Statistical channel parameter estimation 185 3門 hamrer ranging from2to eigenvalue define a distributed scatterer to be an SDS when the ratio of the la eigenvalue by the second largest signal covariance matrix of the Ist-order GAM model has rank two,the later ratio coincides with the condition number in the next subsection hold,that mal cigenvalues that need to be estimated from the sisal covariance matrix estimated from measurement data.Estimation o ral height.Standard e in the distributed noise spectral heigh AS this ratio is at a first approximation proportional to th aedaiofthelargeotheseondlagetcenchgoin definition,we get a criterion that is independent of the SNR. A new Quantitative Criterion for Determining Whether a Scatterer is an SDS n7.1
Statistical channel parameter estimation 185 0 5 10 15 100 101 102 103 104 ηGAM, M = 3 ηGAM, M = 4 ηGAM, M = 5 ηGAM, M = 7 η, M = 3 η, M = 4 η, M = 5 η, M = 7 σφ˜ [ ◦ ] ηGAM, η Fig. 7.1 Graphs of ηGAM and η versus the true AS σφ˜ with M ∈ [3, . . . , 7], φ¯ = 60◦ , and the AS as a parameter ranging from 2 ◦ to 15◦ . The input SNR γ equals 15 dB. The marks “◦” on the graphs denote the points for which ∆η = 0.05. formulation of this statement is that the ratio of the largest signal eigenvalue by the sum of the other signal eigenvalue is large. For simplicity reason, we consider instead the ratio of the largest signal eigenvalue by the second largest eigenvalue. This simplification is sensible as the second largest eigenvalue also dominates the sum of all signal eigenvalues but the two largest ones. Experimental investigations show that this ratio is a monotone decreasing function of the AS. It is infinite when the scatterer is specular, i.e. has AS zero. The above rationale leads us to define a distributed scatterer to be an SDS when the ratio of the largest eigenvalue by the second largest eigenvalue is larger than a certain threshold. We propose to select the threshold in such a way that this ratio computed for the effective model (7.1) and the same ratio computed for the 1st-order GAM model (7.9) are “close”. Notice that since the signal covariance matrix of the 1st-order GAM model has rank two, the later ratio coincides with the condition number of this matrix. Closeness of these two ratios is selected as an indicator of good agreement between both models. The details of the method, especially the translation of the above qualitative statements into quantitative relations is given in the next subsection. Both definitions inherently determine each a criterion, in form of a comparison of a quantity to a threshold, that can be used for deciding from measurement data whether a given distributed scatterer is an SDS or not. This quantity depends on some signal eigenvalues that need to be estimated from the signal covariance matrix estimated from measurement data. Estimation of signal eigenvalues requires an estimate of the noise spectral height. Standard estimates of the noise spectral height are either the smallest eigenvalue or the average of the noise eigenvalues. However, since all estimated eigenvalues consist of the contributions of both signals and noise in the distributedscatterer scenario, the estimation of the noise spectral height is inaccurate. This problem of noise spectral height estimation can be circumvented when the second definition is applied. Remember that this definition relies on the ratio between the largest and second largest signal eigenvalues. It is shown in Appendix II that for sufficiently small AS this ratio is at a first approximation proportional to the squared ratio of the largest to the second largest Gerschgorin radii (GRs) of the estimated signal covariance matrix. These radii exhibit the virtue that they do not depend on noise Wu et al. (1995). Thus, by replacing the ratio of eigenvalues with the ratio of GRs in the criterion induced by the new definition, we get a criterion that is independent of the SNR. A new Quantitative Criterion for Determining Whether a Scatterer is an SDS We first calculate the covariance matrix Σy of y(t) in (7.1) and the covariance matrix ΣyGAM of yGAM(t). We consider the case where Assumptions 1)–4) in Subsection 7.1.2 are satisfied. Furthermore, we assume that the azimuth deviations of the subpaths are von-Mises distributed and that the complex weights of the subpaths have identical
