108 Channel measurements Define the r Expectation (E-)Step: ompute Efiog(0s..dg 1=i+1 Maximization (M-)Step: ComputesargmxA().update No Figure 5.3 Flow graph of the SAGE algorithm
108 Channel measurements Initialize ˆθ 0 Iteration i Define the parameter subset θSi Define the admissible data space XS i Expectation (E-) Step: Compute Λ(θSi ) = E{log f(XS i ; θSi , ¯θSi )|Y = y; θ = ˆθ i} Maximization (M-) Step: Compute ˆθSi = arg max θSi Λ(θSi ), update ˆθ i Convergence Achieved? Yes No i = i + 1 Output ˆθ = ˆθ i Figure 5.3 Flow graph of the SAGE algorithm
Channel measurements 109 the entries inwith indices listed in the complement ofS xs:the hidden-data space selected for @s, a realization of, s:theindcxsetscleciedinthteihiterationoftheSAGEalgorithnm agpeceasodaedwihksanadnislehidendaifthefolomrgondionsahadResirandHern f(u.r5:0)=f(ulr5:0s)f(r5:0) (5.2) The parameter subsets anon.The number of elements in S n be larger than one,depending on the resolution of the equipment,the admissible hidden-data should be defined to embody the sum of the signa oribioof the two paths.In this case.conais more than one element and the maximization step becomes hwsthe lowof the SAGE lgorithm.One iteration of the SAGE algorithm conists of wo major e expectation of the loglikelh nood function o estimates from the previous iteration.Thisx the the M-step.To the in thecase where the parameter vecto nuhemulmrm-ime o the SAGE framework Fleury et al.(1999),i.e. The RIMAX Algorithm The RIMAX n paths and disp tion of t diffuse scatterers The dCCmtpathCcmponcatsinth: This function can be descibedby three parameterthe time ofaal and the delay spread of the dense mutipath nsists of the p of the specular paths and the other set contains the p rameters characterizing the dense multip ath com onents.In the M steps of the algorithm,the grad dient based m hods,such as Gauss-Nev evenverg-Marquardt M q1963 of the Fisher information matrix of the parameter estimates Richter and Thoma (2003).The diagonal elements of the inverse of this matr provide estimates when its para considered to be"unreliable"Richter et a (2003). In the RI rithm.these estimation sche can be exrended to include the estimation of the characteristics of (largely-)distributed diffuse scatterers Richter and Thoma(2003). 5.4 Evaluation of the measurement efficiency secton.we
Channel measurements 109 θS˜: the entries in θ with indices listed in the complement of S, XS : the hidden-data space selected for θS, x S : a realization of XS , S i : the index set selected in the ith iteration of the SAGE algorithm. The space XS associated with θS is an admissible hidden data if the following condition is satisfied Fessler and Hero (1994) f(y, xS ; θ) = f(y|x S ; θS˜)f(x S ; θ). (5.2) The above equation implies that the conditional distribution f(y|x S ; θ) coincides with f(y|x S ; θS˜). In the SAGE algorithm proposed for channel parameter estimation Fleury et al. (1999), the parameter subsets S i contain one element. Thus, the multiple-dimensional maximization in the ML estimation reduces to an onedimensional search in each SAGE iteration. The number of elements in S i can be larger than one, depending on the definition of the hidden-data space XS i . For example, when two paths are closely-spaced with separation below the resolution of the equipment, the admissible hidden-data should be defined to embody the sum of the signal contributions of the two paths. In this case, S i contains more than one element and the maximization step becomes computationally more “expensive”. Fig. 5.3 shows the flow graph of the SAGE algorithm. One iteration of the SAGE algorithm consists of two major steps: expectation (E-) step and maximization (M-) step. In the E-step, the expectation of the loglikelihood function of admissible hidden data for the current parameter vector θS is computed based on the observation and the parameter estimates from the previous iteration. This expectation is an objective function that is maximized with respect to the parameter vector θS in the M-step. To further reduce the complexity, in the case where the parameter vector θS contains more than one entry the coordinate-wise updating procedure Fleury et al. (1999) can be used to estimate these parameter entries sequentially. Thus, the multiple-dimensional optimization problem is solved using 1-dimensional