: TJ630.34; TN911.7 : A : (2011) Joint Estimation of Source Number and DOA for Underwater Passive Target Based on RJMCMC Method

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1 9 6 ¹ Vol. 9 No TORPEDO TECHNOLOGY Dec. 2 RJMCMC Ü ö, n 2,,, (.,, 772; 2.,, 97) : x p x x ¹, ƒ x ¹ u z x ¹(DOA) ½ u x, x n, u Å ƒ (RJMCMC) ¹, n Œ u x Å, ( ) z x ¹ x z, z p x {, RJMCMC ¹ z : z ¹; ; ; ; Å (RJMCMC) : TJ63.34; TN9.7 : A : (2) Joint Estimation of Source Number and DOA for Underater Passive Target Based on RJMCMC Method CHEN Zhao, LIU Zheng-guo 2, WANG Hai-yan, SHEN Xiao-hong, HE Bin (.College of Marine Engineering, Northestern Polytechnical University, Xi an 772, China; 2.China Shipbuilding Industry Corporation, Beijing 97, China) Abstract: Conventional ideband signal processing method of subspace class ideband requires more snapshots to get a better estimation result. We apply a Bayesian high resolution direction of arrival(doa) estimation method based on the time-domain ideband signal model to underater passive target DOA estimation. Compared ith conventional subspace class methods, the proposed method can process signal directly in time-domain and needs remarably less snapshots. We use a time-domain ideband signal model to construct a posterior probability density function(pdf) of the parameters according to Bayes la, and then use the reversible jump Marov chain Monte Carlo(RJMCMC) method for Bayesian estimation to search peas of the posterior PDF. This method can jump among parameter spaces ith different orders, thus can implement joint model order (i.e. source number) and DOA estimation. Simulation results sho that the RJMCMC method exhibits good performance in joint estimation of model order and DOA. Key ords: passive target direction of arrival (DOA) estimation; time-domain ideband model; Bayesian high resolution; model order detection; reversible jump Marov chain Monte Carlo (RJMCMC) zä x p x x r x, n x, 2-3-7; jƒ (697253), g (296238), (NPU-FFR-JC24). (983 ), u,, z x» ¹ ÄÆ..yljszz.cn 45

2 2 2 9 ¹ x r x, z ¹,,, z, x x, ¹ ¹ u x n ¹x ƒ Lasenby Fitzgerald [] ƒ u p ¹ {, ¹ x n, ¹x ¹ Œ, º x [2-3] z x ¹ ÄÆ, z, x ª u z x p, p x x ¹ x p uxz (coherent signal subspace method, CSSM), z ¹x [4] x ƒ, u z x z ¹ x z x n ¹ n Œ, ¹ x, ¹, º u Å (reversible jump Marov chain Monte Carlo, RJMCMC) n x Œ [5-7] u [4] x (maximum aposterior, MAP) x [4]x RJMCMC x  u z x x ¹ x x ¹, ¹ z (direction of arrival, DOA)x { x DOA [4] x u x, p, x i px g,. M Ž (uniform linear array, ULA), K s (t) θ, =,e,k, s (t) x, n l u u l f [ f, f ], f = f +Δf ; =,,, K () u : f f l s (t)x n n ; Δf x s (t) s (n), n nt s (n=,2,e, N, N ), T s φ π/2,, (M )τ M, s (t) m tx τ l τ s ( t m ) h ( m ) s ( n l) m=,,, M (2) φ π/2 Å Fig. Array signal reception model hen incident angle φ π/2 : τ x (inter sensor delay, ISD), τ = dsin( θ ) / c (3) 46 Torpedo Technology.yljszz.cn

3 2 2, Š: RJMCMC x z ¹ 6 : c ; d Ž x Ä; h l (mτ ) mτ x x l Œ, u sinc, (l-d), D s ( n D) h ( l D) s ( n l) (4) l h ( mτ ) = h ( D) = sinc( l D)* ( l D) (5) l l D = mτ / T = mτ * f (6) s s, f s x n, à s =,, K u f ƒ 2* max { f } (7) Äd à (8) ( f max x z n ), d=c/f s c c d λ min = (8) 2 2 fmax fs u (3) (8) τ = dsin( θ) / c d / c= / fs = Ts (9) τ [ Tmax, Tmax ] = [ Ts, Ts], =,,, K», σ 2 x, x z,, t x»t x y(n), y(n)zr M, u (2) s () t s( t τ ) s () t = = s[ t ( M ) τ ] h() h() hl () h( τ) h( τ) hl ( τ) h[ ( M ) τ] h[ ( M ) τ] hl [ ( M ) τ] s ( n) s ( n ) () s[ n ( )] - s ( n) s ( n ) s( t) = [ h( τ), h( τ),..., hl-( τ)] s[ n ( )] = h( τ ) s ( n l) = H( τ ) s ( n) () l K y( n) = s( t) + σ ( n) = K = hl( τ) s( n l) + σ( n) = s( n l) s ( n l) = [ hl( τ), hl( τ),, hl( τk )] + σ( n) sk ( n l) = H ( τ) a( n l) + σ ( n) (2) l : H l (τ)zr M K l ; τ=[τ, τ,e, τ K- ] T zr K z x ; a(n)zr K n x ; (n)z R M n N(, I M ) x z(n)zr M z( n) = y( n) H ( τ ) a ( n l) (3) { y(n) x, l z( n) = H ( τ ) a( n) +σ ( n) (4) φ >π/2x, M, (M-)τ, 2 2 φ >π/2 Å Fig. 2 Array signal reception model hen incident angle φ >π/2 s (t) m tx y( n) = E H ( τ) a( n l) + σ ( n) (5) M l : E M, H l (τ)x Ε M = (6).yljszz.cn 47

