 Research
 Open Access
Joint iterative beamforming and power adaptation for MIMO ad hoc networks
 Engin Zeydan^{1}Email author,
 Didem Kivanc^{2},
 Ufuk Tureli^{2} and
 Cristina Comaniciu^{1}
https://doi.org/10.1186/16871499201179
© Zeydan et al; licensee Springer. 2011
 Received: 30 November 2010
 Accepted: 26 August 2011
 Published: 26 August 2011
Abstract
In this paper, we present distributed cooperative and regretmatchingbased learning schemes for joint transmit power and beamforming selection for multiple antenna wireless ad hoc networks operating in a multiuser interference environment. Under the total network power minimization criterion, a joint iterative approach is proposed to reduce the mutual interference at each node while ensuring a constant received signaltointerference and noise ratio at each receiver. In cooperative and regretmatchingbased power minimization algorithms, transmit beamformers are selected from a predefined codebook to minimize the total power. By selecting transmit beamformers judiciously and performing power adaptation, the cooperative algorithm is shown to converge to a pure strategy Nash equilibrium with high probability in the interference impaired network. The proposed cooperative and regretmatchingbased distributed algorithms are also compared with centralized solutions through simulation results.
Keywords
 MIMO
 ad hoc networks
 game theory
 beamforming
1 Introduction
Multipleinput multipleoutput (MIMO) communication techniques have been shown to boost the capacity and spectral efficiency of wireless communication systems [1, 2]. MIMO wireless systems can sustain more simultaneous transmissions in a reduced area through interference management [3]. When transmission parameters such as transmit power, beamformer selection, frequency, modulation, transmission rate are modified to adapt to the interference environment, MIMO systems gain an additional advantage. Adaptive wireless systems can achieve system efficiency, lower computational complexity, and overhead compared to a centralized system.
Transmit beamforming has been the focus of extensive research in the literature [4–11] and designing optimum signaling at the transmitter can lead to significant improvements for systems operating in varying interference [4, 6, 12–16]. In spatial transmit beamforming, each communicating node's symbol stream is multiplied by a preselected transmit beamforming weight vector for transmission through multiple antennas such that the overall interference due to other multiple nodes is minimized. Adaptive optimizing of transmitter beamforming improves efficiency by steering the beam toward the intended receiver, while placing nulls toward the unintended receivers in order to avoid causing excessive interference to them. Transmitters may adapt their signals using a lowrate feedback from the receiver [17]. A power control mechanism can also be combined with limited rate feedback from the receiver in order to satisfy certain QualityofService (QoS) requirements at the receiver [18–20].
In general, MIMO beamforming techniques in communication systems are addressed in three different systems: pointtopoint, cellular, and ad hoc networks. The great potential of MIMO in pointtopoint communication is shown in [1, 4, 6, 21] and linear precoders (eigencoders) and beamformers have been designed for pointtopoint MIMO links in [5, 7]. In cellular networks, beamforming algorithms minimize the total power and enhance capacity for arrayequipped base stations and single antenna mobile transmitters [8–11]. In ad hoc networks, without a central controller, distributed beamforming techniques increase system throughput and lower energy consumption [12, 22–24]. However, optimization solutions designed for ad hoc networks need careful study, because the environment is interference limited and the performance of MIMO techniques depends significantly on the overhead introduced by the proposed algorithms.
Distributed spatial beamforming algorithms are proposed for multiuser ad hoc MIMO networks in [23, 24] under channel reciprocity conditions. Channel reciprocity holds when the channel matrix at the receiver is the transpose of the channel matrix at the transmitter, this is usually assumed in timedivision duplex (TDD) systems [24]. Bromberg [24] consider the capacity maximization problem and propose a locally enabled global optimization (LEGO) algorithm for distributed beamforming update under Gaussian otheruser interference. Iltis et al. [23] formulate the problem as a noncooperative game for overall power minimization of the network under a constant QoS constraint (i.e., target signaltointerference plus noise ratio (SINR)). The proposed iterative minimum meansquare error (IMMSE) algorithm solves an optimization problem by computing transmit/receive beamformer pairs and transmit powers in a distributed manner [23]. In the IMMSE algorithm, the receive beamformer is the conjugate of the transmit beamformer and the algorithm relies on the channel reciprocity condition. Hence, the IMMSE algorithm does not demand explicit feedback schemes for channel state information (CSI) at the receiver. However, during the updating procedure of the IMMSE algorithm, transmission overhead of training sequences and power control commands are incurred. The amount of overhead increases with iterations, since the algorithm performs transmit/receive beamformer and power updates iteratively. Moreover, if the transmitter and receiver use different channels or frequencies for transmission and reception, i.e., when channel reciprocity is not valid, CSI must be fed back to the transmitter, which necessitates overhead.