searches. This coordinate-wise updating procedure still belongs to the SAGE framework Fleury et al. (1999), i.e. updating each parameter entry can be viewed as one SAGE iteration. The RIMAX Algorithm The RIMAX algorithm Richter (2004), Richter and Thoma (2005) and Richter et al. (2003) can be viewed as an extension of the SS-model-based SAGE algorithm Fleury et al. (1999). The RIMAX algorithm can be used for joint estimation of the parameters characterizing specular propagation paths and dispersion of distributed diffuse scatterers. The contribution of the distributed diffuse scatterers to the received signal is called dense multipath components in the papers. The power delay profile of these components is characterized using a one-sided exponential decaying function. This function can be described by three parameters: the time of arrival and the delay spread of the dense multipath components, as well as the average power of these components. In the RIMAX algorithm, unknown parameters are grouped into two sets. One set consists of the parameters of the specular paths and the other set contains the parameters characterizing the dense multipath components. In the Msteps of the algorithm, the gradient based methods, such as Gauss-Newton or Levenverg-Marquardt Marquardt (1963) algorithm, are implemented. For each specular path, an approximation of the Hessian is computed to be the estimate of the Fisher information matrix of the parameter estimates Richter and Thomä (2003). The diagonal elements of the inverse of this matrix provide estimates of the variances of the estimated parameters. In this algorithm, the variance estimates are used to describe the reliability of the corresponding parameter estimates. A specular path is dropped when its parameter estimates are considered to be “unreliable” Richter et al. (2003). In the RIMAX algorithm, individual dominant path components are treated as contributions of specular paths. In the estimation schemes proposed in this thesis (Chapter ??), the dominant path components are treated as dispersed path components. Similar to the RIMAX algorithm, these estimation schemes can be extended to include the estimation of the characteristics of (largely-)distributed diffuse scatterers Richter and Thomä (2003). 5.4 Evaluation of the measurement efficiency The following words should go to the beginning of the chapter. It is always necessary to evaluate whether the measurements have been taken effectively. In this section, we describe the impact of the inaccurate calibration on the
110 Channel measurements rs Wo will 5.5 Impact of phase noise in a TDM channel sounding system 5.5.1 Introduction Most available channel sounders for measuring mmo spatial channel adopt the time-division-multiplexing mode Examples of these channel soun ii)enough samples of the channels are collected,which offer sufficiently high intrinsic resolution for parameter 中R贤的子发。 ed in the following items.ie.the local phased locked s of the antenna response ng in the near oupling 2004, ete 020056 can be modeled as the zero-mean non-stationary infinite power Wiener process Almers et al (2005).Usually the long erm varying phase corrected or mi ated by ding synchronization devices,such as Clo nd t der Thehor varyn phaseppwthn themwe eriod much smaller than I second,usually noise on the estimation and the techniques proposed to mitigate the impact of the phase noise. 5.5.2 Behavior of the short-term phase noise e for the measurement campaigns in Chapter9 address in Czink(2007),50x32 MIMO matrix has beer duration is 7ms.so for this scenario.the properties of the phase noise with the time less than 1 of interest (5.3) where y(t)dt -9t+）-ot (5.4)
110 Channel measurements estimation results. We will present how to avoid or correct the estimation results for example by introducing some lookup-table based methods or use analytical methods to correct the estimation results. The impact of inaccurate calibration data on the estimation results are described (Käske et al. 2009). The influence of the phase noise on the estimation results is also shown (Taparugssanagorn and Ylitalo 2005) (Taparugssanagorn et al. 2007a) (Taparugssanagorn et al. 2007b). 