4 2 2 9 Hl( τ), ϕ π /2 Hl ( τ ) =, l =,,, L (7) EMHl( τ), ϕ > π /2 z( n) = H ( τ) a( n) +σ ( n) (8).2 n (8), H ( τ ) x», u (8) ZzR M N, Z = [ z(), z(2),, z( N)] (9) x, j N 2 2 σ = σ n= N 2 M /2 2 n= (2π σ) 2σ l( Z a, τ,, ) l[ z( n) a, τ,, ] T = exp [ z( n) H ( τ) a( n)] [() z n H ()()] τ a n (2), { x ¹ u pº, x n j x ƒ π( a, τ, σ, Z) = l( Z a, τ, σ, ) p( a, τ, δ σ) 2 p( τ ) p( σ ) p( ) (2), δ 2 Â, x [4], τ 2 p( τ )~ U[ T, T ] (22) σ x, s 2 2 σ p( σ ) = / (23) x Λ x, Λ x s Λ Λ p ( ) = e (24)!, Ãx τ (22)~ (24) (2), 2 a σ ƒ Λ e π( τ, Z) 2 N /2 ( δ ) 2 T +! { tr[ P ( ) ˆ H τ Rzz] } s MN /2 Λ (25) 2 T Σ H ( τ) = ( + δ ) H ( τ) H ( τ) T T H = I 2 ( )[ ( ) ( )] P H τ H τ H τ H ( τ) ( + δ ) T a ( n) m ( ) ( ) MAP a n = Σ H τ H ( τ) z( n) (26) T = Σ H ( τ) H ( τ)[ y( n) Hl( τ) a( n l)] N ˆ T tr( P H ( τ) R ) ( ) ( ) ( ) zz = z n P H τ z n n= (25) τ x n, n Œ [], x x DOA z Œ ¹, (MCMC) x 2 RJMCMC xz ( ¹) ¹, (25) τ ux RJMCMC x RJMCMC u 3 ƒ x [4] : (update move) (death move) t (birth move) 3 ƒ x n u, d b, u (27), c x n, c =.5 [4], p() (24) Λ x u + b + d = p ( ) d + = c*min, p ( + ) p ( + ) b = c*min, p ( ) (27) RJMCMC x, x 3 ƒ x º () t t x (, τ ) () t () t () t () t = τ τ2 τ τ [,,, ] (28) ) :, τ () t, σ 2 48 Torpedo Technology.yljszz.cn

5 2 2, Š: RJMCMC x z ¹ 6 x º, ( t + τ ) () t 2 ~N( τ, σ ) 2) t : <K max, K max ºx ( 6) U[T s,t s ] τ, x τ () t ( t+ ) ( t) ( t) τ+ = τ,, τ ',, τ (29) 3) : > x τ () t, j x,» m ( t+ ) ( t) ( t) ( t) ( t) τ = τ,, τm, τm+,, τ (3) n * * * π( τ ) q( τ τ ) α( ττ, ) = min, π( ) q ( * τ τ τ ) (3) : τ τ *, q( ) º x {,, n * * π( τ ) α( ττ, ) = min, π( τ ) (32), RJMCMC x x n Å x ¹x x RJMCMC x ), U[,K max ] ; x U[T s,t s ] τ () 2) t, U[,] u, u (27)¹ x n u, d b u<u, u u d +u, t 3) u tx, (29)¹ x n, τ ( t+ ) = τ * ; ( t + τ ) = τ ( t ) 4) z 5) ¹x 3 Ü, Š x z, º x, tx z x z, Ž u n j x 3 [8] :, 2 x, 3 Ž x u [9]x t { z x 3 x 3 z x n Fig. 3 3 Ü Poer spectrum model of underater target radiation noise 4 u { u M= ULA, z BW=.9 N=64 x 3 x, zšx (signal-to-noise ratio, SNR) x θ =6., θ 2 =2. θ 3 =8., x Ä x f z[9, 2 9] Hz, f 2 z[ 3, 3 4] Hz f 3 z[, 3 ]Hz n f s =8 Hz c= 5 m/s, Ä d= c/f s =.87 5 m { ½.yljszz.cn 49