186 Statistical channel parameter estimation magnitude.The(m,m)th entry in reads Abdi and Kaveh (2000) 之vm)=5·a2-2m-mP+2rm-m☑e0心（高） 8mm,(m.m)e(1.....M2 7.30) In the above expression,represents the Kronecker delta.The covariance matrix is derived to be EyoAM =F()diag(o,(+Ixt. (7.31) Then,the first in radii (Grs)of the m calculated as de ihad in A regardless of the distribution of the azimuths of arrival of the subpaths. array elements. 本业se △=(GAM-)/ (7.32) of the between the two models.We consider that the agreement between the 1order △I≤△p (7.33) where the threshold value△is small..e,g.△ =0.05.An example is shown in Fig.7.1 with An =0.05.It can be tished when n .52.63 and 74 forM=3,4.5 and 7 respectively.This observation 刀2(1+△)GAM (7.34) all tha ands to i definition of SDS:a dis is an SDS w definition does not require the estimation ight,as An andn are SNR.Wor the fact that an SDS for aven arr size.it is als an SDS for any smaller anay size hus,if a scatterer is Array Size Adaptation Technique buted scatterer is an SDS whe en Inequality(7.34)is satisfied,in which epancy between the two models depends on the sed ed byth or can be mch attempts to make Inequality(7 the cae hiere te uind amay h UA tinne 4)to hold by pns popnpe a small percentage,e.g. h od function(Eq.(7.l5)with ()replaced byr(a
186 Statistical channel parameter estimation magnitude. The (m, m′ )th entry in Σy reads Abdi and Kaveh (2000) Σy(m, m′ ) = σ 2 α I0(κ) · I0 q κ 2 − π 2(m − m′) 2 + 2jπκ(m − m′) cos(φ¯) + σ 2 w · δmm′ , (m, m′ ) ∈ {1, . . . , M} 2 . (7.30) In the above expression, δmm′ represents the Kronecker delta. The covariance matrix ΣyGAM is derived to be ΣyGAM = F(φ¯)diag{σ 2 α, σ2 β }F H(φ¯) + σ 2 wIM . (7.31) Then, the first two Gerschgorin Radii (GRs) of the matrix Σy are calculated as described in Appendix C and the ratio η between the largest and the second largest GRs is computed. In the same way, we obtain the ratio ηGAM for the matrix ΣyGAM . We show in Appendix C that each ratio is proportional to the square root of the ratio between the largest and the second largest signal eigenvalues computed from the corresponding signal covariance matrix. This relation holds regardless of the distribution of the azimuths of arrival of the subpaths. Fig. 7.1 depicts η and ηGAM versus the AS σφ˜. The input SNR refers to the SNR at the input of the array elements. It is observed that η ≥ ηGAM holds for all considered values of σφ˜ and M. Moreover, η and ηGAM are close when the AS σφ˜ is small. Agreement between the values of these two ratios is selected as an indicator of the good match between the 1 st-order GAM model (7.9) and the effective signal model (7.1). Thus, we define ∆η .= (ηGAM − η)/η (7.32) as a figure of merit of the agreement between the two models. We consider that the agreement between the 1 st-order GAM model and the effective model is good when ∆η ≤ ∆ηth , (7.33) where the threshold value ∆ηth is small, e.g. ∆ηth = 0.05. An example is shown in Fig. 7.1 with ∆ηth = 0.05. It can be seen that Inequality (7.33) is satisfied when η ≥ 35, 52, 63 and 74 for M = 3, 4, 5 and 7 respectively. This observation holds when the NA φ¯ is within the array beam-width. By inserting (7.32) in (7.33), the latter inequality can be recast according to η ≥ (1 + ∆ηth ) · ηGAM. (7.34) We have now all the elements in our hands to introduce the new definition of an SDS: a distributed scatterer is an SDS if the received signals contributed by this scatterer satisfies (7.34). This new definition does not require the estimation of the noise spectral height, as ∆η and ∆ηth are independent of the SNR. Worth being noticed is the fact that if Inequality (7.34) holds for a certain array size, say M′ , it holds for all array sizes less than M′ . Thus, if a scatterer is an SDS for a given array size, it is also an SDS for any smaller array size. Array Size Adaptation Technique As mentioned in the previous subsection, a distributed scatterer is an SDS when Inequality (7.34) is satisfied, in which case the 1 st-order GAM model provides a good fit of the effective signal model. It can be observed from Fig. 7.1 that whether Inequality (7.34) holds or not depends not only on the AS of the distributed scatterer but also on the used array size. Simulation studies presented in Appendix C.0.1 show that due to the discrepancy between the two models, the AS estimator exhibits a bias that depends on the used array size. The bias of the AS estimator can be maintained at a reasonably low value by adjusting the array size properly. Inspired by these observations, we propose a technique called Array Size Adaptation (ASA), which, for a given AS of a scatterer, attempts to make Inequality (7.34) to hold by selecting the array size appropriately. This technique is derived for the case where the used array is a ULA. However, it is applicable in the case of arrays with arbitrary layouts. In the ASA technique, M is adjusted to a value Ma such that the ratio η possibly satisfies the inequality ∆η ≤ ∆ηth. Here, ∆ηth is fixed empirically to be a small percentage, e.g. ∆ηth = 0.05. The selected array size Ma is within the range [Mmin, Mmax], where Mmax is the number of array elements of the physical array, while Mmin depends on the used parameter estimator. For example, when the DML estimator (7.15)–(7.16) derived for the single SDS-scenario (D = 1) is used, the projection matrix ΠF (φ¯) in the log-likelihood function (Eq. (7.15) with ΠB(φ¯) replaced by ΠF (φ¯) )