In order to lower the communication overhead between transmitter and receiver when channel reciprocity does not hold, a scheme to limit feedback by quantizing the transmit beamformer in single user MIMO systems is proposed in [21]. The concept is based on selecting a codeword in a predetermined codebook that is known to both transmitter and receiver. Selecting the transmit beamformer from a predefined codebook reduces overhead in nonreciprocal channels. Moreover, latency is reduced in highly mobile and unstable communication networks and user participation is minimized. In this scenario, the receiver only feeds back the index of the selected transmit beamformer to the transmitter. When there is no channel reciprocity between transmitters and receivers, an iterative limited feedback beamforming algorithm using a predetermined codebook is proposed in [25]. The algorithm maximizes the transmission rate in MIMO multiuser ad hoc networks using sequential discrete transmit beamformer selection updates. In each iteration, each node formulates its best response strategy, which maximizes the received SINR. However, the convergence of the algorithm has not been investigated.
Game theory has enabled efficiency and convergence proofs of some of the important problems in wireless communications such as distributed power control algorithm design [26], joint codedivision multiple access (CDMA) waveform, and power control design [19, 20, 27] and optimum transmission signaling strategies [28, 29]. The application of game theory to distributed beamforming is problematic [23]. Lacatus and Popescu [20] and Popescu et al. [19] study joint CDMA codeword (or sequence) and power adaptation as a noncooperative game. The problem is formulated as a separable game using noncooperative convex games, with corresponding subgames: power control and codeword control game. However, in contrast to our joint optimization problem, the joint optimization of powers and CDMA codewords is investigated only over convex games (i.e., the set of action space is nonempty, compact, and convex [26, 30]), and therefore the decision variables (i.e., the powers and codeword sequences) are continuous, not discrete in these games.
Optimum transmit signaling for rate maximization in MIMO interference systems has been studied using game theory [12–16]. In these papers, the system is modeled as a noncooperative game where every MIMO link is a player and computes against the others by choosing the transmit covariance matrix to maximize its own rate. Liang and Dandekar [13] investigate rate maximization for MIMO ad hoc networks by performing power control. The existence of a Nash equilibrium (NE) solution is shown using concave game analysis. The convergence of the proposed algorithms is not studied. Arslan et al. [14] show that individual mutual information maximization is a concave game [31] in MIMO interference channels, which implies the existence of a NE for arbitrary channel matrices. The equilibrium is provably unique when multiuser interference (MUI) is sufficiently small. Decentralized algorithms using local information provide update strategies to determine the link parameters. As an extension of their work and for more general conditions, the uniqueness of the NE solution is provided in [15]. Scutari et al. [15] provide a unified framework for the noncooperative mutual information maximization problem for MIMO interference systems. A unified set of sufficient conditions guaranteeing the uniqueness of the NE and the convergence of asynchronous waterfilling algorithm is provided for square nonsingular channel matrices. The analysis is based on interpreting the MIMO waterfilling operator as a matrix projection onto the convex and closed set of covariance matrices. In [16], same authors extend their results for arbitrary channel matrices. However, these papers do not address the selection of discrete optimized signaling. The existence (or uniqueness) of the NE solution that is proven in [13, 14] is valid either for convex or concave games or for positive definite covariance matrices that are well defined as a convex and closed set [15, 16]. Cooperative and noncooperative algorithms for joint channel and power allocation chosen from the "discrete" strategy space are studied in [32] in the context of wireless mesh networks. However, the proposed noncooperative algorithm is suboptimal and one of the adaptation parameters (i.e., channel adaptation) is not followed after the first iteration.
