Joint power allocation for OFDM system with cooperation at the transmitters
© Wang et al; licensee Springer. 2012
Received: 20 July 2011
Accepted: 30 April 2012
Published: 30 April 2012
It is known that traditional water-filling (WF) provides a closed form solution for capacity maximization in orthogonal frequency division multiplex (OFDM) system. The solution is derived from a maximum mutual information argument with a single transmitter. Motivated by the novel technology of cooperative communication, we consider a new power allocation problem for OFDM systems with two cooperative transmitters, where each transmitter has an individual power constraint and can obtain their own perfect channel state information (CSI). The transmitters first cooperate by sharing the CSI, and then jointly optimize power allocation in the metric of sum throughput, which can be modeled as a non-convex constrained optimization problem. Through an application of Karush-Kuhn-Tucker conditions, the problem is reformulated as a convex one. Then, the closed form solution is derived with the nature of traditional WF as well as cooperative properties. Based on the derived solution, an iteration algorithm for joint water level is given for the first time, which can be explained as a cooperative WF relative to the traditional WF. Motivated by the deriving process, we extend parts of the conclusion to N-transmitter case. Numerical results are presented to evaluate the optimal power allocation scheme in OFDM cellular system. For comparison, we also evaluate the traditional non-cooperative WF and equal power allocation scheme.
Transmit power allocation combined with rate adaptation is considered as a powerful method to increase the throughput of wireless networks [1, 2]. In an orthogonal frequency division multiplex (OFDM) system, multiple receivers access a single transmitter through orthogonal subcarriers. Under a transmit power constraint at the transmitter, the traditional water-filling (WF) power allocation scheme has been proved to be optimal in the sense of maximizing the sum throughput . The WF solution is derived for a maximum mutual information problem, which is widely used in OFDM system or any other scenarios that can be modeled as that multiple receivers access single transmitter through orthogonal channel [4–6], . The traditional WF solutions are very simple to evaluate since all of them have a single water level and a power constraint. As a consequence, it is quite straightforward to compute them numerically in practice. In order to find the exact value of the water level, iterative WF algorithm has been proposed in many literature to compute the solutions numerically [7, 8].
Recently, the novel technology of cooperative communications has widely been proposed for wireless networks such as cellular networks and wireless ad hoc networks [9–12]. The essential of cooperative communications lies in that by exchanging information some individual independent transmission links or systems can merge into an equal larger link or system. Then, through jointly designing the transmit/receive structure or optimizing the recourse allocation from a global rather than local perspective, various gain can be obtained over the non-cooperative case. Moreover, in practical system, cooperative beam/resource, control, cooperative transmission, relaying, and cooperative MIMO are drawing attention as a solution to achieve high user throughput at the cell edge (and system throughput) in cellular systems. As discussed in standardizing groups of IMT-advanced, these technologies are expected to be essential in the next generation cellular networks.
Motivated by the concept of cooperative communication, this article considers a cooperative power allocation scheme for OFDM systems with individual independent power constraint at each transmitter. The transmitters with their own perfect channel state information (CSI) available first cooperate by exchanging the CSI, and then jointly optimize the power allocation in the metric of sum throughput (capacity). We first focus on 2-transmitter case, and then extend parts of the derived conclusion to general N-transmitter case.
The main contribution of this article is that we obtain the closed form solution for throughput maximization for 2-transmitter case by solving a non-convex constrained optimization problem. The solution turns out to take the form of traditional WF and also combined with some regular cooperative feature. Based on the derived solution, an optimal joint WF (Jo-WF) algorithm is proposed to get the joint Jo-WF level subsequently for the first time. Motivated by the theoretical derivation of the 2-transmitter case, we also extend parts of the conclusion to arbitrary N-transmitter case. Numerical simulation results verify that the proposed Jo-WF power allocation provides a significant sum throughput gain over the traditional non-cooperative WF and equal power allocation (EPA). It is also concluded that when there is no cooperation between the transmitters, traditional WF is just local optimal, and the EPA is near optimal when the transmission power is high enough. Parts of this study appear in a pattern work .
Although the study is analyzed for OFDM system, it is emphasized that the derived solution can be also applied into any other scenarios that can be modeled as that multiple receivers access multiple transmitters through orthogonal channel in the time, space, or code domain. Considering the flexibility of transmitter's category, e.g., base station or relay station, it is known that the derived Jo-WF power allocation scheme can be valid for any cooperative networks such as next-generation cellular networks or ad hoc networks.
2. System model
In order to focus solely on power allocation, we do not explicitly consider subcarrier scheduling here. However, it is noted that the power allocation results presented in this article are valid for any scheduling strategy, as the effect of one such strategy over another is simply to induce different subcarrier statistics for the selected subcarrier .
where N0 represents the power spectral density of AWGN. P nk is the transmit power allocated from the n th transmitter to the k th subcarrier, P n is the power constraint at the n th transmitter, and h nk is the corresponding subcarrier gain between the n th transmitter and the k th subcarrier.
3. Optimal transmit power allocation
Note 1: Before we move on, it is emphasized that the problem (3) is a non-convex optimization problem, which can be reformulated as a convex one when N = 2 through the application of Karush-Kuhn-Tucker (KKT) conditions . In order to solve the problem from the mathematical point of view, in the following part, we first analyze a 2-transmitter case and achieve the closed form solution. Then, based on the derived solution an optimal cooperative power allocation algorithm is presented subsequently. Finally, motivated by the regular theoretical derivation of the 2-transmitter case, we extend the parts of the conclusion to arbitrary N-transmitter case.