Statistical channel parameter estimation 187 s an iden fore.M chosen to be the am the high as possible.In addition.the implemented wh the performance of the SAGE algorithm augmented with the ASA technique. 7.1.6 Simulation Studies Sughts and a von-Mises distribution centered at zero azmuth.TheRx antenna is an 8-element ULA with half-a-wavelength inter in the sequel make use of the pseudo-spectrum (7.21). Estimation frror of the mi Azimuth estimator derived based on the ss model of the absolute normalized estimation error with respectively the As and the number of observation samples N as a parameter.Using (7.6)the pdf ofis approximated according to e+ (7.35) rdin and b are compured using the paramerer sinNand eclose to the ate CCDFs when(Fig7.2a)and wher of large stimtion of the SS-MLazmuth estimator can be reduced by increasing the number of observation 7.1.7 Array Size Adaptation According to the definition provided in Section 7.1.5,the signal contribution of an SDS is accurately approximated with
Statistical channel parameter estimation 187 becomes an identity matrix when M = 2. Therefore, Mmin must be larger than 2 to avoid this situation. This restriction does not apply to the SML estimator (7.18) with D = 1, and therefore, Mmin = 2 is selected in this case. When the selected array size Ma is less than Mmax, the variance of the AS estimate increases due to the reduced array aperture. As the bias and variance jointly influence the accuracy of the estimate, the ASA technique improves the performance of the AS estimator only if the decrease of the bias dominates the increase of the variance. This usually occurs when the AS is large as shown in Subsection ??. To limit the variance increase, the ASA technique is augmented with the following method. In the case where ∆η ≤ ∆ηth holds for more than one selection of the array size, Ma is chosen to be the largest among these values in order to maintain the resolution as high as possible. In addition, the array is split into sub-arrays with aperture Ma. The NA and the AS estimators are applied to the Ma signals at the output of these individual sub-arrays. The final estimates are obtained by averaging the estimates computed from the sub-array output signals. The ASA technique can be implemented in combination with the estimators described in Section 7.2.7 to estimate the parameters of a single SDS. In the multiple-SDS scenario, this technique is applicable in combination with the SAGE algorithm for the estimation of the parameters of individual SDSs. Simulation results in Section 7.1.6 illustrate the performance of the SAGE algorithm augmented with the ASA technique. 7.1.6 Simulation Studies In this section we evaluate by means of Monte-Carlo simulations the performance of the proposed estimators. Except when explicitly mentioned the single-SDS scenario is considered in this section. The considered scenarios are simulated as follows. Each individual SDS consists of L= 50 sub-paths. The sub-path weights and azimuth deviations with respect to the NAs are randomly generated according to Assumptions 1)–5) in Subsection 7.1.2. The NAs of the SDSs ly in the array beam-width. The azimuth deviations are generated according to a von-Mises distribution centered at zero azimuth. The Rx antenna is an 8-element ULA with half-a-wavelength interelement spacing. Each result (point) shown in Fig. 7.2 and Fig. 7.4 is calculated from 2000 simulation runs, while the results depicted in Fig. 7.5 to Fig. 7.11 are calculated from 500 runs each. Note that the MUSIC estimators mentioned in the sequel make use of the pseudo-spectrum (7.21). Estimation Error of the ML Azimuth Estimator derived based on the SS Model We first investigate the accuracy of the approximation in (7.6) in a noiseless (σ 2 w = 0) scenario. Fig.7.2a and Fig.7.2b depict the empirical (estimated) and the approximated complementary cumulative distribution functions (CCDFs) of the absolute normalized estimation error ϕ .