JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering . MUSIC MUSIC TN (2010)

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1 JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering Vol. 28 No. 3 May DOI:.3969/j.issn MUSIC 2. MUSIC Taylor MUSIC MUSIC Taylor TN () Success Probability of Direction-Finding of MUSIC Algorithm with Modeling Errors WANG Ding, WU Ying Institute of Information Engineering, PLA Information Engineering University, Zhengzhou 2, China Abstract: The direction-finding (DF) success probability is an important specification of the eigen-structure algorithm. We derive the DF success probability of the MUSIC algorithm in the presence of modeling errors. A closed-form expression of DF error and its statistical properties are obtained from the first-order Taylor expansion of the spatial spectrum function. Two classes of DF success definitions are given for single-source DF and holistic DF based on these definitions and statistical characteristics of the modeling errors. The calculation formulas of the DF success probability are derived. The theoretical analysis is validated through simulation using a uniform circular array (UCA) and a uniform linear array (ULA). Keywords: direction-finding, modeling errors, MUSIC algorithm, direction-finding success probability, Taylor expansion, quadratic form of Gaussian vector MUSIC [] MUSIC [2-5] [2] MUSIC [3] MUSIC Taylor [4-5] MUSIC MUSIC (No.BSLWCX) wang_ding84@yahoo.com.cn DSP hnwuying22@63.com

2 2 28 [6-7] MUSIC () [6] Taylor [7] MUSIC [8-] MUSIC. [8-9] [] [] MUSIC. MUSIC Taylor MUSIC ) ( )ẋ(θ) ẍ(θ) ( )x(θ) θ 2) [ ] Moore-Penrose 3) I n n i e (n) i ; 4) O n m n m O n n n n 5) δ kl delta k = l, δ kl =, δ kl = MUSIC M D(D < M) k θ k, X(t) = Âs(t) + n(t) () s(t) n(t) ÂÂ = A + Ã, A = [a(θ ) a(θ 2 ) a(θ D )] a(θ k ) Ã = [ã(θ ) ã(θ 2 ) ã(θ D )] ã(θ k ) [ ] E ã(θ k )ã H (θ l ) = δ kl σai 2 M [ ] (2) E ã(θ k )ã T (θ l ) = O M [2,7,-] () ] R = E[ X(t) XH (t) = ÂP ÂH + σni 2 M (3) P = E[s(t)s H (t)] σ 2 n R [] λ λ 2 λ D > λ D+ = λ D+2 = = λ M = σ 2 n (4) û, û 2,, û M, [] span{û, û 2,, û D } = range{â} (5) span{û D+, û D+2,, û M } range{â} 2 ÛN = [û D+ û D+2 û M ], P N = ÛNÛ H N = I M ÂÂ = P A (6) MUSIC D(θ) = a H (θ) P N a(θ) = a H (θ) P A a(θ) (7) (7) P N ( P A ) MUSIC MUSIC 2 MUSIC MUSIC J(θ, Ã) = 2 D(θ) θ = Re{ȧ H (θ) P A a(θ)} (8)

3 3 MUSIC 29 θ k θ k J( θ k, Ã) = J(θ k, O M D ) = (9) J( θ k, Ã) J(θ k, O M D ) + J(θ, O M D) ( θ k θ k ) + θ θ=θk M m= n= θ k θ k Ã Taylor M m= n= J(θ k, Ã) Ã=OM D ã (r) + J(θ k, Ã) Ã=OM D ã (i) () ã Ã m n ã(r) ã (i) (9) () θ k = θ k θ k = M m= n= J(θ k, Ã) Ã=OM D ã (r) + M J(θ, O M D ) θ m= n= J(θ k, Ã) Ã=OM D ã (i) () θ=θk (2) P A η = P A Â ( η Â P Â ) H A η Â Â = e m (M) e (D)T n, { Re { Re J(θ k, Ã) ȧ H (θ k ) P A Â = ie (M) m e (D)T n Ã=OM D = } Ã=OM D a(θ k ) = (2) Re{f H (θ k )e (M) m δ nk } (3) J(θ k, Ã) ȧ H (θ k ) P A Ã=OM D = } Ã=OM D a(θ k ) Im{f H (θ k )e (M) m δ nk } (4) f(θ k ) = PA ȧ(θ k). (3) (4) A a(θ k ) = e (D) k PA a(θ k) = M, J(θ, O M D ) = ȧ H (θ k )PA θ ȧ(θ k) = h(θ k ) θ=θk (5) (3) (5) () θ k = θ k θ k = f (r)t (θ k )ã (r) (θ k ) + f (i)t (θ k )ã (i) (θ k ) h(θ k ) = (6) f (r) (θ k ) f (i) (θ k ) f(θ k ) ã (r) (θ k ) ã (i) (θ k ) ã(θ k ) MUSIC. θ k ã (r) (θ k ) ã (i) (θ k ) ã (r) (θ k ) ã (i) (θ k ) θ k E[ θ k ] = var[ θ k ] = σa f(θ 2 k ) 2 2h 2 (θ k )/2 = σah 2 (θ k )/2, k =, 2,, D (7) (7)h(θ k ) = f(θ k ) 2 2 (6) k l E[ θ k θ l ] = θ k θ l 3 MUSIC 3. MUSIC k θ k θ,. θ. 2 θ k

