Simulation results reported here using the channel generated by [40] whose spatial correlation at the BS is related to AoA distribution and antenna spacings. In addi-tion, the environment surrounding a user is of rich scattering with AoD’s uniformly distributed in [−π, π) making spatial correlation at MSs negligible. This setting accu-rate describes the environment where the BS with large-scale antenna array are mounted on an elevated tower or building. Throughout this section, we assume that there are 8 uniformly distributed users in a circular cell of radius R with the mean AoAs equis-paced within [−60◦, 60◦]. The other simulation parameters are listed in Table 3.1 unless explicitly stated otherwise. We define average received signal-to-noise power ratio as SNRdef= βkkpkk2/T with k the index of farthest MS from the BS, and normalized mean squared error (NMSE) as the MSE between the real and estimated vectors normalized by the former’s dimension and entry variance.
First, in Fig. 3.1 we compare the performance of the proposed LSFC estimator (3.3) with that of a conventional LS estimator [32, Ch. 8]
β =ˆ
(AHA)−1AHvec(Y)2
(3.22)
where A = (1T ⊗ H) PT ⊗ 1M
. As opposed to (3.3), the conventional estimator needs to know SSFCs beforehand, hence a full knowledge of SSFCs is assumed for the
Table 3.1: Simulation parameters
Parameters Values
Operating frequency 2.6 GHz [2]
Cell radius R 100 meters
Pathloss exponent α 3 Shadow fading variance σs 10 dB Number of BS antennas M 100 BS antenna spacing ξ 0.5λ
Number of MSs K 8
latter. Figure 3.1 shows that our proposed estimator outperforms the conventional sig-nificantly even if the latter has full knowledge of SSFCs. It is because the former takes advantage of the noise reduction effect that massive MIMO systems have offered. More-over, as antenna spacing increases, the channel decorrelates and thus the estimation error due to spatial correlation decreases, which verifies Theorem 3.2.1. Figure 3.2 illus-trates the effect of massive antennas to MSE. Owing to the fact that we have assumed perfect SSFC knowledge for the conventional LSFC estimator, MSE decreases with in-creasing sample amount as M increases. Unlike the conventional, the amount of known information does not grow with M for the proposed LSFC estimator. However, it still can be utilized to improve estimation and thus enables the proposed to outperform the conventional. In addition, the proposed is robust to SNR degradation due to noise or user-BS distance increment.
When comparing our proposed estimators with EM-based estimators, Fig. 3.3-3.7 shows the superiority of the proposed estimators. The details of EM approach is as follows: (i)set the initial value of bβ;
(ii)evaluate the LMMSE estimator of SSFC,
vec(H) = Diag\
(ii)evaluate the LMMSE estimator of LSFC, pd
where A =
1T ⊗ bH
PT ⊗ 1M
;
(iv)recursively compute until convergence occur.
Moreover, the modified EM (MEM) approach is obtained by replacing AHA = (HHH) (P?PT)−→ Diag M kpa.s. 1k2· · · M kpKk2 in the EM approach.
The results in Figs. 3.8 and 3.9 demonstrate the large-system performance of the proposed LSFC and full-order SSFC estimator. As can be seen, similar to the results in Fig. 3.2, the accurate LSFC estimates due to large received samples make its compensa-tion prior to the SSFC estimacompensa-tion reliable. Furthermore, such large sample size clearly improves the performance of the SSFC estimators directly. In addition, the proposed LSFC estimator has noise-robustness in the sense that the MSE performance is nearly the same when SNR = 0 dB to 15 dB, which gives similar result as Fig. 3.2.
0 5 10 15
10−6 10−5 10−4 10−3
BS antenna spacing (λ)
NMSE
0 dB 10 dB 20 dB
Proposed Conventional
Figure 3.1: MSE performance of the conventional and proposed LSFC estimator with perfect SSFC knowledge assumed for the former, AS= 15◦.
After discussing some results about the average MSE performance of MSs, we recast to compare the MSE performance of MSs at different locations, or equivalently, different
0 50 100 150 200 10−5
10−4 10−3 10−2
Number of antennas
NMSE
0dB 10dB 20dB
Conventional
Proposed
Figure 3.2: MSE performance of the conventional and proposed LSFC estimator with perfect SSFC knowledge assumed for the former, AS= 15◦.
2 4 6 8 10 12 14 16 18 20
10−5 10−4 10−3 10−2 10−1 100 101 102
AS=7.2o, SNR=10dB
Iteration
NMSE
EM MEM proposed
SSFC
LSFC
Figure 3.3: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of EM-based estimators, where AS= 7.2◦, SNR=10dB, and full modeling order is used. Initial β is chosen as E {β}.b
2 4 6 8 10 12 14 16 18 20 10−5
100 105
AS=7.2o, SNR=10dB
Iteration
NMSE
EM MEM proposed
LSFC SSFC
Figure 3.4: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of EM-based estimators, where AS= 7.2◦, SNR=10dB, and full modeling order is used. Initial β is chosen asb 121K.