=|φˇ|/σφ˜ with respectively the AS σφ˜ and the number of observation samples N as a parameter. Using (7.6) the pdf of ϕ is approximated according to fϕ(z) ≈ 2Γ(N + 1 2 ) √ π Γ(N) · 1 + z 2 −(N+ 1 2 ) . (7.35) The r.h.s. pdf in (7.35) is independent of σφ˜ and so is the CCDF computed from this pdf. This CCDF is used as an approximation of the CCDF of ϕ. For short, it is referred to as the “approximate CCDF” in the sequel. The results presented in Fig. 7.2a and Fig. 7.2b are computed using the parameter setting φ¯ = 0◦ , σφ˜ ∈ [0.5 ◦ , 8 ◦ ], N = 10 and φ¯ = 0◦ , σφ˜ = 2◦ , N ∈ [1, . . . , 50] respectively. We observe that the empirical CCDFs are close to the approximate CCDFs when σφ˜ < 4 ◦ (Fig.7.2a) and when ϕ < 0.5 (Fig.7.2b). These observations are in accordance with the fact that the approximation (7.6) holds for small φˇ. Furthermore, from Fig. 7.2b the CCDFs decrease when N increases. This indicates that in the ID case, the probability of large estimation errors of the SS-ML azimuth estimator can be reduced by increasing the number of observation samples. 7.1.7 Array Size Adaptation According to the definition provided in Section 7.1.5, the signal contribution of an SDS is accurately approximated with the 1 st-order GAM model. In this subsection, we show by means of Monte-Carlo simulations that this approximation depends on the AS and the used array size: when the array size increases, the AS has to decrease for the approximation to hold. In the simulations, the ASA technique is used with the settings ∆ηth = 0.05, Mmin = 3. The solid curves shown in Fig. 7.3 depict the empirical mean µˆMa of Ma with Mmax as a parameter. It can be observed that for small ASs
188 Statistical channel parameter estimation 107 N-1 10 02040608 12 0^ 0.2040608 1214 50 Fig.7.3 The empirical average of M.versus the AS with M as a parameter:N=500,=10dB,and M·08=14 (7.36) providespromion of he dependence berwennThfc haorimlies )can be used as an Performance of the NA Estimators 一2Cm tha he MUSIC an
188 Statistical channel parameter estimation 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−3 10−2 10−1 100 Emp. CCDF, σφ˜ = 1o Emp. CCDF, σφ˜ = 0.5 o Emp. CCDF, σφ˜ = 2o Emp. CCDF, σφ˜ = 4o Emp. CCDF, σφ˜ = 8o Approx. CCDF z ˆP[ϕ ≥ z] = (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−2 10−1 100 z ˆP[ϕ ≥ z] Emp. CCDFs Approx. CCDF N = 1 N = 3 N = 5 N = 10 N = 50 (b) Fig. 7.2 Empirical (Emp.) and approximate (Approx.) CCDFs of the absolute normalized estimation error of the SS-ML azimuth estimator applied in an SDS scenario (a) with σφ˜ as a parameter, N = 10; (b) with N as a parameter, σφ˜ = 2◦ . 0 1 2 3 4 5 6 7 2 4 6 8 10 12 14 16 18 σφ˜ in [ ◦ ] Mmax = 8 Mmax = 12 Mmax = 16 Empirical approximation (7.36) ˆµMa Fig. 7.3 The empirical average µˆMa of Ma versus the AS σφ˜ with Mmax as a parameter: N = 500, γ = 10dB, and φ¯ = 90◦ . µˆMa = Mmax, while µˆMa = Mmin for large ASs (σφ˜ ≥ 5 ◦ ). In between, µˆMa decreases inversely proportionally to σφ˜. The equation µˆMa · σφ˜ = 14 (7.36) provides a good approximation of the dependence between µˆMa and σφ˜. The fact that µˆMa = Mmin for σφ˜ ≥ 5 ◦ implies that the 1 st-order GAM approximation is not accurate in that range of the AS. Equation (7.36) can be used as an empirical criterion for the preselection of the size of ULA arrays given an AS value and vice-versa. Performance of the NA Estimators Fig. 7.4 depicts the empirical CCDF of the absolute normalized estimation error ϕ of the NA estimators. The CCDF of ϕ of the SS ML azimuth estimator is also reported for comparison purpose. The parameter setting is N = 10, σφ˜ = 2◦ and γ = 25 dB. It can be observed that the SML and the DML estimators perform similarly and better than the MUSIC and SS ML estimators. The MUSIC estimator outperforms the SS-ML estimator. The results show that the NA estimators have significantly lower probability of large estimation errors compared to the conventional SS-ML estimator
Statistical channel parameter estimation 189 10 10 10 ig.7.4 Empirical CCDF of theabso s witho=2°，y=25dB,N=10. 10 Fig.7.5 RMSEEs of the NA estimators versus the input,N=50. Fig.7.6 RMSEEs of the NA estimators versus the true AS,N=50
Statistical channel parameter estimation 189 0 0.2 0.4 0.6 0.8 1 10−3 10−2 10−1 100 z ˆP[ϕ ≥ z] SS-ML DML SML MUSIC Fig. 7.4 Empirical CCDF of the absolute normalization error of the NA estimators with σφ˜ = 2◦ , γ = 25 dB, N = 10. −10 −5 0 5 10 15 20 25 10−1 100 101 102 RMSEE of ˆ¯φ [ o ] γ [dB] SS-ML DML SML MUSIC p CRLB(φ¯) Fig. 7.5 RMSEEs of the NA estimators versus the input SNR γ with σφ˜ = 3◦ , N = 50. 0 1 2 3 4 5 6 7 8 9 10 10−2 10−1 100 101 RMSEE of ˆ¯φ [ o ] SS-ML DML SML MUSIC p CRLB(φ¯) σφ˜ [ ◦ ] Fig. 7.6 RMSEEs of the NA estimators versus the true AS with γ = 10 dB, N = 50