156 Deterministic channel parameter estimation Cyele 1 Cycle 2 2 Mi-l …M网g口…g···☐2…M网a2…网a· Rx Figure 6.5 Timing structure of sounding and sensing windows with C2(2）=【e212c2.22j (6.14 C1(1.）=[c11(1,e）c1.21,] (6.15) (6.16 (9=a(),w(tj7 (6.17刀 The equation(6.13)can be recast as following s(t0)=exp(j2 avit0·{[ae1,1c21(2.)c1(1.d)+a1,2c2,1(2jcH2z(1) +a.21c2.2(2jc.(1)+au2.2c2.2(2.c.2(1.】ut-} (6.18) (6.19y 6.3.2 Channel Sounding Technique both and R.The timing structures used in these sounder systems are quire simlar and are depicred in Figure The mth antennae of Array 1 is active during the sounding windows (6.20) i=1 where i denotes the cycle index and t4m=(位-1)Tg+(m1-1)T. no theorn te m The so-called sensing window 9r(t-t.m.m)2m2=1,,M2,m1=1,,M corresponds to the case where ng is used.the s ndowsand the senin windows)eed merely
156 Deterministic channel parameter estimation 1 Tt 2 M1-1 M1 1 2 M2-1 M2 Cycle 1 Cycle 2 1 2 1 2 1 2 Tg Tsc Tcy t Tr Array 1 Switch 1 Tx Array 2 Switch 2 Rx M2-1 M2 M2-1 M2 M2-1 M2 Figure 6.5 Timing structure of sounding and sensing windows with C2(Ω2,ℓ) = c2,1(Ω2,ℓ) c2,2(Ω2,ℓ) (6.14) C1(Ω1,ℓ) = c1,1(Ω1,ℓ) c1,2(Ω1,ℓ) (6.15) Aℓ = αℓ,1,1 αℓ,1,2 αℓ,2,1 αℓ,2,2 = [αℓ,p2,p1 ] (6.16) u(t) = [u1(t), . . . , uM(t)]T . (6.17) The equation (6.13) can be recast as following s(t; θℓ) = exp(j2πνℓt) · αℓ,1,1c2,1(Ω2,ℓ)c T 1,1 (Ω1,ℓ) + αℓ,1,2c2,1(Ω2,ℓ)c T 1,2 (Ω1,ℓ) +αℓ,2,1c2,2(Ω2,ℓ)c T 1,1 (Ω1,ℓ) + αℓ,2,2c2,2(Ω2,ℓ)c T 1,2 (Ω1,ℓ) u(t − τℓ) (6.18) = exp(j2πνℓt) · X 2 p2=1 X 2 p1=1 αℓ,p2,p1 c2,p2 (Ω2,ℓ)c T 1,p1 (Ω1,ℓ) u(t − τℓ). (6.19) 6.3.2 Channel Sounding Technique Most of the widely-used channel sounders, such as the Propsound Czink (2007), the Medav sounder Richter and Thoma (2005), and the rBECS sounder Yin et al. (2012) are equipped with the radio frequency (RF) antenna switches at both Tx and Rx. The timing structures used in these sounder systems are quite similar and are depicted in Figure 6.5. The m1th antenna element of Array 1 is active during the sounding windows1 q1,m1 (t) = X I i=1 qTt (t − ti,m1 + Tg), m1 = 1, ..., M1, (6.20) where i denotes the cycle index and ti,m1 = (i − 1)Tcy + (m1 − 1)Tt. Here q1,m1 (t) is a real function, with value of 1 or 0 corresponding to the active or inactive moments of the m1th window. Let us define the sounding window vector q1(t) .= [q1,1(t), ..., q1,M1 (t)]T . The so-called sensing window qTsc (t − ti,m1,m2 ), m2 = 1, . . . , M2, m1 = 1, . . . , M1 corresponds to the case where 1Remarks: If another ordering of polarization sounding/sensing is used, the sounding windows q1(t) and the sensing windows q2(t) need merely to be appropriately redefined.
