Energy Efficiency with Adaptive Decoding Power and Wireless Backhaul Small Cell Selection Tri Minh Nguyen∗ , Animesh Yadav† , Wessam Ajib‡ and Chadi Assi§ ∗ ÉTS, email:
[email protected], † Memorial University, email:
[email protected] ‡ UQÀM, email:
[email protected] § Concordia University, email:
[email protected]
Abstract—This paper considers the problem of maximizing energy efficiency on the downlink of two-tier wireless backhaul small cell heterogeneous networks, where an interference mitigation strategy that combines reverse time division duplexing and equally orthogonal spectrum splitting is proposed. By enabling the small cell access points with the capability of switching ON/OFF, we develop a joint design of transmit beamforming, power and small cell selection that maximizes the proposed weighted access energy efficiency metric. To better convey the total power consumption model, we assume the adaptive decoding power model at each small cell access point. The formulated problem is combinatorial and non-convex, which is NP-hard in general. Hence, to find a more realistic close-to-optimal feasible solution, we iteratively approximate the non-convex constraints in the formulated problem as second order cone ones based on the first order Taylor convex approximation and error-controlled second order cone approximation. The problem arrived at each iteration is a mixed integer second order cone programming, which can be solved optimally and efficiently by available dedicated solver to achieve the final result at convergence. Numerical results are studied to show the improvement of our proposed model compared to previous works.
I. I NTRODUCTION The 5G networks will rely on network densification to fulfill their promise for providing thousand-fold enhancement of network capacity [1], [2]. With dense network deployment, achieving maximum throughput with minimal energy consumption has recently become an extremely attractive research issue [3]. According to [4], base station (BS) sites consume almost 80% of the total energy; thus, saving energy on large scale networks subsequently leads to greener communications. On the other hand, as more BS are deployed, wireless backhaul (WB) emerges as a cheap and more practical technology for its wired counterpart. With WB, the backhaul data is now transmitted via wireless channels using sufficient amount of transmit power so that a certain level of quality of service in the network is maintained. This encourages a more appropriate transmit design that takes into account WB nature to achieve the optimal energy efficiency (EE) in WB networks. Recently, research for green heterogeneous networks (HetNets) has become a critical requirement towards the evolution of 5G networks [3], [5]. In particular, optimal beamforming and zero-forcing beamforming design for achieving the maximal EE in multicell multiuser HetNets were studied in [6], [7]. When the number of small cells is large, the amount of This work was supported in parts by the Fonds de Recherche du Québec Nature et Technologies (FRQNT), 2015-2016.
power for switching the BS ON/OFF becomes importantly comparable to the transmit power. This motivated the authors of [8], [9] to consider BS sleep/active to improve the achieved EE compared to the fixed circuit power. More crucially, at the small cell access point (SAP), the energy consumed for decoding the collected WB data from the MBS should not be ignored in the WB network since the SAPs have small range of operation and use relatively low transmit power [10]. For the downlink (DL) of WB HetNets, each SAP translates this decoding backhaul message power amount to be proportional to its achievable backhaul rate. Hence, this decoding power is fundamental [10] to the EE optimization problem. In this paper, we consider the two-tier small cell HetNets that operate WB communication (WBC) concurrently with the wireless access communication (WAC). WBCs between multi-antenna MBS and single antenna SAPs are considered among conventional WAC of MBS-macrocell user (MUEs) and SAPs-small cell users (SUEs). By enabling the capability of switching ON/OFF the SAP, we apply the reverse time division duplexing (RTDD) combined with equal spectrum splitting in each time slot to separate transmissions between WBC and WAC for interference mitigation; this is on top of jointly designing the beamforming, power allocation and SAP selection that maximizes our proposed weighted access energy efficiency (WAEE) metric, which is defined as the ratio of the total weighted WA spectral efficiency achieved at the users to the total power consumption. Unlike traditional works (e.g., [6], [7], [11]) where a fixed value of decoding power is assigned at each BS, we consider the adaptive decoding power which is naturally obtained from the occurrence of WBC in our model. The formulated problem is a combinatorial non-convex problem, which is very difficult to solve. Motivated by the objective of attaining high-quality feasible solution, we approximate the non-convex constraints in the formulated problem as second order cone (SOC) ones based on the first order Taylor convex approximation (FOTCA) and error-controlled SOC approximation. The problem obtained at each iteration becomes a mixed integer second order cone programming (MISOCP), which can be solved optimally and efficiently by available dedicated solvers to achieve the final result at convergence. The organization of the paper is as follows. Section II introduces the system model and the WAEE. Section III formulates the optimization problem that maximizes the W on the DL of the WB HetNets and proposes the algorithm to solve it. Section V and VI present our numerical results and
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interference as noise, the achievable rate at the 𝑗th receiver can be given by 𝑅𝑗 (v) = log (1 + Γ𝑗 (v)), where 2 Γ𝑗 (v) = v𝑗𝐻 h𝑗
/
⎞ ∑ v𝑘𝐻 h𝑗 2 + 𝑁0 ⎠ , ⎝ ⎛
(2)
𝑘∈ℱ ∖𝑗
Fig. 1. RTDD resource block configuration.
