Power distribution scheme for the ELs is discussed in this section. The objective is to optimize the overall system performance by balancing the tradeoff between the proposed ELs and the BL. Therefore, we propose to minimize the overall BER of the ELs with the constraint on the BL's capacity loss. Furthermore, by taking the different reliability/coverage of different ELs into account, we introduce the proportional reliability constraints into our system. The benefit of these constraints is that we can flexibly control the reliability of different ELs for different purposes and therefore, ensuring that each signaling is able to achieve its target service quality given sufficient available transmit power and tolerable BL's capacity loss.
The power distribution problem can be expressed mathematically as,
(37)
subject to,
(38)
(40)
where Pb,idenotes the BER upper bound of the i th EL as derived in (29). k
i
denotes the total effective number of transmitted bits of the i th EL. P
E,total
is the total transmit power budget allocated to the ELs. is the set of the predefined values which are used for the proportional reliability constraints. Note that the nonlinear inequality constraint -C
BL
≤ -C makes the optimization problem in (37) nonconvex. Iterative methods, such as Newton-Raphson or quasi-Newton methods can be used to obtain the solutions; however, with a large amount of computational complexity. Fortunately, under certain approximations, the optimization problem can be relaxed to convex problem and therefore, the optimal or near-optimal solutions of problem can be found with low complexity. Due to the MCSK used on ELs, the operating SNR range for the ELs is much lower than that of the BL. Therefore, we analyze the low SNR case where certain approximations can be made.
Under relatively low SNR range, the SER of the ELs P
e,i
, i = 1,2,...,K are already very low and significantly smaller than the variance of channel estimation error . The term in (33) can be neglected and the capacity of BL given by (33) can further be simplified to
(41)
It is straightforward that the approximated capacity is now a concave function w.r.t. P
i
, which makes the optimization problem convex. Its global optimal solution can then be obtained by Karush-Kuhn-Tucker (KKT) conditions [35] as follows,
(42)
(43)
(44)
(45)
where λ, μ, and ϖ
i
are the Lagrange multipliers. λ ≥ 0, μ ≥ 0, and ϖ
i
≥ 0 ∀i ∈ {1,...,K}.
Note that the capacity given by (33) is a monotonously increasing function w.r.t. P
i
and therefore, the constraint in (39) can be reformulated by
(46)
where the right-hand side of the above inequality is denoted by χ. The condition in (44) can then be rewritten by
(47)
From (43) and (47), we note that λ and μ are not allowed to be synchronously nonzero which means
(48)
(49)
when χ ≠ P
total
. Therefore, the optimization problem can be discussed in the following circumstances:
A. When χ < PE,total
With this condition, we can obtain since . According to (43), we must attain λ = 0. Therefore, (42) can furthered be expanded and solved by
(50)
where [x, y]+ ≜ max {x, y} and . Note that when μ = λ = 0, then it is easy to obtain P
i
→ ∞, which is impossible for real implementation. Therefore, this circumstance is not allowed to occur. μ and λ must not be synchronously zero.
B. When χ ≥ PE,total
Similarly, we can obtain that since . Thus, μ = 0 in this circumstance. The solution is thus given by with μ replaced by λ
(51)
Also μ and λ are not allowed to be synchronously zero.
From the optimal power distribution solution for the ELs, it implies that the power level for different ELs depends on the parameters λ, μ, and ϖ
i
, i = 2,..., K. First, λ is the dual variable associated with the total transmit power budget. It is straightforward that a larger transmit power budget will result in a smaller λ and thus a high power level, and vice versa. Second, μ is the dual variable associated with the tolerable capacity loss of the BL. If the BL can accommodate a larger residual interference introduced by the transmission of ELs, μ would be smaller, and therefore a higher power level, and vice versa. For instance, in an extreme case where the BL cannot accommodate any interference or in other words, the capacity loss constraint for the BL is zero, then μ would be approaching infinity and the resultant zero power level indicates that no ELs' transmission is allowed in this condition. Similarly, the analysis can be also applied to ϖ
i
, i = 2,..., K which are associated with the proportional reliability constraints for the ELs.