3.3 Simulations
5.2.3 Proposed Master Problem Optimization
With (5.15), the master problem thus becomes maxFR pre-coder is difficult to solve. However, we can resort to the same diagonalization in Chapter 3 and 4 to find the optimum relay precoder. Let A = PN
S,TIN + σ−2n,dHHSDHSD and B = HHSRFHRHHRD¡
σn,r2 HRDFRFHRHHRD+ σ2n,dIM¢−1
HRDFRHSR. We then have the same opti-mization in (3.26). The details are then omitted here. As a result, the optimum relay precoder
can be found by substituting (3.32) into (3.30). After the optimum relay precoder is determined, the source precoder can then be obtained via (5.15).
We summarize the computational complexity of the linear source and relay precoders in Table 5.1. Comparing to Table 4.1, we can find that the procedures to compute the precoders proposed in this chapter and Chapter 4 are similar except for the GMD in (5.13) and (4.8). Since the THP and unitary precoders are involved in (3.15) and (3.16), the computational complexity of the precoders proposed in Chapter 3 is higher, as shown in the additional steps 7-9 of Table 3.1.
Finally, we summarize and compare the nonlinear transceivers proposed in this dissertation in Table 5.2. As shown in this table, FSof each structure is expressed as an unitary matrix. It is seen that the relay precoders for the THP source precoded system and the linear source precoded system (with MMSE-SIC receiver) are the same. We also summarize physical implications of the precoders optimization in Table 5.3. From the table, we see that the unitary FSis designed to equalize either the MSE or the SNR/SINR of each data substream in the subproblem problem.
The relay precoders decouple the effective channels and allocate the power for each parallel channel to either maximize SNR/SINR or to minimize the MSE.
§ 5.3 Simulations
In this section, we evaluate the performance of the proposed precoded systems studied in this dissertation. As previously, we assume that the CSIs of all links are known at all nodes. The elements of each channel matrix are i.i.d. complex Gaussian random variables with zero-mean and same variance. For the first set of simulations, we let N = R = M = 4, SNRsr = 20 dB, SNRsd = 5 dB, and vary SNRrd. Eight systems are compared, namely (a) un-precoded system with MMSE receiver, (b) linear relay precoded system with MMSE receiver [43], [44], (c) un-precoded system with QR-SIC receiver, (d) linear source and relay precoded system with MMSE receiver (Chapter 2), (e) un-precoded system with MMSE-OSIC receiver, (f) THP
Table 5.1: Complexity of linear source and relay precoders (MMSE-SIC receiver).
Step Operation FLOPs
1 H0SR (3.27) O(N3+ RN2)
2 SVD HRD = UrdΣrdVHrd (3.28) O(MR2+ R3)
3 SVD H0SR= U0srΣ0srV0Hsr (3.29) O(RN2+ N3)
4 Σr (3.32) O(κIr)
5 FR (3.30) O(R3)
6 SVD eH = UHeΣHeVHHe O(MN2+ N3)
7 GMD
qPS,T
σs2NΣHe
σs−1IN
(5.13) O ((2M + N)N2+ N3)
8 FS,opt (5.15) O(N3)
Ir is denoted as the iteration number of the water-filling process (3.32).
source and linear relay precoded system with MMSE receiver (Chapter 3), (g) linear source and relay precoded system with QR-SIC receiver (Chapter 4), and (h) linear source and relay precoded system with MMSE-SIC receiver (Chapter 5). Since the THP source and linear relay precoded system is considered, we adopt the 16-QAM modulation scheme. Fig, 5.1 and Fig. 5.2 show the simulated BLER and BER for the systems mentioned above, respectively. From Fig.
5.1, we can observe that the performance of the linear receivers are limited. The performance of un-precoded system can be improved by the linear relay precoder and can be further enhanced by the linear source and relay precoders. When the SNR of the relay-to-destination link is sufficiently high, the significance of the relay precoder is reduced. This indicates that the relay precoder is not critical. So, the performance of un-precoded and the relay precoded systems is close. Also, since nonlinear receivers can provide higher diversity gain [13], they perform well
Table 5.2: Source and relay precoders in the proposed nonlinear transceivers.
Table 5.3: Optimizations of the source and relay precoders in the proposed nonlinear
in the high SNR regions, even for the un-precoded systems. As expected, the linear source and relay precoded system with the nonlinear receivers are better than the un-precoded systems. Due to the fact that the MMSE-SIC receiver has larger diversity gain, the precoded system with the MMSE-SIC receiver outperforms the precoded system with the QR-SIC receiver. Although the THP source and linear relay precoded system is also a nonlinear system, its BLER performance is inferior to that of the precoded systems with the QR-SIC and MMSE-SIC receivers. This is because the former system is designed by the MMSE criterion, while the latter system by the BLER criterion. However, in terms of the BER, the THP source and linear relay precoded system performs slightly better than the linear source and relay precoded system with the QR-SIC/MMSE-SIC receivers, as shown in Fig. 5.2.