conclusion of the paper. II. S YSTEM M ODEL A. Spatial model This work considers the DL of a two-tier HetNet consisting of one MBS in the macrocell tier and 𝑆𝑎 SAPs in the small cell tier. By adopting WBC, our communication model can generally have two types: WBC from the MBS to the SAPs and WAC from MBS (or SAPs) to MUEs (or SUEs), respectively. The MBS is equipped with 𝑁 antennas to communicate with its 𝑀 macrocell users (MUEs) and 𝑆𝑎 SAPs via WAC and WBC, respectively. For simplicity, we assume that each SAP serves the same number of 𝑆𝑢 SUEs in its small cell. In this work, we consider the system that exploits the RTDD [12] to reverse the uplink (UL) and DL transmissions in two consecutive time slots so that when the MBS transmits its signals to its MUEs on the DL, each SUE transmits its signal to its serving SAP on the UL. A similar transmission protocol applies to the UL of the macrocell and the DL of the the small cells. In addition, access and backhaul communications at one SAP are accommodated in orthogonal spectrum to avoid selfinterference when using in-band half-duplex SAPs. We assume that the spectrum is split equally in each time slot for WBC and WAC as depicted in Fig. 1.
where we denote the set of transmit beamforming as v = {v𝑖 , ∀𝑖 ∈ ℱ}. In the other time-frequency block dedicated for small cell DL, each 𝑖th SAP transmits data to its 𝑆𝑢 intended SUEs in the 𝑖th small cell. By denoting 𝑝𝑖𝑗 as the transmit power from the SAP to the 𝑗th SUE in the 𝑖th small cell and ℎ𝑖𝑘𝑗 as the channel from the 𝑖th SAP to the 𝑗th SUE in the 𝑘th small cell, the received signal within this time-frequency block at the 𝑗th SUE in the 𝑖th small cell is given by ∑ √ √ ℎ𝑖𝑖𝑗 𝑝𝑖𝑘 𝑠𝑖𝑘 𝑦𝑖𝑗 = ℎ𝑖𝑖𝑗 𝑝𝑖𝑗 𝑠𝑖𝑗 + 𝑘∈𝒮𝑢 ∖𝑗
+
∑ ∑
√ ℎ𝑘𝑖𝑗 𝑝𝑘𝑙 𝑠𝑘𝑙 + 𝑧𝑖𝑗 ,
(3)
𝑘∈𝒮𝑎 ∖𝑖 𝑙∈𝒮𝑢
where 𝑠𝑖𝑗 is the message intended for the 𝑗th SUE from the{ SAP }in the 𝑖th small cell with unit average power, i.e., 𝔼 𝑠𝑖𝑗 𝑠∗𝑖𝑗 = 1. Alternatively, 𝑧𝑖𝑗 is a circular symmetric complex AWGN at the SUE from the SAP in the 𝑖th small cell, distributed according to a normal distribution 𝒞𝒩 (0, 𝑁0 ). By treating interference as noise, the achievable rate at each 𝑗th SUE in the 𝑖th small cell can be written as 𝑟𝑖𝑗 (p) = log (1 + 𝛾𝑖𝑗 (p)) where 2
𝛾𝑖𝑗 (p) = 𝑝𝑖𝑗 ∣ℎ𝑖𝑖𝑗 ∣ /𝐼𝑖𝑗 (p) ,
(4)
where p = {𝑝𝑖𝑗 , ∀𝑖 ∈ 𝒮𝑎 , ∀𝑗 ∈ 𝒮𝑢 } is ∑ the set of transmit 2 power of the SAPs and 𝐼𝑖𝑗 (p) = 𝑘∈𝒮𝑢 ∖𝑗 𝑝𝑖𝑘 ∣ℎ𝑖𝑖𝑗 ∣ + ∑ ∑ 2 𝑘∈𝒮𝑎 ∖𝑖 𝑙∈𝒮𝑢 𝑝𝑘𝑙 ∣ℎ𝑘𝑖𝑗 ∣ + 𝑁0 . C. WAEE
B. Signal model We assume the channel is flat over the spectrum and invariant within the coherence time 𝑇𝑐 where 𝑇𝑐 is larger than the duration of one time slot. Based on that, we first consider the transmission at one particular time and frequency block dedicated for the macrocell DL as in Fig. 1. By denoting ℱ = {𝒮𝑎 , ℳ} = {{1, . . . , 𝑆𝑎 } , {𝑆𝑎 + 1, . . . , 𝑆𝑎 + 𝑀 }}, where SAP and MUE indices set is 𝒮𝑎 and ℳ, respectively, the received signal within this resource block at the 𝑗th receiver is ∑ 𝑦𝑗 = v𝑗𝐻 h𝑗 𝑥𝑗 + v𝑘𝐻 h𝑗 𝑥𝑘 + 𝑛𝑗 , (1) 𝑘∈ℱ ∖𝑗
