Rate-Optimized Power Allocation for DF-Relayed OFDM Transmission under Sum and Individual Power Constraints
© Luc Vandendorpe et al. 2009
Received: 10 November 2008
Accepted: 20 May 2009
Published: 29 June 2009
We consider an OFDM (orthogonal frequency division multiplexing) point-to-point transmission scheme which is enhanced by means of a relay. Symbols sent by the source during a first time slot may be (but are not necessarily) retransmitted by the relay during a second time slot. The relay is assumed to be of the DF (decode-and-forward) type. For each relayed carrier, the destination implements maximum ratio combining. Two protocols are considered. Assuming perfect CSI (channel state information), the paper investigates the power allocation problem so as to maximize the rate offered by the scheme for two types of power constraints. Both cases of sum power constraint and individual power constraints at the source and at the relay are addressed. The theoretical analysis is illustrated through numerical results for the two protocols and both types of constraints.
In applications where it is difficult to locate several antennas on the same equipment, for size or cost issues, it has been proposed to mimic multiantenna configurations by means of cooperation among two or more terminals. Cooperation or relaying, also coined distributed MIMO, has gained a lot of interest recently. Cooperative diversity has been studied for instance in [1–3] (and references therein) for cellular networks.
In this paper we consider communication between a source and a destination, and the source is possibly assisted with a relay node. All the channels (source to destination, source to relay and relay to destination) are assumed to be frequency selective and in order to cope with that, OFDM modulation with proper cyclic extension is used. The relay operates in a DF mode. This mode is known to be suboptimum [4, 5]. Decode-and-forward is adopted here as a relaying strategy for its simplicity and its mathematical tractability. Two protocols (P1 and P2) are considered. Each protocol is made of two signaling periods, named time slots. The first time slot is identical for both protocols. During this period, on each carrier, the source broadcasts a symbol. This symbol (affected by the proper channel gain) is received by the destination and the relay. The relay may retransmit the same carrier-specific symbol to the destination during the second time slot. Whether the relay does it or not will be indicated by the optimization problem which is formulated and solved in this paper. The protocol P2 differs from the protocol P1 in that, in the latter, the source does not transmit during the second time slot, irrespective to whether the relay is active or not during the second time slot. For P2, on a per carrier basis, the source sends a new symbol if the relay is inactive. The reason for not having the source and the relay transmitting at the same time is to avoid the interference that would occur in this case, thus rendering the optimization problem somewhat tedious. Moreover in practice source and relay will have different carrier frequency offsets which is likely to require involved precorrection mechanisms. A scenario with interference will be investigated in the future.
For both protocols, whenever it is active, the relay uses the same carrier as the one used by the source. This is an a priori choice made here to make the optimization more tractable. It is however clear that carrier pairing between source and relay is a topic for possible further optimization of the scheme. At the destination, it is assumed that for the relayed carriers, the receiver performs maximum ratio combining of what is received from the source in the first time slot, and what is received from the relay in the second one, for each tone.
OFDM with relaying has already been investigated by some authors. In , the authors consider a general scenario in which users communicate by means of OFDMA (orthogonal frequency division multiple access). They propose a general framework to decide about the relaying strategy, and the allocation of power and bandwidth for the different users. The problem is solved by means of powerful optimization tools, for individual constraints on the power. In the current paper, we restrict ourselves to a single user scenario but we investigate more deeply the analytical solution and its understanding. We study power allocation to maximize the rate for both cases of sum power and individual power constraints. We also compare two different DF protocols and show the advantage of having the source also transmitting during the second time slot. In  the authors consider a setup which is similar to the one we address in this paper but with nonregenerative relays. In , the authors investigate OFDM transmission with DF relaying, and a rate maximizing power allocation for a global power constraint. They briefly investigate the power allocation for the protocol named P1 in the current paper, and a sum power constraint only. On the other hand they investigate optimized tone pairing. In , the authors consider OFDM with multiple decode and forward relays. They minimize the total transmission power by allocating bits and power to the individual subchannels. A selective relaying strategy is chosen. More recently, in  the authors also consider OFDM systems assisted by a single cooperative relay. The orthogonal half-duplex relay operates either in the selection detection-and-forward (SDF) mode or in the amplify-and-forward (AF) mode. The authors target the minimization of the transmit-power for a desired throughput and link performance. They investigate two distributed resource allocation strategies, namely flexible power ratio (FLPR) and fixed power ratio (FIPR).