3.1. 2-Transmitter case
which means that in order to maximize the sum throughput, only some m th receiver is transmitted jointly by the two transmitters, and each other receiver is only transmitted by some single transmitter. The first term on the right-hand side represents the throughput of the receiver transmitted jointly by the two transmitters. The second term represents the sum throughput of the receivers transmitted by single transmitter. Or from mathematics perspective, the optimization problem (4) must be achieved on a specific bound domain.
where λ nk and v n are Lagrange multiplier associated with inequality constraint and equality constraint, respectively. In the following, we will prove that the optimization problem (4) can only be achieved on a specific bound domain by contradiction.
which is almost impossible in practical system. Therefore, in order to maximize the sum throughput, L ≤ 1, that is, at most one receiver can be transmitted jointly by the two transmitters, and each other receiver is transmitted only by some single transmitter, i.e., the optimization problem (4) can be reformulated as (5).
which is consistent with the theorem 1, i.e., only the first receiver is transmitted jointly by the two transmitters.
Theorem 2: The solution to the problem (5) takes the simple form of traditional power WF  results and is also characterized by cooperation.
Proof: We first prove that the optimization problem (5) is a convex optimization problem, and then achieve the solution through the application of KKT conditions.
Note 2: To this point, the solution for is achieved, and according to the third condition in (17), we can also obtain the solution for x nm . From (18), it is observed that the power allocation for the receivers transmitted by single transmitter take the form of traditional WF results. The receivers which have better channel state will get more power allocation and the receivers which have the channel state worse than the water level should not be transmitted. To further get the value of , the water-level v1 and v2 must be obtained. In the following part, we proceed to get the water-level v1 and v2 with an iteration algorithm which is proposed for the first time.
In conclusion, the convex optimization problem (15) can be solved by Algorithm 3. To this point, the corresponding optimal cooperative algorithm is presented.
Note 3: To this point, we propose the iteration algorithm to get the value of the water level v1 and v2. It is noted that, if v1 < γ1mand v2 < γ2m, i.e., there is one receiver transmitted jointly by the two transmitters, we have (21) which reflects the cooperative feature. To further analysis, the power allocation results x1nand x2nare not independent and they restrict each other through the water level v1 and v2. In addition, from (21) and the algorithm solving for water level, the two water levels v1 and v2 can be unified as a single global water level and the power allocation scheme can be explained as a cooperative global WF process relative to the traditional WF power allocation. Moreover, it should be emphasized that, to perform the proposed joint power allocation scheme, each transmitter must obtain all the CSI to further get all the γ k , which can be realized only by cooperation between the transmitters, i.e., exchanging CSI.
3.2. N-transmitter case
In this section, based on the above analysis and theorem, we extend parts of the derived conclusion to arbitrary N-transmitter case through mathematical derivation. Some similar assumption and analysis method are omitted here.
which means that there is no more than one receiver transmitted jointly by more than one transmitter.
which is impossible in practical system. So, the supposition cannot hold, i.e., the theorem 3 is proved.
4. Numerical simulations
R is 1000 m. Non-co R is 600 m. Path loss model adopts Okumura-Hata : l(d) = 137.74 + 35.22 lg(d) in dB, Shadowing's standard deviation is 3.65 dB. For the sake of simplification, we assume ΔB = 15 kHz and the downlink noise N0ΔB at each subcarrier is assumed to be the same as -105 dBm. Assume that the two transmitters have the same power constraint as P. The numerical results are generated by averaging the throughput over 1000 randomly generated users' location realizations.
In this study, we have investigated the power allocation for OFDM system with cooperation at the transmitters. The transmitters first cooperate by exchanging the CSI, and then joint optimal power allocation. To maximize the sum throughput, at most one receiver should be jointly transmitted by the two transmitters, and each other receiver is transmitted by some single transmitter. Then, the closed form solutions to the optimal joint power allocation are achieved in the 2-transmitter case, which turn out to take the form of traditional WF and also combined with some regular cooperative feature. Based on the solution, an optimal joint power allocation algorithm is proposed subsequently for the first time, which can be explained as a joint WF relative to the traditional WF. Motivated by the derivation process in the 2-transmitter case, we extend parts of the conclusion to N-transmitter case. Numerical results verify the optimality of the derived scheme and show throughput gains over traditional non-coordinated WF and EPA.
Algorithm 1. Jo-WF algorithm for water level
i = 1
while v1 > γ i do
i = i + 1
Algorithm 2. Traditional WF algorithm for water level
i = 1
while v1 > γ i do
i = i + 1
Algorithm 3. Optimal cooperative power allocation algorithm
Assume both of x 1mand x 2mare positive, which can only hold if v 1 < γ 1mand v 2 < γ 2m.
Solve (22) with the Jo-WF algorithm in Algorithm 1 to get v 1 and then obtain v 2 through the second equation in (21).
Check whether v 1 < γ 1mand v 1 < γ 2m. If satisfying, go to step 4, otherwise, go to step 5
Get through (18) and x nm through the third condition in (17)
If v n ≥ γ nm , set x nm = 0. Then the problem can easily be solved with the traditional WF algorithm in Algorithm 2.
This study was sponsored by International Scientific and Technological Cooperation Program (2010DFA11060), National Natural Science Foundation of China (61027003), China-EU International Scientific and Technological Cooperation Program (0902).
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