Energy-efficient uplink power control for multiuser SIMO systems with imperfect channel state information
© Jang et al.; licensee Springer. 2014
Received: 30 December 2013
Accepted: 1 October 2014
Published: 12 October 2014
This paper addresses energy-efficient design for uplink multiuser SIMO systems with imperfect channel state information (CSI) at the base station (BS). Since the CSI at the BS is always imperfect due to the channel estimation error and delay, the imperfectness of the CSI needs to be considered in practical system design. It causes interuser interference at the zero-forcing (ZF) receiver and makes it difficult to obtain the globally optimal power allocation that maximizes the energy efficiency (EE). Hence, we propose a non-cooperative energy-efficient uplink power control game, where each user selfishly updates its own uplink power. The proposed uplink power control game is shown to admit a unique Nash equilibrium. Furthermore, to improve the efficiency of the Nash equilibrium, we study a new game that utilizes a pricing mechanism. For the new game, the existence of a Nash equilibrium and the convergence of the best response dynamics are studied based on super-modularity theory. Simulation results show that the proposed schemes can significantly improve the EEs of the mobile users in uplink multiuser SIMO systems.
Multiple-input multiple-output (MIMO) has been considered as one of the key technologies for wireless communication systems due to its potential to achieve high spectral efficiency (SE) as well as increased diversity and interference suppression . For this reason, many research on MIMO has focused on increasing the SE [2–6]. On the other hand, the rapid increase in the wireless data traffic has caused dramatic increase in energy consumption of wireless communications, which results in massive greenhouse gas emission and high operation cost . Thus, energy-efficient communication system design is becoming more important in preserving the environment and reducing operation cost. Moreover, energy-efficient design is also important for prolonging the battery life because the development of battery technology has not kept up with the increasing demand on the energy supply for the mobile communications.
For this reason, recent research on MIMO systems has also considered energy-efficient designs [8–19] as well as spectral-efficient designs. For example, in , a single-user MIMO system, where the MIMO channel is converted to parallel independent subchannels through singular value decomposition (SVD) and then the transmit power is allocated across the subchannels to maximize the energy efficiency (EE) of the system, is considered. In , an energy-efficient precoder design is investigated according to the type of fading, i.e., static, fast, and slow fading. In , power allocation and antenna selection are jointly optimized to maximize the EE. In , the uplink of a MIMO system is considered, and a mechanism for the mobile terminals to switch between MIMO and single-input multiple-output (SIMO) to increase their EE is proposed. In , the downlink of a multiuser MIMO system is considered, and the optimal power allocation that maximizes the EE of the base station (BS), which employs zero-forcing (ZF) beamforming, is designed. In , the EE capacity for uplink multiuser MIMO system is defined, and a low-complexity uplink power allocation algorithm that achieves this capacity is proposed. In , the optimal number of mobile users in uplink multiuser MIMO systems and the optimal power allocation that maximize the EE are discussed. In , the energy-efficient link adaptation for uplink coordinated multi-point (CoMP) systems is investigated.
The above research assumes perfect channel state information (CSI) at the transmitters and/or the receivers. However, CSI is always imperfect due to channel estimation error and delay, and therefore, it is important to consider the impact of imperfect CSI for practical wireless communication system design. So far, only a few research has considered imperfect CSI in EE design for MIMO. In , energy-efficient subcarrier and power allocation in the uplink of a multi-carrier interference network are addressed, where only statistical CSI is available at the transmitters. In , bandwidth, transmit power, active transmit/receive antenna number, and active user number are adjusted to improve the system-wise energy efficiency in the downlink multiuser MIMO systems assuming imperfect CSI at the BS.
In this paper, we address energy-efficient power control of uplink multiuser SIMO systems with imperfect CSI at the BS. The imperfect CSI causes interuser interference at the ZF receiver and makes it difficult to obtain the globally optimal power allocation that maximizes the EE. Hence, instead of using a conventional optimization-theoretic approach, we propose a non-cooperative energy-efficient uplink power control game, where each user selfishly updates its own uplink power to maximize its own EE. It is shown that the proposed uplink power control game admits a unique Nash equilibrium. Furthermore, to improve the efficiency of the Nash equilibrium, we study a new game that utilizes a pricing mechanism. For the new game, the existence of a Nash equilibrium and the convergence of the best response dynamics are studied based on super-modularity theory. Simulation results show that the proposed schemes can significantly improve the EEs of the mobile users in uplink multiuser SIMO systems.
