3.3 Simulations
4.2.2 Antenna Selection
The algorithm developed above assumes that L = N ≤ M. If L < N ≤ M, the precoder matrix is not square and the GMD method cannot be applied. A simple remedy to this problem is to use the antenna selection method. Using the method, we can select L antennas from the N antennas (L ≤ N, R, M ) such that the geometric mean of the equivalent channel in (4.13) is
maximized. Thus, the problem can be formulated as that each column of Pi contain only one nonzero element (with the value of one) indicating the antenna selected. For example, for a 3 × 2 system, we have
P =
After the optimum antenna selection matrix is determined, the optimum source precoder can then obtained by (4.12) and the proposed method can be applied accordingly.
§ 4.3 Simulations
We consider a single relay AF MIMO relay system, and assume that CSI of all links are known at all nodes. Furthermore, the elements of each channel matrix are i.i.d. complex Gaussian random variables with zero-mean and same variance. Without of loss generality, we let the modulation scheme be 4-QAM and use the BER as the performance measure. For the first set of simulations, we let N = L = R = M = 4, SNRsr=SNRrd = 15 dB, and SNRsd
be varied. Seven systems are compared, namely (a) the un-precoded system with the zero forcing (ZF) receiver, (b) the un-precoded system with the MMSE receiver, (c) the un-precoded system with the QR-SIC receiver, (d) the un-precoded system with the MMSE-ordered SIC (OSIC) receiver [13], (e) the linear relay precoded system with MMSE receiver [43], (f) the
precoded system with the GMD source precoder [11], [12], and (g) the linear source and relay precoded system with QR-SIC receiver. It is noteworthy that the simulation conducted in [43]
does not consider the signal received from the direct link. For better performance, we further take the signal into consideration when implementing the MMSE receiver. Fig. 4.2 shows the BER performance comparison. As we can see, the performance of the un-precoded systems is limited. The best un-precoded system is the one with the MMSE-OSIC receiver. Although it is better than the linear relay precoded system in the high SNR region, it is much worse than the GMD source and the linear source and relay precoded systems. The linear source and relay precoded system significantly outperforms the GMD source precoded system; SNR is improved by 4 dB when BER is 10−3. This is because the linear source and relay precoded system takes two precoders into consideration, yielding a higher received SNR at the destination. We then consider another scenario where SNRsr = 15 dB and SNRrd = 0 dB. Now, the link between the relay and the destination becomes poorer. Theoretically, the relay precoder will become less critical in this scenario. Fig. 4.3 shows the performance comparison for all systems.
As expected, the performance of the relay-only precoded system is seriously degraded. The linear source and relay precoded system, however, still has the superior performance. The performance gap between the linear source and relay precoded and the GMD source precoded system becomes somewhat smaller. This is also expected since the role of the relay precoder is less critical, as mentioned.
As discussed, the GMD criterion can maximize the lower bound of the channel free distance [12], and the performance of an ML detector is directly related to the free distance. We can expect that the linear source and relay precoders can also improve the system performance if an ML receiver is used at the destination. Fig. 4.4 shows the performance comparison for the uncoded, the GMD source precoded, and the linear source and relay precoded systems with the ML receiver applied at the destination. Here, SNRsr = SNRrd = 10 dB and L = N = R = M = 2. As shown in this figure, the performance of the linear source and relay precoded system is better than that of the the un-precoded and the GMD source precoded systems. The
GMD source precoder has better performance than un-precoded system since the channel’s free distance is improved by the source precoder. The performance gap between the linear source and relay precoded method and GMD source precoder is due to the additional relay precoder, further enlarging the lower bound of the channel’s free distance.
In conventional (non-cooperative) MIMO systems, spatial multiplexing cannot be applied when L = N > M . However, in a cooperative system, with the aid of the relay, the degree of freedom of the overall system is increased. In other words, even when L = N > M , spatial multiplexing can still be used in cooperative systems. The final set of simulations is to compare the performance of the un-precoded and the linear source and relay precoded systems in this scenario. For the un-precoded schemes, we let L = N = 4, R = M = 2 and the modulation scheme be 4-QAM. Since L = N > M , the proposed system has to conduct antenna selection.
