In Section 4.4, the RR estimation performance with respect to modeling order ˆm? (given in Definition 4.4.6) is analyzed assuming perfectly known bias matrix B or at least [B]``’s. This is not the case in practice. We investigate the case when only the spatial correlation is known and proposed a (channel) rank (modeling order) determination method on which the estimates for mean AoA and RR channel SLFCs are based.
Lemma 4.7.1. The optimal modeling order for an uplink user channel is a decreasing function of the mean AoA to the BS (with respect to the array broadside). That is to say, if 0 ≤ |φ2| < |φ1| ≤ π2, then m?(φ1) < m?(φ2), where m?(φ) denotes the optimal modeling order with mean AoA being φ.
As φ and thus B = QHWH(φ)ΦW(φ)Q are still unknown, except when φ = 0, we propose the iterative modeling order determination (IMOD) algorithm which initializes φ as 0 and accordingly the maximal possible ˆˆ m? = ˆm?(0) and finds the modeling order
ˆ
m? recursively. The following steps are executed sequentially:
1. (Initialization) Initialize ˆφ = 0.
2. (Updating bias matrix) Calculate
B = Qˆ HWH( ˆφ)ΦW( ˆφ)Q (4.56)
and solve (4.35) with B = ˆB to obtain the estimate ˆm?. 3. (Updating mean AoA) With m = ˆm?, find ˆφ via (4.9).
4. (Recursion) Go to Step 2); Terminate and output ˆm? if the stopping criterion is met.
while once the modeling order is determined, we can proceed to the estimation of RR SSFC representations with the mean AoA ˆφ obtained here. Two remarks are given to conclude this section.
Remark 19. The modeling order ˆm? obtained via IMOD algorithm requires much less complexity yet achieves near-optimal performance as compared with m? via exhaustive search [12] or some SVD-based methods [26] [27]. This is because it does not need to know the perfect SSFC matrix (which is impossible) nor the eigen-structure of spatial correlation matrix, and the energy compaction nature of the basis used guarantees fast convergence. In addition, IMOD algorithm is basis-independent while SVD-based meth-ods can only apply to KLT basis case. It is shown in Section 4.8 that the algorithm can achieve convergence within two iterations.
Remark 20. If the system requirement mandates all user channel vectors be estimated with a single modeling order ˆm?, we choose to minimize the error of the user who is the most sensitive to the modeling order mismatch. As suggested by Remark 16, the performance suffers more from insufficient than redundant order.
ˆ
m? = max
k m?k. (4.57)
Remark 21. As a matter of fact, the correlation matrix used in IMOD algorithm is not perfect, we should doing correlation estimation before apply IMOD algorithm. It is worth noting that sufficient samples from different OFDM subcarriers help to improve the estimation accuracy. The spatial correlation matrix can be written as
Φ = 1
γ2 E{YppHYH} − kpk2IM
(4.58) where the term E{YppHYH}def= Ψ can be accurately estimated by either ML estimation
Ψ =b 1 n − 1
Xn
`=1
Y`ppHY`H (4.59)
or shrinkage estimation method introduced in [20], where ` denotes the subcarrier index and n is the number of subcarriers.
4.8 Numerical Results and Discussion
In this section, we present the performance of our SSFC estimator and investigate the effect of modeling order as well as basis matrices. Assume the same simulation scenario as Section 3.4 is used here, wherein parameters are listed in Table 3.1 unless explicitly stated otherwise. Recall that the discussion in this chapter does not vary from user to user, so we focus only on an MS here, and define the average Rx SNR as SNR def= βkpk2/T . Besides, the mean AoA is assumed to be uniformly distributed in [−60◦, 60◦].
We study the SSFC estimation performance with respect to modeling order and basis matrix with estimated or perfectly-known β. Since the spatial correlation increases with reducing AS, the spatial waveform of a user is anticipated to be smoother. As a result, in the case where AS is comparatively small, the estimation performance due to over-modeling a channel not only cannot improve, but also may degrade because the number of parameters to be estimated grows with the same amount of available data. As can be observed in Fig. 4.3, the estimation accuracy with polynomial basis degrades as modeling order increases from 20 to 100 for SNR smaller than 9 dB. Besides, the optimal modeling order increases with SNR, e.g., optimal order at SNR = 5 and 10 dB are respectively 20 and 30. This is because in the low SNR regime, MSE performance is noise-limited as suggested in Section 4.6, while modeling error shows its importance for high SNRs.