The rest of the paper is organized as follows: In Section 2, we describe the system model. In Section 3, we define the EE of each mobile user. In Section 4, a non-cooperative energy-efficient uplink power control game is formulated, the existence and the uniqueness of the Nash equilibrium are discussed, and a pricing mechanism is introduced to improve the efficiency of the Nash equilibrium. The numerical results are reported in Section 5, while the concluding remarks are given in Section 6.
Notations: Superscripts (·) T , (·)∗, and (·) H stand for transpose, complex conjugate, and complex conjugate transpose operations, respectively. Uppercase boldface letters are used to denote matrices, whereas lowercase boldface letters are used to denote vectors. I stands for an identity matrix. ; denotes zero-mean circularly symmetric, complex Gaussian distribution with covariance matrix σ2I. represents expectation. [A] ij signifies the i-th row, j-th column element of matrix A. x ≽ y denotes componentwise inequality between vectors x and y. denotes the projection of a vector x on a subspace .
2 System model
where is the large-scale fading coefficient from the k th user to the BS assumed to be known a priori, is the M × 1 channel vector from the k th user to the BS, x k is the symbol of the k th user, and is the M × 1 zero-mean additive white Gaussian noise (AWGN) vector.
where p k = E[|x k |2] and c k = β k / σ2 are transmission power and the channel-to-noise ratio (CIR) of the k th user, respectively. Then, the instantaneous rate of the k th user is given byar k = log(1 + γ k ).
3 Energy efficiency
To obtain the distribution of γ k , we consider the following properties, which are proved in .
Consider a M × 1 Gaussian random vector and a M × 1 unit norm random vector u (∥u ∥ = 1) which is independent of g. Then, |g T u|2∼Exp(1) where Exp(θ) denotes exponential distribution with mean θ.
Consider a vector space with . Also, define , where , and assume the elements of h are i.i.d complex Gaussian random variables with unit variance. Then, where denotes chi-square distribution with degree of freedom θ.
Since finding a closed form expression of (5) is not easy, we resort to a lower bound of R k obtained in the following theorem.
where s k = (M - K)c k |ρ k |2 and i k = c k (1 - |ρ k |2), for k = 1, …, K.
See Appendix 1. □
Note that the above lower bound assumes the regime where the number of BS antennas exceeds the number of user, i.e., M > K. Many recent researches advocate using sufficiently large number of antennas at the BS to increase EE as well as SE . Using large number of antennas at the BS can reduce the transmit power of the mobile users in the uplink and slow down the battery power consumption. For example, in massive MIMO, the BS employs massive number of antennas, say a hundred or a few hundreds of antennas, to improve the EE of users or BS .
where p = [p1, …, p K ] T denotes the uplink power vector of the users. Finding the optimal p that maximizes the system EE using the conventional optimization theory is difficult because the objective function η is not concave in p. Larger number of users K in the system will result in more local maximums, and searching for the globally optimal power allocation for the users would be a daunting task. Hence, in this paper, we consider a game theoretic approach, where each user finds its own uplink power in a distributed fashionb.
4 Energy-efficient uplink power control based on pricing
where . For more details on the Nash equilibrium, we refer interesting readers to .
The game has a unique Nash equilibrium. The existence and the uniqueness of the Nash equilibrium can be shown by using the results in  because the utility in (8) has the same form as the one used in . Note that  discusses a game in a relay-assisted network where K transmitter-receiver pairs communicate by means of an AF relay. If the direct links between the transmitters and the receivers are ignored, the SINR expression in  has the same form as the SINR in (2). An interesting characteristic in the SINR expressions in (2) and  is that both SINR expressions have a common term in the denominator that depends on the k th user’s signal power (or the k th transmitter’s signal power) p k . While this term is due to the channel estimation error in (2), it is due to the transmit power normalization at the relay in .