4 k Pr{ θ k θ} = Φ( θ/ var[ θ k ]) Φ( θ/ var[ θ k ]) = 2Φ( θ/ var[ θ k ]) (8) Φ(u) = 2π u e x2 /2 dx, [2] 3.2 (8).. 2 max { θ k } θ, k D 2 k l θ k θ l Pr{ max k D { θ k } θ} = Pr{ θ k θ} = [2Φ( θ/ var[ θ k ] ) ] (9) (8) (9) D ( θ k ) 2 θ, 2. 2 θ = [ θ θ 2 θ D ] T, 2 Pr{ θ 2 2 D( θ) 2 }, 2 θ 2 2 (6) θ = F T F T 2... F T D ã ã 2 = F ã (). ã D F k = [f (r)t (θ k )/h(θ k ) f (i)t (θ k )/h(θ k )] T ã k = [ã (r)t (θ k ) ã (i)t (θ k )] T. θ 2 2 = ã T F T F ã = ã T Gã (2) G = F T F. (2) (2) ã σai 2 2MD /2 (2) θ 2 2 θ 2 2 [3-4] 2 π Pr{ θ 2 2 c} = + t Im{e itc ϕ θ 2 2 (t)}dt (22) ϕ θ 2 2 (t) θ 2 2 ϕ θ 2 2 (t), θ 2 2 G D, G G = λ k g k gk T (23) λ k G g k (23) (2) θ 2 2 = ã T Gã = 2 σaλ 2 k (gk T ã) 2 d = λ k χ 2 k() (24) λ k = σaλ 2 k /2, ã = 2ã/σ a χ 2 k () = d k l χ 2 k () χ2 l () E[gk T ãã T g l ] = gk T, k = l g l = δ kl = (25), k l χ 2 k () ϕ χ 2 k ()(t) = ( 2it) /2 [5] θ 2 2 ϕ θ 2 2 (t) = ( 2iλ k t) /2 = ( + 4λ 2 kt 2 ) /4 exp{i arctan(2λ k t)/2} (26)

5 3 MUSIC 293 (26) (22) Pr{ θ 2 2 c}= 2 π + { t (+4λ 2 kt 2 ) /4 sin{ arctan(2λ k t)/2 tc}}dt = 2 π + t sin{α(t)} dt (27) β(t) α(t) = D arctan(2λ k t)/2 tc, β(t) = D (+ 4λ 2 k t2 ) /4 (27) t t + lim t sin{α(t)} cos{α(t)} α(t) = lim t β(t) t β(t) + t β(t) = λ k c (28) t + 2 () F ; 2 G = F T F ; 3 G D λ k, k =, 2,, D; 4 λ k = σaλ 2 k /2, k =, 2,, D; 5 (27) ) (3) 2) 5 Monte Carlo 3) 2 Lobatto [6] db θ =. r/λ =.5, σ a =.3, θ =.3 ( ) 2 2 θ 2 =, σ a =.3, θ = θ 2 =, r/λ =.5, θ = θ 2 =, r/λ =.5, σ a = /( ) Figure Direction-finding success probability versus the azimuth separation of the two sources Figure 2 Direction-finding success probability versus the ratio of radius and wavelength

