Chun-Jung Wu, David W. Lin, and Tzu-Hsien Sang Department of Electronics Engineering and Institute of Electronics
National Chiao Tung University, Hsinchu, Taiwan 30010, ROC
E-mails: cjwu.ee91g@nctu.edu.tw, dwlin@mail.nctu.edu.tw, tzuhsien54120@faculty.nctu.edu.tw
Abstract—We consider the design of a multi-input multi-output (MIMO) transmission system equipped with distributed, single-antenna amplify-and-forward relays. In particular, we consider how to optimize the complex forwarding coefficients of the relays to maximize the system capacity. Existing studies on MIMO relay network designs mostly concentrate on the case of single relay with multiple antennas, whose results are not applicable to distributed networks. We first analyze how the system capacity relates to the relay transmission powers and the channel coefficients. An upper bound on the asymptotic capacity at high relay powers is derived. Aided by the capacity results, we consider the optimization of relay coefficients under two noise conditions: that where the destination noise dominates the total noise and that where the relay noise dominates the total noise. It turns out that no simple analytic solutions can be found for all relay network sizes. Therefore, we propose suboptimal solutions that involve computation of optimal relay coefficients with optimal selection of relays for network sizes that can be solved analytically. The results on diversity order are found to show resemblance to that for single-hop MIMO systems employing antenna selection. Simulation results also verify the superior performance of the proposed technique.
I. INTRODUCTION
There is a recent surge in studies of relay-aided multi-input multi-output (MIMO) transmission due to its potentials in answering to the ever-rising quest for higher mobile data speed. The primary uses of relays, as envisioned by many, are to enhance coverage and capacity. To preserve the degree of freedom available in a MIMO system, in such relay-aided MIMO transmission it is natural to consider using a relay with multiple antennas or using a relay network.
For the case of single relay equipped with multiple antennas, both single-input single-output (SISO) [1] and MIMO [2], [3] end-to-end transmissions have been considered. To fully exploit the advantage that may be offered by the relay, suitable beamforming over the multiple relay antennas is needed for both the backward and the forward MIMO channels. One disadvantage of the single-relay architecture is its lack of spatial or system diversity. When the single relay is in an outage condition due to fading or other reasons, the relaying link breaks. A reasonable alternative, therefore, is to employ geographically scattered multiple single-antenna relays for
This study was conducted under the Wireless Broadband Communications Technology and Application Project of the Institute for Information Industry, which has been subsidized by the Ministry of Economic Affairs of the Republic of China, and under Grant NSC 98-2219-E-009-012 of the National Science Council of the Republic of China.
better diversity. However, the advantage in diversity comes at the cost of greater difficulty in beamforming compared to using a single multi-antenna relay, as subsequent discussion will further demonstrate.
For the case of multiple single-antenna relays, i.e., the case of a distributed single-antenna relay network, the most straightforward design conceivable of the relay gains (i.e., relay coefficients) is to assign identical gains to all relays subject to some power limit. This has been termed the all-pass scheme [4]. Some have studied the ergodic characteristics of capacity under simple designs like the all-pass [5]. But how to effect good designs of the relay gains remains a problem to be further looked into.
In this work, we consider the design of distributed single-antenna amplify-and-forward (AF) relay networks. In partic-ular, we consider the optimization of the relay forwarding coefficients for maximization of the Shannon capacity of the system. We aim at an analytical solution of the problem, although it will be seen later that there are limits to what can be achieved by analysis. We assume that the relays know the relevant channel coefficients (or channel state information, CSI).
First, we examine the dependence of capacity on the relay transmission powers and the channel coefficients. Due to the use of AF, both the signal power and the relay noise are scaled by the relay forwarding coefficients. We show that the capacity increases with the relay output powers, but there is an upper bound to its asymptotic value at the limit of very high relay powers. The analysis also helps to guide the subsequent relay network design. In relay network design, we note that there are two noise terms that impact the system performance, namely, the relay noise and the destination noise. We find that an analytical characterization of the optimal solution and the performance bound can be obtained when either noise dominates the total noise and when the number of relays are of certain particular values. Based on these observations, we thus propose suboptimal designs that optimize the relay forwarding coefficients with proper relay section.