Now, we discuss the techniques on how to improve the efficiency of the Nash equlibrium of the game . In the game , each player only aims to maximize its own EE by adjusting its own power, but it ignores the interference it generates to the other players. Thus, the Nash equilibrium of the game may be inefficient in the Pareto sense . We say that a strategy profile p1 is more efficient than another strategy profile p2 if, for all , η k (p1) ≥ η k (p2) and for some , η k (p1) > η k (p2).
is the utility of the k th player that incorporates pricing factor c > 0. Then, we discuss the existence of a Nash equilibrium of the game . Note that is the sum of two quasi-concave functions, which is not necessarily quasi-concave. Here, we resort to super-modularity theory  to show the existence of a Nash equilibrium. The formal definition of a supermodular game is provided below.
is a compact subset of R
is upper semi-continuous in p
For all p -k ≽ p-k′, the quantity is non-decreasing in .
The game in (13) satisfies conditions 1) and 2), but it violates condition 3). Therefore, the game is not a supermodular game. However, if the strategy spaces of players are modified appropriately according to Theorem 2, the resulting game becomes supermodular.
is a supermodular game.
See Appendix 2. □
Now, we propose an algorithm that finds the pricing factor c that improves the system performance of the game in the Pareto sense, which is summarized in Algorithm 1. The same mechanism in  has been used in Algorithm 1; we first obtain the utilities at the Nash equilibrium of the game with no pricing, i.e., c = 0, which is equivalent to playing the game . Then, the game is played again after incrementing the price by a positive vale Δ c. If the utilities at this new equilibrium with some positive price c improve with respect to the previous instance, the pricing factor is incremented and the procedure is repeated. This process is repeated until an increase in c results in utility worse than the previous equilibrium values for at least one player. We declare the last value of c with Pareto improvement to be the best pricing factor, cbest. As will be shown in our simulation, this technique performs very well in improving the efficiency of the Nash equilibrium.
5 Numerical results
In this section, we present the performance of our energy-efficient uplink power control scheme obtained by simulations. The system parameters used for simulation are as follows: system bandwidth W=10 kHz, noise power spectral density N0 = -174 dBm/Hz, noise power σ2 = N0W = -134 dBm, and maximum transmit power of users pmax = 23 dBm and circuit power of users p c = 115.9 mW.
where the utility of each player is the lower bound of the ergodic rate in (6). It is clear that the Nash equilibrium of the game is p∗ = [pmax ⋯ pmax] T because is monotone increasing in p k .
which implies that the energy-efficient uplink power control game can be regarded as a variation of the spectral-efficient game with a pricing. Since the game applies a penalty to a power consumption, the players in tend to use the power in a conservative way. This reduces the interference to the other players and the average rate of the users can be improved.
We have considered energy-efficient transmit power control for uplink multiuser SIMO systems when the BS has imperfect CSI using a game-theoretic approach. The proposed energy-efficient uplink power control game is shown to have at least one Nash equilibrium. Furthermore, the uniqueness of the Nash equilibrium as well as the convergence of the best response dynamics is shown. To improve the efficiency of the Nash equilibrium, we propose a new game that utilizes a pricing mechanism. For the new game, the existence and the convergence of the best response dynamics is also investigated by using the super-modularity theory. Finally, we propose a simple algorithm to find the pricing factor that improves the system performance in the Pareto sense. From the simulation results, we can see that the proposed energy-efficient power control schemes significantly enhance the EE of the users in uplink multiuser SIMO systems.
a In this paper, metrics are computed in units of nats/s to simplify the expressions and analysis, with 1 nats/s = 1.443 bits/s.
b There are some low-complexity optimization methods that can reduce the complexity of the exhaustive search. For example, monotonic optimization  effectively reduces the feasible region when the object function to be maximized is increasing.
Proof of Theorem 1
where and .
Proof of Theorem 2
Since 1) is decreasing in p k > 0 and 2) w(p k ) < 0 is increasing in p k ≥ pmin,k, it is clear that g(p k ) is non-decreasing in p k ≥ pmin,k.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2012202).
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