Here, we let L = 2, N = 4, R = M = 2 and the modulation scheme be 16-QAM. With the setting, the transmission rates of the un-precoded and precoded systems are the same (8 bits/channel usage). Let SNRsr=SNRrd = 15 dB and SNRsd be varied. Fig. 4.5 shows the performance of the un-precoded and linear source and relay precoded systems. As we can see, the linear source and relay precoded system significantly outperforms the un-precoded systems.
There exists error floors for the un-precoded systems since the noise at the relay link tends to dominate the overall performance when SNRsd is high. Due to the precoding operation, we do not observe the error floor phenomenon in the linear source and relay precoded system.
H
SR#
Source: N antennas
QR-SI C
#
Relay: R antennas
#
Destination: M antennas
H
SDH
RD: First phase : Second phase
F
Ss
y
RF
RNsubstreams
Figure 4.1: Linear source and relay precoded AF MIMO relay system with QR-SIC receiver.
−4 −2 0 2 4 6 8 10 12 14 16 10−8
10−7 10−6 10−5 10−4 10−3 10−2 10−1
SNR (dB)
BER
Un−precoded (ZF) Un−precoded (MMSE)
Linear relay precoded (MMSE) Un−precoded (QR−SIC) Un−precoded (MMSE−OSIC) GMD source precoded (QR−SIC)
Linear source and relay precoded (QR−SIC)
Figure 4.2: BER performance comparison for linear source and relay precoded system with QR-SIC receiver and existing precoded systems (L = N = R = M = 4, SNRsr=SNRrd = 15 dB).
−4 −2 0 2 4 6 8 10 12 14 16 10−5
10−4 10−3 10−2 10−1 100
SNR (dB)
BER
Un−precoded (ZF) Un−precoded (MMSE)
Linear relay precoded (MMSE) Un−precoded (QR−SIC) Un−precoded (MMSE−OSIC) GMD source precoded (QR−SIC)
Linear source and relay precoded (QR−SIC)
Figure 4.3: BER performance comparison for linear source and relay precoded system with QR-SIC receiver and existing precoded systems (L = N = R = M = 4, SNRsr=15, SNRrd=0 dB).
−5 0 5 10 15 20 10−6
10−5 10−4 10−3 10−2 10−1
SNR (dB)
BER
Un−precoded
GMD source precoded
Linear source and relay precoded
Figure 4.4: BER performance comparison for un-precoded, GMD source precoded, and linear source and relay precoded systems with ML receiver at the destination (L = N = R = M = 2, SNRsr=SNRrd=10 dB).
0 5 10 15 20 25 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
BER
Un−precoded (ZF) Un−precoded (MMSE) Un−precoded (QR−SIC) Un−precoded (MMSE−OSIC)
Linear source and relay precoded (QR−SIC)
Figure 4.5: BER performance comparison for linear source and relay precoded system with QR-SIC receiver and un-precoded systems (L = N = 4, R = M = 2 and 4-QAM is used for un-precoded systems; N = 4, L = R = M = 2 and 16-QAM is used for linear source and relay precoded system with QR-SIC receiver).
Chapter 5
Joint MMSE-SIC Transceiver Design with Linear Source and Relay Precoders
As well known in the precoded MIMO systems, the MMSE-SIC receiver outperforms the QR-SIC receiver. It is reasonable to assert that the same result can be obtained for MIMO relay systems. For MIMO systems, the precoder with the MMSE-SIC receiver can be solved by the UCD method. In this chapter, we consider the system where linear precoders are used at the source and the relay, and an MMSE-SIC receiver at the destination. We show that the UCD is not directly applicable in AF MIMO relay systems. However, if the source precoder is constrained to be unitary, the problem can be solved. In Section 5.1, we give the system model for the MMSE-SIC receiver. In Section 5.2, we propose a modified UCD method to derive the source and relay precoders. Finally, we report simulation results in Section 5.3 to conform the effectiveness of the proposed method.