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

Abstract

[ProQuest: [...] denotes non US-ASCII text; see PDF]

This paper introduces an optimum amplify-and-forward (AF) distributed beamforming (DBF) in the presence of cochannel interference (CCI) when only local channel-state information (CSI) is available at each relay. It is shown that the proposed DBF closely achieves the performance obtained with global CSI when interference power toward relays is small or there are a large number of interferers but greatly reduces the complexity and overhead. The proposed DBF provides significant improvements over the conventional DBF designed without considering CCI at the cost of slightly increased complexity and overhead and achieves the capacity scaling of ( 1 / 2 ) log [...] K through K relays, where ( 1 / 2 ) log [...] K corresponds to the maximal capacity scaling when there is no CCI.

Jung-Bin Kim 1,2 and Ji-Woong Choi 3 and Hyuk Choi 4 and John M. Cioffi 1,5

Academic Editor:Aline Roumy

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

Received 13 July 2015; Revised 18 November 2015; Accepted 1 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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-plus-noise ratio (SINR) when [figure omitted; refer to PDF] 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 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 [figure omitted; refer to PDF] through [figure omitted; refer to PDF] relays, where [figure omitted; refer to PDF] 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.

Notations. [figure omitted; refer to PDF] denotes the diagonal square matrix with [figure omitted; refer to PDF] on its main diagonal, [figure omitted; refer to PDF] the complex conjugate, and [figure omitted; refer to PDF] the Hermitian, respectively. [figure omitted; refer to PDF] is the Euclidean norm of the vector [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denotes the [figure omitted; refer to PDF] identity matrix. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] mean the expectation and the variance of a random variable (r.v.). [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes convergence with probability one. For two functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] means that [figure omitted; refer to PDF] , or equivalently [figure omitted; refer to PDF] .

2. System Model

Figure 1 depicts a wireless network that consists of a source, a destination, and [figure omitted; refer to PDF] relays. Let [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] interferers. Hereafter, subscripts [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote the source, the [figure omitted; refer to PDF] th relay, and the destination, respectively, and [figure omitted; refer to PDF] is the index of interferers. Because of the long distance between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , 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.

Figure 1: System model. A source communicates with a destination through [figure omitted; refer to PDF] intermediate relays, where the relays and the destination are affected by a number of [figure omitted; refer to PDF] interferers. The relays operate in a half-duplex mode with AF strategy.

[figure omitted; refer to PDF]

Frequency-flat block-fading channels are assumed, where [figure omitted; refer to PDF] denotes the channel coefficient between node [figure omitted; refer to PDF] and node [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] is the channel coefficient between the [figure omitted; refer to PDF] th interferer and receiving node [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). 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. That is, channel powers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are independent and exponentially distributed r.v.'s whose means are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively.

During the first phase, [figure omitted; refer to PDF] transmits [figure omitted; refer to PDF] with power [figure omitted; refer to PDF] . The received signal at relay [figure omitted; refer to PDF] is corrupted by multiple interfering signals [figure omitted; refer to PDF] 's with power [figure omitted; refer to PDF] 's: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is complex additive white Gaussian noise (AWGN) at relay [figure omitted; refer to PDF] . During the second phase, each relay simultaneously retransmits the signal: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the beamforming weight for relay [figure omitted; refer to PDF] to be optimized. When the normalized amplifying gain is considered as [figure omitted; refer to PDF] the transmission power of [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] .

Aggregate transmit power over all relays is assumed to be constrained by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is complex AWGN and [figure omitted; refer to PDF] 's are the interfering signals during the second phase with powers [figure omitted; refer to PDF] 's. It is assumed that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Using a beamforming weight vector [figure omitted; refer to PDF] , the SINR of the received signal at [figure omitted; refer to PDF] is represented by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

3. Distributed Beamforming with CCI Based on Local CSI

Fact 1.