The paper is organized as follows. The system under consideration is described in Section 2. The rate optimization for a sum power constraint is investigated in Section 3 for the two protocols. The cases of individual power constraints are dealt with in Section 4. Finally numerical results are discussed in Section 5.
2. System Description
During the first time slot, the source sends one modulated symbol on each carrier. During the second time slot, the relay selects some of the modulated symbols that it decodes, and retransmits them. For each relayed symbol, we constrain the relay to use the same carrier as that used by the source for the same symbol. Based on the two signalling intervals, the destination implements maximum ratio combining for the carriers with relaying. As explained earlier, we consider two protocols, called P1 and P2. In protocol P1, the carriers that are not relayed are simply not used in the second time slot (neither by the relay nor by the source). In protocol P2, a new carrier specific modulated symbol is sent by the source in the second time slot on each one of the carriers that are not used by the relay.
3. Rate Optimization for a Sum Power Constraint
We first investigate the case of a sum power constraint. The techniques used in this section will be useful in solving the problem with individual power constraints. It is well known [11, 12] that the optimization with individual power constraints can be solved by reformulating it properly into an equivalent problem with a sum power constraint. All channels gains are assumed to be perfectly known for the central device computing the power allocation. The overhead associated with channel updating is not discussed further in the current paper.
We investigate the two protocols separately.
3.1. Protocol P1
where is the total power budget available for the source and the relay together, and is the total number of carriers. Below, the objective function will be worked out in order to find criteria enabling to decide about the set or to which each carrier has to be assigned.
where is the Lagrange multiplier associated with the global power constraint and is the Lagrange multiplier associated with the decodability (perfect decode and forward) constraint on carrier . The are indicators taking values or and whose optimization will provide the solution for the assignment to sets and .
Let us first investigate whether the decodability constraints are active or not for relayed carriers. For relayed carrier , . If a constraint is inactive, its associated Lagrange multiplier is zero . Assuming this may be the case, setting the and taking the derivative of the Lagrangian with respect to the powers for a relayed carrier leads to
This means that when , the allocation to or to of the carrier may be based on either comparisons in (14) because they are equivalent. And in short, to be relayed, a carrier should fulfil the following two conditions simultaneously: and .
All these derivations basically also show that, after the assignment step, our constrained optimization problem can actually be solved thanks to the seminal waterfilling algorithm, applied to a water container built either from or from . The latter values actually show that the constraint related to the DF operating mode of the relay leads to particular values to be used for the container. More specifically, for the set , these values are modified values with respect to the .
3.2. Protocol P2
where is the set of carriers (or tones) receiving power at the source only, and is the complementary set, that is, carriers receiving power at both the source and the relay. We also denote by the power allocated to a carrier at the source. If this carrier is not relayed, each protocol instant uses .
Again the derivations show that the constrained optimization problem can be solved using the waterfilling algorithm, applied to a water container built either from or from . It is also important to note that for the nonrelayed carriers two identical values have to be used for the water container, corresponding to the two protocol instants. At the end of the waterfilling one checks if any of the relayed carriers receives an amount of power larger than the threshold given by (22). If this happens, the relayed carrier fulfilling this condition and for which the rate increase is the largest one is moved from the set to the set . The waterfilling is applied again. This procedure is iterated till none of the relayed carrier receives an amount of power larger than its associated threshold. In the sequel this procedure will be named the reallocation step.
