Channel measurements 113 ing 8 x 8 Mimo-TDM t and with
Channel measurements 113 Figure 5.7 The DoA and DoD estimates of multiple propagation paths by using 50 × 32 MIMO-TDM sounder without and with applying the sliding window. Figure 5.8 The DoA and DoD estimates of multiple propagation paths by using 8 × 8 MIMO-TDM sounder without and with applying the sliding window. Figure 5.9 The DoA and DoD estimates of multiple propagation paths by using 4 × 4 MIMO-TDM sounder without and with applying the sliding window
114 Channel measurements The original signal model without phase noise considered can be written as Y-2se:o+w (5.7刀 In (5.7)the vector represents the parameters of the th propagation path.The signal component S()is referred to as the observation of the signals in I measurement cycles,i.e. St0)= t:01 tE [O.T:] (5.8) s1,NM+)s(2,N,M+8).s，N,M+) The time seau with. .N and m 1. .M refers to the beginning of the interval when the mth Tx antenna transmits and the nth Rx antenna receives in the ith cycle,i.e. am=-生,+a-生r+m-z (5.9) h不mgkaaaomommeocsaitnoaebu,wihana The phase no can efo)wo uansmit anter cxpj9(t1.1,1+t)} Cxp(t2,1.1+t)} expjo(t1.1.1+t) xpjt12,1+) exptj(t22.1+t)... expijo(t.2,1+t) 亚()= expfjp(ti.N.1+t)) expfjp(N,1+t} expfj(tI.N.+t)) tE[O,T+] (5.10) xpJt1,1,2+t)} exp(jp(t2..2+t)) exp(j(t.1+t)} Lexpjp(t1.N.M+t)}expfjp(t2.N.M+t))...exp(jp(ti.N.M+t)) Including the phase noise components into the signal model in(5.7)yields: ￥（因=∑st8)⊙(0+W) (5.11) =1 thedpod Por he nhe 跏)=∑s(t0)⊙()+o( (5.12) for the PROPSound case,the phase noise observed can be considered as Gaussian distributed.It can be calculated 2ex550t0h1egh8n to approximate(),i.e. 的t)≈exp{j0}×(1+j△(t)+..) (5.13)
114 Channel measurements The original signal model without phase noise considered can be written as Y (t) = X L ℓ=1 S(t; θℓ) + W(t). (5.7) In (5.7) the vector θℓ = [Ω1,ℓ, Ω2,ℓ, τℓ, νℓ, αℓ] represents the parameters of the ℓth propagation path. The signal component S(θℓ) is referred to as the observation of the signals in I measurement cycles, i.e. S(t; θℓ) = s(t1,1,1 + t; θℓ) s(t2,1,1 + t; θℓ) . . . s(tI,1,1 + t; θℓ) s(t1,2,1 + t; θℓ) s(t2,2,1 + t; θℓ) . . . s(tI,2,1 + t; θℓ) . . . . . . . . . . . . s(t1,N,1 + t; θℓ) s(t2,N,1 + t; θℓ) . . . s(tI,N,1 + t; θℓ) s(t1,1,2 + t; θℓ) s(t2,1,2 + t; θℓ) . . . s(tI,1,2 + t; θℓ) . . . . . . . . . . . . s(t1,N,M + t; θℓ) s(t2,N,M + t; θℓ) . . . s(tI,N,M + t; θℓ) , t ∈ [0, Tr] (5.8) The time sequence ti,n,m with i ∈ [1, . . . , I], n ∈ [1, . . . , N] and m ∈ [1, . . . , M] refers to the beginning of the interval when the mth Tx antenna transmits and the nth Rx antenna receives in the ith cycle, i.e. ti,n,m = (i − I + 1 2 )Tcy + (n − N + 1 2 )Tr + (m − M + 1 2 )Tt (5.9) with Tcy, Tr and Tt representing the intervals between two consecutive cycles (within one burst), switch interval between two receive antennas and switch interval between two transmit antennas. The phase noise can be formulated as a matrix Ψ(t), i.e. Ψ(t) = exp{jϕ(t1,1,1 + t)} exp{jϕ(t2,1,1 + t)} . . . exp{jϕ(tI,1,1 + t)} exp{jϕ(t1,2,1 + t)} exp{jϕ(t2,2,1 + t)} . . . exp{jϕ(tI,2,1 + t)} . . . . . . . . . . . . exp{jϕ(t1,N,1 + t)} exp{jϕ(t2,N,1 + t)} . . . exp{jϕ(tI,N,1 + t)} exp{jϕ(t1,1,2 + t)} exp{jϕ(t2,1,2 + t)} . . . exp{jϕ(tI,1,2 + t)} . . . . . . . . . . . . exp{jϕ(t1,N,M + t)} exp{jϕ(t2,N,M + t)} . . . exp{jϕ(tI,N,M + t)} , t ∈ [0, Tr] (5.10) Including the phase noise components into the signal model in (5.7) yields: Yp(t) = X L ℓ=1 S(t; θℓ) ⊙ Ψ(t) + W(t) (5.11) where ⊙ denotes the Hadamard product. For the convenience of following derivation, the vectorized representation of (5.11) is used yp(t) = X L ℓ=1 s(t; θℓ) ⊙ ψ(t) + w(t) (5.12) with yp(t) = Y~ p(t). The vectorization is to rearrange a matrix into a column vector by concatenating all columns of the matrix. It is shown in Section 5.5.2 that when the delay lag is larger than 200 µs, the phase noise ϕ behaves as a white Gaussian random variable. In the case where the interval between two consecutive subchannels is about 510 µs, as for the PROPSound case, the phase noise observed can be considered as Gaussian distributed. It can be calculated from Figure 5.5 that when the delay lag is larger than 5 × 102 √ µs, the deviation in degree of the phase noise is 2π5 ∗ 109 × 100 × 10−6 × 10−7.5 × 180/π = 0.4051 degrees. Therefore, it is reasonable to use Taylor series expansion to approximate ψ(t), i.e. ψ(t) ≈ exp{jϕ0} × (1 + j∆ϕ(t) + . . .) (5.13)
Channel measurements 115 where 「ph.11+)1 9t1.21+t) △p()= -(t1.1.1+t) (5.14) (tI.N.M+t) denotes the deviation of the phase noise from the average of the phase noise.The termexpcan be included into the complex attenuation By dropping the high-order Taylor series component,we obtain (5.15) =∑st:0)+s0)⊙j△p(0+w) (5.16) d~CW(∑st8u),∑∑at)st9▣.Re+2 (5.17刀 =1=1 The above signal model can be used to derive estimators of both path parameters and the entries ofRIn paths,i.e. x(t)=s(t:0)⊙1+j]+w(t) (5.18) where w()standard white Gaussian noise with component variance withbeing nonnegative and satisfying the equality >B,=1. when sounding a subchannnel.Similarly():,and s() ..tal In the expectation step of the ith iteration,the objective function defined as Q:v.R)=logp(0.R-)f(Y=.R.)dz (5.19) is calculated.The probability density function(R)is 1 u48,,)=p-a-s6》严a-s6》 (5.20) where l.I denotes the determinant of the matrix given as argument.and x is the covariance matrix of Ex=s(0t)s(0)R+I. (5.21) Derivation shows tha Q(0:y.Re.-1)=-log(NMDEx,(0c)) -tr{x,(8)-(②x(l--8xy(t-8y(--1xy(- +4xy(-xr(-)} +2e(s(0e)x(6e）-uxr(-}-s(8e)x(8e)-2s(8) (5.22)
Channel measurements 115 where ∆ϕ(t) = ϕ(t1,1,1 + t) ϕ(t1,2,1 + t) . . . ϕ(tI,N,M + t) − ϕ(t1,1,1 + t) (5.14) denotes the deviation of the phase noise from the average of the phase noise. The term exp{jϕ0} can be included into the complex attenuation αℓ, ℓ = 1, . . . , I. By dropping the high-order Taylor series component, we obtain bsy(t) = X L ℓ=1 s(t; θℓ) ⊙ [1 + j∆ϕ(t)] + w(t) (5.15) == X L ℓ=1 s(t; θℓ) + s(t; θℓ) ⊙ j∆ϕ(t) + w(t) (5.16) A realistic assumption is made that during one subchannel, the phase noise does not change significantly. The variable t is then dropped from ∆ϕ(t). The elements of ∆ϕ are independent Gaussian random variables, i.e. ∆ϕ ∼ CN (0, Rϕ) with Rϕ being the covariance matrix of ∆ϕ. It is easy to show that y(t) is Gaussian distributed, i.e. y(t) ∼ CN ( X L ℓ=1 s(t; θℓ), X L ℓ=1 X L ℓ ′=1 s(t; θℓ)s(t; θℓ) H · Rϕ + σ 2 wI) (5.17) The above signal model can be used to derive estimators of both path parameters θℓ and the entries of Rϕ. In Taparugssanagorn et al. (2007c), the SAGE algorithm is derived for estimating θℓ under the assumption that Rϕ is known. In this SAGE