Sum Rate Maximization Based on Joint Power Control and Beamforming in MIMO One Way AF Relay Networks

Aims: Joint Power control and beamforming in MIMO one way AF relay networks by obtaining beamforming weights on relay nodes through maximizing sum rate while the total power consumed in all relay nodes are not greater than the certain predefined threshold. Study Design: Comparative study by simulation in MATLAB software.


INTRODUCTION
Due to the increasing requests for wireless communication services, we need systems which support higher data rates, capacity and quality in a joint state. Limited allocated frequency spectrum forces the wireless networks to use improved protocols and algorithms supporting higher spectral efficiency to achieve higher bit rate [1]. Efficient use of bandwidth and two inherent abilities, achieving higher transmission data rates and offering higher bit rates are important advantages of multi input multi output (MIMO) communication systems. On the other hand, the efficiency of these systems will be decreased by undesired phenomena such as noise, interference and fading. Therefore, in order to improve the performance of these systems, several techniques such as power control, beamforming and space-time coding have been proposed and investigated. The most fundamental problem in a large wireless network which shares a limited frequency band is cochannel interference (CCI) and it is inevitable due to frequency reuse [2]. Also, it can reduce system performance. Therefore, some research works are focused on the methods that reduce CCI. Generally, beamforming and power control are used jointly in CCI wireless systems to increase system performance.
However, implementing multiple transmit antennas in mobile terminals may not be always feasible due to power and space limitations as well as computational complexity. One approach to tackle these practical restrictions is to use relays [3] as virtual antennas helping communication between transmitter and receiver. The main idea of these networks is to allow different nodes to be involved in transmission by making virtual array antennas. This method, avoids the problems related to implementing antenna arrays such as, physical limitations, mutual coupling and so on. Moreover, it makes the possibility of taking diversity and other benefits of MIMO systems and also applying ideas introduced for MIMO systems, beamforming and power control.
It should be mentioned that for long distance communication between transmitter and receiver, the direct path experiences large attenuation which greatly reduces the communication quality. Hence, using relay based networks, transceiver nodes can connect to each other in two time steps. In the first step, transmitter sends the signal to relay and relay processes the received signal. This process in relay node depends on the type of relay. Afterwards, in the second step, the relay sends the processed signal to the receiver.
Relays can be classified based on the type of signal processing operation. For this purpose, several schemes have been proposed. Some of them are amplify-and-forward (AF), decode-andforward (DF), compute-and-forward (CF) and estimate-and-forward (EF). Among these schemes, the most effective and popular one is AF. AF relay applies linear signal processing to the signal and forwards it to the receiver. AF relays are called non regenerative because decoding is not performed on them. Although they have no effect on the decreasing of error in system [4], AF relay offers less complexity and higher processing speed with respect to other schemes.
Several algorithms are introduced and evaluated in joint beamforming and power control for MIMO relay networks. In [5], joint power control and receiver beamforming in MIMO relay networks are used to reduce CCI. In [6], in order to improve the performance of system, power control and beamforming at both transmitter and receiver are implemented. In usual wireless communications systems just a single source and a single destination communicate with each other through multi relays [7][8][9]. In the most of investigations, due to the high attenuation, the direct path between the source and the destination is ignored. In contrast, in some studies such as [3], the direct path is also considered but this path is the dominant one. The proposed method in [7] cannot be used when the number of pairs of source -destination is more than one. In [6,10] the consumed power is minimized while the required signal to interference plus noise ratio (SINR) of each path is guaranteed. They used semidefinite relaxation technique to turn the problem into a semidefinite programming (SDP) problem which can be solved by using interior point methods. In [10], pair of source -destination and multiple relays which all nodes have just a single antenna are considered. In contrast, a system has been studied in [6,11] which includes pair of sourcedestination and multiple relays that all of them are equipped with multiple antennas. For a network with pair of transmitter -receiver with multiple relays considering various receiver filters, the beamforming of the transmitter and relays are optimized in [12]. Ref. [13] proposes the optimal joint source and relay power allocation to maximize the end-to-end achievable sum rate. Transmit pre-coders, relay matrices and receive decoders are optimized in [14] to maximize the achievable sum rate.
In this paper, a network consisting of multi pairs of transmitter -receiver that communicate with each other using multi non regenerative relays is considered. In this investigation, an algorithm is suggested which finds an optimal beamforming of relays and maximizes sum rate subject to relay power constraint.
The rest of the paper is organized as follows. In the Section 2, system model is described. Then, we formulate the end-to-end sum-rate maximization problem and present our proposed approach. Simulation results for the different modes of the system are carried out to evaluate the performance of the proposed algorithm in Section 3. Finally, Section 4 presents conclusions and introduces some suggestions for further works.

