Generic channel models 55 cm(0c)=e-j2(m-1)(As/X)sin(0) .e-j2(m-1)(/)sin(0) (3.34) So Equation (3.33)can be written to be sm(0e)≈2Ye·e-2(m-1△/)im(0e）-e·j2x(m-1） (As/X)cos(0e))e-j2(m-1)(/)() The above expression can be written in matrix notation as s(6)=D(a)B, (3.35) where。,÷a.l,B is a column vector B.÷be,vgT,andD(a.)is2×Mmatrix defined as follows D(ae)÷a(a)b(ael, with a(de)=[cm(0e):m=1,....MT, Written in matrix notation,the received signal is x=8(8e)+0 =D(a。)a。+w with zero mean and variance,and they are independent with the complex attenuations.When the comple )it can be easily 卧d=吃d=d=0 E购=E∑Yeid=∑EhdE9u=0, =1 =1 =写 =吃a21-nii,例 =L2听=呢哈 c,we ue呢dee呢the beanbe boo恤ti
Generic channel models 55 and noticing that when the field pattern of the antenna elements is assumed to be isotropic, the steering vector and its derivative can be written as cm(θc) = e −j2π(m−1)(∆s/λ) sin(θc) , ∂cm(θc) ∂θc = (−j2π(m − 1)(∆s/λ) cos(θc)) ·e −j2π(m−1)(∆s/λ) sin(θc) . (3.34) So Equation (3.33) can be written to be sm(θc) ≈ γc · e −j2π(m−1)(∆s/λ) sin(θc) − ψc · j2π(m − 1) ·(∆s/λ) cos(θc))e −j2π(m−1)(∆s/λ) sin(θc) . The above expression can be written in matrix notation as s(θc) = D(θ¯ c)βc, (3.35) where θ¯ c .= [θc], βc is a column vector βc .= [γc, ψc] T , and D(θ¯ c) is 2 × M matrix defined as follows D(θ¯ c) .= [a(θ¯ c) b(θ¯ c)], with a(θ¯ c) .= [cm(θc); m = 1, . . . , M] T , b(θ¯ c) .= [∂cm(θc) ∂θc ; m = 1, . . . , M] T Written in matrix notation, the received signal is x = s(θc) + w = D(θ¯ c)βc + w. The nominal incident angle θc is constant across all the snapshots. But the angle spreads ˜θc,ℓ, ℓ = 1, . . . , Lc could have different constellations from snapshot to snapshot. We assume that ˜θc,ℓ, ℓ = 1, . . . , Lc follows the Gaussian distribution with zero mean and variance σ 2 θ , and they are independent with the complex attenuations. When the complex attenuation αc,ℓ, ℓ = 1, . . . , Lc is assumed to be random variable with N (0, 1 Lc σ 2 γc ), it can be easily shown that the elements in the vector βc are also random variables. Their first and second moments can be computed as following, E[γc] = E[ X Lc ℓ=1 γℓ] = X Lc ℓ=1 E[γℓ] = 0, E[ψc] = E[ X Lc ℓ=1 γℓ ˜θℓ] = X Lc ℓ=1 E[γℓ]E[ ˜θℓ] = 0, E[γcγ ∗ c ] = E[ X Lc ℓ=1 γℓ X Lc ℓ ′=1 γ ∗ ℓ ′ ] = X Lc ℓ=1 E[γℓγ ∗ ℓ ] = σ 2 γc , E[ψcψ ∗ c ] = E[ X Lc ℓ=1 γℓ ˜θℓ X Lc ℓ ′=1 γ ∗ ℓ ′ ˜θ ∗ ℓ ] = X Lc ℓ=1 E[γℓγ ∗ ℓ ]E[ ˜θℓ ˜θ ∗ ℓ ] = Lc 1 Lc σ 2 γc σ 2 θ = σ 2 γc σ 2 θ . For notational convenience, we use σ 2 ψc to denote σ 2 γc σ 2 θ in the subsequent. It can be also shown similarly that E[γcψ ∗ c ] = 0
56 Generic channel models R=ElxaM] =D(0e)ElBe:]D(0e)+ =c(0)P.c(8)H+2I where R-匠] 3.4.4 First-order Taylor expansion model II Another approximate model using the first order Taylor expansion can be written as ≈e·cn(0。+) -ee-s((() (3.36) where is redefined to be+,and (is the angle spread.Written in vector notation we obtained s(0)=e·c(ae), (3.37) where=001.The received signal can be written in vector notation as x=c(a)he+w ee of the hed R=Ezx] =cae)Ehe11c(ae+2】 =a2 c(0)c(6)+a2I. 3.5 Time-evolution model kkerna and Herben (2007)the tir e-variant characteristics of path parameters are ed estimation methods t.This'（realistic) c出c子 ermore,due to alpaths.These performance of algorithms and the of the channe e te therefore of great importance to use appropriate algorithms to estimate
56 Generic channel models The covariance matrix of the received signal x which consists of the signal of a single source, can be calculated to be R = E[xxH] = D(θ¯ c)E[βcβ ∗ c ]D(θ¯ c) H + σ 2 wI = c(θc)Pcc(θc) H + σ 2 wI where Pc = σ 2 γc 0 0 σ 2 ψc . 3.4.4 First-order Taylor expansion model II Another approximate model using the first order Taylor expansion can be written as sm(θc) ≈ γc · cm(θc) + ψc · ∂cm(θc) ∂θc ≈ γc · cm(θc + ψc γc ) = γc · e −j2π(m−1)(∆s/λ) sin(θc+j∆θc) , (3.36) where θc is redefined to be θc + R{ ψc γc }, and ∆θc .= I{ ψc γc } is the angle spread. Written in vector notation we obtained s(θc) = γc · c(θ¯ c), (3.37) where θ¯ c = [θc, ∆θc]. The received signal can be written in vector notation as x = c(θ¯ c)γc + w. The parameter θc and ∆θc are assumed to be deterministic over realizations. The covariance matrix of the received signal x is written as R = E[xxH] = c(θ¯ c)E[γcγ ∗ c ]c(θ¯ c) H + σ 2 wI = σ 2 γc c(θ¯ c)c(θ¯ c) H + σ 2 wI. 3.5 Time-evolution model Recently, the temporal behavior of propagation paths have gained much attention Czink et al. (2007b); Kwakkernaat and Herben (2007). In Kwakkernaat and Herben (2007) the time-variant characteristics of path parameters are considered as an additional degree of freedom for path clustering. In Czink et al. (2007b), the evolution of clusters of paths is illustrated using measurement data. In these works, the path evolution characteristics are obtained indirectly from the path parameter estimates computed from individual observation snapshots. The used estimation methods, e.g. the SAGE algorithm Czink et al. (2007b) and the Unitary ESPRIT Kwakkernaat and Herben (2007), are derived under the assumption that the path parameters in different observation snapshots are independent. This (unrealistic) assumption results in a “loss of information” in the estimation of the path evolution in time. Furthermore, due to model-order mismatch and heuristic settings in these algorithms, such as the (usually fixed) dynamic range, a timevariant path may remain undetected in some snapshots. As a result, a time-variant path can be erroneously considered as several paths. These effects influence the performance of clustering algorithms and the effectiveness of the channel models derived based on these results. It is therefore of great importance to use appropriate algorithms to estimate the temporal characteristics of paths directly. In recent years some methods have been proposed to track time-variant paths for multiple-input multiple-output (MIMO) channel sounding Chung and Böhme (2005); Richter et al. (2006); Salmi et al. (2006). In Chung and Böhme