2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100 102 104
AS=7.2o, SNR=0dB
Iteration
NMSE
EM MEM proposed SSFC LSFC
Figure 3.5: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of EM-based estimators, where AS= 7.2◦, SNR=0dB, and full modeling order is used. Initial β is chosen asb 121K.
2 4 6 8 10 12 14 16 18 20 10−5
100 105
AS=15o, SNR=10dB
Iteration
NMSE
EM MEM proposed
LSFC SSFC
Figure 3.6: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of EM-based estimators, where AS= 15◦, SNR=10dB, and full modeling order is used. Initial β is chosen asb 121K.
2 4 6 8 10 12 14 16 18 20
10−6 10−4 10−2 100 102 104
AS=15o, SNR=0dB
Iteration
NMSE
EM MEM proposed SSFC LSFC
Figure 3.7: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of EM-based estimators, where AS= 15◦, SNR=0dB, and full modeling order is used. Initial bβ is chosen as 121K.
0 20 40 60 80 100 120 140 160 180 200
Figure 3.8: MSE performance of the proposed LSFC and SSFC estimator versus number of BS antennas and received SNR, where AS= 7.2◦, and full modeling order is used.
0 20 40 60 80 100 120 140 160 180 200
Figure 3.9: MSE performance of the proposed LSFC and SSFC estimator versus number of BS antennas and received SNR, where AS= 15◦, and full modeling order is used.
LSFC. Consider Fig. 3.10, the red lines with × represents the nearest MS while the blue lines with ◦ represents the farthest MS from the BS. Assume the pilot power is such that the received SNR from farthest MS at BS equals to 10.22dB for left figure and 20.22dB for right figure. It can be seen clearly that the pilot power has little effect on the MSE performance of LSFC estimator due to the robustness of our estimator to noise. Nevertheless, when considering the full-order SSFC estimator, the nearest MS indeed has better MSE performance.
0 50 100 150 200
10−5 10−4 10−3 10−2 10−1 100
Number of antennas
MSE/β
Nearest (36.71dB) Farest (10.22dB)
0 50 100 150 200
10−4 10−3 10−2 10−1 100
Number of antennas
MSE/β
Nearest (SNR=46.71dB) Farest (SNR=20.22dB) SSFC
LSFC
SSFC
LSFC
Figure 3.10: MSE performance of the proposed LSFC and SSFC estimator versus number of BS antennas with different user location, hence different received SNR at BS (indicated in the legend), where AS= 15◦, and full modeling order is used.
Chapter 4
Estimation of Small-Scale Fading Coefficients
Since the SSFC estimation scheme is valid for any user-BS link, for the sake of brevity, we omit the user index k in the ensuing discussion.
4.1 Reduced-Rank Channel Modeling
In [12], two analytic correlated MIMO channel models were proposed. These models generalize and encompass as special cases, among others, the Kronecker [9, 40], virtual representation [11] and Weichselberger [10] models. They often admit flexible reduced-rank representations. Moreover, if the angle spread (AS) of the transmit signal is small, which, as reported in a recent measurement campaign [2], is the case when a large uniform linear array (ULA) is used at the BS, one of the models can provide angle of arrival (AoA) information. In other words, since the ASs from uplink users in a massive MIMO system are relatively small (say, less than 15◦), the following RR model is easily derivable from [12, Proposition 1]
Lemma 4.1.1 (RR representations). The channel vector seen by kth user can be repre-sented by
h = Q(I)mc(I) (4.1)
or alternately by
h = W(φ)Q(II)m c(II) (4.2)
where Q(I)m, Q(II)m ∈ RM ×m are predetermined (unitary) basis matrices and c(I), c(II) ∈ Cm×1 are the channel vectors with respect to bases Q(I)m and Q(II)m for the user k-BS link and W(φ) is diagonal with unit magnitude entries. The two equalities hold only if m = M and become approximations if m < M . Furthermore, if the AS is small, [W(φ)]ii = exp
−j2π(i−1)ξλ sin φ
with ξ and λ being the antenna spacing and signal wavelength and φ is the mean AoA.
This Lemma suggests that the mean AoA (which is approximately equal to the incident angle of the strongest path) of each user link is extractable if the associated AS is small. Having a large aperture, a massive MIMO antenna array may provide fine AoA resolution and a channel estimator based on the model (4.2) can therefore offer accurate mean AoA information [1].
Remark 8. The estimated mean AoAs can be used by the BS to perform downlink beamforming. The use of predetermined basis matrices, as the virtual representation [11], avoids the need of the spatial correlation information required by [10].
Remark 9. For large-scale ULAs, the spatial correlation can be high, small modeling order m may be sufficient to capture the spatial variance of the SSFCs.