Similar trend is also observed with DCT basis in Fig. 4.5. Figure 4.6 shows that for 15◦ angle spread and SNR smaller than 8 dB, the MSE performance deteriorates as modeling order goes up from 30 to 100.
All the three cases mentioned above verify Remark 15 in previous discussion, however, it shall be noted that as polynomial basis is used and the angle spread is 15◦, the best
MSE performance happened when full modeling order is used as depicted in Fig. 4.4.
This is owing to the fact that when AS goes large from 7.2◦ to 15◦, the spatial correlation dropped and hence steepen the spatial waveform, for which higher modeling order (in this case, 100) is needed. In other words, mcdefined in Definition 4.4.6 gets large toward 100 when spatial correlation reduced, which also verifies Remark 4.4.7. Furthermore, Section 4.4.2 has shown that DCT-II outperforms polynomial basis in sense of energy compaction capability, thus, a smaller optimal modeling order must be found in Fig. 4.6 compared to that in Fig. 4.4, which is equal to 100.
For the sake of finding the optimal modeling order more precisely, we use the exhaus-tive search method similar in [12], which needs the exact SSFC information for all users, to calculate the NMSE for all possible modeling orders, and depict them in Fig. 4.7 and Fig. 4.8 for a fixed SNR, 10dB. The simulation results are obviously consistent with that in Fig. 4.3 to 4.6. Moreover, we observe that when angle spread is 7.2◦ (large spatial correlation), the optimal modeling order when using DCT-II basis, which is about 15, is only slightly inferior to that when using polynomial basis, which is around 25, as claimed in [35]; however, the performance gap between using respectively the two bases becomes large when angle spread is 15◦ (small spatial correlation), that is, 30 for the former and 100 for the latter.
We then turn into investigating the impact of different SNR on modeling order, and then validate Lemma 4.6.1. When polynomial basis is used, Fig. 4.3 and Fig. 4.7 shows that when angle spread equals to 7.2◦, the optimal modeling order is about 20 as SNR = 9 dB while larger than 20 as SNR > 9 dB. As DCT-II basis is used, Fig. 4.5 and 4.7 shows that the optimal modeling order at SNR = 5 and 16 dB are respectively 15 and 25 when angle spread is 7.2◦; besides, Fig. 4.6 and 4.8 shows that the optimal modeling order at SNR = 10 and 16 dB are respectively 30 and 40 when angle spread is 15◦. This is because in low SNR regime, MSE performance is noise-dominant, while modeling error shows its importance for high SNRs as we have discussed.
Note that the proposed SSFC (and SSFC-2) estimators with order 100 gives the identical performance that a conventional LS estimator in (4.31) can offer as presented in Lemma 4.4.3, and depicted in Fig. 4.7-4.8. Although, in the case where AS= 15◦ and polynomial basis is used, (4.10) and (4.9) fail to give better performance than that of the conventional LS estimator as depicted in Fig. 4.8, the proposed estimator offers direct performance-complexity and/or performance-feedback rate trade-off.
Succeedingly, we inspect the effect of modeling order on spatial waveform of ULA as DCT-II basis being chosen, mean AoA being 21π and AS being 7.2◦. Fig. 4.9-4.12 show the spatial waveform (real part) of ˆh(I), ˆh(II), and exact h when modeling order is 5, 15, 80 and 100, respectively. It is clear that modeling order of 15, being correspond to the optimal modeling order, is sufficient to capture all channel behavior, while modeling order of 80 and 100 over-model the channel behavior, and modeling order of 5 is not enough to represent the spatial waveform. Furthermore, the SSFC-2 estimator is always insufficient to model the channel aside from using full modeling order.