4. Rate Maximization for Individual Power Constraints
First, note that for the optimum power allocation with individual power constraints, it might happen that constraint (28) is inactive for certain values of channel parameters, but constraint (4) will always be active. In other words, at the optimum, the full available power will always be used at the source, while some of the power available at the relay may not be used. This can be explained using simple intuitive arguments. Assume a solution is found such that is not fully used. The rate can be further increased by allocating the remaining source power to a carrier in set or in set . For the relay power, things may be different. For instance, it may even happen that all carriers are allocated to the set in which case the relay does not transmit at all. One way to take this particular case into account is to perform a first optimization (called first step hereafter), trying to allocate the source power in an optimum way, not considering the constraint on the relay power. After this allocation process of the source power, one has to check whether the relay power is sufficient or not. If it is sufficient, then the optimum solution corresponds to this particular situation in which the full relay power is not used. If not, it can now be assumed that the relay power constraint is satisfied with equality at the optimum, and the full iterative method explained below should be used. Let us first describe the first step.
4.1. First Step
Again, we analyze the two protocols separately.
4.1.1. Protocol P1
Hence the problem can be solved by means of a waterfilling procedure, where the container is built from values in set , and values in set . With such an allocation procedure, the minimum power required at the relay is given by where . If this value is below the power available at the relay, the problem is solved. This would correspond to a situation where the relay is located far away from the source, and, in a sense, not very useful for the protocol used here. Otherwise one has to investigate the situation where both power constraints are active (saturated), which is of most interest.
4.1.2. Protocol P2
So the conclusions are similar to those drawn for protocol P1. The problem can again be solved by means of a waterfilling procedure, where the container is built from the values , and the values in set . However it has to be noted that for the values related to set those values have to be used twice because of the two time slots. Besides that, the reallocation procedure has to be implemented: it has to be checked whether any of the carrier allocated to set receives an amount of power above a certain threshold. If this happens, carriers have to be moved from set to set , and the waterfilling has to be applied till this no longer happens, as explained above. The value to be used for the threshold is similar to (22), where has to be used instead of .
4.2. Second Step
A second step is needed unless the power used at the relay by the procedure described in the first step is below the available relay power. Two Lagrange multipliers, ad , now have to be used for the power contraints. One element in the direction of the solution lies in the observation  that the rate only depends on the products of powers and (possibly modified) channel gains. Hence allocating power to a carrier with gain provides the same rate as allocating power to a carrier with gain . Let us assume for the moment that the optimum and are known. The allocation rules proposed above to define the sets and should be revisited with gains modified as: ; and . The equivalent powers under consideration are now and .
4.2.1. Protocol P1
It is interesting to compare this Lagrangian with the one given by (5). Actually they both have the same structure. The first difference is that (5) is based on 's and 's while (36) is based on 's and 's. Assuming that and are known, and thanks to the use of the modified gains and powers, the individual power constraints give rise to a single sum power constraint. The associated Lagrange multiplier now has to be equal to .
and vice-versa. Therefore, the allocation procedure of the carriers turns out to be equivalent to that in the sum power case, with properly modified channel gains.
There is however one important exception to this rule which is related to the particular case where the equality holds. It has been assumed previously that this particular case needs not being investigated as it is very unlikely to happen. This applies for the sum power constraint. However, in the case of individual power constraints, the procedure is now working with the modified values which are no longer given but depend on the Lagrange parameters and . It may happen (and has been encountered for some of the channels randomly generated) that the optimal values of these Lagrange parameters are such that the equality is exactly met on some carriers (usually at most one). This particular situation needs a few additional developments and adjustments which have been presented in  and will not be repeated here.