SYSTEM MODEL
The system consists of two pairs of sourcedestination with R relays. It is assumed that there is no direct path between the source and the destination as shown in Fig. 1. Each relay transmits an amplitude-and phase-adjusted version of its received signal. In other words, relay multiplies its received signal by a complex weight and then transmits it to the receiver.
Assuming that the coefficient matrix of the channel between pth source and rth relay and also between rth relay and pth destination are denoted by f rp and g rp , respectively, the received signal at rth relay can be obtained by (1) [10]. v where s p is the information symbol transmitted by the pth source and v r is the additive zero mean noise at the rth relay node.
Using the following assumptions throughout this section, (1) can be rewritten in the form of (2).
The rth relay multiplies its received signal by a complex weight coefficient * r w . Thus, signals transmitted by all relays can be expressed as (3).

Fig. 1. A system consists of R relays and 2 source-destination pairs
and t are R × 1 vectors whose rth entry is the signal transmitted by the rth relay. Denoting the vector of the channel coefficients from the relays to the kth , the received signal at kth destination can be written as: where k n is the zero-mean noise at the kth destination with a variance of 2 n σ . Also, it is assumed that the channel coefficients between sources and relays, the channel coefficients between relays and destinations, the source signals, the relay noise and the destination noises are jointly independent.

Sum Rate Maximization Considering Consumed Power Constraint
In general, there are two ways to control relays' power. Let max r , x P be the maximum transmit power at rth relay and max x P be the maximum sum transmit power at all the relays. Considering power control, the individual relay power constraints are [11]: where the sum of relays' powers constraint is: Our goal is to achieve the weight of optimum beamforming { } R 1 r * r w = , in such a way that sum rate became maximized while the sum of consumed powers in all relay should not be greater than the certain predefined threshold. Thus, the optimization problem can be written as: Now, the total consumed power can be rewritten of the form (9): is the correlation matrix of the received signal at the relay, ,..., , ( Using (2) and assumptions 1-4, Eq. (10) is given:  (11) Note that the total relay transmit power depends not only on the variances of the source-relay channel coefficients but also on the relay noise powers [10]. According to the variables that we described in the previous equations, for the system shown in Fig. 1, sum rate can be achieved from (12) Due to transferring data in two periods of time, 1/2 coefficient is appeared in (12). According to theorem 1 expressed in [14], then: is the destination beamforming.
In our investigation, destinations have singleantenna, so 1 = k B and the condition of theorem 1 in [14] is satisfied.
TSTLR is the sum power of desired signals to total leakage power ratio, i.e., The kth receiver noise power can be rewritten as: [ ] The kth desired signal power is given by: ,..., , It is obvious that k h contains the total path gains from the kth source to its corresponding destination via R relays. Also, with using (4) and denoting , the interference power is given by [10]: By using (16), (17) and (19), we can rewrite (14) as (21) is not a convex optimization problem and may not have a low-complexity solution. Hence, it makes a difficult problem. To fix this problem, there are three ways, minimizing the denominator, maximizing the numerator, and maximizing the difference between the numerator and the denominator.
We exploit a semi-definite relaxation approach to solve a relaxed version of (23). To do so, let us define The first approach is introduced in [11], namely TL. In simulations, TL method is called algorithm 1 and the two other proposed methods are named algorithm 2 and 3, respectively. As the next problem, the rank constraint in (24) is not convex. Using semi-definite relaxation technique, we remove this non-convex constraint and solve the following optimization problem: The optimization problem (25) is indeed convex because all constraints are in the form of linear and SDR. So, it can be efficiently solved using interior point based software tools such as the convex optimization (cvx) MATLAB toolbox [15] which produces a feasibility certificate if the problem is feasible. cvx can solve standard problems such as linear programs (LPs), quadratic programs (QPs), second-order cone programs (SOCPs), and semidefinite programs (SDPs); but compared to directly using a solver for one or these types of problems, cvx can greatly simplify the task of specifying the problem.
In SDP mode, cvx applies a matrix interpretation to the inequality operator, so that linear matrix inequalities (LMIs) and SDPs may be expressed in a more natural form.
By solving the optimization problem (25), the matrix opt X , is not necessarily of rank one. It means that the minimum value of the relaxed problem (25) only provides a lower bound on the minimum value of the original problem (24). Proof of this is available in [10]. As it is shown in [16], we can always find a rank-one solution to the relaxed problem (25) as long as 3 ≤ d . Otherwise, one might resort randomization techniques to obtain a suboptimal rank-one solution. In these techniques, the optimal matrix opt X is used to generate several suboptimal weight vectors, from which the best solution will be selected [17][18][19].

RESULTS AND DISCUSSION
In this section, the impact of various factors on the performance of the algorithms is shown. In all simulations, it is assumed that maximum sum power consumption at relays is equal to unit ( 1 max = x P ), the channel coefficients f and g are generated as identically independent distribution (i.i.d) complex Gaussian random variables with variances 2 f σ and 2 g σ , respectively. It is also assumed that all transmitters have equal unit powers. All simulation results are averaged over 500 independent channel realizations. All MATLAB simulation codes are run on a PC with RAM=4GB, Processor: Intel (R) Core ™ i5-2400 CPU @ 3.10GHz, System Type: 64bit.

