Interference Control via AP Suppression and Power Allocation . 70

4.1 Downlink Interference Control of LAA-LTE for Coexistence with Asym-

4.1.5 Interference Control via AP Suppression and Power Allocation . 70

4.1.5.2 Problem Formulation Let us consider the power allocation on subcarriers in the eNB. Our interference control scheme is to control the power on each subcarrier of the eNB for increasing the interference level to AHAPs while maximizing the average sum-rate of the UE associated with the eNB.

Note that the channel matrices H_eNB,APα and H_APα,UE are functions of the random vari- ables τ and τ1, respectively. Assuming that the difference of the starting points of the OFDM symbols of LAA-LTE and Wi-Fi is uniformly distributed, i.e., τ₁ ∼ Uniform[0,3.3 µs] and τ ∼Uniform[−T_total^W , T_total^L −T_total^W ], we get the SINR on subcarrier l averaged over the timing difference as

SINR_l=

p_L,l

h^(l)_eNB,UE

2

P

α∈A_AH N_FFT^W

P

w=1

p_W,wγ_AP^w,l

α,UE+N₀ ,

whereγ_AP^w,l

α,UE is defined by γ_AP^w,l

α,UE,E_τ1

[H_APα,UE]_(w,l)

2

=E_τ1 h

F_LH˜_APα,UEK_AP,UEi

(w,l)

2

=π^T_α,wEτ1

n θ_lθ^H_l

o π^∗_α,w, where π_α,w = row_w

F_LH˜_APα,UE

and θ_l = col_l K_AP,UE

; here, row_i(·) and col_j(·) denote the i-th row vector and j-th column vector of a matrix, respectively, and ·^H and ·^∗ denote the transpose conjugate and conjugate operations, respectively. Similarly, (61) averaged overτ can be rewritten as

J_α,

N_FFT^W

X

w=1 N_FFT^L

X

l=1

E_τ (

p_L,l h

F_WH˜_eNB,APαK_eNB,APi

(l,w)

2)

=

N_FFT^W

X

w=1 N_FFT^L

X

l=1

p_L,l·φ^T_α,lEτ

n ψ_wψ^H_w

o φ^∗_α,l,

whereφ_α,l =row_l

F_WH˜_eNB,APα

andψ_w =col_w K_eNB,AP

. Though a closed-form calculation ofE_τ1n

θ_lθ^H_l o

andE_τn

ψ_wψ^H_wo

is difficult, they can be readily obtained via numerical integra- tion and from the definitions of KAP,UE in (52) and KeNB,AP in (58) for uniformly distributed τ and τ₁.

Hence, the problem is formulated as

pL,cmax1...c|A|

N_FFT^L

X

l=1

log













 1+

p_L,l

h^(l)_eNB,UE

2

P

α∈A N_FFT^W

P

w=1

c_α·p_W,w·γ_AP^w,l

α,UE+N₀













 ,

s.t (p_L)^T·1_NL

FFT ≤P_L, (62a)

p_L0, (62b)

J_α ≥Γ_ED,∀α∈ A_E, (62c)

(1−c_α)·J_α≥(1−c_α)· Γ_ED,∀α∈ A_AH, (62d)

c_α ∈ {0,1},∀α∈ A_AH, (62e)

c_α = 0,∀α∈ A_E, (62f)

where Γ_ED is the ED threshold of Wi-Fi, 1N is an all-ones vector that has N ones, cα is an integer parameter to represent which AP is active in the eNB’s transmission. Ifc_α= 1, theα-th AP can start a transmission in the eNB transmission, andcα = 0 otherwise.

Eqs. (62a) and (62b) are the maximum power constraint and the non-negative power con- straint, respectively. In our scheme, the eNB controls the power on the subcarriers to increase the interference to AHAPs, which is represented in (62d). However, the interference control can cause reduced interference received at EAPs, which can change some of EAPs into AHAPs.

Hence, we need the constraint (62c) to ensure that the eNB remains exposed to the current EAPs while changing the current AHAPs to EAPs.

The formulation is a combinatorial problem due to binary parameters, cα,∀α ∈ A_AH. To make the problem tractable, it is decomposed into two parts, the AHAP selection module and the power allocation module. The first module is to determinecα,∀α∈ A_AH; that is, it determines which AHAPs are suppressed. In the second module, the transmit power for each subcarrier is obtained for givenc_α.

4.1.5.3 AHAP Selection Module The goal is to suppress as many AHAPs as possible to help the eNB avoid receiving interference from them. To this end, we first attempt to suppress all the AHAPs, i.e.,c_α = 1,∀α∈ A_AH. With that choice, the formulation can be rewritten as

maxpL

N_FFT^L

X

l=1

log







 1 +

p_L,l

h^(l)_eNB,UE

2

N₀









, (63a)

s.t (pL)^T·1_N^L

FFT ≤PL, (63b)

p_L0, (63c)

Jα ≥ΓED, α∈ A_E∪ A_AH. (63d)

(a) Polyhedra exists. (b) No polyhedra exists.

Figure 33: Depending on the channel on each subcarrier, the examples for the existence of the polyhedra are shown when two subcarriers are used in LAA-LTE.

