154 Deterministic channel parameter estimation 6.2 Maximum-likelihood method The maximum-likelihood estimation is considered as a standard method for estimating channel parameters based or certain predefined models.In this subsec ion,this method is briefly introduced by using the channel mode where the A narro (6.7刀 with spectral height of N The DoA is a unit vector uniquely determined by azimuth of arrival (AoA) and elevation of arrival 6 0,as =[sin()cos(),sin(@)sin(),cos()]. (6.8) 日=(a1.21,a2.22...a.2w (6.9) vtheohe ators of the model pa arameters can be calculated by maximizing the log-likelihood function of the A(e)=logp[eh. (6.10) npyin thethat the obe twhe The estimators can be obtained as 合L=argm8xA(⊙) (6.11) oblem to be solved in(6.11)is practically prohibitive due to the complexity of exhaustive searching mum-likelihood estimator can be evaluated by using the Cramer-Rao bound.In Fleury eta(1999).the derivation of the Cramer-Rao bound is provided.The readers are referred to this paper for the details 6.3 The EM and SAGE algorithms arrival of indi propagat paths.In Fleury et al.(2002a,b)the SAGE algorithm was e tended to include the initialization-and mproved SAGE (SAGB)algorithm.The acronym ISI stresses the fact that the ntiaizti effect of polariz 154 Deterministic channel parameter estimation 6.2 Maximum-likelihood method The maximum-likelihood estimation is considered as a standard method for estimating channel parameters based on certain predefined models. In this subsection, this method is briefly introduced by using the channel model where the propagation paths are only characterized by their directions of arrival (DoAs). Notice that this simplified model can be replaced by other more complicated channel models when deriving the ML estimators of the parameters of the model. A narrowband 1 × N single-input multiple-output (SIMO) scenario is considered where the propagation paths between the Tx and Rx have different DoA Ω. Following the nomenclature in Yin et al. (2003a), the narrowband representation of the impulse responses h ∈ CN of the SIMO channel can be written as h = X L ℓ=1 αℓc(Ωℓ) + w, (6.7) where ℓ is the index of specular propagation paths, L represents the number of paths, αℓ and Ωℓ are respectively, the complex attenuation and the direction of arrival (DoA) of the ℓth path, w represents the standard white Gaussian noise with spectral height of No. The DoA Ωℓ is a unit vector uniquely determined by azimuth of arrival (AoA) φ ∈ [−π, π] and elevation of arrival θ ∈ [0, π] as Ω = [sin(θ) cos(φ),sin(θ) sin(φ), cos(θ)]. (6.8) In (6.75), c(Ω) represents the array response at a given DoA. The parameters of interest for estimation in the above mentioned generic channel model (6.75) are Θ = (α1, Ω1, α2, Ω2, . . . , αN , ΩN ). (6.9) The ML estimators Θb ML of the model parameters can be calculated by maximizing the log-likelihood function of the Θ given the observations of h. Λ(Θ) = log p[Θ|h, (6.10) where p(θ) denotes the likelihood function of Θ. Applying the assumption that the noise component w is white Gaussian, it can be shown that Λ(Θ) = −N log(2πσw) − 1 2Nσ2 w xˆ [i+1] ℓ − X L ℓ=1 αℓc(Ωℓ) 2 . The estimators Θb ML can be obtained as Θb ML = arg max Θ Λ(Θ). (6.11) The optimization problem to be solved in (6.11) is practically prohibitive due to the complexity of exhaustive searching in multiple dimensions. The performance of the maximum-likelihood estimator can be evaluated by using the Cramer-Rao bound. In Fleury et al. (1999), the derivation of the Cramer-Rao bound is provided. The readers are referred to this paper for the details. 6.3 The EM and SAGE algorithms The space-alternating generalized expectation-maximization (SAGE) algorithm was applied for channel parameter estimation in Fleury et al. (1999) for estimating the delay, Doppler frequency, direction (azimuth and elevation) of arrival of individual propagation paths. In Fleury et al. (2002a,b) the SAGE algorithm was extended to include the estimation of the directions of departure of paths and used to estimate path parameters with the measurement data collected with multi-antenna array installed in both the transmitter and the receiver. This SAGE algorithm is called initialization-and-search-improved SAGE (ISI-SAGE) algorithm. The acronym ISI stresses the fact that the initialization and search procedures of the SAGE algorithm are optimized to speed up its convergence and enhance its capability of detecting weak paths. In order to accurately model the dispersion characteristics of the channels that incorporate the effect of polarizations, the ISI-SAGE algorithm has been extended in Fleury et al. (2003); Yin et al. (2003b) to include estimation of the polarization matrices of waves propagating from the Tx site to the Rx site in a MIMO system