180 Statistical channel parameter estimation 7.1.4 Parameter Estimation using the 1*t-order GAM Model In this section,we introduce the 1-order GAM model and derive estimators of the model pa deterministic and stochastic ML methods,as well as a novel MUSIC algorithm.An As estimator based on these parameter estimators is also proposed. The 1t-order GAM Model The GAM model Asztely et al.(1997)makes use of the fact that the deviations are small with high probability. In the 1"-order GAM model,the first-orde Taylor series expansi n the re it sig al of the We regaed the model and the effective signal model is provided in Section.1.5. measure of the et al.(1997) 0≈ycAM=∑au0[c(可+rc'(例+w =a(0c(o)+8t)c(⑥)+w(④)， (7.9) where In matrix notation,(.9)reads VGAM(t)=F()E()+(t) (7.10) with F(o)=c(o)c(o)and (t)=a(t).B(t)T. The autocorrelation functions of o(t)and B(t)is calculated to be respectively, R付stoa()o"@+-】=. and Rg(r)E3()3(t+r】=a(r) (7.11) (7.11)that 0后=0月02 (7.12) deviatio s.th e ai cton spread can b 1975) SDS is propor Fleury(2 (7.13) e d In a scenario with DSDSs,(7.9)extends to v0≈aAu(e饼2oa0a+(e0c(网+o0 =B()y)+w(), (7.14180 Statistical channel parameter estimation 7.1.4 Parameter Estimation using the 1 st-order GAM Model In this section, we introduce the 1 st-order GAM model and derive estimators of the model parameters using standard deterministic and stochastic ML methods, as well as a novel MUSIC algorithm. An AS estimator based on these parameter estimators is also proposed. The 1 st-order GAM Model The GAM model Asztély et al. (1997) makes use of the fact that the deviations φ˜ d,ℓ are small with high probability. In the 1 st-order GAM model, the first-order Taylor series expansion of the array response is used to approximate the effective impact of the SDSs on the received signal. We regard a distributed scatterer as an SDS when its contribution to the output signal of the Rx array is closely approximated using the 1 st-order GAM model. A measure of the fit between the approximation model and the effective signal model is provided in Section 7.1.5. We first consider a single-SDS scenario. The function c(φ¯ + φ˜ ℓ) in (7.1) can be approximated by its first-order Taylor series expansion at φ¯. Inserting this approximation for each c(φ¯ + φ˜ ℓ) in (7.1) yields the 1 st-order GAM model Asztély et al. (1997) y(t) ≈ yGAM(t) .= X L ℓ=1 aℓ(t) c(φ¯)+φ˜ ℓc ′ (φ¯) + w(t) =α(t)c(φ¯) + β(t)c ′ (φ¯) + w(t), (7.9) where c ′ (φ¯) .= dc(φ) dφ   φ=φ¯ . In matrix notation, (7.9) reads yGAM(t) = F(φ¯)ξ(t) + w(t) (7.10) with F(φ¯) .=[c(φ¯) c ′ (φ¯)] and ξ(t) .=[α(t), β(t)]T. The autocorrelation functions of α(t) and β(t) is calculated to be respectively, Rα(τ) .= E[α(t)α ∗ (t + τ)] = X L ℓ=1 Raℓ (τ) and Rβ(τ) .= E[β(t)β ∗ (t + τ)] = σ 2 φ˜ · Rα(τ), (7.11) where σ 2 φ˜ .= E[φ˜2 ℓ ]. Note that E[φ˜ ℓ] = 0 according to Assumption 1). The parameter σ 2 φ˜ is the second-central moment of the azimuth deviation. By denoting the variances of α(t) and β(t) with σ 2 α and σ 2 β respectively, we conclude from (7.11) that σ 2 β = σ 2 φ˜ · σ 2 α. (7.12) This equality can also be obtained using the results given in (Shahbazpanahi et al. 2001, (49)–(51)). We refer to the parameter σφ˜ as the AS of the SDS. Note that as shown in Fleury (2000), the natural figure for characterizing direction dispersion is the direction spread. However, in a scenario with horizontal-only propagation and small azimuth deviations, the direction spread can be approximated by σφ˜ expressed in radian Fleury (2000). For example, in the case where the azimuth power spectrum of an SDS is proportional to the von-Mises probability density function Mardia (1975) fφ˜ ℓ (φ) = 1 2πI0(κ) exp{κ cos(φ − φ¯)}, (7.13) where κ denotes the concentration parameter and I0(·) represents the modified Bessel function of the first kind and order 0, the approximation is close provided κ ≥ 7, i.e. σφ˜ ≤ 10◦ Fleury (2000). In a scenario with D SDSs, (7.9) extends to y(t) ≈ yGAM(t) .= X D d=1 αd(t)c(φ¯ d) + βd(t)c ′ (φ¯ d)  + w(t) = B(φ¯)γ(t) + w(t), (7.14)