6 Figure 3 Direction-finding success probability versus the standard deviation of the modeling errors /( ) 4 Figure 4 Direction-finding success probability versus the angle error tolerance /( ) 5 Figure 5 Direction-finding success probability versus the azimuth separation of the two sources Figure 6 Direction-finding success probability versus the ratio of element spacing and wavelength db () θ =. 5.5 σ a =.3, θ = θ 2 =, σ a =.3, θ =.3 (.5) 7 2 θ 2 =,.5 θ = θ 2 =,.5 σ a = Figure 7 Direction-finding success probability versus the standard deviation of the modeling errors 8 ). 2) 2

7 3 MUSIC ) 2. 4). 5). 6). 7) /( ) 8 Figure 8 Direction-finding success probability 5 versus the angle error tolerance MUSIC Taylor MUSIC.. : [] Schmidt R O. Multiple emitter location and signal parameter estimation[j]. IEEE Transactions on Antennas and Propagation, 986, 34(3): [2] Swindlehurst A, Kailath T. A performance analysis of subspace-based methods in the presence of model error part I the MUSIC algorithm[j]. IEEE Transactions on Signal Processing, 992, (7): [3] Friedlander B. A sensitivity analysis of the MU- SIC algorithm[j]. IEEE Transactions on Acoustics, Speech and Signal Processing, 9, 38(): [4]. MUSIC [J]. 5, 27(): Dong Xiaohui, Ke Hengyu, Dong Zhifei. An analysis of FO-MUSIC algorithm in the presence of sensor perturbations[j]. Modern Radar, 5, 27(): (in Chinese) [5]. MUSIC [J]. 8, 36(2): 5-7. Liu Jian, Yu Hongqi, Huang Zhitao, Zhou Yiyu. Performance analysis of the MUSIC algorithm for noncircular signals with modeling errors[j]. Acta Electronica Sinica, 8, 36(2): 5-7. (in Chinese) [6] Hamza R, Buckley K. An analysis of weighted eigenspace methods in the presence of sensor errors[j]. IEEE Transactions on Signal Processing, 995, 43(5): -. [7] Ferr ol A, Larzabal P, Viberg M. On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: case of MUSIC[J]. IEEE Transactions on Signal Processing, 6, 54(3): 7-9. [8]. MUSIC[J]., 28(6): 5-7. Su Weimin, Gu Hong, Ni Jinlin, Liu Guosui, Zhang Guangyi. A statistical performance analysis of the MUSIC algorithm in the presence of amplitude and phase perturbation[j]. Acta Electronica Sinica,, 28(6): 5-7. (in Chinese) [9]. [J]., (5): Su Weimin, Gu Hong, Ni Jinlin, Liu Guosui. An analysis of the effects of multi-channel amplitude and phase errors on the spatial spectrum and resolving performance[j]. Progress in Natural Science,, (5): (in Chinese) [] Weiss A J, Friedlander B. Effects of modeling errors on the resolution threshold of the MUSIC al-

8 gorithm[j]. IEEE Transactions on Signal Processing, 994, 42(6): [] Ferr ol A, Larzabal P, Viberg M. On the resolution probability of MUSIC in presence of modeling errors[j]. IEEE Transactions on Signal Processing, 8, 56(5): [2]. [M]. 2: Sheng Zhou, Xie Shiqian, Pan Chengyi. Probability theory and mathematical statistics[m]. Beijing: Higher Education Press, 2: (in Chinese) [3] Imhof J P. Computing the distribution of quadratic forms in normal variables[j]. Biometrika, 96, 48(2): [4] Dugina T N, Martynov G V. Computing the distribution function of the ratio of quadratic forms in normal variables[j]. Journal of Mathematical Sciences, 99, 53(6): [5]. [M]. :, 6: Zhang Runchu. Multivariate statistical analysis[m]. Beijing: Science Press, 6: (in Chinese) [6]. MATLAB[M]. :, 6: Zhang Zhiyong. Proficient in MATLAB[M]. Beijing: Beijing University of Aeronautics and Astronautics Press, 6: (in Chinese) ( : )

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