158 Deterministic channel parameter estimation 6.3.3 SAGE Algorithm Log-likelihood function of the complete/hidden data First,the complete/hidden data is defined as (6.31) It can be shown that the log-likelihood function of given the observation X(t)=(t)reads A(0)2Rs(:0)z(t)dt-Is(t:0)Pdt (6.32) After certain manipulations,it can be shown that A(:)x 2R(of f(B))-IPTeD(Sa..Sh)ot (6.33) The calculation ofG and Gwill be elaborated in the following Computation of G From(6.29)we can write -/会2am (6.34 Inserting(6.30)into(6.34)yields -三三/ano2m0 w.WYx.U.加0 X(r:小 (6.35) with X()denoting the M2x M dimensional matrix with entries X.ma.m(Te,ve)=exp(-j2vdtm(t:T)ze(t)dt =exp(-j2xvet)qz.ma(t)qn.m(t-rt)u"(t-r)ze(t)dt We may notice from the timing structure of the sounding and sensing system that,in each cycle,i.e.i=1,2...,only s principle yie 92m()q1,mt-T)u'(t-T)= u化-T）；tem2m4mam:+T 0 ；otherwise 158 Deterministic channel parameter estimation 6.3.3 SAGE Algorithm Log-likelihood function of the complete/hidden data First, the complete/hidden data is defined as Xℓ(t) = s(t; θℓ) + p βℓ r No 2 q2(t)W(t). (6.31) It can be shown that the log-likelihood function of θℓ given the observation Xℓ(t) = xℓ(t) reads Λ(θℓ; xℓ) ∝ 2R Z s(t; θℓ) ∗xℓ(t)dt | {z } G1  − Z |s(t; θℓ)| 2dt | {z } G2 . (6.32) After certain manipulations, it can be shown that Λ(θℓ; xℓ) ∝ 2R{α H ℓ f(θ¯ ℓ)} − IP Tsc · α H ℓ D˜ (Ω2,ℓ, Ω1,ℓ)αℓ. (6.33) The calculation of G1 and G2 will be elaborated in the following. Computation of G1 From (6.29) we can write G1 = Z X 2 p2=1 X 2 p1=1 sp2,p1 (t; θℓ) ∗xℓ(t)dt = X 2 p2=1 X 2 p1=1 Z sp2,p1 (t; θℓ) ∗xℓ(t)dt. (6.34) Inserting (6.30) into (6.34) yields G1 = X 2 p2=1 X 2 p1=1 Z sp2,p1 (t; θℓ) ∗xℓ(t)dt = X 2 p2=1 X 2 p1=1 Z α ∗ ℓ,p2,p1 exp(−j2πνℓt)c H 2,p2 (Ω2,ℓ)U ∗ (t; τℓ)c ∗ 1,p1 (Ω1,ℓ)xℓ(t)dt = X 2 p2=1 X 2 p1=1 α ∗ ℓ,p2,p1 c H 2,p2 (Ω2,ℓ) Z exp(−j2πνℓt)U ∗ (t; τℓ)xℓ(t)dt | {z } .=Xℓ(τℓ,νℓ) c ∗ 1,p1 (Ω1,ℓ), (6.35) with Xℓ(τℓ, νℓ) denoting the M2 × M1 dimensional matrix with entries Xℓ,m2,m1 (τℓ, νℓ) = Z exp(−j2πνℓt)U ∗ m2,m1 (t; τℓ)xℓ(t)dt = Z exp(−j2πνℓt)q2,m2 (t)q1,m1 (t − τℓ)u ∗ (t − τℓ)xℓ(t)dt. We may notice from the timing structure of the sounding and sensing system that, in each cycle, i.e. i = 1, 2, ..., I, only when the mth 2 transmitter antenna and the mth 1 receiver antenna are active, the product between q2,m2 (t) and q1,m1 (t) gives non-zero result. Applying this principle yields q2,m2 (t)q1,m1 (t − τℓ)u ∗ (t − τℓ) = ( u ∗ (t − τℓ) ; t ∈ [ti,m2,m1 , ti,m2,m1 + Tsc] 0 ; otherwise