where h𝑗 ∈ℂ𝑁 ×1 is the channel state vector which includes the fading gain and pathloss components and v𝑗 ∈ℂ𝑁 ×1 is the beamforming vector from the MBS to the 𝑗th receiver. 𝑥𝑗 is the message } for the 𝑗th receiver with unit average power, { intended i.e., 𝔼 𝑥𝑗 𝑥∗𝑗 = 1 and 𝑛𝑗 is a circular symmetric complex additive white Gaussian noise (AWGN) at the 𝑗th receiver, which follows a normal distribution 𝒞𝒩 (0, 𝑁0 ), where 𝑁0 is the noise power over the allocated spectrum. Treating
In this subsection, we introduce the expression to compute the WAEE of the considered system. Let us first define the total power consumption model, denoted as 𝑃tot (v, p, b): ∑ 2 ad 𝑃tot (v, p, b) = 𝜅𝑚 ∥v𝑖 ∥ + 𝑃𝑚 + 𝑃 dec 𝑚 𝑖∈ℱ
+ 𝜅𝑠
∑ ∑ 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢
𝑃 circ
∑𝑚 ( ) dec 𝑏𝑖 𝑃𝑠ad + 𝑃𝑠,𝑖 , 𝑝𝑖𝑗 +
(5)
𝑖∈𝒮𝑎
where 𝜅𝑚 , 𝜅𝑠 > 1 are the constants which account for the inefficiency of the power amplifier at the MBS and each ad SAP when they transmit. In the macrocell power term, 𝑃𝑚 is defined as the power for dissipating the transmitting filter, mixer, frequency synthesizer and digital-to-analog converter dec the power and keeping the MBS circuit active, while 𝑃𝑚 ad for decoding at the MBS. For simplicity, we assume that 𝑃𝑚 remains constant at all time. Moreover, since we consider the DL of the MBS, we also assume the MBS always decodes the dec is same amount of information at all time slots so that 𝑃𝑚 also constant. On the other hand, to address the SAP selection, we introduce the binary variables 𝑏𝑖 = {0, 1} , ∀𝑖 ∈ 𝒮𝑎 to represent the status of the SAP, where 𝑏𝑖 = 1 means that the
𝑖th SAP is active to receive backhaul data and transmit to its SUEs and 𝑏𝑖 = 0 otherwise. It is worth to note that when 𝑏𝑖 = 1, an amount of power 𝑏𝑖 𝑃𝑠ad is consumed at the 𝑖th SAP. Then, we must ensure the relationship that when 𝑏𝑖 = 0, v𝑖 = 0 and 𝑝𝑖 = 0 for 𝑖 ∈ 𝒮𝑎 . This condition can be easily met by introducing the following two convex constraints: 2 ∥v𝑖 ∥ ≤ 𝑏𝑖 𝑃max and 𝑝𝑖𝑗 ≤ 𝑏𝑖 𝑝max . More importantly, the dec , should be considered more carefully in decoding term 𝑃𝑠,𝑖 the small cell tier. Since each 𝑖th SAP receives backhaul data from the MBS via the wireless channel, it should decode the received signals before transmitting them to its SUEs. Thus, based on the consensus that the decoding power used for this process is proportional to the achievable rate [10] 𝑅𝑖 at each 𝑖th SAP, we formulate the decoding power as a linear function of the achievable rate, e.g., dec (v) = 𝛼𝑖 𝑅𝑖 (v) , ∀𝑖 ∈ 𝒮𝑎 . 𝑃𝑠,𝑖
(6)
With the above analysis, we propose an expression for computing a WAEE, defined as the ratio between the weighted sum achievable rate at all the MUEs and SUEs over the total power consumed at the MBS and SAP to achieve the above rate. In particular, the WAEE (computed in bits/Joule/Hz) can be expressed as ∑ ∑ ∑ 𝜔𝑅 𝑖∈ℳ 𝑅𝑖 (v) + 𝜔𝑟 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 𝑟𝑖𝑗 (p) . 𝜂 (v, p, b) = 𝑃tot (v, p, b) (7) III. WAEE O PTIMIZATION P ROBLEM A. Problem formulation The constrained optimization problem of maximizing the WAEE can be formulated as follows: max
𝜂 (v, p, b) ∑ 𝑟𝑖𝑗 (p) , ∀𝑖 ∈ 𝒮𝑎 , s.t. 𝑅𝑖 (v) ≥
v,p≥0,b
∑
(8a) (8b)
𝑗∈𝒮𝑢 2
∥v𝑘 ∥ ≤ 𝑃max ,
(8c)
𝑝𝑖𝑗 ≤ 𝑏𝑖 𝑝max , ∀𝑖 ∈ 𝒮𝑎 ,
(8d)
𝑘∈ℱ
∑
𝑗∈𝒮𝑢 2
∥v𝑖 ∥ ≤ 𝑏𝑖 𝑃max , ∀𝑖 ∈ 𝒮𝑎 , 𝑏𝑖 = {0, 1} , ∀𝑖 ∈ 𝒮𝑎 ,
(8e) (8f)