Deterministic channel parameter estimation 157 .The mth Tx antenna is active; The m2th Rx antenna is sensing. where 4m2m1=(-1)+(m1-1+(m2-10, The sensing window for the math Rx dual antenna is given by the real function (6.21) We can define the sensing window vector q2()=21),92.M2 as well as (6.22) Transmitted signal Making use of the sounding window vector (t),we have the explicit transmitted signal u(t)by concatenating the inputs of the M elements of Array 1 u)）=g()u( (6.23) Received signal The signal at the output of Switch 2 can be written as (6.24) with s(t:0)=exp(j2)()C2(D2)A.C(SL)q(t-T)u(t-T). 6.25) Implementing (6.18),we can rewrite s(t:0)=exp(j2mvt)q(t)s(t:0:) 2=11=】 (6.26) (2002a)to incorporate polarization. ng the same approa U(传：T）=q2()q(④'u(t-Ti. (6.27 With this definition,(6.26)can be further written as (6.28) We can also expresss(:)as (6.29 where 8)兰exp(j2mwut)cn(n2.t)Uncp(1） (6.30) an expression similar to (7)in Fleury et al.(2002a)
Deterministic channel parameter estimation 157 • The m1th Tx antenna is active; • The m2th Rx antenna is sensing, where ti,m2,m1 = (i − 1)Tcy + (m1 − 1)Tt + (m2 − 1)Tr. The sensing window for the m2th Rx dual antenna is given by the real function q1,m2 (t) = X I i X M1 m1=1 qTsc (t − ti,m2,m1 ). (6.21) We can define the sensing window vector q2(t) .= [q2,1(t), ..., q2,M2 (t)]T . as well as q2(t) = X I i=1 X M2 m2=1 X M1 m1=1 qTsc (t − ti,m2,m1 ). (6.22) Transmitted signal Making use of the sounding window vector q1(t), we have the explicit transmitted signal u(t) by concatenating the inputs of the M1 elements of Array 1 u(t) = q1(t)u(t). (6.23) Received signal The signal at the output of Switch 2 can be written as Y (t) = X L ℓ=1 q T 2 (t)s(t; θℓ) + r No 2 q2(t)W(t), (6.24) with s(t; θℓ) = exp(j2πνℓt)q T 2 (t)C2(Ω2,ℓ)AℓC1(Ω1,ℓ) T q1(t − τℓ)u(t − τℓ). (6.25) Implementing (6.18), we can rewrite s(t; θℓ) = exp(j2πνℓt)q T 2 (t)s(t; θℓ) = exp(j2πνℓt) · X 2 p2=1 X 2 p1=1 αℓ,p2,p1 q T 2 (t)c2,p2 (Ω2,ℓ)c T 1,p1 (Ω1,ℓ)q1(t) · u(t − τℓ). (6.26) Expression (6.26) is the extension of the first equation in (7) in Fleury et al. (2002a) to incorporate polarization. Following the same approach as in this paper, we define the M2 × M1 sounding matrices U(t; τℓ) = q2(t)q1(t) Tu(t − τℓ). (6.27) With this definition, (6.26) can be further written as s(t; θℓ) = exp(j2πνℓt) X 2 p2=1 X 2 p1=1 αℓ,p2,p1 c T 2,p2 (Ω2,ℓ)U(t; τℓ)c1,p1 (Ω1,ℓ). (6.28) We can also express s(t; θℓ) as s(t; θℓ) = X 2 p2=1 X 2 p1=1 sp2,p1 (t; θℓ), (6.29) where sp2,p1 (t; θℓ) .= αℓ,p2,p1 exp(j2πνℓt)c T 2,p2 (Ω2,ℓ)U(t; τℓ)c1,p1 (Ω1,ℓ), (6.30) an expression similar to (7) in Fleury et al. (2002a)
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