Power minimization using distributed algorithms with transmit beamformer selection is challenging especially in ad hoc networks. Unlike power control games, in beamforming games there is no natural ordering of the actions [23]. In MIMO ad hoc networks operating in MUI environments, the interference at each user depends on the transmission parameters of the other users. The beamforming decision of each user reshapes the interference emitted to other links, in ways that may be difficult to predict. Changing the beamforming vector may reduce interference on some links, while other links may suffer from higher interference. The affected nodes will then change their own beamforming vectors, setting off an cascade of changes in the network. Moreover, if the node pairs belong to different regulation entities, the noncooperative node pairs may only want to minimize their own transmit power rather than the overall power.
The analysis for the selection of actions from the "discrete" codebook set and convergence analysis is still missing for joint transmit beamforming and power adaptation in the literature. To the best of authors' knowledge, the problem of joint discrete transmit beamforming and power adaptation has not been formalized in multiuser MIMO ad hoc networks. In this paper, we study a decentralized approach for optimizing the transmit beamformer and power levels using local information and reasonable computational burden. We consider total power minimization under a constant received target SINR constraint. Our contributions in this paper are twofold: First, we study an efficient cooperative beamforming algorithm for global power minimization problem with convergence analysis. For the cooperative algorithm, the amount of information to be exchanged between nodes will grow with the number of iterations. Second, we study a noncooperative regretmatching learning algorithm which jointly selects transmit beamformer and power to minimize the total power consumed by the network. The noncooperative solution reduces the amount of overhead by using only local information. We compare the performances of our proposed algorithms with the optimal global solution which is found by exhaustively searching the entire feasible strategy space.
The rest of this paper is organized as follows. Section 2 outlines the system model used in the paper. The optimization problem and its game theoretical interpretation are presented in Section 3. The cooperative wireless ad hoc network and noncooperative counterpart are investigated Sections 4 and 5, respectively. The performance evaluation of the proposed algorithms is provided in Section 6. Finally, Section 7 concludes the paper.
2 System model and concepts
where H_{m,i}denotes the T × T MIMO channel between the i th transmitting node and the m th receiving node and is assumed to be quasistatic and P_{ m } is the power of the m th transmitting node. The additive white Gaussian noise terms ${n}_{m}\in {\u2102}^{T}$ have identical covariance matrices σ^{2}I_{ T } where σ^{2} is the noise power and I_{ T } is the T × T identity matrix. We note that different covariance matrices for noise will not affect the performance of the proposed algorithms. Note that the first term of the righthand side of (1) is the desired signal, whereas the second term is the interference from the other transmitting nodes.
where Θ_{ m } and P_{ m } are the transmit beamformers and powers of nodes other than m.
where ${\widehat{w}}_{m}={R}_{m}^{1}{H}_{m,m}{t}_{m}$. The resulting received SINR at the m th receiving node due to desired transmitter of m th node pair is
${\Gamma}_{m}=\frac{{P}_{m}{w}_{m}^{H}{H}_{m,m}{t}_{m}{}^{2}}{{{\sum}_{i\ne m}P}_{i}{w}_{m}^{H}{H}_{m,i}{t}_{i}{}^{2}+{\sigma}^{2}},$ (4)
where w _{ m }^{2} = t _{ m }^{2} = 1 for all m.
The proposed distributed algorithms attempt to achieve a target SINR by adjusting transmit powers. To construct a distributed iterative limited feedback beamforming scheme, let us first consider the case when there is only one node pair in the wireless network. The receiver selects the transmit beamformer from the codebook Δ_{1} as
$\begin{array}{c}\hfill {t}_{1}^{\u2020}=\mathsf{\text{arg}}\phantom{\rule{2.77695pt}{0ex}}\underset{{t}_{1}\in {\Delta}_{\phantom{\rule{0.3em}{0ex}}1}}{\text{max}}\phantom{\rule{2.77695pt}{0ex}}{\Gamma}_{1},\hfill \end{array}$ (5)
where γ_{0} is the target SINR value.
Consider now the case where N node pairs coexist in the wireless network. Note that for each node pair m, the value of received SINR, i.e., Γ_{ m } , is a function of (Θ, P). Therefore, the transmit power of one node pair depends not only on the transmit beamformer it selects, but also on the transmit power and beamformer selection of other nodes in the network. Furthermore, in beamforming, if user i ≠ m changes its transmit beamformer t_{ i } to increase its own SINR Γ_{ i } , it can either increase or decrease Γ_{ m } , the SINR of link m, depending on the relative positions of the nodes. Therefore, designing an optimal distributed algorithm which converges to a set of beamformers to minimize the overall transmit power while meeting target SINRs for all node pairs is not a straightforward task.
