Amplify-and-Forward Distributed Beamforming with Local CSI in the Presence of Interferences

1Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA 2Broadcasting & Telecommunications Media Research Laboratory, ETRI, Daejeon 305-700, Republic of Korea 3Department of Information & Communication Engineering, Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 771-873, Republic of Korea 4School of Computer Science, University of Seoul, Seoul 130-743, Republic of Korea 5Department of Computer Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Cooperative relaying has attracted a great deal of attention because of its appealing properties for both performance and various applications.Among various schemes, cooperative beamforming is being widely considered because it achieves optimal diversity-order performance and capacity scaling by maximizing the received signal-to-noise ratio (SNR).An upper bound on capacity scaling of dual-hop relay networks was provided in [1], in which the capacity scaling was achieved using the consequence of receive and transmit matched filtering at each relay in distributed way.However, an optimum design for beamforming weight was not taken into account in [1].An optimal distributed beamforming (DBF) to maximize the received SNR was proposed in [2], which showed that the optimal performance is achieved only with local channel-state information (CSI) obtained at each relay.The results of [2] were extended to two-way relaying in [3]; near optimum joint DBF was introduced with which the maximal capacity scaling and full diversity order were achieved.
The above-mentioned works do not consider the impact of cochannel interference (CCI) that is one of the major limiting factors on the performance of wireless communication systems.Recently, [4] introduced optimal beamforming that maximizes the received signal-to-interference-plusnoise ratio (SINR) when  sources perform DBF toward a relay and the destination is corrupted by CCI.However, the impact of CCI was considered only at the destination.Although there is an abundance of research on cooperative beamforming with a variety of scenarios, the distributed approach based on local CSI considering CCI has not yet been thoroughly investigated.
This paper investigates the optimum DBF based on local CSI when the relays and the destination are affected by CCI.The proposed DBF has very small complexity and overhead compared to the cooperative beamforming obtained with global CSI.More details provided in this paper are summarized as follows: (i) An optimal amplify-and-forward (AF) DBF weight is proposed in the presence of CCI at both the relays and 2 Mathematical Problems in Engineering the destination when only local CSI is available at each relay.
(ii) The proposed DBF is shown to achieve nearly the performance obtained with global CSI when there are a large number of interferers or interference power toward relays is small.
(iii) The DBF has a capacity scaling of (1/2) log  through  relays, where (1/2) log  corresponds to the maximal capacity scaling when there is no CCI.
Numerical results verify that the proposed DBF represents significant improvements over the conventional DBF designed without considering CCI at the cost of slightly increased overhead and complexity.This paper is organized as follows: Section 2 introduces the system model for DBF protocol.Section 3 presents the optimum DBF weight, and its capacity scaling law is derived.Finally, the numerical results are presented in Section 4, and concluding remarks are given in Section 5.

System Model
Figure 1 depicts a wireless network that consists of a source, a destination, and  relays.Let S  = {1, 2, . . ., } be a set of the relays.Each node has a single antenna and the relays operate in half-duplex mode with AF strategy.All the relays and the destination are affected by  interferers.Hereafter, subscripts , , and  denote the source, the th relay, and the destination, respectively, and  is the index of interferers.Because of the long distance between  and , there is no direct link between them.It is assumed that the activities of interferers change slowly, and, therefore, each node is affected by the same interferers during two phases.
Frequency-flat block-fading channels are assumed, where ℎ , denotes the channel coefficient between node  and node  (,  ∈ {, S  , }) and  , is the channel coefficient between the th interferer and receiving node  ( ∈ {S  , }).Channel reciprocity is assumed and each node has the receivers' CSI.The channel coefficients are modelled by independent but not identically distributed (i.n.i.d.) complex Gaussian r.v.'s.During the first phase,  transmits   with power   .The received signal at relay  is corrupted by multiple interfering signals  ,1 's with power   's: where   ∼ N  (0, 1) is complex additive white Gaussian noise (AWGN) at relay .During the second phase, each relay simultaneously retransmits the signal: where   denotes the beamforming weight for relay  to be optimized.When the normalized amplifying gain is considered as the transmission power of   becomes |  | 2 .Aggregate transmit power over all relays is assumed to be constrained by where   is the maximum transmission power available at each relay.The assumption makes the DBF more practical at the network point of view.With the constraint, the total used power remains constant regardless of the number of relays .It is an effective way to constrain the interference to other nodes in the network.Moreover, under the assumption, the transmission power cannot be shared among different nodes, which may not be practical.The received signal at  is given by where where

