D ) D x x J P x R D D R D ) Q.E.D. Te BWE is te MLE only for te deterministic model it a single snapsot. For multiple snapsots, te model is [ d ) + d )]a k, ic is akin to d )a,k + d )a,k ere a,k β a,k, te fully correlated multipat problem toug for many snapsots, a stocastic-model ML solution may be preferred). x D P x R x y R yy y WPB-9

10 MLE: COERENT VS. NONCOERENT COMBINATION Te MLE for knon polarization coerently combines in polarization., ) ) + ) W ) α ) x N α ) Te MLE for unknon polarization noncoerently combines. * * J x N y Witener R yy z J Te polarization estimate is gien by R But because te pase must be estimated, it cannot be exploited to aciee te same leel of gain as in coerent combination for knon polarization. WPB- γ tan / ), yy y η / )

11 MLE PROCESSING FLOW If adaptie beamforming is performed before oter coerent processing stages, ten cannels per beam must be carried. N Pre- Beamforming Coerent Processing N Beamforming - it R yy itening zl Ver Post-Beamforming Coerent Processing Post-Beamforming Coerent Processing z If an explicit polarization estimate is desired, it sould be computed after te last stage in te coerent processing cain. Te improed SNR ill improe te quality of te estimate. In te example aboe, te by data z after te post-beamforming coerent processing stage is used. Te estimate is R yy z WPB-

12 SIMPLE EXAMPLE: BASIC ARRAY CONFIGURATION Underlying teory olds for a general array, it arbitrary element polarization responses and element locations. Tis includes elements tat are separated by many aelengts, as often found in space telescope applications. To proide an intuitie feel for te concepts inoled, oeer, consider a ery simple example. linear segments of dipoles, 3 elements eac, alf-ae spaced. 45 degree slope. Dipoles oriented along segment axes. Various segment separations considered. Direction ectors for broadside o azimut, o eleation) are T T d )[ ], d )[ ] 8 Element Locations 6 z aelengts) x aelengts) WPB-

13 SIMPLE EXAMPLE: RESPONSE OF ORIZONTAL & VERTICAL BEAMFORMERS Separate te to segments orizontally by 4 aelengts default config.). T orizontally polarized source at broadside, x )[ ] At broadside, d x peaks and d x. As scan off in azimut, ertical beamformer output is nonzero een toug source is orizontally polarized. Witener accounts for cross-coupling of beamformer. For tis example, te itener is simply te identity matrix for o eleation cut. orizontal -) and Vertical - -) Witened Beamformer Output 5 5 db Azimut degrees) WPB-3

14 SIMPLE EXAMPLE: MLE PERFORMANCE INDEX Solid plot is MLE performance index. Dased plots are responses of itened) orizontal and ertical beamformers. Te MLE performance index is te enelope of te to responses. Te mainlobe is idened, resulting in larger RMSE angle estimation tan for knon orizontal) polarization. 5 5 db Azimut degrees) WPB-4

15 SIMPLE EXAMPLE: POLARIZATION ESTIMATE VS. AZIMUT MLE polarization estimate aries it angle to maximize performance index. Witin te mainlobe te polarization estimate takes on many alues. For example, en ertical output is zero, estimate is orizontal polarization. 3 5 ZL RCP VER LCP ZL Pol. 5 db Azimut degrees) WPB-5

16 WPB-6 CRB ON AOA ESTIMATION CRB ON AOA ESTIMATION For te single snapsot) obseration model x ξ) + n, ere ξ is te M by unknon real-alued parameter ector and n is complex-gaussian it zero mean and coariance R, te M by M Fiser information matrix is gien by ere te m t column of te N by M matrix is ξ) /ξ m. For te unknon-polarization problem ae Te 6 columns of are gien by ] R [ Re al Γ )a ) d e sin ) d cos )a, d j γ γ η + )a d e sin j )a ) d e cos ) d sin - )a, d j a )e, d a )a ) d e sin ) d cos )a ) d e sin ) d cos j j a j j j γ η γ γ γ γ γ φ γ φ γ φ η η η η For te knon-polarization problem, only te first four columns are used.)

17 CRB ON AOA ESTIMATION CONT.) Te CRB is gien by te elements of B Γ - -. Denoting [ B ] m,n by b mn, one as b and b are te CRB on RMSE for φ and estimation, respectiely. b is te correlation coefficient beteen te φ and estimation. / b b ) Example equi-probablilty contour Scatter plot example Eleation Eleation Azimut Azimut To obtain CRB for te rotated coordinate system, compute te eigenalues and eigenectors ordered so tat λ λ ) of b b M b b λ and λ are te CRB on RMSE along te major and minor axes, respectiely. β tan [ u ] / [ u ] ) is te ellipse orientation, ere M u λ u. Direct numeric computation is preferred. Wile analytic expressions can be obtained using a partitioned matrix inerse approac, te resultant expressions do not lend great insigt due to teir complexity. To understand AOA result, study sape of corresponding MLE perf index and its indiidual components i.e., itened zl and er beamformer responses). WPB-7