algorithm, the admissible hidden data is defined to be the contribution of individual propagation paths, i.e. xℓ(t) = s(t; θℓ) ⊙ [1 + jψ] + w′ (t). (5.18) where w′ (t) ∈ CNM×I denotes standard white Gaussian noise with component variance σ 2 w′ = βℓσ 2 w, with βℓ being nonnegative and satisfying the equality P L ℓ=1 βℓ = 1. In real measurements, the discrete samples of xℓ(t) and y(t) are considered for parameter estimation. For notation convenience, we use y to represent y = [y(t);t = t1, . . . , tD] with D being the total number of samples in delay domain when sounding a subchannnel. Similarly, xℓ = [xℓ(t);t = t1, . . . , td], and s(θℓ) = [s(t; θℓ);t = t1, . . . , td]. In the expectation step of the ith iteration, the objective function Q(θℓ; y, Rϕ, θb[n−1]) defined as Q(θℓ; y, Rϕ, θb[n−1]) = Z log p(xℓ|θℓ, Rϕ)f(xℓ|Y = y, Rϕ, θb[n−1])dxℓ (5.19) is calculated. The probability density function p(xℓ|θℓ, Rϕ) is p(xℓ|θℓ, Rϕ) = 1 πNM |Σxℓ | exp{−(xℓ − s(θℓ))HΣ −1 xℓ (xℓ − s(θℓ))} (5.20) where | · | denotes the determinant of the matrix given as argument, and Σxℓ is the covariance matrix of xℓ: Σxℓ = s(θℓ)s(θℓ) H ⊙ Rϕ + σ 2 wI. (5.21) Derivation shows that Q(θℓ; y, Rϕ, θb[i−1]) = − log π NMD|ΣXℓ (θℓ)| − tr ΣXℓ (θℓ) −1 ΣXℓ (θˆ [i−1] ℓ ) − ΣXℓY (θˆ[i−1])ΣY (θˆ[i−1]) −1ΣXℓY (θˆ[i−1]) H + µXℓ|Y (θˆ[i−1])µXℓ|Y (θˆ[i−1]) H + 2re{s(θℓ) HΣXℓ (θℓ) −1µXℓ|Y (θˆ[i−1])} − s(θℓ) HΣXℓ (θℓ) −1 s(θℓ) (5.22)
116 Channel measurements where 2x,Y(-=s-∑-鬥oR+o2I (5.23) y-=区-a-門on+2 (5.24) ur-=o+8xY-y8y(--'(-立s-" (5.25) The estimate of is updated in the maximization step by solving the optimization problem 明=ag盟x{Q(8:y.Re.0n-} (5.26) Performance evaluation The performance of the derived SAGE algorithm has been evaluated by simulations in a single-path scenario graphs of three objective functions are c cessing no hase-nois of th e standard ompared,which are obtained byhedGaor ith phase e data,and the modine zraph with that obtained by the function for mi This that the modified 应心心益气平 algorithms has bee 人 al tha modified SAGE is applied. 5.6 Impact of the phase noise in MIMO channel parallel sounding 5.6.1 Background information easily observed that the random pha e noise with different values xistsin the data collected inindividual subchannels For example,for the line-of-sight (LOS) nario,the dominant propagation path is ob rved to take different values noise caused by the different in the transmitter and the receiver.this inter-subchanne phase of the path parameers,especially the angular parameters which are requred noise on the estimation of the azimuths of ario where only the azimutha esented here.we see that the crease when the variance of the inter-subchannel phase noise increases.This is consistent with we also p e a pre-whitening method to mitigate the impact of the phase noise iio of the angular paramererhodemon the eeee of the propod