6 2 2 9 Table Parameter values of simulation SNR/dB M N L K σ 2 δ 2 f s/hz Λ θ/( ) τ /s [6., 2., 8.] [.37, 2.599, 3.863]* -5 u x {, u RJMCMC z ¹ (ide band Bayesian estimation method,wbbem) [4] x n, RJMCMC u 2 4 ö Fig. 4 Source number estimation result of a single running 4 ¹x u, z, 55 ¹ u { {,, 3 ¹ 3 5 x ¹ u {,, 3 ¹ 6 u MAP x, u, 3 x, x ¹xz» 7~ 9 u x (MC) { RJMCMC x 7 MC { ¹ x u {, {, 83 ¹ 3, 7, ¹ 2 Fig. 5 5 Direction of arrival(doa) estimation result of a single running 7 Fig. 7 ö Source number estimation result of independent Monte Carlo simulations 6 ö Fig. 6 Source amplitudes restoration result of a single running 8 ö 2 Fig. 8 DOA estimation results hen source number estimation is 2 42 Torpedo Technology.yljszz.cn

7 2 2, Š: RJMCMC x z ¹ 6 2 ö Table 2 Source number estimation results and root mean square error(rmse) of DOA estimation under different signal-to-noise ratios 2 3 SNR ¹ 3 /db ¹x ¹x ¹x x RMSE/( ) RMSE/( ) RMSE/( ) ö 3 Fig. 9 DOA estimation results hen source number estimation is 3 8 ¹ 2 x DOA ¹ u {, 7 {, ¹ 3 2, x ¹ 3 x ¹ u { z x 3 x x ¹ x, x DOA ¹ x  x ¹ z x{ o  9 ¹ 3 x DOA ¹, 2, 2 3 x DOA ¹ u { x DOA ¹ ( ) { x DOA ( u ), ( u ) 2 x ¹ [6.,.9 ]; 2 3 x ¹ [.9,8.2 ] u ¹ 3 x, x 2, u RJMCMC x ¹ ¹x x ux ( 2 ), x ¹ p x ; u x ( ), x ¹   u z ¹,, x x ¹ [4] x x ¹ z ¹, z x u x p x i z, x, z p;, x x ¹, z i ;, ¹x ( ) x x, ¹, x ¹x z» u RJMCMC xz z x n Œ x 3 z x ¹ { { WBBEM ¹ z, ¹ z,, x ¹ : [] Lasenby J, Fitzgerald W J. A Bayesian Approach to High Resolution Beamforming[J]. Radar and Signal Processing, IEEE Proceedings F, 99, 38(6): [2] Huang J G, Sun Y. A Ne Gibbs Sampling DOA Estimator Based on Bayesian Method[J]. Acoustic,.yljszz.cn 42

8 2 2 9 Speech and Processing, 23(5): [3] Sun Y, Liu K W, Huang J G, et al. Bayesian DOA Estimator by Gibbs Sampling[J]. Computer Engineering and Applications, 22, 38(2): [4] William N, Reilly J P, Kirubarajan T, et al. Wideband Array Signal Processing Using MCMC Methods[J]. IEEE Transactions on Signal Processing, 25, 53(2): [5] Andrieu C, Doucet A. Joint Bayesian Model Selection and Estimation of Noisy Sinusoids Via Reversible Jump MCMC[J]. IEEE Transactions on Signal Processing, 999, 47(): [6] Green P J. Reversible Jump Marov Chain Monte Carlo Computation and Bayesian Model Determination[J]. Biometria, 995, 82(4): [7] Larocque J R, Reilly J P. Reversible Jump MCMC for Joint Detection and Estimation of Sources in Colored Noise[J]. IEEE Transactions on Signal Processing, 22, 5(2): [8],,, Š. x {[J]. u, 26(4): 7-2. [9] William N, Reilly J P, Kirubarajan T. Wideband Array Signal Processing Using Sequential Monte Carlo Methods[R]. Department of Electrical and Computer Engineering, McMaster University, 23: -28. ( : ) z. x.,,. 2, 9(3). 2. z., n, o, Š. 2, 9(). 3. z TOP xz y ¹., u, n l, Š. 2, 8(2). 4. zz g { ¼.,, n, Š. 29, 7(6). 5. gz x z.,. 29, 7(4). 6. z., n,, Š. 29, 7(4). 7. n., n, o. 29, 7(). 8. x., n, Â. 28, 6(6). 9. z g»., u, n l, Š. 28, 6(5).. z x. r,, Š. 28, 6(3).. u z x., Â, y. 28, 6(3). 2. z z x ¹.. 28, 6(). 3. Â u., n,. 27, 5(6). 4. :,, Â. 27, 5(6). 5. n.,,. 26, 5(). 6. ¹x y., š,. 26, 5() 7. Tiger SHARC DSP x»¹. n,,, Š. 26, 5(4). 8. Â pxz ¹.,, n. 26, 5(3). 9. x u.,, n, Š. 26, 5(2). 2. ƒ z ¹».,. 25, 4(4). 2. z» x¼.,, n. 25, 4(4). 422 Torpedo Technology.yljszz.cn