Massive MIMO with BD Precoding Scheme in Single-Cell Systems
3.4 Numerical Results and Discussions
Tt?1, K
. (3.32)
For BD-STBC, when Pt is finite, P → ∞, the second derivative of the ¨Cipup with respect to Tt2 is given by concave with respect to Tt2. By letting the first derivative of (3.20) to be zero, we have
ln(1 + vTt2) = T v + 1
1 + vTt2 − 1. (3.34)
From (3.34), the optimum training length Ttopt2 to maximize ¨Cipup under BD-STBC is given by
Ttopt
In this section, both simulated and analytical SE results of BD-PS and BD-STBC over fading channels are presented. We assume that the transmitted signals are modulated with OFDM.
The parameters are chosen based on long-term evolution standards where an OFDM symbol duration is approximated as 71.4 µs. We choose the channel coherence time to be Tc= 1 ms, then T = 196 symbols is the coherence interval. The simulation results are computed based on
Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems 10,000 independent channel realizations.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15
x f λ(x)
ηK = 20 ηK = 200 ηK = 1000
Figure 3.2: Empirical eigenvalue distribution of GkGHk when η = 2.
We first investigate the EED of GkGHk as shown in Fig. 3.2. As we have mentioned in Section 3.2.1, the eigenvalues of GkGHk converge to E{λm} = 1 − K−1ηK as ηK increases. In particular, with η = 2, when ηK increases from 20 to 1000, the eigenvalues of GkGHk converge to E{λm} ≈ 0.5. This result validates the accuracy of the matrix approximation of eΛk as ηK is large.
Next we demonstrate the accuracy of the derived closed-form SE expressions of BD-PS and BD-STBC. Figure 3.3 presents the achievable spectral efficiencies for BD-PS, BD-STBC, BD-based RZF [82] and BD-based SVD [82] in the case of perfect CSI. We assume that N = 60, K = 10 and each user is equipped with M = 3 antennas. It is shown that the proposed BD-PS performs best due to high array gain. In Fig. 3.4 the analytical and simulated SE results of the proposed BD-PS and BD-STBC schemes are demonstrated when the CSI is imperfect.
As observed from Figs. 3.3 and 3.4, the analytical results of these schemes are very close to
Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems
Figure 3.3: Spectral efficiencies versus SNR for BD-PS, BD-STBC, BD-based RZF [82] and BD-based SVD [82] when T = 196, N = 60, K = 10, M = 3, L = 2 and CSI is perfect.
Figure 3.4: Spectral efficiencies of the BD-precoding and BD-STBC schemes in the case of imperfect CSI when T = 196, N = 60, K = 10, M = 3, and L = 2.
Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems the simulation results for both cases of perfect CSI and imperfect CSI (σ2e = 0.1, 0.3, 0.5). The SE of BD-PS is proportional to N , whereas the SE of BD-STBC is proportional to M and L.
In addition, N is much larger than M . Hence, the system SE of BD-PS is higher than that of BD-STBC. Figure 3.5 illustrates the SE of BD-PS when N increases for both cases of perfect CSI and imperfect CSI. When N increase, ηK is increased for fixed values of K and M , the eigenvalues are close to the deterministic value. The analytical results of BD-PS match well with the corresponding simulated results, especially at large N . Therefore, we will use these analytical SE expressions for further analysis.
50 100 150 200 250 300
0 20 40 60 80 100 120 140 160 180 200
Number of BS antennas (N)
Capacity (bits/s/Hz)
analytical (perfect CSI) simulation (perfect CSI) analytical (imperfect CSI) simulation (imperfect CSI) σe2= 0.1, 0.3, 0.5
Figure 3.5: System SE of BD-PS versus the number of BS antennas when T = 196, K = 10, M = 3 and SNR = 10 dB.
In Fig. 3.6, we show the capacities of BD-PS and BD-STBC, i.e., Cp and ¨Cup, versus the number of users in the case of perfect CSI, respectively. We assume that the downlink SNR threshold is γth1 = γth2 = 0 dB. In order to determine the optimal numbers of users K1opt and K2opt for maximizing the cost function of the corresponding P1 and P2, we need to define the maximizers K1? in Theorem 1 and K2? in Theorem 2, respectively. The maximizers K1? and K2?
Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems
0 5 10 15 20 25
0 20 40 60 80 100 120
Number of users
Spectral efficiency (bits/s/Hz)
BD−STBC (SNR = 10 dB) BD (SNR = 10 dB) BD−STBC (SNR = 20 dB) BD (SNR = 20 dB) Maximum points
Figure 3.6: SEs of the BD-precoding and BD-STBC schemes versus the number of users when T = 40, M = 2, L = 2, N = 60 for the case of perfect CSI.
can be obtained from the proposed algorithm shown in Table 3.2 and they are presented in Table 3.3. Furthermore, the columns named as “Fig. 3.6” in Table 3.3 indicate the maximum points which are presented in Fig. 3.6. It has been shown in Table 3.3 that the optimal numbers of users in Theorems 1 and 2 are similar to those obtained from Fig. 3.6. We observe that the optimum Kopt increases as the transmit power P increases for a given parameter setting, such as M = 2, N = 60, and T = 40.
The SE comparisons of these schemes for the optimum number of users Kopt and the case with K = T /2 are shown in Fig. 3.7. Note that K = T /2 is greater than Kopt due to the conditions for the concavity of the optimization problems P1 and P2. In Fig. 3.7, the system SE for the optimum Kopt is always better than that of the case with K = T /2 for different parameter settings over the whole range of SNR.
In Fig. 3.8, we investigate the system SE for different lengths of training sequence Tt with Pt= 3 dB. The optimum training lengths for maximizing the system SE with PS and
BD-Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems
0 2 4 6 8 10 12 14 16 18
0 20 40 60 80 100 120
SNR (dB)
Spectral efficiency (bits/s/Hz)
BD−STBC (K=K
opt) BD−STBC (K=T/2) BD (K=K
opt) BD (K=T/2)
Figure 3.7: SE comparisons of the precoding schemes for the optimum number of users Kopt and the case with K = T /2 for M = 2, L = 2, N = 60 and T = 40.
Table 3.3: The optimal number of users
Parameter settings K1? for BD-PS K2? for BD-STBC Proposed algorithm Fig. 3.6 Proposed algorithm Fig. 3.6
SNR =10 dB 11 11 10 10
SNR =20 dB 14 14 15 15
STBC, i.e., Ttopt1 and Ttopt2 , respectively. From Fig. 3.8, we observe that there is a small system SE loss of Topt as compared to that of Tt= K (it means Tt1 = Tt2 = K) at low SNR region.
However, when SNR is greater than 11 dB, the system SE with Toptoutperforms that of Tt= K.
This is because Topt is typically obtained for the case P Pt. Therefore, in order to improve the system performance, the optimum Topt should be analyzed. The closed-form expressions of Topt presented in Section 3.3.2, which are easily obtained from the basic system parameters, can help the system design engineers to define the optimum training length efficiently.
Chapter 3: Massive MIMO with BD Precoding Scheme in Single-Cell Systems
Figure 3.8: The proposed system SE with different lengths of training sequence Tt when T = 196, K = 10 and Pt= 3 dB.
3.5 Summary
In this chapter, we have investigated the SE performance of massive MIMO systems with BD-precoding and BD-STBC schemes. Although the analytical results have been obtained under a large-scale assumption, they have been shown to be tight and accurate for the large-scale and conventional MIMO systems in both cases of perfect CSI and imperfect CSI (i.e., M = 2 and 3 have been used to obtain the analytical and simulation results). It has been shown that BD-PS outperforms BD-STBC. In addition, we have studied the optimal users and training lengths for maximizing the SE of these schemes, respectively. The closed-form expressions of the optimum training lengths for the proposed system have also been derived for high SNR region. From the simulation results, the accuracy of our analysis on the optimum training lengths and the optimum number of users for these schemes have been successfully demonstrated.