Fig. 5.3 and 5.4 show the BLER and BER performances of the aforementioned transceivers with using 4-QAM modulation for each substream. As we can see, the THP source and linear relay precoded system with MMSE receiver is much worse. This is because it only workable for higher order modulation (m ≥ 16), as Section 3.1.1 described.
For the second set of simulations, we still compare the performance of aforementioned precoded systems. However, we let SNRsd = 5 dB, SNRrd = 20 dB, and vary SNRsr. Fig 5.5 and Fig. 5.6 show the simulation results for the BLER and the BER, respectively. As we can see, the relay precoded system with the MMSE receiver outperforms the un-precoded systems with the linear and nonlinear QR-SIC receivers along the increase of the SNR. This is because the performance is dominated by the links of the source-to-destination and the relay-to-destination when SNRsr is high. As a result, the additional relay precoder can improve the overall link quality. Unlike the previous case, the performance of the un-precoded system with the MMSE-OSIC receiver is inferior to the source and relay precoded system with the MMSE receiver. This is because when the SNR of the source-to-relay link is sufficiently high, the MIMO relay system is degenerated to the MIMO system. As a result, the significance of the relay precoder is increased. Also, as expected, the precoded systems with the nonlinear source precoder or with the nonlinear receivers outperform the un-precoded systems and the precoded
systems with linear precoders and linear receivers.
The MLD receiver is known to be optimal. It is then interesting to know its performance in MIMO relay sytsems. For the third set of simulations, we compare the performance of the un-precoded system with the ML receiver and that of un-precoded systems. Since the computational complexity of the ML receiver is high, we only consider the system with N = R = M = 2.
Let SNRsd= 5 dB, SNRrd= 20 dB, and vary SNRsr. Fig. 5.7 shows the BLER comparison. As shown in this figure, the un-precoded system with ML receiver outperforms other un-precoded systems. However, it is poorer than all the precoded systems we proposed.
In real-world applications, CSIs have to be transmitted to the location where the precoders are calculated. Thus, quantization and transmission errors may arise. We refer this phe-nomenon as imperfect CSI. In the final set of simulations, we compares the performance of all un-precoded/precoded systems when CSIs are not perfect. As that in the previous works [54], the imperfect channel ˆH is related to the true channel H via the equation of ˆH =
√1 − ρH +√
ρ∆H, where ∆H models the channel error and the coefficient ρ characterizes the magnitude of the error. The elements of ∆H are modeled as i.i.d. Gaussian distributions with zero mean and same variance. Fig. 5.8, shows the simulation results. Here, we let SNRsd
= 5 dB, SNRsr = 25 dB, SNRrd = 20 dB, and ρ be varied. As we can see, the performance of the precoded systems degrades as ρ increases, especially for the THP source precoded system with MMSE receiver, and the linear source precoded system with MMSE-SIC receiver. The linear source precoded system with QR-SIC is less affected. Note that the CSIs are assumed perfectly known at the destination for un-precoded systems, and their BLERs are not affected by the value of ρ.
5 10 15 20 25 30 35 40 10−2
10−1 100
SNR (dB)
BLER
Un−precoded (MMSE) Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.1: BLER performance comparison for un-precoded and precoded systems (16QAM, N = R = M = 4, SNRsr=20, SNRsd=5 dB).
5 10 15 20 25 30 35 40 10−3
10−2 10−1 100
SNR (dB)
BER
Un−precoded (MMSE)
Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.2: BER performance comparison for un-precoded and precoded systems (16QAM, N = R = M = 4, SNRsr=20, SNRsd=5 dB).
−5 0 5 10 15 20 10−6
10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
BLER
Un−precoded (MMSE)
Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.3: BLER performance comparison for un-precoded and precoded systems (4-QAM, N = R = M = 4, SNRsr=20, SNRsd=5 dB).
−5 0 5 10 15 20 10−7
10−6 10−5 10−4 10−3 10−2 10−1
SNR (dB)
BER
Un−precoded (MMSE)
Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.4: BER performance comparison for un-precoded and precoded systems (4-QAM, N = R = M = 4, SNRsr=20, SNRsd=5 dB).
0 5 10 15 20 25 30 35 40 10−2
10−1 100
SNR (dB)
BLER
Un−precoded (MMSE) Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.5: BLER performance comparison for un-precoded and precoded systems (16QAM, N = R = M = 4, SNRsd=5, SNRrd=20 dB).
0 5 10 15 20 25 30 35 40 10−3
10−2 10−1
SNR (dB)
BER
Un−precoded (MMSE) Linear relay precoded (MMSE) Un−precoded (QR−SIC)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.6: BER performance comparison for un-precoded and precoded systems (16QAM, N = R = M = 4, SNRsd=5, SNRrd=20 dB).
