28 3 5 JOURNAL OF APPLIED SCIENCES Electronics and Information Engineering Vol. 28 No. 3 May DOI:.3969/j.issn.255-8297..3. MUSIC 2. MUSIC Taylor MUSIC MUSIC Taylor TN9.7 255-8297()3-289-8 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 9--2-3-4 (No.BSLWCX) E-mail: wang_ding84@yahoo.com.cn DSP E-mail: hnwuying22@63.com
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 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 à Taylor M m= n= J(θ k, Ã) Ã=OM D ã (r) + J(θ k, Ã) Ã=OM D ã (i) () ã à 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  ( η  P  ) H A η   = e m (M) e (D)T n, { Re { Re J(θ k, Ã) ȧ H (θ k ) P A  = 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
292 28 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). 2. 3 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)
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). 2 2 4 ) (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 =.3 3 2 θ 2 =, r/λ =.5, θ =.3 4 2 θ 2 =, r/λ =.5, σ a =.3 2 2 3 3 /( ) Figure Direction-finding success probability versus the azimuth separation of the two sources 3.2.3.4.5.6.7.8.9...2 2 Figure 2 Direction-finding success probability versus the ratio of radius and wavelength
294 28 3..2.3.4.5.6.7.8 3 Figure 3 Direction-finding success probability versus the standard deviation of the modeling errors 3..2.3.4.5.6.7.8.9. /( ) 4 Figure 4 Direction-finding success probability versus the angle error tolerance 3 3 3 /( ) 5 Figure 5 Direction-finding success probability versus the azimuth separation of the two sources 3..25.3.35..45. 6 Figure 6 Direction-finding success probability versus the ratio of element spacing and wavelength db () θ =. 5.5 σ a =.3, θ =.3 6 2 θ 2 =, σ a =.3, θ =.3 (.5) 7 2 θ 2 =,.5 θ =.3 8 2 θ 2 =,.5 σ a =.3 3..2.3.4.5.6.7.8 7 Figure 7 Direction-finding success probability versus the standard deviation of the modeling errors 8 ). 2) 2
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