The resulting performance of the proposed designs shows close resemblance to that of MIMO systems with antenna selection [6], [7], which have been developed to reduce the complexity of the MIMO transceiver without sacrificing the diversity order by activating only a subset of the transmitter or the receiver antennas. In [7], the outage diversity with 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications
Fig. 1. MIMO system with distributed relays.
antenna selection is shown to be similar to a full-complexity MIMO system (i.e., without antenna selection). Further ap-proximations to capacity distribution and performance loss of antenna selection have been conducted in [8] and [6]. Thanks to this analogy we may predict that a relay-selection system with optimal setting of relay forwarding coefficients would have similar performance characteristics to the better-studied MIMO antenna-selection systems.
In what follows, Sec. II describes the system model and con-siders its Shannon capacity. Sec. III discusses the optimization of relay coefficients. Sec. IV presents and elaborates on some numerical results. And Sec. V is the conclusion.
II. SYSTEMMODEL ANDCAPACITYANALYSIS
We consider a distributed MIMO relay system composed of one source terminal, one destination terminal, and L single-antenna relays. The source and the destination terminals are both equipped with M antennas. The relays simply amplify the received signals from the source (with possibly different complex gain coefficients) and transmit the amplified signals to the destination. Hence the destination terminal would receive both the amplified signals and the amplified relay noises. To preserve the degree of freedom provided by the source and the destination terminals, we assumeL ≥ M. Fig. 1 illustrates the system model.
Let x ∈ CM and y ∈ CM denote the signals transmitted from the source terminal and received by the destination terminal, respectively, where C denotes the set of complex numbers. Let GH ∈ CL×M be the matrix of MIMO channel coefficients between the source terminal antennas and the relays, where superscriptH denotes Hermitian transpose. Sim-ilarly, let F ∈ CM×Lbe the channel matrix between the relays and the destination terminal antennas. The received signals at the relays are assumed to be subject to additive complex circular white Gaussian noise (AWGN) nR∼ CN (0, σ2RIL), where IL denotes the L × L identity matrix. Likewise, the received signals at the destination are interfered by AWGN nD∼ CN (0, σ2DIM).
Upon receiving the (composite) signal from the source,
the ith relay applies a complex gain r(i) ∈ C. Let r = [r(i), · · · , r(L)]T represent the gain vector of the relay net-work, where superscript T stands for matrix transpose. As-sume that no direct link exists between the source and the destination. The end-to-end transmission behavior can be written as
y = F RGHx + F RnR+ nD (1) where R = diag(r) with diag(r) denoting a diagonal matrix formed of the elements of vector r.
In any practical design, the transmission powers of the source terminal and the relays are limited. Therefore, we assume that the source transmits independent streams over its M antennas with equal power σx2. On the other hand, assume that the relay network is subject to a total power limit PR. Hence we have
where· denotes the 2-norm of a vector and girepresents the ith column of G, which is a vector of the complex conjugates of the channel coefficients between the source antennas and theith relay.
A. System Capacity
The noise vector F RnR + nD in (1) received at the destination is in general spatially correlated. To find the system capacity, consider using a noise whitening filter W−1/2at the destination, where W is the autocorrelation matrix of the noise given by
W = E{(F RnR+ nD)(F RnR+ nD)H}
= σD2IM+σ2R(F R)(F R)H. (2) Let H F RGH denote the noise-free equivalent end-to-end channel matrix. The system capacity is then a function of R as [9]
C(R) log det(IM+σx2HHW−1H) (3) where det(·) denotes the matrix determinant and log stands for base-2 logarithm. The optimization problem can be stated as B. Power Scaling and Capacity
The inequality power constraint (5) naturally prompts one to think: is it possible to simplify the constraint by considering only the equality therein without impacting the optimality of the solution? Or, alternatively, given a certain r that satisfies (5) with inequality, will the system capacity be increased by scaling r to reach equality in (5)? Intuitively, the answer may
seem to be a no-brainer as increasing the transmission power should be beneficial to the signal-to-noise ratio (SNR) and thus the capacity. But mathematically, due to the presence of some matrices in (3) and (1), a solid proof nevertheless requires a little work. We give the proof in Appendix A.
(Some intermediate results in the proof will also have use later in system design.) For convenience, we state the result as a theorem.
Theorem 1 (Capacity scaling): When the (complex) relay gains R are scaled by s ∈ C with |s| > 1, C(sR) > C(R).
Therefore, we confirm that scaling up of the relay gains can increase system capacity. Hence we may simplify the optimization constraint to
PR=
L i=1
(σR2 +σ2xgi2)|r(i)|2. (6)
That is, the relays should transmit at the maximum allowed total power.