When [figure omitted; refer to PDF] is positive definite Hermitian, the following modified Rayleigh-Ritz theorem holds for any row vector [figure omitted; refer to PDF] [2, Proposition 1]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvalue of [figure omitted; refer to PDF] and the equality holds when [figure omitted; refer to PDF] for any nonzero constant [figure omitted; refer to PDF] .

When there is no limit on available CSI at each relay, that is, global CSI is available, the optimal beamforming weight vector [figure omitted; refer to PDF] that maximizes the received SINR [figure omitted; refer to PDF] in (5) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The proof is as follows. The received SINR [figure omitted; refer to PDF] in (5) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . From Fact 1, the optimal vector [figure omitted; refer to PDF] in (8) is obtained, where the value of [figure omitted; refer to PDF] is chosen to meet the aggregate the power constraint [figure omitted; refer to PDF] .

However, using [figure omitted; refer to PDF] is not realistic for DBF. To calculate [figure omitted; refer to PDF] in a distributed way, [figure omitted; refer to PDF] should be delivered to each relay, but it requires a significant burden because (1) acquiring [figure omitted; refer to PDF] causes very high complexity since all the individual channel coefficients of interference channel [figure omitted; refer to PDF] 's must be estimated and (2) sharing [figure omitted; refer to PDF] causes large overhead. Therefore, using [figure omitted; refer to PDF] in DBF is impractical, especially when [figure omitted; refer to PDF] or [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] that maximizes the received SINR [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

Proof.

To calculate the weight coefficient [figure omitted; refer to PDF] at each relay with only local CSI, [figure omitted; refer to PDF] in (9) must be a diagonal matrix, and relay [figure omitted; refer to PDF] needs to be able to estimate [figure omitted; refer to PDF] without communication between relays. Therefore, [figure omitted; refer to PDF] must be replaced by [figure omitted; refer to PDF] From Fact 1, [figure omitted; refer to PDF] is obtained by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Because [figure omitted; refer to PDF] is a diagonal matrix, its inverse is easily obtained from [figure omitted; refer to PDF] , and closed-form [figure omitted; refer to PDF] is obtained as in (11) and (12).

Each relay calculates [figure omitted; refer to PDF] in a distributed way with only local CSI [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are delivered from the destination (to calculate [figure omitted; refer to PDF] with very small overhead, several methods are available such as training-sequence-based channel estimation [5-7]). In this sense, [figure omitted; refer to PDF] is called a DBF vector with local CSI. Therefore, [figure omitted; refer to PDF] induces very small overhead. Moreover, calculating [figure omitted; refer to PDF] causes low complexity, because each relay estimates not [figure omitted; refer to PDF] 's but corresponding aggregate interference plus noise power ( [figure omitted; refer to PDF] ), which is much easier to estimate [8, 9]. Nevertheless, [figure omitted; refer to PDF] still shows excellent performance as follows: (1) [figure omitted; refer to PDF] achieves nearly the optimum performance of [figure omitted; refer to PDF] when [figure omitted; refer to PDF] is large enough or interference power toward relays is small and (2) [figure omitted; refer to PDF] achieves the capacity scaling of [figure omitted; refer to PDF] , which corresponds to the maximal capacity scaling of cooperative relaying without CCI.

Corollary 2.

When the number of interferers [figure omitted; refer to PDF] is sufficiently large, it becomes [figure omitted; refer to PDF] , and, therefore, [figure omitted; refer to PDF] achieves the optimum performance of [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] is limited and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

When interference power toward relays is small, it is obvious that [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] closely achieves the performance of [figure omitted; refer to PDF] .

Theorem 3.

When [figure omitted; refer to PDF] with any finite [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the ergodic capacity with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] converges to [figure omitted; refer to PDF] .

Proof.