Unfortunately, however, the problem may be infeasible. To be specific, we can rewrite the constraint (63d) as

J_α= (p_L)^T·v_eNB,APα ≥Γ_ED, (63e) where

v_eNB,APα =h

N_FFT^W

X

w=1

φ^T_α,1E_τn

ψ_wψ^H_wo φ^∗_α,1,

. . . ,

N_FFT^W

X

w=1

φ^T_α,NL FFTE_τn

ψ_wψ^H_wo φ^∗_α,NL

FFT

iT

.

Hence, all the constraints are closed half-space for p_L. If the set constituted by those constraints is not empty, i.e., feasible, the feasible set is the convex hull of a polyhedra. However, in Fig. 33, when the channel gainsPN_FFT^W

w=1 φ^T_α,lE_τn

ψ_wψ^H_wo

φ^∗_α,lfrom each LAA-LTE’s subcarrier to all Wi-Fi’s subcarriers are small, the set can be empty, which means that there can be no feasible set. Moreover, the more AHAPs an eNB tries to suppress, the lower the probability that a feasible set exists. Consequently, the case of empty feasible set indeed happens, since the eNB could have small channel gain to at least one AHAP.

Hence, the AHAP selection module needs to select which AHAPs should be suppressed in the eNB’s transmission, which is executed as follows. The module first creates a list including all possible AHAPs’ combinations. The number of AHAPs is|A_AH|, and hence there are(2^|AAH|−1) cases of AHAP sets. Let us denote the i-th AHAP set by B_i. The AHAP set list is then constructed as follows:

• Each element of the list consists of table index i, AHAP-index set B_i, and the sum of interference from the AHAPs in B_i to the UE associated with the eNB.

• The elements are sorted in descending order by the sum interference. If there are elements with the same interference value, the elements additionally are sorted by the number of AHAPs inB_i in descending order.

According to the above construction rules,B₁ will include the AHAPs’ combinations generating the highest sum-interference to the UE; namely,B₁ includes all the AHAPs. On the other hand, B₍₂|AAH|−1) includes a single AHAP causing the lowest interference to the UE. For given i, the proposed AHAP selection module setscα = 0, α∈ B_i in (15). In addition,cα= 1, α∈ A_AH\ B_i. That is, the module attempts to suppress the AHAPs in B_i while doing nothing for the rest of the AHAPs, i.e., AHAPs in A_AH\ B_i. Subsequently, the module checks if this choice is feasible when solving the problem (15). Specifically, fromi= 1toi= (2^|AAH|−1), the module attempts to solve the following feasibility problem:

Find p_L (64a)

s.t (p_L)^T1_NL

FFT≤P_L, (64b)

p_L0, (64c)

(p_L)^Tv_eNB,APα ≥Γ_ED, α∈ A_E∪ B_i. (64d) Starting fromi= 1, if the module fails to solve (64), it increasesiby 1 and repeats the feasibility check forB_i with updatedi. If the module succeeds in solving (64) for someˆi <2^|AAH|, it stops finding the feasible AHAP set, and the final feasible AHAP set is obtained as Bˆ=B_i. If there is no feasible case from the list, the eNB allocates an equal powerP_L/N_FFT^L to each subcarrier.

Via the AHAP selection module, the eNB can select as many AHAPs as possible, which can be suppressed together with the consideration of the interference from AHAPs to the UE.

The feasibility problem (64) is a convex problem, and thus can be evaluated by the semidef- inite programming (SDP). Though the number of searches in worst-case is (2^|AAH|−1), we will show that the number of actual searches is smaller than 10 in most cases as shown in Section 4.1.7.1.

4.1.5.4 Power Allocation Module For obtainedc_α,∀α∈ A, the original problem can be rewritten as

maxpL

N_FFT^L

X

l=1

log













 1+

pL,l

h^(l)_eNB,UE

2

P

α∈A N_FFT^W

P

w=1

cα·pW,w·γ_AP^w,l

α,UE+N0















, (65a)

s.t (p_L)^T1_NL

FFT ≤P_L, (65b)

p_L0, (65c)

(p_L)^Tv_eNB,APα ≥Γ_ED, α∈ A_E∪B.ˆ (65d)

Algorithm 1 Proposed Interference Control Algorithm Initialization

1) Set OFDM parameters in Table 8, the thresholdΓ_ED, the maximum transmit power,P_L and P_W,

and the transmit power vectorp_W,j=P_W/N_FFT^W ,∀j.

2) Obtain the channel matricesH_eNB,APα and H_APα,UE from the channel impulse responses.

3) Go to the AHAP Selection Module.

procedureAHAP Selection Module

1) Make the combination list based on AHAPs.

2) FindBˆfrom Table indexi= 0.

repeat

• i=i+ 1.

• Solve the feasibility problem (64) withB_i. untilThe feasibility problem is solved or

ireaches N_List.

3) Once the feasibility problem is solved,Bˆ=B_i, or if not,Bˆ=∅. Go to the power allocation module.

procedurePower Allocation Module if B 6=ˆ ∅then

•Solve the power allocation problem (65) withB,ˆ from whichp^∗_L is obtained.

else

• Each element ofp^∗_L is set to P_L/N_FFT^L . Terminate the algorithm.

The reformulated problem is a convex problem with respect to the variablep_Lsince the objective function is the sum of concave functions and the constraints constitute a polyhedra, a convex set. Therefore, this problem can be evaluated by the semidefinite programming to obtain the solution p^∗_L.

Finally, we propose an overall interference control algorithm in Algorithm 1, where NList is the last index of the combination list.