So far, we have assumed that and were known. In fact there is a single pair for which the two power constraints are simultaneously fulfilled. To find this pair, the following algorithm is proposed. The idea is to scan all possible assignments to sets and . For carriers such that , as discussed above, the carrier will be assigned to set . For the other carriers, with , relaying may be considered. Equation (41) says that the assignment of a carrier candidate for relaying depends on the ratio . By sorting the carriers candidates for relaying by decreasing order of the ratios , all possible assignments can be considered. As a matter of fact, if a single carrier gets relayed it will be the first one in the sorted set. If two get relayed, it will be the first two, and so forth. Therefore, by considering all possible sets of first carriers in this sorted set, all possible assignments can be investigated. We have as many situations to consider as we have carriers being candidates to be relayed. For each situation, the assignment to sets and is fixed. For a fixed assignment, the optimization problem to be solved is convex. The corresponding dual problem is also convex. The dual problem can be solved by taking the derivatives of the dual objective with respect to and , and equating these derivatives to zero. The values of and solving these equations can be entered in the primal problem, and the optimum power values can be obtained. The problem is that the equations to find the optimum and are nonlinear. They can be solved for instance in an iterative manner.
This Newton-Raphson procedure is thus to be repeated for each one of the possible assignments.
4.2.2. Protocol P2
Equations (49) and (51) also show that the powers are given by a waterfilling procedure with a common water level or a common power constraint, and containers defined by these equations. The problem is again equivalent to the sum power case and the procedure defined for the maximisation problem in Section 3.2 can be reused. The have to be replaced by , and the by . The comments about the allocation of the carrier to set or are the same as in the case of protocol P1. Recall also that the reallocation step has to be implemented. The Newton-Raphson procedure for the updating of and is similar to that used for protocol P1.
In order to illustrate the theoretical analysis, numerical results are provided and discussed. The number of carriers is set to . Channel impulse responses (CIR) of length are generated. The taps are randomly generated from independent zero mean unit variance circular complex gaussian distributions. Hence the power delay profile is flat. All taps have a unit variance for all links. From these CIRs, FFT are computed to provide the corresponding ( ). We set .
With protocol P1 and individual power constraints, the bit rate achieved is bits for a duration of 2 OFDM symbols. Compared to the same protocol with the sum power constraint, the observed rate loss is due to the values chosen here for the individual power constraints ( - ) which are rather different from the values devoted to the two categories of carriers by the sum power case (136-64). For individual power constraints and protocol P2, the total rate is 318 bits per 2 OFDM symbols duration. The loss incurred compared to the sum power case can be explained in a manner identical to that discussed for protocol P1. And again the advantage of this protocol compared to P1 is visible.
In this paper we considered an OFDM point to point link enhanced by means of a relay. When a symbol is received by the relay on a certain tone, it may be relayed to the destination on the same tone. We have investigated the problem of power allocation to the source and to the relay in order to maximize the rate of the whole transmission for a global power constraint and for individual power constraints at the source and at the relay. Two protocols have been considered; the second one makes a better use of the second time slot whenever the relay is inactive. It is assumed that the destination implements MRC between what is received from the source and what is received from the relay, for each tone. The DF operating mode of the relay puts an additional constraint on the design. The carrier classification (whether a carrier has to be relayed or not) has first been investigated for the sum power case. The power allocation problem has been shown to be of the waterfilling type with a specific construction of the container. It has also been shown how the problem for individual constraints could be recast into an equivalent waterfilling problem by using the technique of equivalent powers and equivalent channels. It has been proposed to find iteratively the two Lagrange multipliers in this second case by means of a Newton-Raphson method implemented for each possible carrier assignment. Numerical results have been provided to illustrate the schemes and have shown the advantage of protocol P2 over protocol P1.
Future work will be devoted to the cases of multiple relays, nonperfect channel state information and a refinement of the power allocation across the two signaling intervals. Moreover, coding will also be included in the transmission scheme and taken into account. Besides these topics, the (peak to average power ratio) PAPR might also be a problem to be considered. PAPR issues are well known with OFDM transmission and are likely to be impacted by power allocation.
The authors would like to thank the Walloon Region DGTRE Nanotic-COSMOS project, the FP6 project COOPCOM and the FP7 Network of Excellence NEWCOM++ for their financial support. Parts of this work have been reported in IEEE SCVT 2007, IEEE ICC 2008 and ISWPC 2008.
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