The comparison of ˆh(I) and ˆh(II) are illustrated in Fig. 4.7-4.8. As being proved in Section 4.5, no matter what modeling order we used and how angle spread is, the MSE performance of SSFC-2 estimator is indeed much worse than that of SSFC estimator (ex-cept for the case when full modeling order is used, they both equivalent to conventional LS estimator).
Fig. 4.13-4.18 show the diagonal entries of bias matrix, when KLT, DCT-II and polynomial basis is used respectively. The solid line represents diagonal distribution of B (SSFC estimator being used) while the dotted line represents diagonal distribution of B (SSFC-2 estimator being used). Firstly, we focus on the solid lines in the figures, when˜ using the KLT basis, diagonal terms larger than η = 1 are indeed concentrated at (low) indices smaller than 15 and 29 (which are exactly the optimal modeling orders found by IMOD algorithm) when AS is 7.2◦ and 15◦, respectively; thus, energy compaction nature of optimal KLT basis is verified. Nonetheless, when DCT-II basis is used, diagonal terms
larger than η = 1 are also concentrated at indices smaller than 15 and 29 when AS is 7.2◦ and 15◦, respectively, which guarantees that the energy compaction capability of DCT-II basis is really near-optimal. On the other hand, the optimal modeling order of polynomial basis, is slight larger than that of DCT-II basis (23 > 15) when AS is 7.2◦, yet much larger than DCT-II basis (100 29) when AS is 15◦; this observation is consistent with what we have in Fig. 4.7-4.8. Secondly, we turn to inspect the dotted lines in the figures, whose energy compaction parts are frequency-shifted to some medium orders as we shown in Remark 17. Recall we have proves in Section 4.7 that the energy compaction property of each solid line becomes more apparent while the mean AoA increases, we depict this phenomenon in Fig. 4.19-4.21. However, dotted lines in these figures also illustrate the fact that the energy compaction part is frequency-shifted to higher orders when the mean AoA increases, we have also shown this in Remark 17. After that, by comparing the solid lines and dotted lines, we can make the same conclusion as Fig.
4.7-4.8, that is to say, SSFC estimator outperforms SSFC-2 estimator a lot.
Fig. 4.22 investigate the convergence speed of our IMOD algorithm given DCT-II basis is used, SNR= 10 dB and η = 1. It is clear that IMOD algorithm converge within two iterations regardless of how large the AS is. Furthermore, we can seen in Table 4.1 that the convergence is quite accurate on account of the fact that when η = 0.5 ∼ 1, IMOD algorithm achieves almost the same optimal modeling order compared with the exhaustive search (or brute force) methods, which is optimal, used in [12]. Besides, the iteration number needed for convergence seems to have little to do with the choice of η.
Therefore, we assume η = 1 henceforth.
Remember that
NMSEm(ˆh)
M = m
M T · SNR + 1
Mtr (DmB) . (4.60)
To compare the NMSE given different BS antenna number, M , we want to emphasize here that the bias term M1tr (DmB) is almost equal for any M if we choose the optimal modeling order, m?, by IMOD algorithm. Thus, a simplified metric to measure the
NMSE of different M is mM?. Table 4.2-4.3 presents the effect of large system to optimal modeling order found by IMOD algorithm given DCT-II and KLT basis is used, and verifies Corollary 4.4.2 since the NMSE metric, mM?, becomes smaller when the number of BS antennas M gets larger. Moreover, we see again that the DCT-II basis gives almost the same optimal modeling order as KLT basis whatever M we used. On the other hand, [18, Lemma 2.] has derived the upper bound of NMSE metric, mM?, as M → ∞, being 0.0187 and 0.0386 when AS= 7.2◦ and AS= 15◦, respectively.