The property of the WB networks in the optimization problem (8) is introduced in constraint (8b) to imply that each SAP backhaul achievable rate should always exceed the corresponding DL rate at each SAP via WAC. The backhaul rate in this case becomes a limiting factor for the access rate at the SAP. Even if there is good channel conditions on the DL of the SAP, without sufficiently good WB link, this SAP cannot achieve higher rate than that regulated by the backhaul rate, and thus restrains the optimal WAEE. (8b) also poses a complicated coupled relationship between v and p by nonlinear non-convex functions, e.g., 𝑅𝑖 (v) , 𝑟𝑖𝑗 (p), which makes it more difficult to solve compared to conventional minimum constant SINR constraint used in MU-MISO networks. In addition, (8c)-(8d) are the maximum budget power constraints at the MBS, SAPs,
in which the MBS and SAP cannot transmit more than 𝑃max and 𝑝max , respectively. The following proposition is provided to characterize the relationship between the achievable rate of the WBC and WAC. Proposition 1. Given the WAEE constrained optimization as in (8), all the inequalities (8b) become equalities at optimality. Proof: We prove Proposition 1 by contradiction. Assuming that at the optimal solution v★ , p★ , (8b) occur at strict inequalities at some index 𝑖 ∈ 𝒮𝑎 . Then, we can find a new ˜ 𝑖 = 𝜖𝑖 v𝑖★ , 𝜖𝑖 < 1 so that 𝑅𝑖 (v) will slightly decrease value v to occur at equality. However, the reduction of v𝑖★ leads to the improvement of the objective function (8a), which contradicts the assumption of optimality. This completes the proof. It is obvious to see that (8) is a combinatorial non-convex problem which is NP-hard in general. This is because of the appearance of 𝑅𝑖 (v) and 𝑟𝑖𝑗 (p) as function of variables v and p and the introduction of binary variables b. In addition, 𝑅𝑖 (v) and 𝑟𝑖𝑗 (p) are neither convex nor concave functions with respect to their variables since we consider the effect of interference in each rate formula. Further, even if we relax the binary variables to be continuous within the interval [0, 1], the relaxed problem is still non-convex, which is still very difficult to solve. Moreover, 𝑅𝑖 (v) appears in the denominator of (8a) which adds a non-convex factor to the optimization problem so that previous known methods for EE problem in [6], [7], [11], [13] cannot be directly employed here. Towards this end, we take advantage of the result ∑ in Proposition 1 where (p) , ∀𝑖 ∈ 𝒮𝑎 at optimality, the event 𝑅𝑖 (v) = 𝑗∈𝒮𝑢 𝑟𝑖𝑗∑ dec occurs to instead compute 𝑃𝑠,𝑖 as a function of 𝑗∈𝒮𝑢 𝑟𝑖𝑗 (p), ∑ dec = 𝛼𝑖 𝑗∈𝒮𝑢 𝑟𝑖𝑗 (p) . Hence, we propose another e.g., 𝑃𝑠,𝑖 optimization problem which is given as follows: ∑ ∑ ∑ 𝜔𝑅 𝑖∈ℳ 𝑅𝑖 (v) + 𝜔𝑟 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 𝑟𝑖𝑗 (p) max (9a) v,p≥0,b 𝑃˜tot (v, p, b) s.t. (8b) − (8d).
(9b) ∑
2 circ = 𝜅(𝑚 𝑖∈ℱ ∥v𝑖 ∥ + 𝑃𝑚 where 𝑃˜tot (v, p, b) )+ ∑ ∑ ∑ ∑ ad 𝜅𝑠 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 𝑝𝑖𝑗 + 𝑖∈𝒮𝑎 𝑏𝑖 𝑃𝑠 + 𝛼𝑖 𝑗∈𝒮𝑢 𝑟𝑖𝑗 (p) . The purpose of visiting this problem will be helpful in the next section to facilitate the development of the low complexity algorithm.
