54 Generic channel models The deviation of he direcion ofrival is calculated aswithdenoting the nominaldirectoof For simplicity reason,we focuson one cluster scenario.So thebo of thethuteris -w (3.30) eom密aeedsyaogeomesaud n=sn(e)+wn (3.31) d sm(。)=ecm(0e, (3.32) whereand are respectively the complex attenuation and the incident angle of the cth spectral wave. compare th performance of the estimatorusin the proposed approximate models with the one mate m 3.4.3 First-order Taylor expansion modelI The first order Taylor expansion with respect to the spread of the parameter is a6。+d≈ao)+daal where represents the nominal value and denotes the spread.Applying this principle to (3.30),we obtain an approximation model for slightly distributed sources as -低+i 侧+u2 (3.33) ndthe ne pdof theted 54 Generic channel models The deviation of the direction of arrival is calculated as ˜θc,ℓ = θc,ℓ − θc with θc denoting the nominal direction of arrival of all rays. For some applications, ˜θc,ℓ are assumed to be constant for multiple channel realizations. However they can be also assumed to be random variables, following approximately Gaussian distribution N ∼ (0, σ2 θ ). For simplicity reason, we focus on one cluster scenario. So the contribution of the cth cluster is sm(θc) = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ), (3.30) Assuming that the transmitted signal u(t) is unitary one, the received signal originating from the slightly distributed source and additive white Gaussian noise can be described as xm = sm(θc) + wm = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ) + wm, (3.31) where w is complex circularly symmetric additive white Gaussian noise with variance of σ 2 w. Traditionally the slightly distributed source is approximated with spectral wave. We call this approximation model as spectral wave model (SWM). The signal contribution of spectral wave at the output of the mth antenna can be written as sm(θc) = γc · cm(θc), (3.32) where γc and θc are respectively the complex attenuation and the incident angle of the cth spectral wave. Here we propose another two models that are used to approximate the slightly distributed source. In the simulation study section, we compare the performance of the estimators using the proposed approximate models with the one using the SWM model. 3.4.3 First-order Taylor expansion model I The first order Taylor expansion with respect to the spread of the parameter is a(φo + φ˜) ≈ a(φo) + φ˜ ∂a(φo) ∂φo , where φo represents the nominal value and φ˜ denotes the spread. Applying this principle to (3.30), we obtain an approximation model for slightly distributed sources as sm(θc) = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ) = X Lc ℓ=1 γc,ℓ · cm(θc + ˜θc,ℓ) ≈ X Lc ℓ=1 γc,ℓ · cm(θc) +X Lc ℓ=1 γc,ℓ · ˜θc,ℓ · ∂cm(θc) ∂θc , (3.33) where θc is the nominal direction of arrival and ˜θc,ℓ is the angle spread of the ℓth wave in the cth distributed source. Introducing parameters γc = X Lc ℓ=1 γc,ℓ, ψc = X Lc ℓ=1 γc,ℓ · ˜θc,ℓ