We than consider the impact of imperfect (estimated) spatial correlation matrix to our IMOD algorithm. The energy distribution of rotated channel vector, which is just the diagonal distribution of bias matrix B, is depicted in Fig. 4.23-4.30 when the number of subcarriers is 100, 50, 30 or 2 and AS is 7.2◦ or 15◦, respectively. It is clear that when the number of subcarriers larger than the true optimal modeling order, than KLT and DCT-II basis yields almost the same estimated dominant rank as the true one. However, when the number of subcarriers reduced to 2, the estimated dominant rank equals to the number of subcarriers while remains the same (29 for 15◦ AS and 15 for 7.2◦ AS) when KLT and DCT-II basis are used respectively. The reason why this is so is due to the fact that when DCT-II basis is used, we can find out the largest frequency components even if the number of samples (subcarriers) are small, this phenomenon can also be observed when using polynomial basis; on the other hand, KLT basis is irregular with the number of nonzero terms on the diagonal of B less or equals to the number of samples (subcarriers), hence, when using KLT basis, we need sufficient samples to accurately estimate the dominant rank, which is called the “sample-deficient problem”. It is worth mentioning that common SVD-based methods [26] [27] is equivalent to the KLT basis case, thus, they also suffer from the sample-deficient problem when number of subcarriers are smaller than the true optimal modeling order. Last but not least, the shrinkage correlation estimation seems to be unnecessary when doing rank determination, since from the above simulation results, it performs not better than ML
estimation while a heavy-tail problem is discovered, which may influence the accuracy of rank determination. We summarize in Table 4.4 and Table 4.5 the effect of imperfect spatial correlation matrix estimated by ML estimator (4.59) to optimal modeling order found by IMOD algorithm when using KLT, DCT-II or polynomial basis.
Finally, Fig. 4.31 shows the effectiveness of the large aperture of the ULA to resolve user AoAs even when the ASs of different MSs overlap.
0 2 4 6 8 10 12 14 16 18 20
10−4 10−3 10−2 10−1 100
SNR (dB)
NMSE
AS=7.2o
m=20 m=30 m=40 m=100
Perfect LSFC
Figure 4.3: MSE performance of the proposed SSFC estimator versus received SNR and modeling order with estimated and perfect LSFC, where AS= 7.2◦, and polynomial basis is used.
0 2 4 6 8 10 12 14 16 18 20 10−3
10−2 10−1 100
SNR (dB)
NMSE
AS=15o
m=50 m=60 m=80 m=100
Perfect LSFC
Figure 4.4: MSE performance of the proposed RR SSFC estimator versus received SNR and modeling order with estimated and perfect LSFC, where AS= 15◦, and polynomial basis is used.
0 2 4 6 8 10 12 14 16 18 20
10−4 10−3 10−2 10−1 100
SNR (dB)
NMSE
AS=7.2o
m=15 m=25 m=35 m=100
Perfect LSFC
Figure 4.5: MSE performance of the proposed SSFC estimator versus received SNR and modeling order with estimated and perfect LSFC, where AS= 7.2◦, and DCT-II basis is used.
0 2 4 6 8 10 12 14 16 18 20 10−4
10−3 10−2 10−1 100
SNR (dB)
NMSE
AS=15o
m=30 m=40 m=50 m=100
Perfect LSFC
Figure 4.6: MSE performance of the proposed RR SSFC estimator versus received SNR and modeling order with estimated and perfect LSFC, where AS= 15◦, and DCT-II basis is used.
0 10 20 30 40 50 60 70 80 90 100
10−2 10−1 100
Modeling order
NMSE
AS=7.2o
DCT basis poly. basis
hˆ(I)
hˆ(II)
Figure 4.7: MSE performance of the proposed SSFC estimators, ˆh(I) and ˆh(II), versus modeling order when using respectively DCT-II and polynomial basis, where AS= 7.2◦, SNR= 10 dB, and imperfect LSFC is used.
0 10 20 30 40 50 60 70 80 90 100 10−2
10−1 100
Modeling order
NMSE
AS=15o
DCT basis poly. basis
hˆ(I)
hˆ(II)
Figure 4.8: MSE performance of the proposed SSFC estimators, ˆh(I) and ˆh(II), versus modeling order when using respectively DCT-II and polynomial basis, where AS= 15◦, SNR= 10 dB, and imperfect LSFC is used.
0 10 20 30 40 50 60 70 80 90 100
−2.5
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
Spatial waveform, order=5
Antenna index
Real amplitude
Exact hˆ(II) hˆ(I)
Figure 4.9: Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I) and ˆh(II), compared with true (exact) spatial waveform when DCT-II basis being chosen, where AS= 7.2◦, modeling order=5, SNR= 10 dB, mean AoA= 21π, and imperfect LSFC is used.