116 Channel measurements where ΣXℓY (θˆ[i−1]) = s( ˆθ [i−1] ℓ ) X L ℓ=1 s( ˆθ [i−1] ℓ ) H ⊙ Rϕ + βℓσ 2 wI (5.23) ΣY (θˆ[i−1]) = X L ℓ=1 s( ˆθ [i−1] ℓ ) X L ℓ=1 s( ˆθ [i−1] ℓ ) H ⊙ Rϕ + σ 2 wI (5.24) µXℓ|Y (θˆ[i−1]) = s(θℓ) + ΣXℓY (θˆ[i−1])ΣY (θˆ[i−1]) −1 y − X L ℓ=1 s(θˆ [i−1] ℓ ) . (5.25) The estimate of θℓ is updated in the maximization step by solving the optimization problem θˆ [i] ℓ = arg max θℓ Q(θℓ; y, Rϕ, θb[n−1]) . (5.26) The multi-dimensional optimization problem in (5.11) can be solved by element-wise maximization of the objective function as described in Fleury et al. (1999). Performance evaluation The performance of the derived SAGE algorithm has been evaluated by simulations in a single-path scenario in Taparugssanagorn et al. (2007c). The covariance matrix of the phase noise Σϕ is obtained by using realistic measurement data. In the Fig. 3 of Taparugssanagorn et al. (2007c) which is also shown below as Figure ??, the graphs of three objective functions are compared, which are obtained by the standard SAGE algorithm processing nophase-noise data, the standard SAGE with phase-noise data, and the modified SAGE algorithm with phase-noise data. The curvature of the three graphs at the location of correct estimates are compared. It is apparent that the modified SAGE algorithm being used to process phase-noise-distorted data exhibits similar graph with that obtained by the standard SAGE processing the clean data. This demonstrates that the modified SAGE algorithm has the whitening function for mitigating the phase noise impact on the propagation path parameter estimation. The comparison of the root mean square estimation error (RMSEE) curves obtained by Monte-Carlo simulations among different SAGE algorithms has been illustrated in Fig. 4 in Taparugssanagorn et al. (2007c). The results show that the RMSEEs for the path parameters, which are Doppler frequency, azimuth of departure and azimuth of arrival, are all reduced when the modified SAGE is applied. 5.6 Impact of the phase noise in MIMO channel parallel sounding 5.6.1 Background information Huawei researchers have been constructing a channel measurement system consisting of a signal generator and an oscilloscope, which are used as the transmitter and the receiver, respectively. From the results sent by Huawei, it can be easily observed that the random phase noise with different values exists in the data collected in individual subchannels. For example, for the line-of-sight (LOS) scenario, the dominant propagation path is observed to take different values in delay. The deviation is up to dozens of delay samples. This problem may attribute to the existence of random phase noise caused by the different front-end chains in the transmitter and the receiver. Intuitively, this inter-subchannel phase noise can influence the estimation of the path parameters, especially the angular parameters which are required to be estimated by using the output signals from multiple subchnanels. In this chapter, we focus on the impact of the inter-subchannel phase noise on the estimation of the azimuths of arrival of propagation paths. The signal model presented here is for a simplified scenario where only the azimuthal parameters of paths are considered. However, the results obtained can be generalized to the cases where the estimation of the delays, Doppler frequencies are of interest. From the theoretical analysis presented here, we see that the estimation errors increase when the variance of the inter-subchannel phase noise increases. This is consistent with the simulation results obtained. From the description of the error, we also propose a pre-whitening method to mitigate the impact of the phase noise on the estimation of the angular parameters. Simulation results shown demonstrate the effectiveness of the proposed method.
