Statistical channel parameter estimation 181 NAEstimators In this subsection,the standard deterministic and stochastic ML estimation methods as well as a novel MUSIC algorithm are applied using the 1-order GAM model to derive estimators of the NAs of SDSs Deterministie ML (DML)NA Estimator ML estimator of is calculated as Krim and Viberg (1996) (7.15) The parametersy(t).t=t, are estimated as (Y(t))pMB()'y(t),t=h.....tx. (7.16) Stochastic ML (SML)NA Estimator respectively.Let be the vector containing the p to be estimated: (7.17 The MLestimator of is a solution to the maximization problem Krim and Viberg (1996) SsML arg max{-In(lyGaMll -tr[(GAM)]), (7.18) where the covariance matrix of GAM()in (7.14)reads GM=B()R,B(+2I (7.19 Here.denores maris and i)is the covariance maris of. The maximization op )n(retivdimensionl and e search procedures prohibits the implementation of opML and th SAGE algorithm Fleury et al.(1999);Yin and Fleury (2005)provides w-complexity approximation of these ML estimators MUSIC NA Estimator The standard MUSIC algorithm Schmidt(1986)derived based on the SS model(7.3)uses the pseudo-spectrum llc(o) 7.20) Here..e denotes the Frobenius norm and E is an orthonormal basis of the estimated noise subspace calculated The azimuths of the D scatterers are estimated to be the arguments of the pseudo-spectrum corresponding Statistical channel parameter estimation 181 where B(φ¯) .= [c(φ¯ 1), c ′ (φ¯ 1), . . . , c(φ¯D), c ′ (φ¯D)] and γ(t) .= [α1(t), β1(t), . . . , αD(t), βD(t)]T. Under Assumptions 3)–5) in Subsection 7.1.2 the elements in the vector γ(t) are uncorrelated. NA Estimators In this subsection, the standard deterministic and stochastic ML estimation methods as well as a novel MUSIC algorithm are applied using the 1 st-order GAM model to derive estimators of the NAs of SDSs. Deterministic ML (DML) NA Estimator The DML NA estimator based on the 1 st-order GAM model can be derived similarly to the SS-ML azimuth estimator (7.4). Assuming that the weight samples αd(t) and βd(t), t = t1, . . . , tN , d = 1, . . . , D in (7.14) are deterministic, the ML estimator of φ¯ is calculated as Krim and Viberg (1996) φ ˆ¯ DML=arg max φ¯ {tr[ΠB(φ¯)Σˆ y]}. (7.15) The parameters γ(t), t = t1, . . . , tN are estimated as (\γ(t))DML=B(φ ˆ¯) †y(t), t = t1, . . . , tN . (7.16) Stochastic ML (SML) NA Estimator The SML azimuth estimator derived based on the SS model was introduced in Jaffer (1988). We obtain the SML NA estimator based on the 1 st-order GAM model in a similar manner. Making use of the assumptions 1)–5) in Section 7.1.2 and invoking the central limit theorem, the weight samples αd(t) and βd(t), t = t1, . . . , tN , d = 1, . . . , D are uncorrelated complex circularly-symmetric Gaussian random processes with variances σ 2 αd and σ 2 βd respectively. Let Ω be the vector containing the parameters to be estimated: Ω .= [σ 2 w, φ¯ d, σ2 αd , σ2 βd ; d = 1, . . . , D]. (7.17) The ML estimator of Ω is a solution to the maximization problem Krim and Viberg (1996) Ωb SML = arg max Ω {−ln[|ΣyGAM |] − tr (ΣyGAM ) −1Σˆ y }, (7.18) where the covariance matrix ΣyGAM of yGAM(t) in (7.14) reads ΣyGAM = B(φ¯)RγB(φ¯) H + σ 2 wIM . (7.19) Here, IM denotes the M × M identity matrix and Rγ = diag(σ 2 α1 , σ2 β1 , . . . , σ2 αD , σ2 βD ) is the covariance matrix of γ(t). Here, diag(·) denotes a diagonal matrix with diagonal elements listed as argument. The maximization operations in (7.15) and (7.18) require respectively a D-dimensional and a (3D + 1)-dimensional search. The high computational complexity of these search procedures prohibits the implementation of φ ˆ¯ DML and Ωb SML in real applications. As an alternative, the SAGE algorithm Fleury et al. (1999); Yin and Fleury (2005) provides with a low-complexity approximation of these ML estimators. MUSIC NA Estimator The standard MUSIC algorithm Schmidt (1986) derived based on the SS model (7.3) uses the pseudo-spectrum fMUSIC(φ)= kc(φ)k 2 F kc(φ)HEwk 2 F . (7.20) Here, k · kF denotes the Frobenius norm and Ew is an orthonormal basis of the estimated noise subspace calculated from Σˆ y. The azimuths of the D scatterers are estimated to be the arguments of the pseudo-spectrum corresponding to its D highest peaks.