0 5 10 15 20 25 30 35 40 10−2
10−1 100
SNR (dB)
BLER
Un−precoded (MMSE) Linear relay precoded (MMSE) Un−precoded (QR−SIC) Un−precoded (ML)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.7: BLER performance comparison for un-precoded and precoded systems (16QAM, N = R = M = 2, SNRsd=5, SNRrd=20 dB).
0 0.05 0.1 0.15 0.2 0.25 0.3 10−2
10−1
ρ
BLER
Un−precoded (MMSE) Linear relay precoded (MMSE) Un−precoded (QR−SIC) Un−precoded (ML)
Linear source and relay precoded (MMSE) Un−precoded (MMSE−OSIC)
THP source and linear precoded (MMSE) Linear source and relay precoded (QR−SIC) Linear source and relay precoded (MMSE−SIC)
Figure 5.8: BLER performance comparison for un-precoded and precoded systems with imper-fect CSIs (16QAM, N = R = M = 2, SNRsd=5, SNRsr=25 dB, SNRrd=20 dB).
Chapter 6
Joint MMSE Transceiver Design with Quality-of-Service (QoS) Constraints
In previous chapters, we address the transceiver designs in MIMO relay systems maximizing the system performance under the power constraints. In this chapter, we consider the transceiver design minimizing the transmission power under QoS constraints. The transceiver structure considered here is the same as that proposed in Chapter 2, in which the linear precoder is used at the source (and the relay), and the linear MMSE receiver at the destination. Since there is a one-to-one mapping between the BER and the MSE, we use the MSEs of signal streams as our QoS constraints. We first consider the precoders design in two-hop systems and then general MIMO relay systems. Our formulation leads to an optimization problem that the constraint function is a highly nonlinear function of the precoders, either in the two-hop or general MIMO relay systems. To overcome the problem, we first propose new precoder structures which can simplify the optimization in the two-hop system. The proposed structures can translate the matrix-valued optimization problem into a scalar-valued one, facilitating the derivation of the optimum solution. For general MIMO relay systems, the problem becomes more involved since the direct link is included. Based on the proposed precoder structure, however, we can derive an MSE upper bound. Using the upper bound as the constraint function, the original optimization
problem can be greatly simplified, and the solution can be obtained by the primal decomposition approach. In Section 6.1, we first give the system model and the related optimization problem.
After that, we derive the source and relay precoders in the two-hop MIMO relay and then the general MIMO relay systems, respectively, in Section 6.2 and 6.3. Finally, we evaluate the performance of the proposed precoders in Section 6.4.
§ 6.1 System Model and Problem Formulation
§ 6.1.1 MMSE Receiver with Linear Source and Relay Precoders
We consider the same transceiver in Chapter 2. Recall (2.4), the received signal can thus be expressed as
Based on (6.1), the MMSE equalization matrix Gopt, as shown in (2.8), can be expressed as Gopt = σ2sFHSHH¡
σs2HFSFHSHH + Rn¢−1
. (6.3)
The resultant minimal MSE and MSE matrix can then be expressed as Jmin = trn¡
and
ER = FHSHHSRFHRHHRD¡
σn,r2 HRDFRFHRHHRD+ σ2n,dIM¢−1
HRDFRHSRFS. (6.7)
As we can see from (6.4)-(6.5), the MMSE and MSE matrix are the functions of FS and FR. Also, ED denotes the MSE component due to the direct link and ERis contributed by the relay link. If only the relay link is considered (which is also known as the two-hop MIMO relay system), we can set ED = 0 [43]- [44].
§ 6.1.2 Problem Formulation
To start with, we first let the QoS constraints be defined in terms of the MSEs at the receiver, i.e.,
E(i, i) := E£
|si− ˆsi|2¤
≤ ρi, 1 ≤ i ≤ L, (6.8)
where E(i, i) is the ith component of E and ρi is the MSE constraint for the ith data stream.
Here we note that 0 ≤ ρi < σs2since E [|si− ˆsi|2] <= E [|si|2] = σs2.
Our task is to design the source and relay precoders such that the transmission power is minimized and designated MSE constraints are satisfied. To proceed, let us define the power consumption at the source and the relay, respectively, as
PS,T = tr¡ E£
FSssHFHS¤¢
(6.9)
and
PR,T = tr
³ E
h
FR(HSRFSs + nR) (HSRFSs + nR)HFHR i´
= tr¡ FR¡
σs2HSRFSFHSHHSR+ σn,r2 IR¢ FHR¢
. (6.10)
With (6.5)-(6.10), the joint source/relay precoders design problem can be formulated as Taking a closer look at (6.11), we readily find that the MSE matrix E involves a series of matrix multiplications and inversions and is a complicated function of FSand FR. Also, the problem is not a convex optimization problem. Therefore, the exact solution to (6.11) is almost impossible to derive. In the next section, we propose a new method to solve the problem