3 Optimization problem and game theoretical interpretation
subject to Γ_{ m } ≥ γ_{0}, w _{ m } = t _{ m } = 1,
P_{min} < P_{ m } ≤ P_{max}, ∀m ∈ {1, 2, ..., N}, where P_{min} and P_{max} are the minimum and maximum transmit powers, respectively. We consider the above problem as a normal form game ∏ which can be mathematically defined by the triplet $\prod =\u27e8\mathcal{N},\mathcal{C},{\left\{{U}_{m}\right\}}_{m=1}^{N}\u27e9$ where $\mathcal{N}=\left\{1,2,\dots ,N\right\}$ is the finite set of players of the game, $\mathcal{C}={C}_{1}\times {C}_{2}\times \cdots \times {C}_{N}$ represents the set of all available actions for all the players and ${\left\{{U}_{m}\right\}}_{m=1}^{N}:\mathcal{C}\to \mathbb{R}$ is the set of utility functions that the players associate with their strategies. The actions c_{ m } ∈ C_{ m } for a player m are the selection of transmit powers P_{ m } ∈ [P_{min}, P_{max}] and the transmit beamformer t _{ m } ∈ Δ_{ m }.
where $\left({c}_{m}^{\prime},{c}_{m}\right)$ refers to the strategy profile in which the action of user m is changed from c_{ m } to ${c}_{m}^{\prime}$ while the actions of all the other players in the game remain the same. In the following sections, we will discuss the scenarios where the node pairs are cooperative and noncooperative respectively in order to search for the best results and provide convergence guarantees.
4 Cooperative and noncooperative beamforming for MIMO ad hoc networks
4.1 Optimal (centralized) solution
where R_{ m } is a function of (Θ _{ m }, P _{ m }) as shown in (2). A naive approach for solving the problem is to investigate all strategy profiles Θ = [t_{1}, ..., t _{ m }, ...,t_{ N } ] ^{ T } exhaustively (note that for a given fixed strategy profile Θ, the corresponding power profile P can be computed using (10) for each individual node pair m). In order to compute (9), the centralized agent evaluates the total network power for ${\Upsilon}^{N}$ possible beamforming vector combinations. For example, for a network size with 10 node pairs where each user has to select from a codebook of size $\Upsilon =16$ beamformers, the search space is 16^{10} strategy profiles. Consequently, finding the centralized transmit beamformer is cumbersome in largescale wireless ad hoc network. To alleviate the complexity problem, while maintaining good performance results, we propose two decentralized power minimization algorithms using cooperative and noncooperative techniques.
4.2 Cooperative power minimization using beamforming
In other words, we model the game as an identical interest game which is a special case of potential games [34]. It is easy to verify that all identical interest games have at least one pure NE, which will represent any action profile that maximizes U_{network}(Θ, P) [14, 32]. We analyze a cooperative power minimization algorithm (COPMA) which can converge to the optimal NE with arbitrarily high probability. This method is analogous to the decentralized negotiation method called adaptive play[14]. The key characteristic of COPMA is the randomness deliberately introduced into the decisionmaking process to avoid reaching a local solution. In COPMA, the choices of players (in our case transmit beamformer selections) lead the system to the optimal NE solution with arbitrarily high probability.