5.5 Impact of phase noise in a TDM channel sounding system 5.5.1 Introduction Most available channel sounders for measuring MIMO spatial channel adopt the time-division-multiplexing mode. Examples of these channel sounders are the PROPSound, the RUSK sounder, and the sounder designed by the CRC, Canada. The propagation channels between any pair of the Tx antenna and Rx antenna are measured sequentially in timeslots. During the measurements, two switches are applied in the Tx and Rx for connecting the RF transceiving chains to the specific Tx and Rx antennas respectively in the dedicated timeslots. As indicated in (Taparugssanagorn et al. 2007a), the switching rate should be selected to satisfy two criteria: i) all subchannels between the Tx antennas and the Rx antennas are measured within the channel coherence time; ii) enough samples of the channels are collected, which offer sufficiently high intrinsic resolution for parameter estimation. In the TDM-sounding scheme, phase noise can be generated in the following items, i.e. the local phased locked oscillators at the Tx and the Rx, the switches at the Tx and the Rx, and the calibration errors of the antenna responses. The latter errors attribute to the low antenna gain due to the directional characteristics of the antennas, the coupling among antennas, and the impact of the scatterers appearing in the near field in the vicinity of the antenna array. Phase noise generated by the local phase locked oscillators in the Tx and the Rx can have impact on the MIMO channel capacity estimation (Baum and Bölcskei 2004; Pedersen et al. 2008b; Taparugssanagorn and Ylitalo 2005), and on channel parameter estimation (Taparugssanagorn et al. 2007a,b). The long-term slowly varying phase noise can be modeled as the zero-mean non-stationary infinite power Wiener process Almers et al. (2005). Usually the longterm varying phase noise can be corrected or mitigated by adding synchronization devices, such as the Rubidium Clock, in the Tx and the Rx, and thus, it is not necessary to consider the long-term varying phase noise in the MIMO channel sounder. The short-term varying phase noise, appearing within the time period much smaller than 1 second, usually has significant impact on the high-resolution parameter estimation. For this reason, we concentrate on investigating the behavior of short-term varying phase noise. In the following subsections, we will show the impact of the phase noise on the estimation and the techniques proposed to mitigate the impact of the phase noise. 5.5.2 Behavior of the short-term phase noise The short-term phase noise, observed in less than 1 s, may be modeled as an autoregressive integrated moving average (ARIMA) process Taparugssanagorn et al. (2007a). Normally, for a MIMO channel sounder, a measurement cycle is usually less than 1 s. For instance, for the measurement campaigns in Chapter 9 address in Czink (2007), 50 × 32 MIMO matrix has been adopted for 5.25 GHz. A subchannel is sounded within a period of 510µs. Thus, for one cycle of measurement, the time is about 8.42 ms. In the case where 4 cycles are combined and the data is processed as one snapshot, the time duration is 67 ms. So for this scenario, the properties of the phase noise with the time less than 100 ms are of interest to investigate. The Allan variance is applied to characterize the time-domain statistical behavior, which is calculated as σ 2 y (τ) = E (¯yk+1 − y¯k) 2 2 (5.3) where y¯k = 1 τ Z tk+τ tk y(t)dt = φ(tk + τ) − φ(tk) 2πfcτ . (5.4)
Channel measurements 111 早 一 点 Figure 5.4 The measurement setting for measuring the phase noise The term()in (5.4)represents the instantaneous normalized frequency deviation from the carrier frequencyf which is computed as 1,d =2m不。d (5.5) mn thar the of the phaeThe of c2m0=2w-2m2amr24an）-2oa+oa识 1 -2 (5.6) and denoting an dmer rein tothe tora number of am of p ase no el so same fixed atremuaror ThathRx eachquipp with individual clock operate in the way as in real eviation,1.e.y m1 is computed an iustrated in Figure 5.5.In this figure,both r ed onar o th and th of0 for the lan variance of the phase noise isidentical with that obrained froma whit 5.5.3 Mitigation of the impact on the high-resolution parameter estimation A method proposed in Taparugssanagorn et al.(2007d)for mitigating the impact of phase noise on the parameter ultiple secut napshots of r ement data as the observ on of the function is to average over multiple channel observations,such that the impact of the phase noise can be compromised cvcles of data.similan 8 and 5.9 respectively.It can be observed nois thm is derved t sed n the sinasod icity,th dified the assumption t r than the is referred to as the inverse of delay spread of the channe TheTDM channel bandwidth.Here I soun