With [figure omitted; refer to PDF] , the received SINR at [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and ( [figure omitted; refer to PDF] ) follows from the fact that [figure omitted; refer to PDF] for sufficiently large [figure omitted; refer to PDF] . The ergodic capacity with [figure omitted; refer to PDF] is given by [10]: [figure omitted; refer to PDF] where the factor [figure omitted; refer to PDF] denotes the rate loss because of the half-duplex constraint of relays. Because [figure omitted; refer to PDF] satisfies the Kolmogorov conditions as shown in the Appendix, the following theorem can be applied [11, Theorem [figure omitted; refer to PDF] ]: [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

4. Numerical Results

In this section, [figure omitted; refer to PDF] is compared with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the weight vector of a conventional DBF that maximizes the received SNR when there is no CCI [2]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Comparing with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] requires only a slight increase in overhead and complexity in order to estimate [figure omitted; refer to PDF] at the corresponding relay and to feed back [figure omitted; refer to PDF] from the destination. It is assumed that the relays are located in the middle of the source and the destination, and, therefore, [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] . For comparison purposes, simulation results for [figure omitted; refer to PDF] are also plotted. According to the location of interferers, three cases are considered as follows.

Case 1.

The distances between relays-interferers and destination-interferers are the same, and, therefore, the relays and the destination are affected by the same average interfering power with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Case 2.

The interferers are closely located to the destination with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Case 3.

The interferers are closely located to the relays with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Figure 2 plots the ergodic capacity for Cases 1 and 2, and Figure 3 for Case 3, with parameter values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] dB. For all cases, the figures show that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] achieves remarkable performance gains over [figure omitted; refer to PDF] ; when [figure omitted; refer to PDF] , 21%, 20%, and 29% gains are obtained for Cases 1, 2, and 3, respectively. Moreover, [figure omitted; refer to PDF] closely achieves [figure omitted; refer to PDF] for Case 2 because interference power toward relays is small, but [figure omitted; refer to PDF] is superior to [figure omitted; refer to PDF] for Cases 1 and 3 at the cost of greatly increased overhead and complexity.

Figure 2: Comparison of ergodic capacity: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] dB, and [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] : Case 1 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and Case 2 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 3: Comparison of ergodic capacity: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] dB, and [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] : Case 3 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

As [figure omitted; refer to PDF] increases, however, [figure omitted; refer to PDF] closely achieves [figure omitted; refer to PDF] for all cases as shown in Figures 4 and 5, in which the ergodic capacity is plotted for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] dB, and [figure omitted; refer to PDF] dB. The figures shows that [figure omitted; refer to PDF] achieves nearly [figure omitted; refer to PDF] for all cases and also represents remarkable performance gains over [figure omitted; refer to PDF] , greater than 21% for all cases when [figure omitted; refer to PDF] .

Figure 4: Comparison of ergodic capacity: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] dB, [figure omitted; refer to PDF] dB, and [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] : Case 1 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and Case 2 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 5: Comparison of ergodic capacity: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] dB, [figure omitted; refer to PDF] dB, and [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] : Case 3 with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

5. Conclusions

This paper has proposed the optimal AF DBF [figure omitted; refer to PDF] in the presence of CCI when only local CSI is available at each relay. With slight increased overhead and complexity, [figure omitted; refer to PDF] efficiently reduces the impact of CCI and yields significant improvements over [figure omitted; refer to PDF] . Using [figure omitted; refer to PDF] is more attractive when interference power toward relays is small or there are a large number of interferers where [figure omitted; refer to PDF] achieves nearly the same performance as [figure omitted; refer to PDF] .

Acknowledgments

This work was partly supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2015R1A2A2A01008218), Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (no. B0101-15-0557, Resilient Cyber-Physical Systems Research), and the Robot Industry Fusion Core Technology Development Project of the Ministry of Trade, Industry & Energy of Korea (10052980).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Appendix

Appendix

Lemma A.1.

For any finite [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with large [figure omitted; refer to PDF] , [figure omitted; refer to PDF] in (18) satisfies the Kolmogorov conditions: [figure omitted; refer to PDF]

Proof.

Let [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] . [figure omitted; refer to PDF] 's are exponentially distributed r.v.'s which mean [figure omitted; refer to PDF] and variance [figure omitted; refer to PDF] are bounded. Therefore, the Kolmogorov conditions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are satisfied. Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] also satisfies the Kolmogorov conditions.