Motivated by Song et al. [32], COPMA can be implemented distributively as follows: Assume that each node pair m in the network has an unique ID_{ m } and maintains two variables ${P}_{m}^{current}$ and ${P}_{m}^{updated}$which are the transmit power of the m th node pair prior to and after the change of transmit beamformer, respectively. The node pairs can be chosen randomly or in a roundrobin order for updating of the transmit beamformers. Whenever a node pair changes its strategy, it broadcasts a vector $\left[I{D}_{m},{P}_{m}^{current},{P}_{m}^{updated}\right]$ via a backbone network. After that, all the other node pairs $i\in \mathcal{N}\backslash m$ will set ${P}_{i}^{current}={P}_{i}^{updated}$, recalculate ${P}_{i}^{updated}$ as the new transmit power and send the vector $\left\{I{D}_{i},{P}_{i}^{current},{P}_{i}^{updated}\right\}$ to the updating node pair m. Finally, the m th node pair will decide whether the new transmit beamformer should be kept or changed with some probability which depends on p_{current} and p_{updated} which are the total transmit power in the network prior to and after the random change of the transmit beamformer, respectively. Note that since p_{current} and p_{updated} are calculated by each node pair independently, the unique ID of each node provides a checklist to accurately add up transmit powers. For this paper, we assume that unique node IDs are built into each node and in network timing synchronization is perfect so that power updates are always received in the correct round. The detailed description of COPMA is provided as follows:
Initialization: For each transmitting and receiving pair m, the initial index of transmit beamformers for all node pairs is selected as one and the initial transmit powers are set as ${P}_{m}={P}_{max},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\forall m\in \mathcal{N}$.
 1.
Set t _{ m }(n) = t _{ m }(n  1), $\forall m\in \mathcal{N}$. Calculate ${P}_{m}^{current}$ as in (10) $\forall m\in \mathcal{N}$.
 2.
To update node pair m, randomly choose a transmit beamformer, ${t}_{m}^{updated}\in {\Delta}_{m}$ and calculate the transmit power required when the updated transmit beamformer is used, ${P}_{m}^{current}$ as in (10). Then, broadcast a data vector $\left[I{D}_{m},{P}_{m}^{current},{P}_{m}^{updated}\right]$ to all other node pairs $i\in \mathcal{N}\backslash m$.
 3.After receiving the data vector, for each i,

If P_{ i } changes (due to change in interference perceived at the i th receiver), every other node pair $i\in \mathcal{N}\backslash m$ sets ${P}_{i}^{current}={P}_{i}^{updated}$ and calculates its new transmit power from (10) and sets it to ${P}_{i}^{updated}$.

If P_{ i } does not change, ${P}_{i}^{current}$ and ${P}_{i}^{updated}$ remain unchanged.

 4.
Node pair m computes the current total network power as ${P}_{current}={\sum}_{m=1}^{N}{P}_{m}^{current}$ and updated total network power as ${P}_{updated}={\sum}_{m=1}^{N}{P}_{m}^{updated}$ with ${t}_{m}^{updated}$ based on the received power values from all other node pairs $i\in \mathcal{N}\backslash m$.
 5.For a smoothing factor τ > 0, set ${t}_{m}\left(n\right)={t}_{m}^{updated}$ for the m th node pair with probability$\frac{1}{1+exp\left(\left({P}_{updated}{P}_{current}\right)\u2215\tau \right)}.$(13)
 6.
The m th node pair broadcasts a notifying signal that contains the decision about whether the new transmit beamformer is kept. If not kept, every other node pair $i\in \mathcal{N}\backslash m$ keeps ${P}_{i}^{updated}={P}_{i}^{current}$
Until : Predefined number of iteration steps n = κ.
Note that step5 of the updating rule implies that if ${t}_{m}^{updated}$ yields a better performance, i.e., (P_{updated} P_{current}) < 0, the m th node pair will change to updated beamformer ${t}_{m}^{updated}$ with high probability. Otherwise, it will keep the current transmit beamformer with high probability. Note also that the tradeoff between COPMA's performance and convergence speed is controlled by the parameter. τ. Large τ represents extensive space search with slow convergence, whereas small τ represents restrained space search with fast convergence. The smoothing factor τ is selected to be a function of the number of iterations n such that as n increases, τ ↓ 0. For example, we chose τ inversely proportional to n^{2} in our simulations. The longterm behavior of COPMA is characterized in the following theorem.
