PerformanceAnalysisofAngular-Smoothing.doc

豆丁网精品论文performance analysis of angular-smoothingbased root-music for an l-shapedacoustic vector-sensor arrayyougen xu zhiwen liudepartment of electronic engineering, beijing institute of technology, beijing, prc 100081 abstracteigenstructure-based direction-of-arrival (doa) estimation algorithms such as multiple signal classification (music), root-music, estimation of signal parameters via rotational invariance techniques (esprit), encounter great difficulty in the presence of perfectly correlated incident signals. for an array composed of a number of translational invariant subarrays such as a uniform linear scalar-sensor array, this problem can be solved by spatial smoothing. an array of identically oriented acoustic vector-sensors can be grouped into four coupled subarrays of identical grid geometry, respectively corresponding to the pressure sensors and differently oriented velocity-sensors. these four subarrays are angular invariant dependent only on signals direction cosines and an angular smoothing can be exploited for source decorrelation. in this paper, the performance of root-music incorporated with angular smoothing for correlated source direction finding with an l-shaped acoustic vector-sensor array is analyzed in terms of the overall root mean-square errors (rmse) of doa estimates. we derive the analytical expression of the rmse and compare it with simulation results and that of spatial smoothing for a rectangular pressure-sensor array instead.keywords: antenna arrays, array signal processing, direction-of-arrival estimation, acoustic array1 introduction1.1 acoustic vector-sensorsacoustic vector-sensors have attracted increasing interest for subspace-based direction-finding8-17. a “complete” acoustic vector-sensor such as a vector-hydrophone and a microflown consists of a pressure sensor and a collocated triad of three orthogonal velocity sensors 8. these four sensors together measure the scalar acoustic pressure and all three components of the acoustic particle velocity vector of the incident wave-field at a given point. an “incomplete” acoustic vector-sensor consists of a subset of the above four component-sensors, e.g., a three-component hydrophone formed from two orthogonally oriented velocitythis work was supported by the national natural science foundation of china under grant no.60272025, the university basic research foundation of beijing institute of technology under grant no.bit-ubf-200301f07 and no. bit-ubf-200501f4202, and the specialized research fund for the doctoralprogram of higher education under grant no. 20040007013.hydrophones and a pressure hydrophone 13.it is assumed that the acoustic wave is traveling in a quiescent, homogeneous, and isotropic fluid, and isfrom a source of azimuth and elevation , where 0 q ). this proposed angular smoothing scheme represents a counterpart of polarisation smoothing 18 for an array of identically orientedelectromagnetic vector-sensors.although developed here for a very regular l-shaped acoustic vector-sensor array to apply computationally efficient root-music, the angular smoothing algorithm is applicable for an array of identically oriented acoustic vector-sensors but with any arbitrary grid geometry. in addition, unlike classical spatial smoothing,the angular smoothing can be accomplished without spatial-aperture loss. since it is impossible thatux = 1 ,uy = 1anduz = 1may hold simultaneously, this angular smoothing algorithm can still avoidsignal cancellation in the presence of totally uncorrelated signals. these attractive features are analogous to those of the polarisation smoothing algorithm in the context of electromagnetic applications 18.it is important to note that the angular-smoothing scheme generally can only process at most three coherentsignals with distinct doas even with a very large number of n xandn y . this shortcoming may bealleviated by incorporating the conventional spatial smoothing. however, such a consideration is beyond the scope of this paper.1.3 the angular smoothing-based root-musicafter angular smoothing, (8) is equivalent to the covariance matrix with desired rank property of an l-shaped array of conventional scalar sensors (e.g., the pressure hydrophones), and hence two parallel root-music can be applied to the two legs of such arrays to obtain the corresponding two directioncosines.x t tthdefinej = jn ,1, j withj = o1, , i1 ,jthen column ofi , andnx nynx m ,n my y xyj = in , on ,n 1 . then we constructr x = jx r (jx )h ,r y = jy r (jy )h, and further eigen-decomposethem to obtain r = e (e )+ e (e )and