Channel measurements 111 Figure 5.4 The measurement setting for measuring the phase noise The term y(t) in (5.4) represents the instantaneous normalized frequency deviation from the carrier frequency fc, which is computed as y(t) = 1 2πfc · dφ(t) dt (5.5) with φ(t) denotes the instantaneous phase variation. Assuming that the sampling rate of the phase 1 T . The samples of the Allan variance at τ = mT can be estimated as ˆσ 2 y (mT ) = 1 2(N − 2m)(2πfcmT ) 2 N X−2m i=1 (φ(ti+2m) − 2φ(ti+m) + φ(ti))2 , (5.6) with m = 1, . . . , N−1 2 and N denoting an odd number referring to the total number of samples of the phase. An example of measuring the phase noise is illustrated as follows. Fig. 5.6 depicts the measurement setting used to measure the phase noise of a single-input single-output channel sounder. An RF cable connects the Tx and the Rx with a 50 dB fixed attenuator. The Tx and the Rx each equipped with individual clock operate in the same way as in real field measurements. The Allan deviation, i.e. σˆy(mT ), is computed and illustrated in Figure 5.5. In this figure, both the sample Allan variance and their asymptotic characteristics computed from the measured phase noise sequence are depicted. Furthermore the curves computed based on proposed models are also illustrated. It can be observed that an ARMA model with model parameters computed based on the sample Allan variance can be used to describe the behavior of the Allan variance. Furthermore, the short term phase noise component predominates within the range of τ ∈ [0, 200µs]; for τ > 200µs, the Allan variance of the phase noise is identical with that obtained from a white phase noise, as suggested in Characterization of frequency and phase noise (1986); for τ > 1s, the phase noise can be described using random walk models. 5.5.3 Mitigation of the impact on the high-resolution parameter estimation A method proposed in Taparugssanagorn et al. (2007d) for mitigating the impact of phase noise on the parameter estimation performance is to consider multiple consecutive snapshots of measurement data as the observation of the same channel. This method is called “sliding window”. For the considered TDM-based sounding system, this sliding window solution is extended to the spatial domain, i.e. by considering more antennas. In general, the sliding window function is to average over multiple channel observations, such that the impact of the phase noise can be compromised to certain degree. Fig. 5.7 (a) and (b) depict respectively the comparison of the estimation results obtained by using the SAGE algorithm with and without using the sliding window solution over 20 cycles of data. Similarly results obtained with 8 × 8 and 4 × 4 MIMO channel matrices are also illustrated in Figure 5.8 and 5.9 respectively. It can be observed that by using the sliding window function, it is possible to reduce the probability of generation of artifact estimates. Another method introduced in Taparugssanagorn et al. (2007b) is to modify the specular-path SAGE algorithm Fleury et al. (1999) to include a whitening function based on the known covariance matrix of the phase noise. The modified SAGE algorithm is derived based on the signal model introduced in Section 3.3. For simplicity, the bidirection-delay-Doppler frequency generic specular path model is modified to a narrowband channel model based on the assumption that the signal bandwidth is much smaller than the channel bandwidth. Here, the channel bandwidth is referred to as the inverse of delay spread of the channel. The TDM sounding scheme is considered
112 Channel measurements 0-16 10 Time] Figure 5.5 Allan deviation 3m 21 21m RX antenna TX antenna Top view Figure 5.6 The measurement setting for investigating the impact of phase noise
112 Channel measurements Time lag τ [µs] Allan Variance Sample Allan variance Sample Allan variance, asymptote Estimated ARMA process Estimated ARMA process, asymptote 101 102 103 10−16 10−14 10−12 Figure 5.5 Allan deviation Figure 5.6 The measurement setting for investigating the impact of phase noise