5.6 Numerical results
6.3.2 Solution to the max-min fairness problem for N = M
If N = M, each transceiver pair is a member of a unique group and, therefore, each MAC phase is assigned to a single pair. It is assumed that m ≡ Qm and, consequently, transceiver Tm and
transceiver TM+m are assigned to the mth MAC phase. Note that the constraints of the problem in
(6.15) are not affected if the weight vector wmis multiplied by the complex number ejφm, where φm
consequence, wHh
m=wHhM+mis also real-valued according to (6.4) and (6.8). Since each weight
vector wmis assigned exclusively to the mth transceiver pair, and, using the definitions
˜ w,hwT, 1iT, ˜ hm, h hTm, 0 iT , m=1, . . . 2M , ˜
Qm,I+N,diag([Qm,I+N, σ2])1/2, m=1, . . . 2M ,
˜
Dr ,diag([Dr, 0]), r=1, . . . R, D˜ ,diag([D, 0]),
we reformulate the problem in (6.15) to the equivalent problem
max ˜ w,γ γ s.t. ℜ{ ˜w Hh˜ m} ≥√γk ˜Qm,I+Nw˜k, m=1, . . . , 2M ℑ{ ˜wHh˜m}=0, m=1, . . . , 2M ˜ wHD˜rw˜ ≤ pr,max, r=1, . . . , R, ˜ wHD ˜˜w≤ PR,max, [w˜]L·R+1=1. (6.18)
Note that the feasibility problem of computing a weight vector w for some fixed γ that satisfies the constraints of (6.18) is convex and belongs to the class of second order cone problems. Similar to bisection search combined with the SDR method explained in Chapter 3.3.2, the maximum objective value of (6.18) can be found by a simple bisection search on γ.
We remark that, in general, the assumption that wHh
mis real-valued restricts the feasible set
and may lead to a suboptimal solution if N < M .
6.4
Simulation results
A network consisting of 15 relays is considered, where the noise and channel vectors {fm}Mm=1 and
{gm}Mm=1 are generated in each of the 100 simulation runs as complex zero mean circular Gaussian
distributed random variables with unit variance. The proposed algorithm is initialized with a random vector that is created as a complex zero mean circular Gaussian distributed random variable and choose ǫ=10−4.
In the first example, we consider the case of L = 1. In this case, our scheme comprises two time slots and all transceivers are members of a single group. We set pr,max = PR,max/10 for
r=1, . . . , R and the transmit powers of the transceivers {Pm}2Mm=1 are 10dB above the noise power.
We compare the rate achieved by the proposed two time slot scheme to the rate achieved by the one-directional multi-user p2p scheme proposed in [24], using two transmissions, resulting in four
CHAPTER 6. MULTIUSER BI-DIRECTIONAL RELAY NETWORKS 93 0 5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 Proposed scheme, N= 1, M = 2 Four time slot scheme of [24], M= 2 Proposed scheme, N= 1, M = 3 Four time slot scheme of [24], M= 3
Available relay transmit power PR,max (dBm)
R at e (b it s/ ti m e sl ot /H z)
Figure 6.2: First example, N =1. Minimal rate versus the available relay transmit power.
2 3 4 5 6 −5 0 5 10 Upper bound SDR Proposed algorithm Algorithm of [33]
Number of transceiver pairs M
S IN R (d B )
12 15 18 21 24 27 30 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 N = 2, (2, 2), best allocation N = 2, (2, 2), random allocation N = 1 N = 2, (3, 1), best allocation N = 2, (3, 1), random allocation N= 3, (2, 1, 1), best allocation N= 3, (2, 1, 1), random alloc. N= M R at e (b it s/ ti m e sl ot /H z)
Available relay transmit power ˜PR,max(dBm)
Figure 6.4: Second example. Minimal rate versus the relay transmit power.
time slots. The rate is defined as 1/2 · log2(1+SINR)for the proposed two time slot scheme and
1/4 · log2(1+SINR) for the one-directional multi-user p2p scheme, where the prefactors 1/2 and
1/4 take the number of time slots into account. For the one-directional p2p scheme, we consider the joint optimization of two weight vectors of the two sequential transmissions. As the optimization problem of maximizing the minimum SINR has the same structure as (6.6), the proposed algorithm is used to compute the weight vectors. Note that the transceiver powers and the bounds on the sum power and the individual relay powers are set twice as high as the corresponding values for the two time slot scheme since the number of time slots is doubled.
In Figure 6.2, the minimum rate is depicted versus the maximum relay transmit power PR,max
for M =2 and M =3 transceiver pairs. The proposed two time slot scheme outperforms the four time slot scheme for both M =2 and M =3 transceiver pairs.
