3.9 Energy Efficient Power Allocation
3.9.2 Proposed Solution
Let f1(x) = x
TAx
xTEx, and f2(x) = x
TDx
xTFx. The functions f1 and f2 can be put into the form of
Rayleigh quotients as:
f1(y1) = y1TMy1 yTy 1 , f2(y2) = y2TNy2 y2Ty2 , where M = E−12A(E− 1 2)T, and N = F− 1 2D(F− 1 2)T, with y1 ,(E 1 2)Tx and y2 ,(F 1 2)Tx.
As mentioned earlier, the Rayleigh quotient has a maximum value at the dominant eigen- mode of the numerator matrix, and the vector achieving this maximum is the corresponding eigenvector. Let y1∗ and y2∗ be the vectors maximizing the functions f1 and f2 respectively.
We propose choosing the solution of (3.69) to be in the subspace spanned by the two vectors y∗1 and y∗2. Then the solution can be represented as a linear combination as,
x = y1∗+ ρ y∗2. (3.70) Substituting (3.70) into (3.69), will result in the new scalar-valued optimization problem over a single real variable ρ. The optimum ρ is obtained by a 1-D search to maximize the objective function (3.69). Simulation results provided in the next section show that the proposed solution can achieve at least a local optimum, with an acceptable overall performance.
3.10
Numerical Results
In this section, we present the performance of the proposed energy-efficient power allocation algorithm, and compare it to “Equal Power Allocation”. It is to be noted that in the equal power allocation all relays use the same transmission power PR= min{Pmax, Imax/∑Ki=1|fRi,P|
2}.
The noise variance at all nodes, σ2, is set to 1.
In Fig. 3.14, we show the energy efficiency of three power allocation schemes: the proposed algorithm maximizing the energy efficiency, the power allocation in [115] which maximizes the SNR, and the equal power allocation. The simulation results are obtained using 106 transmissions. It is shown that, for a 2-relay network, the energy efficiency is almost the same for the three power allocation schemes. However, for larger networks, the gain starts to increase. For small to moderate secondary source transmission power PS, the proposed
power allocation offers higher efficiency compared to the SNR-maximizing allocation. The performance gap increases as the network size increases which is justified by the increased degrees of freedom in the system. However, for high secondary source power, both schemes show almost the same performance, and the efficiency starts to decrease. This is due to the dominance of PS in the denominator of (3.62) for high secondary source powers.
The corresponding outage probability of the secondary system is given in Fig. 3.15, for interference threshold set to 10 dB, and the individual relays’ power Pmax is set to 0 dB.
The SNR threshold is −3 dB. The results in Fig. 3.15 suggest that for small-size networks, the energy efficient power allocation has almost no effect on the outage probability. As the network size increases a slight performance loss is presented by the energy efficient algo- rithm, compared to the SNR-maximizing allocation. For example, for an 8-relay network, only 0.1 dB loss occurs in the PS at 10−3 probability of outage. The effect of the inter-
ference threshold Imax on the energy efficiency and the corresponding outage probability
is shown in Figs. 3.16,3.17, respectively. The results are obtained at the secondary source transmission power PS = 3 dB, and the individual relays’ power Pmax is also set to 3 dB. It
is shown that for small interference threshold values, the proposed algorithm and the SNR- maximizing algorithm have the same efficiency as satisfying Imax is the limiting constraint in
−5 0 5 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Secondary Source Power, P
S (dB)
Energy Efficiency
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=8
K=6
K=2 K=4
Figure 3.14: The energy efficiency of the secondary network vs the Secondary source trans- mission power PS for different network size (K). The interference threshold is set to 10 dB
and the individual relays’ Power Pmax is set to 0 dB.
−4 −2 0 2 4 6 8 10−4 10−3 10−2 10−1 100
Secondary Source Power, P
S (dB)
Outage Probability P
out
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=2
K=4 K=6
K=8
Figure 3.15: The outage probability of the secondary system Pout vs the secondary source
transmission power PS for different network size (K).The interference threshold is set to 10
rithm achieves higher efficiency. As Imax increases, more tolerance is given in the secondary
network, which allows more power to be used, and hence, the energy efficiency decreases due to the consumption of more power. Fig.3.17 shows the corresponding outage probability. Almost no performance degradation between the two power allocation schemes, and both schemes outperform the equal power allocation.
Next, we show the effect of the individual relay power constraint, Pmax, on the energy
efficiency in Fig. 3.18. The secondary source uses a relatively low transmission power of PS = 0 dB and the interference constraint Imax is set at 10dB. It is shown that, the energy
efficiency is lower for small values of Pmax. This results from the constraint in (3.64) being the
dominant constraint, and it is satisfied at the upper limit for all relays (i.e. PRi = Pmax∀i).
As Pmax increases, more freedom in the power allocation is obtained, which increases the
SNR, and consequently increases the energy efficiency. However, for a fixed source power PS, the energy efficiency will eventually saturate at a certain level. The larger the network
size K, the higher is the energy efficiency saturation level. Fig. 3.19 shows the outage probability of the system (for the same setup of Fig. 3.18) at SNR threshold equals −3 dB. The outage probability of the three power allocation schemes is the same for small values of Pmax.
Increasing Pmax results in almost no performance degradation between the proposed power
allocation scheme and the SNR-maximizing power allocation, and both schemes outperform the equal power allocation.
−2 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Interference Threshould, I max (dB) Energy Efficiency
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=2 K=8 K=8 K=6 K=4 K=4 K=6
Figure 3.16: The energy efficiency of the secondary network vs the interference threshold Imax for different network size (K). The secondary source transmission power PS is set to 3
dB and the individual relays’ Power Pmax is set to 3 dB.
−2 0 2 4 6 8 10 12 14 16 18 20 10−4 10−3 10−2 10−1 100 Interference Threshould, I max (dB) Outage Probability P out
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=2
K=4
K=6 K=8
Figure 3.17: The outage probability of the secondary system Poutvs the interference threshold
Imax for different network size (K). The secondary source transmission power PS is set to 3
−6 −4 −2 0 2 4 6 8 0 0.5 1 1.5 2
Individual Relay Power Constraint, P
max (dB)
Energy Efficiency
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=6 K=8 K=4 K=8 K=2 K=6
Figure 3.18: The energy efficiency of the secondary network vs the individual relay power constraint Pmax for different network size (K). The secondary source transmission power PS
is set to 0 dB. The interference threshold Imax is set to 10 dB.
−6 −4 −2 0 2 4 6 8 10−6 10−5 10−4 10−3 10−2 10−1 100
Individual Relay Power Constraint, P
max (dB)
Outage Probability P
out
Proposed Energy−Efficient Power Allocaion SNR−Maximizing Power Allocation Equal Power Allocation
K=2
K=4
K=8
K=6
Figure 3.19: The outage probability of the secondary system Pout vs the individual relay
power constraint Pmax for different network size (K). The secondary source transmission
power PS is set to 0 dB. The interference threshold Imax is set to 10 dB, and the SNR