18 AGREEMENT BETWEEN MLE AND CRB orizontally polarized source at broadside. No noise. Contours at -.5 db for eac performance index bot indices ae te same peak). Ellipse orientation and axial ratio agree it te respectie CRBs. Monte Carlo MLE RMSE performance agrees it te respectie CRBs. Knon Polarization: MLE Perf Index 5 4 db Unknon Polarization: MLE Perf Index 5 4 db Eleation degrees) Eleation degrees) Azimut degrees) Azimut degrees) -3 WPB-8

19 CRB RMSE ELLIPSES: UNKNOWN VS. KNOWN POLARIZATION orizontally polarized source. Source at broadside for first plot, at 3 o azimut and 3 o eleation for second plot. Solid plot is for unknon polarization, dased plot is for knon polarization. Te unknon polarization RMSE ellipse circumscribes te knon polarization RMSE ellipse..6 Source at Broadside Source at 3 o Azimut, 3 o Eleation Eleation degrees).. Eleation degrees) Azimut degrees) Azimut degrees) WPB-9

20 PERFORMANCE IMPACT OF ESTIMATING POLARIZATION: RMSE VS. SEPARATION FOR LINEAR SEGMENTS Circularly polarized source at broadside. Vary te orizontal separation beteen te segments. For knon polarization, te RMSE decreases as separation increases. For unknon polarization, te RMSE is independent of separation. For zero separation, te RMSE is te same. Minor Axis RMSE Major Axis RMSE RMSE degrees) Unknon Pol Knon Pol RMSE degrees) Unknon Pol Knon Pol Segment Separation aelengts) Segment Separation aelengts) WPB-

21 PERFORMANCE IMPACT OF ESTIMATING POLARIZATION: RMSE VS. SEP FOR MORE TAN LINEAR SEGMENTS Circularly polarized source at broadside. Add anoter linear segment, identical to one of te oters, but it fixed location miday beteen te oter to. Vary te orizontal separation beteen te outer segments. Te RMSE for unknon polarization no decreases it increasing array separation, but is still larger tan te RMSE for knon polarization. For linear segments of crossed dipoles 4 segments total), te RMSE is te same. Minor Axis RMSE Major Axis RMSE RMSE degrees) Unknon Pol Knon Pol RMSE degrees) Unknon Pol Knon Pol Segment Separation aelengts) Segment Separation aelengts) WPB-

22 Eleation degrees) Azimut degrees) WPB- IMPACT OF ASSUMED POLARIZATION: CIRCULARLY POLARIZED BEAMFORMER, ZL SOURCE Assumed-polarization performance using default config..5 db mismatc loss.8 o bias.6 of peak-to-null idt of unknon polarization performance index) Standard deiation commensurate it tat of correctly assumed polarization, but increased by loss due to mismatc 6% increase) Unknon-Pol Perf Index db Eleation degrees) Assumed-Pol Perf Index Azimut degrees) Contours at -.5 db from on peak x true AOA, o mean of estimated AOA Polarization Plot Beamformer polarization True source polarization db

23 Vertical pol component is 6dB don it respect to zl component, it 9º pase offset γ 6.6 o, η 9 o ) Assumed-polarization performance. db mismatc loss.49 o bias. of peak-to-null idt of unknon pol perf index) Standard deiation commensurate it tat of correctly assumed polarization, but increased by loss due to mismatc % increase) Eleation degrees) Azimut degrees) WPB-3 IMPACT OF ASSUMED POLARIZATION: ZL BEAMFORMER, ELLIPTICALLY POLARIZED SOURCE Unknon-Pol Perf Index db Eleation degrees) Contours at -.5 db from on peak x true AOA, o mean of estimated AOA Polarization Plot Beamformer polarization Assumed-Pol Perf Index Azimut degrees) True source polarization db

24 SUMMARY Model General 3-dimensional array consisting of elements it arbitrary polarization responses Found to be linear in polarization Maximum Likeliood Estimator MLE) Performance index need only be searced oer azimut and eleation) Problem analogous to te single-snapsot to-source problem Te BWE is te MLE for a single snapsot, toug not for multiple snapsots To deriations proided incremental and from first principles) Te MLE uses noncoerent combination in polarization, as compared to coerent for knon polarization Processing flo exemplified and explicit polarization estimation proided Example to proide intuitie feel for MLE estimator WPB-4

25 SUMMARY CONT.) Cramer-Rao Bound CRB) Agreement beteen MLE and CRB asymptotic efficiency), for unknon and knon polarization - Monte Carlo - Contour cut of performance indices Performance impact due to estimation of polarization - RMSE for unknon pol. case > RMSE for knon pol. case - Obsere RMSE ellipses circumscribed - For linear segments of dipoles, RMSE is independent of segment separation for unknon polarization Assumed Polarization Angle of arrial AOA) estimate standard deiation commensurate it tat for correctly assumed polarization, but increased by SNR loss due to polarization mismatc Potentially biased AOA estimate WPB-5