In Figure 6.3, we compare the proposed algorithm to the SDR-based Algorithm 1 of Chapter 3.3.1 and the theoretical upper bound which is obtained as a byproduct of the SDR method. Figure 6.3 depicts the minimum SINR versus the number of transceiver pairs for fixed PR,max=10dBm. The
minimum SINR decreases with the increase of M . For M =2 and M =3, the proposed algorithm and the SDR-based algorithm of [33] perform similarly and achieve almost the upper bound on the minimum SINR obtained by the SDR-based algorithm. Therefore, the performance of this scheme
CHAPTER 6. MULTIUSER BI-DIRECTIONAL RELAY NETWORKS 95
cannot be substantially improved by a rank-two beamforming scheme as proposed in Chapter 4. For
M =5 and M =6, the proposed algorithm performs better than the that of [33]. We observe that the SINR increases significantly with a growing number of user pairs.
In the second example, M =4 transceiver pairs are considered. We set pr,max =PR,max/4 for
r=1, . . . , R and the transmit powers of the transceivers {Pm}2Mm=1=10 ·(L+1). We compare the
rate achieved by the proposed schemes which is defined as 1/(L+1)· log2(1+SINR), where the
prefactors take the number of time slots into account.
In Figure 6.4, the minimum rate is depicted versus the maximum relay transmit power ˜PR,max
where PR,max= (L+1)P˜R,max. We indicate the allocation of the transceivers by (a, b, . . .), where
we have one group of a transceivers, one group of b etc.. As the allocation of the transceivers into the groups is not unique (except for the cases L=M and L=1) and as the allocation may influence the performance, we display the allocation that achieves the highest rate (best allocation) and a random allocation of the transceivers. Note that the best allocation is found by testing all possible combinations.
6.5
Conclusion
In this chapter, bi-directional distributed beamforming for the p2p communication of multiple transceiver pairs is considered. In contrast to a single transceiver pair of Chapter 5, the multi-user environment leads to MUI. As a consequence, the gain of the two time-slot scheme of Figure 1.1 (c) to the four time slot scheme of Figure 1.1 (a) in terms of transmission gain is lower as compared with the single transceiver pair case. We have proposed a class of novel TDMA schemes to provide a trade-off between spectral efficiency and the number of degrees of freedom in the beamformer design. The simulation results demonstrate the impact of the choice of the TDMA scheme. Furthermore, an iterative algorithm based on inner approximations is proposed that yields a performance close to the upper bound obtained by SDR.
In this chapter, we have considered a relatively small number of users where the considered rank- one beamforming scheme performs already close to the theoretical bound that cannot improved by using higher rank beamforming schemes. For a large number of users, MUI excessively impairs the communication. In this case, it is desirable to separate the various transmit signals not only in the spatial domain by using beamforming but also in the time domain by using multiple access schemes.
Chapter 7
Differential beamforming
7.1
Introduction
In the previous chapters, we have assumed that the CSI is perfectly known and solved power min- imization problems or max-min fairness problems to compute the relay weight vectors. Moreover, we have assumed that the computation of weight vectors and power scaling factors is performed by a central processing node. In this chapter, we propose a novel DBF scheme that computes the relay weight without explicit knowledge of the CSI. To establish relay communication without CSI, differential techniques [114]–[123] and non-differential techniques [124], [125] for single-antenna re- lays have been proposed. The developed DBF scheme exploits implicit CSI contained in previously received signal vectors to perform receive and transmit beamforming. Note that the DBF scheme does not require a central processing node as each parameter is derived at its respective node. This is different to the previous chapter, where we have assumed that all CSI is accumulated at one node to compute the network parameters.
We consider the bi-directional communication between two terminals in the four time slot scheme illustrated in Figure 1.1 (a). In contrast to Chapter 5, where multiple single-antenna relays form a distributed beamforming system, a single multi-antenna AF relay is regarded.
In our theoretical analysis, we derive an expression for the approximated SNR achieved by the proposed DBF scheme for time-invariant channels. Our simulations demonstrate that the latter approximation is highly accurate in the medium to high-power regime. Based on this approximation, we propose to distribute the total power among the terminals and the relay such that max-min fairness is achieved. The proposed power allocation scheme requires only the knowledge of the noise powers at the RS and the terminals but no CSI. In the simulations, we consider time-variant channels using the model of [138] and compare the BER of the proposed DBF scheme with state of
the art approaches known from the literature for different terminal velocities. The simulation results demonstrate that the proposed DBF scheme dramatically outperforms the differential single-antenna relaying scheme of [123] as well as the multi-antenna relaying scheme of [133], which requires the knowledge of the second order statistics of the CSI.