IV. L OW COMPLEXITY FOTCA- BASED ALGORITHM By observing that the problem in (9) is a combinatorial non-convex problem, we first transform it into another equivalent form that is more tractable. This tractability reveals more apparent form of the hidden non-convex factors that causes the difficulties in solving (9). Then, an unified FOTCA technique is employed to convexify all the existing nonconvex functions. The approximated problem now arrives at the form of generalized mixed integer convex optimization problem, which is still very hard to solve. To solve it, we propose to approximate the exponential cone constraints by SOC constraints so that the mixed integer convex optimization problem can be rewritten as a MISOCP, which can be solved more efficiently by dedicated modern MISOCP solvers like
MOSEK. The following proposition is provided to transform (9) into a more equivalent tractable form to develop the low complexity algorithm to achieve the solution. Proposition 2. By introducing slack variables 𝑡𝑚 𝑖 ≥ 0, ∀𝑖 ∈ ℱ, 𝑠𝑖𝑗 ≥ 0, 𝑡𝑠𝑖𝑗 ≥ 0, ∀𝑖 ∈ 𝒮𝑎 , ∀𝑗 ∈ 𝒮𝑢 , 𝜏 ≥ 0, problem (9) is equivalent to the following optimization problem ⎛ ⎞ ∑ ∑ ∑ max ⎝𝜔𝑅 (10a) 𝑡𝑚 𝑡𝑠𝑖𝑗 ⎠ ⋅ 𝜏 𝑗 + 𝜔𝑟
and (10f) are on the greater side of the inequalities and share 2 the same form of function 𝑓 (𝑦, 𝑥) = ∣𝑦∣𝑥 , ∀𝑦 ∈ ℂ𝑁 , 𝑥 ∈ ℝ+ . In particular, we can rewrite (10f) and (10b) as ∑ ( ) v𝑘𝐻 h𝑖 2 + 𝑁0 , ∀𝑖 ∈ ℱ, 𝑓 v𝑖𝐻 h𝑖 , 𝜈𝑖 ≥ (13) 𝑘∈ℱ ∖𝑖
𝑓 (1, 𝜏 ) ≥ 𝑃˜tot (v, p, b, s) .
(14)
(10d)
At this point, we apply a generic first order Taylor approximation to approximate all the 𝑓 (⋅, ⋅) function in (10b) and (10f). For example, the function 𝑓 (𝑦, 𝑥) can be approximated around the point of 𝑦 (𝑛) , 𝑥(𝑛) by (𝑛) 2 ) 2R (𝑦 (𝑛) 𝑦 ) ( 𝑦 (𝑛) (𝑛) = − (𝑛) 2 𝑥. (15) 𝐹 𝑦, 𝑥, 𝑦 , 𝑥 (𝑛) 𝑥 (𝑥 )
𝑠𝑖𝑗 ≥ log (1 + 𝛾𝑖𝑗 (p)) ≥ 𝑡𝑠𝑖𝑗 , (𝑖, 𝑗) ∈ (𝒮𝑎 , 𝒮𝑢 ) , (10e) 𝐻 2 v h𝑖 𝑖 ≥ 𝜈𝑖 , ∀𝑖 ∈ ℱ, (10f) ∑ 𝐻 2 v 𝑘 h 𝑖 + 𝑁0
Finally, we return at the non-convex constraint (10e). By explicitly extracting the formula of 𝛾𝑖𝑗 (p)and write log (1 + 𝛾𝑖𝑗 (p)) as the difference of two logarithmic functions, we can rewrite this constraint into two separate constraints as
Π
𝑗∈ℳ
𝑖∈𝒮𝑎 𝑗∈𝒮𝑢
˜tot (v, p, b, s) ≤ 1 , s.t. 𝑃˜ 𝜏 log (1 + 𝜈𝑖 ) ≥ 𝑡𝑚 𝑖 , ∀𝑖 ∈ ℱ, ∑ 𝑡𝑚 𝑠𝑖𝑗 , ∀𝑖 ∈ 𝒮𝑎 , 𝑖 ≥
(10b) (10c)
𝑗∈𝒮𝑢
𝑘∈ℱ ∖𝑖
(8c) − (8f)
(10g)
2 circ ˜tot (v, p, b, s) = 𝜅𝑚 ∑ where 𝑃˜ + 𝑃𝑚 + [ ∥v𝑖 ∥ ∑ ] ∑ ∑ ∑ 𝑖∈ℱ ad 𝜅𝑠 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 𝑝𝑖𝑗 + 𝑃 + 𝑠 𝑏 , 𝑖 𝑖𝑗 𝑠 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 Π = {v, p ≥ 0, 𝝂 ≥ 0, s ≥ 0,}t ≥ 0, 𝜏 ≥ 0}, { 𝑠 ; s = t = 𝑡𝑚 , 𝑡 , ∀𝑖 ∈ ℱ, ∀𝑗 ∈ 𝒮𝑎 , ∀𝑘 ∈ 𝒮𝑢 𝑖 𝑗𝑘 {𝑠𝑗𝑘 , ∀𝑗 ∈ 𝒮𝑎 , ∀𝑘 ∈ 𝒮𝑢 }; and 𝝂 = {𝜈𝑗𝑘 , ∀𝑗 ∈ 𝒮𝑎 , ∀𝑘 ∈ 𝒮𝑢 }.