Distributed Beamforming with CCI Based on Local CSI
Fact 1.When T is positive definite Hermitian, the following modified Rayleigh-Ritz theorem holds for any row vector x [2, Proposition 1]: where is the largest eigenvalue of (T /2 ) −1 (s  s)(T 1/2 ) −1 and the equality holds when x = sT −1 for any nonzero constant .
When there is no limit on available CSI at each relay, that is, global CSI is available, the optimal beamforming weight vector w  that maximizes the received SINR   in ( 5) is given by where The proof is as follows.The received SINR   in (5) becomes where [n  n] = I  .From Fact 1, the optimal vector w  in (8) is obtained, where the value of  = √   / ‖ aV −1 ‖ 2 is chosen to meet the aggregate the power constraint   .
However, using w  is not realistic for DBF.To calculate w  in a distributed way, V should be delivered to each relay, but it requires a significant burden because (1) acquiring V causes very high complexity since all the individual channel coefficients of interference channel  , 's must be estimated and (2) sharing V causes large overhead.Therefore, using w  in DBF is impractical, especially when  or  is large.To mitigate this problem, the following theorem introduces a simple DBF when only local CSI is available at each relay.
Theorem 1.When only local CSI is available at each relay, the optimal beamforming weight vector w  that maximizes the received SINR   is given by where u is From Fact 1, w  is obtained by where Because Ṽ is a diagonal matrix, its inverse is easily obtained from [ Ṽ−1 ] , = 1/[ Ṽ] , , and closed-form w  is obtained as in ( 11) and (12).
Each relay calculates  , in a distributed way with only local CSI ℎ , , ℎ , , and ∑  =1   | , | 2 when (∑  =1   | , | 2 + 1) and ‖u‖ 2 are delivered from the destination (to calculate ‖u‖ 2 with very small overhead, several methods are available such as training-sequence-based channel estimation [5][6][7]).In this sense, w  is called a DBF vector with local CSI.Therefore, w  induces very small overhead.Moreover, calculating w  causes low complexity, because each relay estimates not  , 's but corresponding aggregate interference plus noise power (∑  =1   | , | 2 + 1), which is much easier to estimate [8,9].Nevertheless, w  still shows excellent performance as follows: (1) w  achieves nearly the optimum performance of w  when  is large enough or interference power toward relays is small and (2) w  achieves the capacity scaling of (1/2) log , which corresponds to the maximal capacity scaling of cooperative relaying without CCI.
Corollary 2. When the number of interferers  is sufficiently large, it becomes Ṽ ∼ V, and, therefore, w  achieves the optimum performance of w  .
When interference power toward relays is small, it is obvious that Ṽ ≈ V, and w  closely achieves the performance of w  .Theorem 3. When  → ∞ with any finite   ,   , and   , the ergodic capacity with w  , (w  ) ≜ {(w  )} converges to (1/2) log .
Proof.With w  , the received SINR at  becomes where and () follows from the fact that w  Ṽw   /w  Vw   ∼ 1 for sufficiently large .The ergodic capacity with w  is given by [10]: where the factor 1/2 denotes the rate loss because of the half-duplex constraint of relays.Because   satisfies the Kolmogorov conditions as shown in the Appendix, the following theorem can be applied [11,Theorem 1.8.D]: Therefore, and (w  ) ∼ (1/2) log .

Numerical Results
In this section, (w  ) is compared with (w V ), where w V is the weight vector of a conventional DBF that maximizes the received SNR when there is no CCI [2]: where  Figure 2 plots the ergodic capacity for Cases 1 and 2, and Figure 3 for Case 3, with parameter values of  = 5 and   =   =   = 10 dB.For all cases, the figures show that (w  ) > (w  ) > (w V ) and w  achieves remarkable performance gains over w V ; when  = 40, 21%, 20%, and 29% gains are obtained for Cases 1, 2, and 3, respectively.Moreover, w  closely achieves (w  ) for Case 2 because interference power toward relays is small, but w  is superior to w  for Cases 1 and 3 at the cost of greatly increased overhead and complexity.
As  increases, however, w  closely achieves (w  ) for all cases as shown in Figures 4 and 5, in which the ergodic capacity is plotted for  = 30,   =   = 10 dB, and   = 5 dB.The figures shows that w  achieves nearly (w  ) for all cases and also represents remarkable performance gains over w V , greater than 21% for all cases when  = 40.

Conclusions
This paper has proposed the optimal AF DBF w  in the presence of CCI when only local CSI is available at each relay.With slight increased overhead and complexity, w  efficiently reduces the impact of CCI and yields significant improvements over w V .Using w  is more attractive when interference power toward relays is small or there are a large Proof

Figure 1 :
Figure 1: System model.A source communicates with a destination through  intermediate relays, where the relays and the destination are affected by a number of  interferers.The relays operate in a halfduplex mode with AF strategy.
1 ℎ 1, / 1 Proof.To calculate the weight coefficient   at each relay with only local CSI, V in (9) must be a diagonal matrix, and relay  needs to be able to estimate [V] , without communication between relays.Therefore, B must be replaced byB = diag [  ∑ and (1/) ∑  =1    * ,  , ∼ {0, for  ̸ = ;   , for  = }.Therefore, [A  BA] , =     ℎ * , ℎ , V , w  requires only a slight increase in overhead and complexity in order to estimate ∑  =1   | , | 2 +1 at the corresponding relay and to feed back ∑  =1   | , | 2 + 1 from the destination.It is assumed that the relays are located in the middle of the source and the destination, and, therefore,  2 , =  2 , = 1, for all .For comparison purposes, simulation results for (w  ) are also plotted.According to the location of interferers, three cases are considered as follows.