Channel measurements 117 tigating the impact of the phase noise on the estimation.In Section 5.6.6, nted which e中000eu0中ad3m,、,、on aspud3m10 5.6.2 Signal model In the case where only one propagation path exists between the transmitter and the receiver,the received signaly() at the output of a antennas can be written as =ac)⊙p)+n(), (5.27刀 hiapa hpoducthe phae oe nd (enohe nuth a is the ph or vector p(t)can be written as p(t)=[expj(t)].exp{j()).....expj(t)}]. (5.28) the me e we e thar where (t).. fo)=20西epeo6-ml (5.29) series expansion to p(t),and get the first-order Taylor series expansion approximation: p≈p)+△op 5.30) where p(u)=[exp{ju}.exp(ju).....exp(ju] (5.31) and Ad is the vector containing the deviation between and i.e. △)=1()-4,2()-4，，M0-4uT (5.32) Sincep(t)has the form in (5.28), can be calculated as =u ap(t) =jexp{iμ} (5.33) withμbeing alμM×1 vector.Inserting(5.33)into(5.3o),we obtain p(t)≈p(m)+△(t)⊙jexp{μ =exp{i+jexp{i△(), (5.34 Inserting (5.34)into (6.75)yields for an approximation of u(t): t)≈ac(eo[exp{i+jexp{ip△p(t+n() =ac()exp{i+jexp{ira[c(0)⊙△+n(t) 5.35) It can be obs ed that the ved signal c contribu
Channel measurements 117 The rest of the chapter is organized as follows. In Section 5.6.2, the signal model is presented. In Section 5.6.3, the impact of the inter-subchannel phase noise on the estimation is derived. In Section 5.6.5, we proposed the method for mitigating the impact of the phase noise on the estimation. In Section 5.6.6, simulation results are presented which illustrate the impact of the phase noise, and evaluation of the performance of the proposed method is also conducted by using simulation. Conclusive remarks are given in Section 5.6.7. 5.6.2 Signal model In the case where only one propagation path exists between the transmitter and the receiver, the received signal y(t) at the output of M antennas can be written as y(t) = αc(θ) ⊙ p(t) + n(t), (5.27) where α represents the complex amplitude of the path, c(θ) denotes the array response at the azimuth θ, ⊙ is the Hardmard product, p(t) represents a vector containing the phasor attributed to the phase noise, and n(t) denotes the white Gaussian noise. The phasor vector p(t) can be written as p(t) = [exp{jφ1(t)}, exp{jφ2(t)}, . . . , exp{jφM(t)}], (5.28) where φ1(t), . . . , φM (t) are the random phases in the mth subchannel at the time instance t. We assume that φ1(t), . . . , φM (t) are i.i.d. von-Mise distributed random variables, i.e. f(φ) = 1 2πI0(κ) exp{κ cos(φ − µ)}, (5.29) where µ is the measure of location where the distribution is clustered about, and κ is a measure of concentration. We now consider the case where the concentration of φ around µ is high. In such a case, we may apply the Taylor series expansion to p(t), and get the first-order Taylor series expansion approximation: p(t) ≈ p(µ) + ∆φ(t) ⊙ ∂p(t) ∂φ φ=µ , (5.30) where p(µ) = [exp{jµ}, exp{jµ}, . . . , exp{jµ}]. (5.31) and ∆φ is the vector containing the deviation between φm and µ, i.e. ∆φ(t) = [φ1(t) − µ, φ2(t) − µ, . . . , φM (t) − µM ] T. (5.32) Since p(t) has the form in (5.28), ∂p(t) ∂φ φ=µ can be calculated as ∂p(t) ∂φ φ=µ = j exp{jµ} (5.33) with µ being all µ M × 1 vector. Inserting (5.33) into (5.30), we obtain p(t) ≈ p(µ) + ∆φ(t) ⊙ j exp{jµ} = exp{jµ} + j exp{jµ}∆φ(t). (5.34) Inserting (5.34) into (6.75) yields for an approximation of y(t): y(t) ≈ αc(θ) ⊙ [exp{jµ} + j exp{jµ}∆φ(t)] + n(t) = αc(θ) exp{jµ} + j exp{jµ}α[c(θ) ⊙ ∆φ(t)] + n(t). (5.35) It can be observed that the received signal can be approximated by the original signal contributed by the wave propagating along the path, and the signals that are the former weighted by the deviations of the instantaneous phase shifting from the mean of the phase