r = e (e )+ e (e ), wheree ,s ssn n nx x xx h x x x hy y y y h y y y h xenns ss n n nscalled the x-leg signal matrix and span a q -dimensional signal subspace, comprises the q eigenvectorscorresponding to the q largest eigenvalues ofr x , whereas xis composed of the remaining n qsneigenvectors associated with the smallest eigenvalues and called the x-leg noise matrix, xand xaretwo diagonal matrices whose diagonal entries are respectively the q largest eigenvalues and the n qsnssmallest eigenvalues. the y-leg counterpartse y ,e y , yand yhave the same definitions. letn nnn nnp x = e x (e x )h ,p y = e y (e y )h, and bx (, ) = 1,e j x , .,e j (nx 1)x t(9)by (, ) = 1,e j y , .,e j (ny 1)y tx h x xy h y ythen,b (q , q )pn b (q , q ) = 0 ,b (q , q )pn b (q , q ) = 0 , and hence the two root-musicpolynomials are as follows: fx x t x x() = b (1 / ) pn b ()qy y(10)y y t y y f() = b (1 / ) pn b ()qx xwhose q roots of unit modulus are respectivelyv x = expj 2d u/ andvy = expj 2d u/ ,whereq = 1, 2 , .,q. then we haveu = cos sin = (2d )1 (v x )and1yx ,qqqx quy,q = sin q sin q = (2dy )q(vq )( () qdenotes angle operation), and (q , q )can ultimately beobtained via couplingvx ,q q = 1andvy,q q = 1 .zuvector-sensor dyydx xfigure 1. the concerned sensor-source geometry2 overall root mean-square errors of direction cosine estimatespractically, we have no access to the true rykand hence true r due to the existence of noise and finitesample support. commonly, the true rycan be replaced by the following sample covariance matrix: ry =1 y(t )yh (t )(11)k kk k = 1where k is the number of independent snapshots, and perturbation inr = 4 j r jtresults inthe error of ultimate doa estimates.m = 1 m m y min this section, we derive the overall root mean-square errors (rmse) of the direction cosine estimateswhich is defined asqx ,qy,q = e (| u|2 ) + e (| u|2 )(12)q = 1by using root-music described above.the derivations below are primarily along the lines of 21, and will be focused on the x-leg only. the discussion for the y-leg is similar. first, rewrite the null spectrum along the x-axis as x h x xf (x ) = b (x )pn b (x ) . by differentiating the perturbedf (x ) , an estimate off (x ) , with respect tox , and using a first-order approximation, it can be derived that 21e(| u|2 ) = 2 1 x ,q 2d 2k (bx ()h px bx ()2 x x ,q n x ,q 2x ,q 4 (13) m n (x ,q rm,n x ,q )(x ,q rn,m x ,q ) hh4m,n =1h h x x+ rem n (x ,q rm,n x ,q )(x ,q rn,m x ,q )wherexx ,q = 2dx ux ,q / m,n =1,b (x q ) = b (x ) / x | = ,andx x h t2,x h x tx x ,qrm,n = (rn,m )= jm rx ,0 jn + n (m n)in, in whichrx ,0 = j ars a(j ) ,() is the kroneckerdelta function,x ,q = p n b (x ,q ) ,x q = j br s b(j ) b (x ,q ), where() denotes thex x x h x t # x #,moore-penrose pseudo-inverse operation (a# = u1uh, given a s eigendecompositiona = uuh ). note thath j r jt = o, h = 0 , then (13) reduces tox ,q m x ,0 n1,n x 4x ,q x ,q e(| u|2 ) = 2 2 2 (h )(h rx )x ,qx ,q m =1n m x , q2 x ,qx ,q m,m x ,q 4ax ,q= ( 2 2a 2) h 2 rx (14)sx ,q n x ,qx ,q m =1m m,m x ,q = h 2 h rx x ,qx ,q x ,qsrecall thatjx br bh (jx )t # = (bx )+ h r 1 (bx )+ , wherebx = jx b , “+” is the left pseudo-inversewhich is defined by(bx )+ = (bx )h bx 1 (bx )h . hence,x ,q = jx br bh (jx )t # bx ( )x ,qs x ,qs= (bx )+ h r 1 (bx )+ bx ( )s n ,q= (bx )+ h r 1ix(15)and therefore,x =qq = 1e(| ux ,q |2 )q x ,qx ,q x ,q=h 2q = 1 h rx q=ih(h 2 r 1 (bx )+ rx (bx )+ h r 1 ) i(16) n x ,qx ,qsq = 1sn x ,q q =ihh 2 r 1 (bx )+ rx (bx )+ h r 1 i x n x ,q x ,q s sn ,q q = 1qx = tracehx qx wherehx = diagh 2 , h 2 , ., h 2 . similarly,y = qe(| u|2 ) = tracehy qy , which can bex ,1x ,2x ,qq = 1y,q derived in a similar way and hence we do not repeat. then, =x + y , which cl y is dependent onearlthe signal to noise ratio (snr), element spacing, and the number of snapshot, and is also influenced by thetweights = 1, 2 , 3 , 4 . increasing the snr, the array aperture, and/or data length will decrease .n ydydx y4xpressure sensorfigure 2. rectangular scalar-sensor array for spatial smoothing scheme3 theoretical/experimental analyses and comparisons3.1 further analysis and simulation verification of (16)in the interest of simple expositions, the analyses presented below will be only focused on two coherentssignals, which, however, may be directly extended to the case of more signals. letg2 = g , then2 = 1 , andg1 = 1 ,4 4 a2 (m) a (m)a (m) g r s = 4 m 1m =1 m 1 2 m =1 4 (17) a (m)a (m) g a2 (m) | g |2m =1m 1 2 m 2 m =1 and4 4 2 a2 (m) 2 a (m)a (m) g m 1m 1 2 4 rx = bx m =1m =1 (bx )h + 2 2 i 4 4 m n n x 2 a (m)a (m) g2 a2 (m) | g |2 m =1 (18) m 1 2 m 2 2m =1m =1 n= bx r (bx )h + 2 is n n