Proof The proof of Theorem 1 follows similar arguments as presented in [14, 32, 35].
where Θ_{ ij } and Θ_{ lp } differ in exactly one transmit beamformer selection, i.e., Θ_{ ij } ≠ Θ_{ lp } for i = l or j = p, τ is the smoothing factor of COPMA and P(Θ _{ ij }) is the minimum total network power required to reach target SINR γ_{0} for both users at state Θ_{ ij } calculated using (10) for each user. If Θ_{ ij } and Θ_{ lp } are different in more than one position, then ℙ_{τ} (Θ _{ lp } Θ _{ ij }) = 0. In addition, ℙ_{τ} (Θ _{ ij } Θ _{ ij }) > 0 is always true. Therefore, for any fixed τ > 0, the induced Markov chain is irreducible and aperiodic.
where ${\Theta}^{\u2020}\in {\Upsilon}^{2}$. The result validates that COPMA converges to the optimal state with arbitrarily high probability for twoplayer (N = 2) case and the analysis can easily be extended for general multiplayer (N > 2) cases as well. ■
With the above theorem, the transmit beamformer and power level selections are shown to reach the optimal NE solution with arbitrarily high probability.
One disadvantage of cooperativebased algorithms is that the communication overhead incurred to calculate the total network power increases with the number of iterations. In the next section, we study a noncooperative learning algorithm using local information with less computations.
5 Regretmatchingbased joint transmit beamformer and power selection game (RMSG)
In this section, our goal is to obtain a distributed learning algorithm for joint transmit beamformer and power selection scheme in MIMO ad hoc networks that requires only local information for updates. We will use a utility function for noncooperative users. Note that the interaction among N "selfish" node pairs can be defined as noncooperative power minimization game where each node pair is attempting to find their own transmit beamformers to minimize their corresponding transmit powers. In the noncooperative joint iterative beamforming and power adaptation, the N node pairs care only about their own power minimizations exclusively, rather than accounting for the overall network power. Each player's utility function depends on the choice of the transmit beamformer and its own power, as well as on the other users' selections for transmit powers and beamformers via the perceived interference. Note that the noncooperative distributed beamforming algorithms for multiuser MIMO ad hoc networks lack the quality of "strategic complementarities" [36] that is found in power controlonly games [26]. It is thus not clear how to design an ordered set of actions for noncooperative beamforming games. Instead, we study a noncooperative learning algorithm called the regretmatching adaptive algorithm from [37], in which the players choose their actions based on their regret for not choosing particular actions in the past. The steadystate solution of the regretmatchingbased learning algorithm exhibits "no regret" and the probability of choosing a strategy is proportional to the player's "regret" for not having chosen other strategies.
At every step k > 1, each user m updates its own average regret vector ${R}_{m}^{{\stackrel{\u0304}{t}}_{m}}\left(k\right)$ for every strategy in ${\stackrel{\u0304}{t}}_{m}$.
RegretMatchingbased joint transmit beamformer and power selection game (RMSG) algorithm
Initialization: For each transmitting and receiving pair m, the initial transmit beamformers are selected with equal probability, the initial transmit powers are p_{ m } = p_{max} and the initial average regret vector is ${R}_{m}^{{\stackrel{\u0304}{t}}_{m}}\left(1\right)=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall m\in \mathcal{N}$. 

Iterations: 
For = 1, 2,..., κ 
For m = 1, 2,..., N 
 Update the average regret vector ${R}_{m}^{{\stackrel{\u0304}{t}}_{m}}\left(k\right)$ using the recursion in (21) 
 Update the probability distribution ${\phi}_{m}^{{\stackrel{\u0304}{t}}_{m}}\left(k\right)$ in (22) and select the transmit beamformer t_{ m }(k) based on updated ${\phi}_{m}^{{\stackrel{\u0304}{t}}_{m}}\left(k\right)$. 
 Calculate the new transmit power p_{ m } based on selected t_{ m }(k) using (10). 
Next m 
Next k 
Every finite strategy game has a mixed strategy Nash equilibrium [30]. Using a good learning algorithm, any finite game can be shown to converge to a mixed strategy Nash equilibrium. Regretmatchingbased selection is distributed and requires limited information exchange between the users if the utility function is properly selected. The timeaveraged behavior of regretmatching game converges almost surely (with probability one) to the set of coarsecorrelated equilibrium [34, 38]. Therefore, the joint transmit beamformer and power selections converges to a mixed strategy equilibrium solution. In fact, in our joint transmit beamformer and power selection game, the average regret of a user using regret matching becomes asymptotically zero, which is confirmed by our simulations.