Proof: The sketch of the proof can be made by following similar steps for the proof in Proposition 1 to prove the equalities at (10b), (10c), (10d),(10e) and (10f). The details are omitted here for brevity. We observe that the problem in (10) is non-concave due to the non-convex objective function and the non-convex constraints (10b), (10e), and (10f). First, we note that the objective function (10a) is in the form of the product of two linear functions, which are neither a convex nor concave functions. However, this form has a hidden convexity, which can be revealed as follows. By introducing a slack variable 𝑞 ≥ 0, we can easily rewrite (10) as max 𝑞 2
Π,𝑞≥0
s.t. 𝑔 (t) ⋅ 𝜏 ≥ 𝑞 2 ,
(11a)
) ( 2 𝑠𝑖𝑗 + log (𝐼𝑖𝑗 + 𝑁0 ) ≥ log 𝑝𝑖𝑗 ∣ℎ𝑖𝑖𝑗 ∣ + 𝐼𝑖𝑗 + 𝑁0 , (16a) ) ( 2 𝑡𝑖𝑗 + log (𝐼𝑖𝑗 + 𝑁0 ) ≤ log 𝑝𝑖𝑗 ∣ℎ𝑖𝑖𝑗 ∣ + 𝐼𝑖𝑗 + 𝑁0 , (16b) It is obvious to state that both (16a) and (16b) are non-convex 2 𝑆𝐼 constraints. Thus, by denoting 𝑔𝑖𝑗 (p) = 𝑝𝑖𝑗 ∣ℎ𝑖𝑖𝑗 ∣ + 𝐼𝑖𝑗 + 𝑁0 𝐼 and 𝑔𝑖𝑗 (p) = 𝐼𝑖𝑗 +𝑁0 , we (apply the)first order ( 𝐼 Taylor ) approxi𝑆𝐼 (p) , log 𝑔𝑖𝑗 (p) around the mation to approximate log 𝑔𝑖𝑗 point p(𝑛) as ( (𝑛) ) 𝑆𝐼 𝑆𝐼 p 𝑔𝑖𝑗 (p) − 𝑔𝑖𝑗 ( ) p, p = log p + , 𝑆𝐼 (𝑛) 𝑔𝑖𝑗 p (17a) ( ) ( ( )) 𝑔 𝐼 (p) − 𝑔 𝐼 (p(𝑛) ) 𝑖𝑗 𝐼 ( 𝑖𝑗 ) p(𝑛) + , 𝐺𝐼𝑖𝑗 p, p(n) = log 𝑔𝑖𝑗 𝐼 𝑔𝑖𝑗 p(𝑛) (17b)
𝐺𝑆𝐼 𝑖𝑗
(
(𝑛)
)
(
𝑆𝐼 𝑔𝑖𝑗
(
(𝑛)
))
By applying the approximations (15) and (17) into their corresponding non-convex constraints in (10b), (10d) and (10f), problem (9) can be approximated at the (𝑛 + 1)th iteration as max 𝑞
Π,𝑞≥0
(11b)
(10b) − (10g). (11c) ∑ ∑ ∑ 𝑠 where we write 𝑔 (t) = 𝜔𝑅 𝑗∈ℳ 𝑡𝑚 𝑗 + 𝜔𝑟 𝑖∈𝒮𝑎 𝑗∈𝒮𝑢 𝑡𝑖𝑗 for convenience. Now, the current optimization problem is a maximization of new variable 𝑞 2 , which still remains the nonconvex property. However, we can equivalently rewrite this objective function as the maximization of a linear (or concave) function 𝑞 [14]. Moreover, (11b) can be rewritten into a second order cone (SOC) constraint as √ 2 𝜏 + 𝑔 (t) (𝜏 − 𝑔 (t)) ≥ 𝑞2 + . (12) 2 4 Next, by closer looking at the constraints (10b) and (10f), we find that the factors that cause the non-convexity of (10b)
s.t. (𝜏 + 𝑔 (t)) /2 ≥ ∥𝑞, (𝜏 − 𝑔 (t)) /2∥ ) ( 𝐹 1, 𝜏, 𝜏 (𝑛) ≥ 𝑃˜tot (v, p, b, s) , ) ( (𝑛) (𝑛) ≥ 𝐹 v𝑖𝐻 h𝑖 , 𝜈𝑖 , v𝑖 , 𝜈𝑖 ∑ 2 v𝑘𝐻 h𝑖 + 𝑁0 , ∀𝑖 ∈ ℱ,
(18a) (18b) (18c)
(18d)
𝑘∈ℱ ∖𝑖
𝑠𝑖𝑗 + log (𝐼𝑖𝑗 + 𝑁0 ) ≥ ( ) (𝑛) 𝐺𝑆𝐼 p, p , ∀𝑖 ∈ 𝒮𝑎 , ∀𝑗 ∈ 𝒮𝑢 , 𝑖𝑗 ) ( 2 log 𝑝𝑖𝑗 ∣ℎ𝑖𝑖𝑗 ∣ + 𝐼𝑖𝑗 + 𝑁0 − 𝑡𝑖𝑗 ≥ ( ) 𝐺𝐼𝑖𝑗 p, p(𝑛) , ∀𝑖 ∈ 𝒮𝑎 , ∀𝑗 ∈ 𝒮𝑢 , (10c), (10d), (10g).