Next, one may wonder if the capacity could increase without bound if the total relay transmission power tends to infinity.
Intuitively, the answer may appear to be another no-brainer because, from (1), the quality of the source-to-relay links should place a cap on the amount of information rate that the system can support, however much the relay transmission power can be. But again, a solid mathematical proof requires a few lines of reasoning. Again for convenience, we state the result as a theorem below and prove it in Appendix B.
Theorem 2 (Asymptotic capacity with high relay power):
As|s| → ∞, C(sR) is upper-bounded by log det[IM+ (σx
σR)2GGH]
and it approaches the upper bound if and only if G and F R span the same row space.
With Theorem 2, it is verified that C(R) is upper-bounded irrespective of the power level of the relays.
III. RELAYNETWORKDESIGN FORTWOTYPES OFNOISE
DOMINATION
We now consider how to design the relay coefficients for maximization of the system capacity. Directly solving (4) seems difficult because the diagonal nature of matrix R rules out conventional beamforming-based solutions [2]. While algorithms can always be developed to facilitate a solution of the problem via computation, an analytical solution may give more insights to the nature of the optimizing solutions. The latter is the approach taken in this work. For this, note first that the system performance depends on two noise terms, namely, the relay noise vector nR and the destination noise vector nD. The problem becomes much more tractable when one of them dominates. In what follows we concentrate on these two simplified scenarios and attempt at the corresponding solutions of optimal relay gains.
A. Destination Noise-Dominating: Relay Selection and Power Allocation
Hence the capacity is nearly that of anM ×M MIMO system with an end-to-end channel matrix H at SNR = (σx/σD)2. Even in this simplified condition, a simple general solution that maximizes CD(R) for L > M is, to the best of our knowledge, not available. Nor is the efficient technique for maximization of matrix determinant through convex optimiza-tion under linear matrix inequality constraints [11] applicable.
However, we find that an analytical solution can be obtained for L = M under high SNR. To proceed, therefore, we first develop the solution for this condition. The solution can be applied to the conditionL > M to select good relays.
Letρi(HHH) denote the ith largest eigenvalue of HHH.
WithL = M and under a high end-to-end SNR, we have CD(R)≈ log det[(σx
σD)2HHH]
= log[(σx
σD)2det(F FH) det(GGH) det(RRH)]. (8) So we need to find R to maximize det(R) subject to the power constraint (6). Equivalently we may do
maxr
Employing the Lagrange multiplier technique leads to the relay power allocation
|ropt(i)| =
PR
Mσx2gi2. (11) When L > M, we can do relay selection by choosing M relays out of theL such that (8) is maximized.
It is of interest to note that, under the above solution, the product of the last two determinant terms in the right-hand-side (RHS) of (8) is upper-bounded as
det(GGH) det(RRH) = det(GG H)
igi2 ( PR
Mσx2)M
≤ ( PR
Mσx2)M. (12) The last inequality can be shown by considering QR de-composition of G. Let RG denote the triangular matrix in the decomposition. Then det(GGH) =
i|RG(i, i)|2 where RG(i, i) is the ith diagonal element of RG. Apparently,
|RG(i, i)|2 ≤ rGi2 where rGi denotes the ith column of
RG. ButrGi2=gi2 because RG and G are related by a unitary transform. Thus the result. For convenience, define loss factor
B. Relay Noise-Dominating: Relay Selection and Distributed Beamforming
where V ∈ CL×M is the matrix of right singular vectors of F R with its ith column corresponding to the ith largest singular value of F R. Comparing with (30) in Appendix B, we see that the situation is similar to that with very high relay transmission power. Hence by Theorem 2, CR(R) is maximized if the relay setting can be such that F R spans the same row space as G. Note that, contrary to the case where σD σR, in the present case the relay power distribution is not critical to the performance; only the row space of F R matters. In other words, we have a problem of beamforming under a distributed relay network that should try to align the row space of F R with that of G.
To proceed, let fHi denote the ith row of F . Let VN ∈ CL×(L−M) be a matrix of basis vectors for the orthogonal complement of the row space of G; that is, GVHN = 0. And Then r is an optimal solution that maximizes CR(R) if rTΦ = 0. The existence of such a solution would require M(L − M) < L if Φ is of full rank.
