Optimal Power Allocation under Average Input Power

In this section, the optimization problem (3.9a)-(3.9b) can be solved using the results from Subsection 3.3.2.1. After the unique optimum average power value P∗

t for the

power-unconstrained problem is calculated, it needs to be compared with the input average power limit Pmax. If Pt∗ ≤Pmax, it means that the system has enough power

to support the optimal tradeoff performance found in Subsection 3.3.2.1. Otherwise, P∗

t ≥ Pmax means that Pmax is too small to support the power allocation strategy

(3.14)-(3.18b) and the system has to operate at the maximum available power Pmax.

Therefore, the operational input average power value becomes min(P∗

t, Pmax).

Hence, the power-constrained EE-EC tradeoff problem in (3.9a)-(3.9b) simplifies to an EC-maximization problem with two input power constraints, yielding

max Pr≥0 − 1 θTfB lnE_γh_{(1 +Prγ)}−α(θ)i _(3.20a) subject to: Pr ≤ P∗ t Kℓ , (3.20b) Pr ≤ Pmax Kℓ . (3.20c)

The optimal power allocation to solve (3.9a)-(3.9b) is according to (3.14), wherein, the optimal ν∗ _{is found such that K}

ℓPr |ν=ν∗= min(P_t∗, P_max). To summarize, the

Pseudocode of the optimal power allocation algorithm to solve (3.9a)-(3.9b) is illus- trated in Table 3.1.

Furthermore, the optimal power allocation strategy (3.14)-(3.18b) has the follow- ing properties:

Properties 1. 1. For every given weight value, the proposed optimal solution (3.14)- (3.18b)is sufficient for the Pareto optimal set of the original EE-EC MOP Q1. 2. The proposed optimal solution includes the optimal power allocation strategy for the link-layer EE-maximization problem (whenw1 = 1) and also the one for the

Table 3.1: Optimal Power Allocation Algorithm for a Single-User Single-Carrier Sys- tem

Input: Initialization parameters w1 importance weight of EE

Pnorm normalization factor

ΨEE normalization value of EE, e.g., ΨEE = EE|Pt=Pnorm ΨEC normalization value of EC, e.g., ΨEC=Ec |Pt=Pnorm θ exponential decay rate of the QoS violation probability Pmax input average power limit

Pc circuit power

m Nakagami fading parameter ǫ power amplifier efficiency

Kℓ pathloss and noise factor, e.g., Kℓ =PLσ_n2

Tf fading block duration

B channel bandwidth

Step 1:

Create (3.16), using closed-form expressions given in (3.18a) and (3.18b). Find ν∗ _{which solves (3.16) using root-finding functions, e.g., fzero in Matlab.}

Insert ν∗ in (3.14) to calculateP_r∗ and then get P_t∗, using P_t∗ =Kℓ×Pr∗.

Insert ν∗ _{in (3.18a) to calculateP}∗

r and then getPt∗, using Pt∗ =Kℓ×Pr∗. Step 2:

If Pmax > Pt∗

Calculate Ec using (3.5) and EE using (3.6), by applyingPt∗ and Pt∗.

Else

Create P∗

t =Pmax and use Pr∗ =

Pmax

Kℓ

to update ν∗ _{by solving (3.18a).}

Insert ν∗ _{in (3.14) to calculate P}∗

r and then get Pt∗, using Pt∗ =Kℓ×Pr∗.

Calculate Ec using (3.5) and EE using (3.6), by applyingPt∗ and Pt∗.

End

Output: P∗

t, Pt∗, Ec,EE

3. Whenθ→0, EC is equivalent to the ergodic capacity. For the weighted physical- layer EE-SE tradeoff problem, the optimum power allocation strategy is the tra- ditional water-filling approach, with the water level to be chosen so that the maximum tradeoff performance can be achieved [98].

4. When θ _{→ ∞}, EC is equivalent to the zero-outage capacity, and the optimum power allocation strategy is to maintain a constant received signal-to-noise ratio (SNR), at a level that maximizes the tradeoff performance [106].

In more detail, we first note that with a predetermined importance weight, the unique optimal solution of Q8 is sufficient for the optimal solution of the weighted

tradeoff problem Q7 [62, 63]. Then, by applying Lemma 1, Theorem 5 and Theorem 6, one can show that the optimal power allocation strategy (3.14)-(3.18b) for every determined weight value, is sufficient for the Pareto optimal set of the original EE-EC

MOPQ1.

Furthermore, the optimal solution (3.14)-(3.18b) is similar to the optimal power allocation strategy of the link-layer EE-maximization problem in [98], with a different value of the optimal scaled-Lagrangian-multiplier ν∗_{. Specifically, when w}

1 = 1, the

proposed optimal solution (3.14) reduces to the one developed in [98]. It means that the optimal solution in [98] is a special case of the optimal power allocation strategy for the weighted EE-EC tradeoff problem in this chapter. Specifically, in [98], the optimal operational average power equals to min(P_EE∗ , Pmax), while the optimal average power

varies between [P_EE∗ , Pmax], for a typical EE-EC tradeoff problem.

When θ→0, by following similar steps, the optimal power allocation strategy for the weighted tradeoff problem can be derived as

Pr = 1 ρ − 1 γ + , (3.21)

which is the well-known water-filling approach and ρ can be found from the KKT condition E_γ " ln γ ρ +# −ρ ǫPcr +Eγ " 1 ρ − 1 γ +#! +ǫ(1−w1) ΨEC w1ΨEEr ! = 0. (3.22) When θ → ∞, a system with extremely stringent delay requirement is considered, which means in this case, EC is equivalent to the zero-outage capacity [98].