0 10 20 30 40 50 60 70 80 90 100
Figure 4.10: Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I)and ˆh(II), compared with true (exact) spatial waveform when DCT-II basis being chosen, where AS= 7.2◦, modeling order=15, SNR= 10 dB, mean AoA= 21π, and imperfect LSFC is
Figure 4.11: Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I)and ˆh(II), compared with true (exact) spatial waveform when DCT-II basis being chosen, where AS= 7.2◦, modeling order=80, SNR= 10 dB, mean AoA= 21π, and imperfect LSFC is used.
0 10 20 30 40 50 60 70 80 90 100
Figure 4.12: Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I) and hˆ(II), compared with true (exact) spatial waveform when DCT-II basis being chosen, where AS= 7.2◦, modeling order=100, SNR= 10 dB, mean AoA= 21π, and imperfect LSFC is used.
Figure 4.13: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and KLT basis is used.
0 10 20 30 40 50 60 70 80 90 100 0
2 4 6 8 10 12
AS=15o
Index [B]
hˆ(II) hˆ(I)
Figure 4.14: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 21π, SNR= 10 dB and KLT basis is used.
0 10 20 30 40 50 60 70 80 90 100
0 5 10 15
AS=7.2o
Index [B]
hˆ(II) hˆ(I)
Figure 4.15: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and DCT-II basis is used.
0 10 20 30 40 50 60 70 80 90 100
Figure 4.16: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 21π, SNR= 10 dB and DCT-II
Figure 4.17: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and polynomial basis is used.
0 10 20 30 40 50 60 70 80 90 100 0
0.5 1 1.5 2 2.5 3 3.5
AS=15o
Index [B]
hˆ(II) hˆ(I)
Figure 4.18: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 21π, SNR= 10 dB and polynomial basis is used.
0 10 20 30 40 50 60 70 80 90 100
0 2 4 6 8 10 12
AS=15o
Index [B]
hˆ(II) hˆ(I)
Figure 4.19: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 3π21, SNR= 10 dB and DCT-II basis is used.
Table 4.1: Convergence speed and accuracy of proposed IMOD algorithm for different choice of η assume DCT-II basis is used and SNR= 10dB.
AS = 15◦ η = 0.1 η = 0.5 η = 1.0 ˆ
m? 32 29 29
Iteration Number 2 2 2
Exhaustive Search 30 30 30
AS = 7.2◦ η = 0.1 η = 0.5 η = 1.0 ˆ
m? 19 16 15
Iteration Number 1 2 1
Exhaustive Search 15 15 15
0 10 20 30 40 50 60 70 80 90 100
0 2 4 6 8 10 12
AS=15o
Index [B]
ˆh(II) ˆh(I)
Figure 4.20: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 5π21, SNR= 10 dB and DCT-II basis is used.
0 10 20 30 40 50 60 70 80 90 100 0
2 4 6 8 10 12 14
AS=15o
Index [B]
hˆ(II) hˆ(I)
Figure 4.21: Diagonal distribution of the bias matrix, B, with respect to the SSFC estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 7π21, SNR= 10 dB and DCT-II basis is used.
Full10 Initalized 1 2 3
20 30 40 50 60 70 80 90 100
Iteration
Modeling order
AS=7.2o AS=15o
Figure 4.22: Convergence speed of proposed IMOD algorithm. Iteration number “Full ” represents the full modeling order and “Initialized ” means the initialization. Assume DCT-II basis is used, SNR= 10 dB and η = 1.
10 20 30 40 50 60 70 80 90 100
Figure 4.23: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 100. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.24: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 50. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.25: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 30. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.26: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 7.2◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 2. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.27: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 15◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 100. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.28: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 15◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 50. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
10 20 30 40 50 60 70 80 90 100
Figure 4.29: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 15◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 30. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.
Figure 4.29: Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II, and polynomial basis, where AS= 15◦, mean AoA= 21π, SNR= 10 dB and number of subcarriers is 30. Assume imperfect spatial correlation matrix estimated by ML or shrinkage [20] method is used here.