Recall that we have assumed L ≥ M. When L = M + 1, the inequality M(L − M) < L holds and the existence of a maximizing solution r is guaranteed. There is more than one way to solve for the optimal r, which we will not elaborate here. When L > M + 1, the upper bound on CR(R) (as indicated in Theorem 2) is unreachable. But there still exists an optimal r that maximizes CR(R), only that a closed-form solution is not available. A suboptimal design is to consider all possible combinations of M + 1 relays and pick the combination with the best performance. Note that the determination of the best relay combination does not require solution of the optimal r in each case. It can be achieved by comparing the performance upper bounds given in (31). Then the optimal r for the best relay selection can be derived based on the above-discussed methods forL = M + 1.
12 14 16 18 20 22 24 26 28
Fig. 2. MIMO system with distributed relays in destination noise-dominating condition.
IV. NUMERICALRESULTS
To verify and further analyze the performance of the pro-posed technique, we simulate relay-aided MIMO transmission.
The channel matrices F and G are both M × L complex Gaussian matrices with i.i.d. entries distributed according to CN (0, 1). Besides the relay-selection systems proposed in Sec. III, for benchmarking purpose we also consider the all-pass systems as a conceivable example of simple network designs. In addition, a single-hop MIMO system based on antenna selection is also simulated for comparison of the associated diversity orders. We illustrate the performance of different schemes in terms of the cumulative distributions functions (CDFs) of their capacities.
Figs. 2 and 3 show the system performance in the desti-nation noise-dominating condition. In Fig. 2, we see that as the relay noise increases, the capacity drops (i.e., the CDF curve shifts to the left). As expected, the all-pass design results in much worse performance than the proposed design.
Examining the slopes of the CDF curves, we see that the system with distributed relays shows the same diversity order as the single-hop system with antenna selection. Fig. 3 shows the performance under different numbers of relay. We see that the diversity order grows with the size of the relay network, as can also be appreciated by comparing the slopes of the corresponding CDF curves with that for single-hop antenna-selection systems. In both Figs. 2 and 3, there are capacity gaps between relay systems and corresponding antenna-selection systems, which arise due to the loss factorLF.
Now consider the performance under the relay noise-dominating condition. Fig. 4 shows how the performance varies with different relative noise levels. A characteristical difference between the system performance in this condition and that shown in Fig. 2 is the apparent loss of diversity order as the destination noise becomes large. This should be the
16 18 20 22 24 26 28
L= 6 ant selection L= 6 relay L= 9 ant selection L= 9 relay L=12 ant selection L=12 relay
Fig. 3. MIMO system with distributed relays in destination noise-dominating condition: performance with different relay network sizes.
consequence of mismatch between the actual noise condition (that the relay noise is not dominating) and the assumed condition (that the relay noise dominates) under which the solution is obtained. Nevertheless, with sufficiently dominating relay noise (such as when σR/σD= 10 or 100), the diversity performance becomes close to that of an antenna-selection system, which in turn has a similar performance in diversity order as a full MIMO system (the “full 3× 6 MIMO” curve in the plot), only a lower beamforming gain. In addition, the capacity of the all-pass design still falls significantly below that of the proposed design mostly. In Fig. 5, we see that the performance of the proposed design is tightly upper-bounded by the corresponding antenna-selection system (assuming equal number of antennas to select from as there are relays), and the diversity order again grows with the size of the relay network in a similar way to an antenna-selection system.
V. CONCLUSION
We considered the design of distributed relay networks to aid MIMO transmission, where a major point in considering a distributed relay network instead of a single multi-antenna relay was the potential to effect a better diversity performance.
An analytical solution was attempted, though it turned out that such analytical solutions could only be obtained for a limited number of conditions. Nevertheless, these analytical results facilitated suboptimal designs that could exploit the diversity order afforded by the relay network to a similar degree to that of a single-hop MIMO antenna-selection system. The proposed suboptimal designs employed relay selection with optimal setting of relay gains. Simulation results were presented which demonstrated the diversity order performance of the proposed designs and its superior performance compared to the simple all-pass design.
Fig. 4. MIMO system with distributed relays in relay noise-dominating condition.
Fig. 5. MIMO system with distributed relays in relay noise-dominating condition: performance with different relay network sizes.
APPENDIXA PROOF OFTHEOREM1
First, it is clear in (3) that C(sR) = C(|s|R). Without loss of generality we assumes ∈ R+ (the set of positive real
First, it is clear in (3) that C(sR) = C(|s|R). Without loss of generality we assumes ∈ R+ (the set of positive real