Deterministic channel parameter estimation 159 Therefore (p-jant)e (n(t. (6.36) =1,m2m1 Using parameter substitutiontin(6.36)yields Xm2.m(e,以)= exp(-j2m2.m+t)u（传m2m+-Tr6.m2m+td -ep(-j2rvdimm) 1 exp(-j2mvet)u'(ti.mzm+t-Te)re(ti.ma.m+)dt'. (6.37) Here u(t)is a periodic function with period T and ta is integer times of T,therefore u(m+)- u(t).Inserting this expression in(6.37)and substituting twitht,we obtain Xi.mm()= ep(-2mvtm2,m）u'化-Tep(-2mve)zt+tma,mdt Equation(6.35)can be reformulated as =月 Cai2nfn(a） (6.38) beg@=c.x.emd成=nut aVec(Af)=[a,1,a12,at,21,a,2.2 (6.39 [c(2dX(,c1,(,小 fa)= (6.40) c52n2)x(,w)c12n10 we obtain for G GI=ap f(0t) (6.41)Deterministic channel parameter estimation 159 Therefore Xℓ,m2,m1 (τℓ, νℓ) = X I i=1 Z ti,m2,m1 +Tsc ti,m2,m1 exp(−j2πνℓt)u ∗ (t − τℓ)xℓ(t)dt. (6.36) Using parameter substitution t ′ = t − ti,m2,m1 in (6.36) yields Xℓ,m2,m1 (τℓ, νℓ) = X I i=1 Z Tsc 0 exp(−j2πνℓ(ti,m2,m1 + t ′ ))u ∗ (ti,m2,m1 + t ′ − τℓ)xℓ(ti,m2,m1 + t ′ )dt′ = X I i=1 exp(−j2πνℓti,m2,m1 ) · Z Tsc 0 exp(−j2πνℓt ′ )u ∗ (ti,m2,m1 + t ′ − τℓ)xℓ(ti,m2,m1 + t ′ )dt′ . (6.37) Here u(t) is a periodic function with period Tsc, and ti,m2,m1 is integer times of Tsc, therefore u(ti,m2,m1 + t ′ − τℓ) = u(t ′ − τℓ). Inserting this expression in (6.37) and substituting t ′ with t, we obtain Xℓ,m2,m1 (τℓ, νℓ) = X I i=1 exp(−j2πνℓti,m2,m1 ) Z Tsc 0 u ∗ (t − τℓ) exp(−j2πνℓt)xℓ(t + ti,m2,m1 )dt. Equation (6.35) can be reformulated as G1 = X 2 p2=1 X 2 p1=1 α ∗ ℓ,p2,p1 c H 2,p2 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,p1 (Ω1,ℓ) ∗ , = X 2 p2=1 X 2 p1=1 α ∗ ℓ,p2,p1 fp2,p1 (θ¯ ℓ) (6.38) where fp2,p1 (θ¯ ℓ) .= c H 2,p2 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,p1 (Ω1,ℓ) ∗ and θ¯ ℓ = [Ω1,ℓ, Ω2,ℓ, τℓ, νℓ]. Defining αℓ .= Vec(AT ℓ ) = [αℓ,1,1, αℓ,1,2, αℓ,2,1, αℓ,2,2] T , (6.39) f(θ¯ ℓ) =     c H 2,1 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,1(Ω1,ℓ) ∗ c H 2,1 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,2(Ω1,ℓ) ∗ c H 2,2 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,1(Ω1,ℓ) ∗ c H 2,2 (Ω2,ℓ)Xℓ(τℓ, νℓ)c1,2(Ω1,ℓ) ∗     , (6.40) we obtain for G1 G1 = α H ℓ f(θ¯ ℓ). (6.41)