Abstract
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.
.
(resp., 
) the amplitude of the symbol sent by the source (resp., the relay) on carrier 
in the first (resp., second) time slot, and by 
(resp., 
) the complex channel gain for tone 
between source (resp., relay) and destination. The noise sample corrupting the transmission on tone 
during the first time slot is 
, and 
during the second period. These two noise samples are zero-mean circular Gaussian, white and uncorrelated with the same variance 
. Denoting by 
the unit variance symbol transmitted over tone 
, after proper maximum ratio combining at the destination, the decision variable obtained at the 
th output of the FFT (Fast Fourier transform) is given by
and 
.
is the set of carriers (or tones) receiving power at the source only, and 
the complementary set, that is the set of carriers receiving power at both source and relay. These sets are not known in advance and must be characterized in an optimal way. In [
and the rate on the link 
. The power allocation which maximizes (3) is first investigated for a sum power constraint
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.
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 
.
, 
. If a constraint is inactive, its associated Lagrange multiplier is zero [
and taking the derivative of the Lagrangian with respect to the powers for a relayed carrier leads to
. This imposes a constraint on the current channel state, which is almost certain not to happen. Hence, except in very marginal cases, the decode-and-forward constraint has to be saturated. This means
.
, the total amount of power 
allocated to that carrier will be given by 
. Therefore the Lagrangian can be written as:
, 
and 
, while for 
, 
with 
.
and be allocated to set 
. By opposition, when
.
, because of the 
, the rate obtained by allocating the carrier to the set 
will always be higher than the rate obained if the carrier were allocated to 
. It is worth noting that, if 
, the inequality between 
and 
is equivalent to the inequality between 
and 
. As a matter of fact, with the definition of 
,
, 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 
.
,
such that
. For carriers 
and to be served with power,
.
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 
.
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 
.
the rate evolves as
, and a total allocated power 
(over the two instants), the rate evolves as
we have that 
for any value of 
. On the contrary, when 
we have 
. However this is only valid for 
where
, even when 
, the power is better used by allocating the carrier to set 
. Let us define the following Lagrangian, with a Lagrange multiplier 
associated with the global power constraint, and taking into account the saturation of the DF constraints:
the derivatives of this Lagrangian with respect to the power, we get for 
,
stands for 
. Similarly, for 
,
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.
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.
may not be active. This means that there is enough relay power such that for a relayed carrier 
, 
can always be made large enough to have
may be different from 
. Here we investigate a solution for the case where the relay power is not saturated and the related Lagrange multiplier is then 
. The corresponding Lagrange function can be written as:
carrier 
should be allocated to set 
. In the reverse case, it should be allocated to set 
. Once the assignment is known, taking the derivative with respect to 
with 
and equating it to 
, it comes
in the other set, 
, we get
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.
with 
and equating it to 
, it comes
in the other set, 
,
, 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 
.
ad 
, now have to be used for the power contraints. One element in the direction of the solution lies in the observation [
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 
.
'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 
all the results derived in Section 3 apply to our problem with individual power constraints, and to the powers and the gains that have been properly normalized. In particular it can be concluded that for the carriers using the relay, the decode-and-forward constraint will be saturated, leading to 
. Hence 
and 
should be allocated simultaneously leading to a total power denoted by 
where
, we also have
should be placed in set 
if
and 
it turns out that the selection rule when 
amounts to choosing 
when 
or when
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 [
, the rate gain and optimality conditions are given by
, the gain and optimality conditions are given by
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 (
. 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.
and 
correspond to the two power constraints that have to be fulfilled. Hence any classical method known to find the roots of a function (here the derivatives with respect to 
and 
) can be used. A typical method used is the so-called "subgradient method" where the correction to the Lagrange variables 
and 
at step 
is made proportionally to the error on the constraints. Here we try to improve this classical method by using a Newton-Raphson algorithm where the first derivative of the objective function (here the objectives are the constraints) is also used. A Newton-Raphson approach is known to have quadratic convergence, and to always converge for a convex objective function. At iteration 
, the power prices 
and 
are updated according to
, the rate gain and optimality conditions are given by
, the gain and optimality conditions are given by
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.
. 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 
.
for the sum power constraint, and to 
and 
for the case of individual power constraints. Figure 
(solid curve), 
(dash-dotted), 
(dashed) in dBW of the channels. Figure 
and the possible additional split whenever relevant among source power (solid line) and relay power (dashed). The 
s indicate whether the relay is active (
at the top of the figure) or not (
in 
). In this case, the power used by the source is 
and that by the relay is 
. The total rate obtained here is 
bits per a duration of 2 OFDM symbols. If preferred, this rate 
(bits) per 2 OFDM symbols may readily be converted to a spectral efficiency by computing 
(bits/sec/Hz) where 
is the roll-off factor. Figure 
bits for a duration of 2 OFDM symbols. It is also interesting to mention that in this case, the power allocated to the source for the channel realization under consideration is 
and to the relay, the remainder meaning 
. Compared to protocol P1, the gain is noticeable and is clearly due to the better exploitation of the second time slot.
in dBW.
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.
. For each value of 
the results reported are obtained by averaging over 250 channel realizations. The CIRs associated with the 
, have a variance of 20 dBs above those associated with the 
and the 
. The results obtained with the optimized power allocation are contrasted against uniform power allocation. For protocol P1 with uniform power allocation, the carrier allocation to sets 
and 
is performed as in the optimized case. The power available is uniformly divided between the 
carriers. For the carriers to be relayed, the per carrier power is further split between source and relay according to the ratio associated with the saturation of the decodability constraint (7). For protocol P2, the allocation of the carrier to set 
or 
is based on the comparison of 
with 
. If 
carriers are allocated to set 
and 
to set 
the total power is divided by 
in order to take into account the use of the two time slots for the carriers in 
. At this point the reallocation step is implemented and some carriers may be moved from 
to 
. For the carriers remaining in set 
the power is further split among source and relay according to the ratio associated with the saturation of the decodability constraint (7). Figure 
between uniform power allocation and optimized allocation is also reported in Figure 
and protocol P2, all carriers will be allocated to set 
(because of the reallocation step). Because each carrier is used over the two time slots, the rate grows with a slope 
for P2 whereas the slope is only 
with P1. The rate gain results (Figure 
. Clearly and as expected, the benefit of the optimized power allocation decreases with 
. For high values of 
the optimized power allocation tends to become a uniform one.
(dBW) for the two protocols and uniform and optimized power allocation for the sum power constraint. Taps of the 
have a variance 20 dBs above those associated with the 
and the 
.
(dBW) for the two protocols and the sum power constraint. Taps of the 
have a variance 20 dBs above those associated with the 
and the 
.
, have a variance of 10 dBs (instead of 20 dBs) above those associated with the 
and the 
. These results lead to similar conclusions. 
(dBW) for the two protocols and uniform and optimized power allocation for the sum power constraint. Taps of the 
have a variance 10 dBs above those associated with the 
and the 
.
(dBW) for the two protocols and the sum power constraint. Taps of the 
have a variance 10 dBs above those associated with the 
and the 
.