Note that by using the above utility function, each user selects the transmit beamformer t_{ m } ∈ Δ_{ m } to maximize its own "normalized" SINR, ${t}_{m}^{H}{H}_{m,m}^{H}{R}_{m}^{1}{H}_{m,m}{t}_{m}$. The average regret in the recursion formula (21) can be updated locally as the best transmit beamformer is being selected.
6 Simulation results
In this section, we investigate the performance results of centralized optimization, COPMA, and noncooperative regretmatching (RMSG). We assume that the wireless ad hoc network has N homogeneous pairs where each pair has one transmitter node and one receiving node. Each entry in the channel matrix H_{ m,k } ∀_{ m }, $k\in \mathcal{N}$ is assumed to be independent identically distributed complex Gaussian distribution with zero mean and unit variance. We consider a radio propagation channel with pathloss exponent ν = 4. This implies that the fading power is attenuated by ${d}_{m}^{4}$ where d_{ m } is the distance between transmitter and receiver for m th node pair. The target SINR γ_{0} is selected to be 10 dB. We assume that channels do not vary during the iterations. If channel conditions vary during an iteration, this will change the optimization problem and the proposed algorithms' performance degrades. However, depending on the network configuration and the parameters of the algorithm, like the smoothing factor τ for COPMA, the network optimizer can set the convergence steps to be as small as possible while trading against performance degradation in timevarying channels. The Grassmannian codebook of [21] is used for the simulation results. The codebook size is selected to be $\Upsilon =16$ with T = 3 antennas for all users. P_{max} = 100 mW (20 dBm) and P_{min} = 1 mW (0 dBm) in our simulations. We assume six different transmit power levels: 1, 5, 20, 30, 50, and 100 mW motivated by the IEEE 802.11b standard in [39]. Note that the transmit powers are selected from this discrete power level set which corresponds to ceiling function of (10). The selected network topologies are assumed to be feasible for the given power levels [40]. The noise power is σ^{2} = 3.16 × 10^{13} W (95 dBm) which corresponds to approximate thermal noise power for a bandwidth of 20 MHz.
6.1 Comparison of centralized optimization, COPMA, and RMSG for N= 4 node pairs
RMSG algorithm discussed in Section 5 minimizes the total transmit power in the network defined by (9) using the utility function (23) in a noncooperative manner. Figure 3 also shows how the total power in the network varies over 120 iterations using RMSG. Note that RMSG yields inferior performance compared to COPMA in terms of the achieved overall power. However, the updating procedure is noncooperative and requires less overhead as the iterations continue. The total network power converges to a value of 135 mW on the 68th iteration whereas the centralized algorithm's solution requires 65 mW total network power. Steady state is reached when all the users select a transmit beamformer index with probability one.
Probability mass function (p.m.f): In this subsection, we take a look at the probability mass function (p.m.f) ${\phi}_{m}^{{\stackrel{\u0304}{t}}_{m}}$ of the RMSG algorithm calculated in (22).
6.2 Comparison of COPMA and RMSG for N= 10 node pairs
For the RMSG algorithm, the total power converges to a total network power of 0.2250 W from the 1,296th iteration. The joint selection of transmit beamformer indices and transmit powers reaches steady state when no user in the network deviates from its chosen strategy. The majority of users reach a steady state within 115 iterations. However, one user takes longer than 1,000 iterations to reach steady state.
7 Conclusion
In this paper, we have considered both cooperative and noncooperative joint power control and beamforming in MIMO ad hoc networks using a game theoretic approach. Under constant SINR requirements, the joint transmit beamformer and power selection algorithms were studied in the context of total network power minimization. We first considered a cooperative case where all users collaborate with each other in order to minimize the overall power of the network. The game was formulated as an identical interest game, and a decentralized algorithm COPMA with high probability of convergence was proposed and analyzed. To reduce the required overhead incurred by the cooperative algorithm, we have also proposed a noncooperative solution which requires only local information. For our proposed noncooperative algorithm, users update their probabilities of choosing a transmit beamformer and power based on the "regret" of not choosing the other strategies. Numerical results illustrate the convergence properties of the proposed algorithms and their performance in terms of overall power minimization in the network.
Note
This paper is presented in part at the IEEE Global Communications Conference 2010 (Globecom'10), Miami, FL, December 610, 2010.
Declarations
Authors’ Affiliations
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