Comparison of Sum Rate Maximization Methods
In the first experiment, three methods to maximize TSTLR criterion are compared in the case of the two source-destination pairs (d=2), 20 relays (R=20) and dB 10 . It can be seen in Fig. 2 that the third method offers the highest sum rate with respect to two others.
Also, the required simulation time to run the above mentioned algorithms are obtained. It is assumed that R=20, d=2, dB 10 Algorithm 3 is considered as the reference one. The required times for algorithms 1 and 2 with respect to the third one are 1.112 and 1.003, respectively. In other words, the complexity of the proposed algorithm (algorithm 3) is lower than the two other methods (algorithm 2 and TL). Therefore, in subsequent experiments the third method is examined. . It can be seen that increasing the number of relays is the reason for increasing the achievable sum rate. It is obvious that higher number of relays offers additional higher diversity. It should be noted that increasing the number of relays also introduces more computational complexity. Another important point is that although we achieve higher sum rate for a specific noise variance using higher number of relays, the sum rate difference for the same difference number of relays are not the same. For example, in dB 5

The Effect of the Number of Relays in the Performance of the Proposed Algorithm
, the difference between sum rate for R=10 and R= 20 is 0.775 (b/s/Hz), for R=20 and R=30 is 0.503 (b/s/Hz) and for R=30 and R=40 is 0.404 (b/s/Hz). It is due to this fact that the main goal of beamforming is creating vectors with same phases (coherent vectors) at receiver. Since the relays' powers are limited, increasing the number of vectors will increase the sum rate. Moreover, by comparing cases containing 10 and 20 relays with each other, we find that although the number of vectors increases, the length of each vector decreases. While the number of relays increases, we tend close to a saturation case because relay's power is fixed and the total length of the vectors cannot be more than a certain amount.
Therefore, it is important to make a tradeoff between the performance and complexity based on the system requirements and the available resources.

The Effect of Channel Variance
In two following experiments, the effect of the quality of uplink and downlink channels is examined. According to [1] by increasing channel variance or equivalently the quality of channel, average received signal power is increased. In other hand, a larger variance of channel coefficients indicates a better channel or equally amplification. A careful inspection of Figs. 4 and 5 reveals that the effect of channel variance of either hop is not homogeneous in general, but the results clearly demonstrate that the proposed algorithm performs better as the channel quality improves.

Different uplink qualities
In Fig. 4, average achievable sum rate is plotted versus 2 n σ where R=20, d=2 and ଶ sum rate will be increased.

Different downlink qualities
In the case of R=20, d=2 and dB 10 2 = f σ for different values of 2 g σ , the average achievable sum rate versus 2 n σ is plotted in Fig. 5. It can be seen that increasing the quality of downlink causes increasing the achievable sum rate.

The Effect of Transmitter-receiver Pairs
In this experiment the effect of the number of source-destination pairs on the performance of the algorithm is investigated. In other words, we study the effect of channel interferences on the proposed algorithm. By increasing the number of source-destination pairs the interfering signal received at each destination node is also increased. The performance of the algorithm for different d is illustrated in Fig. 6 for R=20, dB 10 Fig. 6 shows that the higher the number of pairs offers the higher sum rate for certain 2 n σ . As can be seen from this Figure, for lower 2 n σ , the effect of the number of pairs is more evident and when the channel became noise dominant the effect of the transmitterreceiver pairs in increasing the sum rate will be decreased. In other words, MIMO channel tends to SISO channel.

CONCLUSION
A network which consists of d transmitterreceiver pairs and R relay nodes is considered. The problem of multiple peer-to-peer communication is studied by considering AF scheme. In our approach, first, the sources transmit their information symbols to the relay network. Then, each relay transmits an amplitude-and phase-adjusted version of its received signal. The amplitude and phase adjustments are performed by multiplication of the relays received signal by a complex weight. The optimal weight vector plays the roles of receive and transmit beamformer at the same time. This vector is obtained through the maximizing of sum rate under constraints of total transmit relays' power. Semidefinite relaxation was used to convert this optimization problem into a semidefinite programming problem and solved the resulting convex problem efficiently using interior point methods.
Two new algorithms have been proposed to maximize total signal to total leakage ratio (TSTLR). Simulation results show that lowcomplex algorithm maximizes sum rate while the total consumed power in all relay nodes is not greater the certain predefined threshold. Also, when the number of relays increases the achievable sum rate increases. It was also shown that when the quality of uplink and downlink channels improves, higher sum rate can be achieved. Moreover, by increasing the number of source -destination pairs the sum rate increases, too. At last, it was shown that when uncertainty of channels increases the achievable sum rate decreases.
As we mentioned before, for is not necessarily of rank one. So, one might resort randomization techniques to obtain a suboptimal rank-one solution. In these techniques, the optimal matrix opt X is used to generate several suboptimal weight vectors, from which the best solution will be selected.
In this paper we assume that the second order statistics of the channel coefficients (rather than their instantaneous values) are available. Robust designs should be considered in the case of imperfect CSI for real communication systems because do not exist unlimited feedback and/or accurate channel estimation.

FUTURE WORK
In this paper, we assumed that all nodes have a single antenna. As the next work, we can consider the case that the nodes have multiple antennas. Another suggestion is to add individual relay power constraints and finding its effect on