The contributions of chapter are:
• We propose a novel DBF scheme for bi-directional AF relaying that does not require CSI at any node.
• We derive an approximation expression for the SNR at each terminal which is accurate for medium and high transmit powers.
• We derive a simple power allocation scheme that achieves almost optimum performance. • We test our proposed DBF scheme under realistic high mobility scenarios using numerical
simulations and demonstrate that the proposed scheme outperforms existing schemes.
7.2
Signal model
We consider a wireless single carrier network with two single-antenna transceivers T1and T2and one
multi-antenna AF relay comprising NR antennas. We assume frequency flat fading channels and
that direct link between the transceivers is not available. We further assume that reciprocity holds for transmissions from the terminals to the RS and vice versa. We consider in our derivations a block fading channel model in which the channels remain constant during two consecutive transmission blocks. This assumption is however relaxed in the simulations, where we consider both time-variant and time-invariant channels.
The bi-directional communication between T1 and T2 is organized in consecutive blocks, where
in each block a four time slot protocol according to Figure 1.1 (a) is used.
In the first time slot of the nth transmission block, terminal T1transmits the signal√P1v1(n)to
the relay, where P1is the transmit power of terminal T1and v1(n)is the transmitted data symbol of
the nth transmission block. We denote f and g as the R × 1 vectors containing the channel coefficients characterizing the transmission in the nth and(n− 1)th transmission block between terminal T1and
the relay and between the relay and terminal T2, respectively. Then, the signals received at the relay
in the first time slot can be expressed as the vector x(n, 1) = [x1(n, 1), . . . , xNR(n, 1)]
T, given by
CHAPTER 7. DIFFERENTIAL BEAMFORMING 99
where η(n, t) = [η1(n)(n, t), . . . , ηNR(n, t)]
T and where η
r(n, t) denotes the noise at the rth relay
antenna in the nth transmission block at time slot t ∈ {1, 2, 3, 4}. The received signal vector x(n, 1)
is weighted by the NR× NR beamforming matrix W(2)and the resulting R × 1 signal vector
t(n, 2) =W(2)x(n, 1) (7.2) is transmitted to terminal T2in the second time slot of the nth transmission block. The corresponding
received signal at terminal T2 is then given by
y2(n) =gTt(n, 2) +ν2(n) =gTW(2)x(n, 1) +ν2(n), (7.3)
where ν2(n)is the noise at terminal T2.
Let us assume that the noise in the network can be modeled as a spatially and temporally uncorrelated random processes with zero mean and variance E{|ηr,t(n)|2}=σR2, E{|ν2(n)|2}=σ22,
E{|ν4(n)|2}=σ21, ∀r, m, t. Moreover, we assume without any loss of generality that E{|v1(n)|2}=
E{|v2(n)|2}=1.
We first regard the ideal case that full CSI is available at the relay. The optimization problem of designing W(2)such that the SNR at terminal T2is maximized has been treated in [132]. It has
been shown that the ideal AF beamforming matrix is given by
W⋆(2) =c2g∗fH, (7.4)
where c2=
q
PR,2/ σ2R+P1kfk2 kfk2kgk2 is a power scaling factor to ensure an average transmit
power of PR(2)at the relay. Note that W⋆(2)contains the vectors f and g that are respectively the
matched filter solutions for receive and transmit beamforming at the relay. The ideal matrix leads to a maximum SNR of
SNR⋆(2) = P1PR(2)kfk2kgk2
σ2Rσ22+σ22P1kfk2+σR2PR(2)kgk2
. (7.5)
In the third and the fourth time slot, the communication from terminal T2 to terminal T1 is accom-
plished. In the third time slot, terminal T2 transmits the signal √P2v2(n)to T1, where P2 denotes
the transmit power of terminal T2 and v2(n) is the transmitted data symbol. Similar as in (7.1),
the received NR× 1 signal vector at the relay in the third time slot of the nth transmission block is
given by
x(n, 3) =pP2v2(n)g+η(n, 3). (7.6) Following the relaying procedure for the second time slot given in (7.2) - (7.5), the relay weights x(n, 3)by the NR× NR beamforming matrix W(4)and transmits the NR× 1 signal vector t(n, 4),
given by t(n, 4) =W(4)x(n, 3)to terminal T1 in the fourth time slot. The ideal relay weight matrix
W(4)⋆ is given by W(4)⋆=c
4f∗gH, where
c4 =
q
PR(4)/ σ2R+P2kgk2kgk2kfk2 is a constant and PR(4)is the transmit power of the relay.
Utilizing the ideal relay beamforming matrices requires the perfect knowledge of the instanta- neous channel vectors f and g. Here, we address the problem of choosing the beamforming matrices in the case that CSI is not available.