(18e)
(18f) (18g)
We observe that without constraints (10c), (17a), and (18f), (18) is a MISOCP problem. This is because these convex constraints contain the form log (𝑥) ≥ 𝑦, which can be equivalently reformulate as 𝑥 ≥ 𝑒𝑦 and reveals (18) as a general mixed integer convex programming. At this point, we are motivated to approximate (10c), (17a), and (18f) with corresponding SOC constraints like in [15] so that (18) becomes a MISOCP and can be solved much more efficiently by a dedicated MISOCP solver like MOSEK. Note that the accuracy of this approximation can be achieved within a controlled error since we approximate 𝑥 ≥ 𝑒𝑦 by 2−𝑚 𝑦 + 5/3 √ 19 , ≤ 𝜅1 , 2 72 ( ) 2 1 + 2−𝑚 𝑦 , (𝜅2 − 1) ≤ 𝜅2 + 1, √ (19) 𝜅1 , 𝜅2 / 24 ≤ 𝜅3 , ∥2𝜅2+𝑖 , (𝜅3+𝑖 − 1)∥ ≤ 𝜅3+𝑖 + 1, 𝑖 = 1, . . . , 𝑚, 𝜅3+𝑚 + 1 ≤ 𝑥 where 𝜅𝑖 , ∀𝑖 = 1, . . . , 3+𝑚 are the newly introduced variables and 𝑚 is the given parameter that control the accuracy of the approximation. For a given sufficiently large 𝑚, the system of constraint }(19) can approximate the hypograph of { (𝑥, 𝑦) ∈ ℝ2 ∣𝑥 ≥ 𝑒𝑦 to an arbitrary accuracy 𝜖. Here, we choose 𝑚 = 8. From [15], when 𝑥 ∈ [0, 𝑥 ¯] and 𝑥 ≥ 𝑒𝑦 , there exists {𝑥, 𝑦, 𝜅1 , . . . , 𝜅3+𝑚 } that satisfies (19). In addition, } { if 𝑥 ∈ [0, 𝑥 ¯] and the solution set (𝑥, 𝑦) ∈ ℝ2 ∣𝑥 ≥ 𝑒𝑦 is able to be extended by {𝑥, 𝑦, 𝜅1 , . . . , 𝜅3+𝑚 } for some 𝑚 that satisfies (19), then 𝑒𝑦 − 𝜖 ≤ 𝑥 ≤ 𝑒𝑦 + 𝜖. After applying this SOC approximation, we are able to solve (8) by applying Algorithm 1. Note that the protocol of Algorithm 1 is based on the availability of MOSEK to iteratively solve the MISOC problem (18). In addition, v(𝑛) , 𝝂 (𝑛) , p(𝑛) , 𝜏 (𝑛) in (18) are not the variables of the optimization problem but the parameters that are iteratively updated by the optimal solution after each iteration. The pseudo code presenting the algorithm to solve (9) is summarized in Algorithm 1. Algorithm 1 Iterative Low Complexity Algorithm. 1: Initialize starting points of v(𝑛) , 𝝂 (𝑛) , p(𝑛) , 𝜏 (𝑛) ; 2: Set 𝑛 := 0; 3: repeat 4: Solve the convex problem in (18) to achieve the optimal
solution v★ , p★ , 𝝂 ★ , s★ , t★ , 𝜏 ★ , 𝑞 ★ ;
5: Set 𝑛 := 𝑛 + 1; 6: Update v(𝑛) = v★ , 𝝂 (𝑛) = 𝝂 ★ , p(𝑛) = p★ , 𝜏 (𝑛) = 𝜏 ★ ; 7: until Convergence
0.15
AEE (bits/Joule/Hz)
Algorithm 1 Dinkelbach approach
0.12
S=4
0.09
S=2
0.06 0.03 0.00
1
200
400
Total iteration number
Fig. 2. Convergence of the proposed algorithm.
600
Convergence Analysis: The proof of convergence is briefly presented here for the sake of completeness. Let 𝑓 (𝑛) denote the optimal objective value and Ω(𝑛) denote the set of optimal solutions at the 𝑛th iteration of Algorithm 1. Due to the linear approximation in (15), (17a) and (17b), the updating rules in Algorithm 1 ensures that Ω(𝑛) is a feasible solution to the problem (18) at step 𝑛 + 1. This subsequently leads to the results of 𝑓 (𝑛+1) ≥ 𝑓 (𝑛) , which means that Algorithm 1 generates a non-decreasing sequence of objective values. Due to the bounded power constraints, Algorithm 1 guarantees to converge. V. N UMERICAL R ESULTS In this section, we show the performance of the proposed algorithm. We consider the case where 𝑆 = 2, 4 or 6 small cells, while we assume 𝑀 = 2 MUEs, 𝑁 = 4 transmit antenna, 𝑃max = 50 dBm, and 𝑝max = 30 dBm. Furthermore, circ ad = 30 dBm, 𝑃𝑠,𝑖 = 30 for the power model, we choose 𝑃𝑚 dBm, 𝜅𝑚 = 𝜅𝑠 = 5 and 𝛼𝑖 = 1, ∀𝑖 ∈ 𝒮𝑎 [11], [13]. Throughout the numerical section, we compare the network performance between the following schemes: ∙ A: maximization of WAEE using adaptive decoding power model without SAP selection. ∙ B: maximization of WAEE using fixed decoding power. Note that Scheme B with fixed decoding power assignment and no SAP selection can be seen similar to the traditional works in the literature [6], [7], [11], [13]. ∙ C: maximization of WAEE using adaptive decoding power model as in (6) and SAP selection. ∙ D: maximization of weighted access sum rate (WASR). The problem of maximizing WASR is similar to (9), except that the objective function is only the numerator of (9a). The solution of the WASR problem is achieved through Algorithm 1 and then used to evaluate the achieved WAEE by plugging it into (7). In Fig. 2, we show the convergence of the objective function in (9) by applying Algorithm 1 with 𝑆 = 2 or 4. In addition, we also compare these algorithms with the traditional approach using Dinkelbach method to solve the fractional nonlinear problem [6], [7], [13]. Note that with Dinkelbach method, the fractional objective function in (9) is transformed into a subtractive form with additional fixed parameter, e.g., 𝜃(𝑡) . The process of solving the new problem with subtractive form can be briefly described in two folds: first, we fix 𝜃(𝑡) and apply the similar convex approximation in Section IV to iteratively solve the approximated problem and update corresponding parameters; second, we update 𝜃(𝑡) and repeat the first step. As shown in the figure, we observe that the result of the objective functions from Dinkelbach approach and Algorithm 1 keep increasing and converge after few iterations. However, the Dinkelbach approach requires more iterations to converge to the same solution as with Algorithm 1. The reason is that in our proposed algorithm, we aim to directly tackle the fractional objective function without introducing new parameter, which helps in saving more time to achieve the final result. This shows the efficiency of our proposed method to achieve final solution with much less computational complexity than traditional approach.
0.15
towards that SAP for WBC. Note that in all cases, Scheme D uses most of the maximum budget power at the MBS 𝑃max and each SAP 𝑝max to maximize the ASR. This explains the low WAEE performance of Scheme D in Fig. 3.
Scheme A dec Scheme B - P = 30 dBm Scheme C Scheme D s,i
AEE (bits/Joule/Hz)
0.12
0.09
VI. C ONCLUSION
0.06
0.03
0.00 10
25
30
35
Pmax (dBm)
40
45
50
Fig. 3. Achieved WAEE w.r.t. 𝑃max at different schemes. 40.0
P (dBm)
30.0
Scheme A; Scheme C;
Scheme B - P Scheme D
dec s,i
= 30 dBm
20.0
to t
10.0 0.0
4
Number of small cell
This paper studies the joint design of transmit beamforming, power allocation and SAP selection that maximizes the WAEE on the DL of the two-tier WB small cell HetNets. An important impact of decoding power and SAP selection at each small cell is considered. To overcome the difficulty in solving the non-convex problem, a low complexity algorithm based on FOTCA and error controlled SOC is proposed, in which an approximated problem is iteratively solved optimally by MISOCP dedicated solver until convergence. Compared to traditional nonlinear fractional programming method, our proposed algorithm performs a much better convergence behavior. Moreover, rather than using fixed decoding power in conventional power model, our proposed EE model with adaptive decoding power achieves a better WAEE since more power usage is conserved by appropriately addressing the importance of adaptive decoding power in the considered model.
6
Fig. 4. Achieved 𝑃˜tot w.r.t. number of small cell 𝑆.
In Fig. 3, we show the achieved WAEE with respect to 𝑃max by applying Algorithm 1 for Schemes A, B, C, and D. From the figure, we observe that when 𝑃max increases, WAEE values of Schemes A, B, and C increase in the beginning and then saturate. Another observation is that the achieved WAEE from Scheme C outperforms other scenarios. This is because when there are some small cells achieving low rate circ to stay ON, while they still need to maintain power 𝑃𝑠,𝑖 it is better to switch OFF these SAPs to save the circuit power so that less total power is consumed to achieve high WAEE. In addition, the solution from Scheme A provides better WAEE than Scheme B. This can be explained as when the SAP is unaware of the amount of power needed for the decoding process, it should apply redundant power to make sure sufficient information will be decoded. Hence, this results in power usage inefficiency and degrades the achieved WAEE. On the other hand, the achieved WAEE of Scheme D shows a very low performance compared to others. This is because with the WASR maximization problem, the MBS and SAPs attempt to use sufficient power to attain the maximum sum rate; thus when 𝑃max becomes large, WAEE will start to decrease. To further investigate the behavior of overall power consumption, in Fig. 4, we evaluate 𝑃˜tot with the solution achieved by applying Algorithm 1 to solve problem (9) to compare the performance of Schemes A, B, C, and D. The result is shown at two level of 𝑆 = 4 or 6 at 𝑃max = 50 dBm. By observing Fig. 4, we see that Scheme C always consumes less power than Scheme A, B and D. This is due to the fact that in Scheme C, the inefficient SAPs are switched off to save not only transmission power but also the circuit power, decoding power and the transmit beamforming power from the MBS
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