4.2 Downlink MU-MIMO LTE-LAA for Coexistence with Asymmetric Hidden
4.2.4 Proposed MU-MIMO LTE-LAA System
4.2.4.5 Beamforming Vector Design
have
FG
khWLi,eNBk2·l(r) | khWLi,eNBk2
(80)
=P
G≤ khWLi,eNBk2·l(r) | khWLi,eNBk2
=P
khWLi,eNBk2·l(R)≤ khWLi,eNBk2·l(r) | khWLi,eNBk2
=P
l(R)≤l(r) | khWLi,eNBk2
=FR
r | khWLi,eNBk2
, (81)
whereFG(·) is the cumulative density function (cdf ) of the short-term fadingG. SinceGfollows the Gamma distribution, from (80), the pdf of R is obtained as follows.
fR
r|khWLi,eNBk2
= µNT
khLWi,eNBk2·l(r)NT−1
·eµkhLWi,eNBk2·l(r)
Γ (NT) .
Recall that the eNB is placed at the origin(0,0), and the AP is located at(x, y). By transforming the polar coordinate form to a Cartesian coordinate form, we finally have
fX,Y
x, y| khWLi,eNBk2
=
fRp
x2+y2| khWLi,eNBk2
px2+y2 . (82) Since the APs are distributed within the coverage of the eNB with radius of reNB, we need to restrict the domain of fX,Y. For the truncated distribution, we can derive
fX,Y
x, y| khWLi,eNBk2, r≤reNB
=
fX,Y
x, y| khWLi,eNBk2 R
B(o,reNB)fX,Y
x, y| khWLi,eNBk2 dxdy
= (p
x2+y2)−1µNT
khWLi,eNBk2·l(r)NT−1
·eµkhWLi,eNBk2·l(r) Γ (NT)R
B(o,reNB)fX,Y
x, y| khWLi,eNBk2 dxdy
,
which proves the lemma.
In Section 4.2.5, the derived transmission probability shall be numerically evaluated, which turns out to be 97-percent accurate.
where B ⊂ AAH is the index set of chosen AHAPs which we want to change into EAPs. The best possible case is the case where the problem PB1 is feasible with the choice of B = AAH such that all the AHAPs can be changed into EAPs deferring their transmissions. However, this choice may make the problem PB1 infeasible in satisfying the constraint (83c). Thus, we need to carefully chooseB so that the problem is feasible while the SINRs of the UEs can be further improved. How to determine Bshall be discussed in Section 4.2.4.5.4 with the consideration of the feasibility ofPB1.
To combine the variables into a single vector in the problem PB1, we define the augmented beamforming vectorwe ∈CNT·N
L
sub·NUE×1 as
we,[(w1,1)T, . . . ,(w1,NUE)T, . . . ,(wNL
sub,1)T,. . . ,(wNL
sub,NUE)T]T.
Then, each beamforming vector can be expressed bywl,u=El,uw. Here,e El,u∈CNT×(NT·NsubL ·NUE) consists of all zeros except for the (l−1)·NUE·NT+ (u−1)·NT+ 1
-th to the (l−1)·NUE·NT+u·NT - th columns, which are equal to theNT×NT identity matrix.
By using w, the probleme PB1 can be transformed into max
we min
l,u
(w)e HAl,uwe
(w)e HDl,uwe (84a)
s.t (w)e HEel,uwe = 1,∀l, u (84b) (w)e HeJαwe ≥ΓED,∀α∈ AE∪ B, (84c) where
Al,u=pLl,u(El,u)Hh(l)eNB,u(h(l)eNB,u)HEl,u, Dl,u=X
j6=u
pLl,j(El,j)Hh(l)eNB,u(h(l)eNB,u)HEl,j+
(X
α∈AAH
NsubW
X
w=1
(1−cα)E[eα]·pWα,w|h(w,l)α,u |2+N0)IN
TNsubL NUE
NsubL NUE , Eel,u= (El,u)HEl,u,
eJα =X
u∈U NsubL
X
l=1 NsubW
X
w=1
pLl,u(El,u)Hh(l,w)eNB,α(h(l,w)eNB,α)HEl,u,
whereIN is theN ×N identity matrix.
By adding an auxiliary variable θ, the problem (84) can be rewritten as max
we θ (85a)
s.t (w)e HAl,uwe ≥θ(w)e HDl,uw,e ∀l, u, (85b) (w)e HEel,uwe = 1,∀l, u (85c) (w)e HeJαwe ≥ΓED,∀α∈ AE∪ B, (85d)
For given θ, we finally have
PB2 :Find Wf (86a)
s.t tr
Al,uWf
≥θtr
Dl,uWf
,∀l, u, (86b)
tr
Eel,uWf
= 1,∀l, u, (86c)
tr
eJαWf
≥ΓED,∀α∈ AE∪ B, (86d)
Wf 0, (86e)
rank
Wf
= 1, (86f)
where tr(·) denotes the trace operation, Wf =w(e w)e H, and Wf 0 implies that Wf is positive semidefinite.
4.2.4.5.2 Problem of the Conventional Rank-1 Approximation The problemPB2 is a convex problem with respect to the variable Wf without (86f), and can be solved by the semidefinite programming (SDP) to obtain the solution W. However, the solutionW without (86f) in general becomes a full rank matrix. The conventional rank-1 approximation [85] suggests to obtain the final solution satisfying the rank-1 constraint by approximating the solution as Wf∗ =λ1q1(q1)H where λ1 is the largest Eigenvalue of W and where q1 is the corresponding Eigenvector of λ1. Then, Wf∗ becomes the rank-1 matrix that is closest to W. However, in our problem, the condition (86c) cannot be met after this approximation, as we show in the following lemma.
Lemma 3 If the rank ofWis higher than 1, the approximated solutionWf∗ =λ1q1(q1)Hcannot satisfy the constraint (86c).
Proof 3 Since the matrix W is positive semidefinite by the constraint (86e), we have the Eigen decomposition of W as W =Prank(W)
i=1 λiqi(qi)H, where λi is thei-th Eigenvalue and qi is the corresponding Eigenvector. From Wf∗ = λ1q1(q1)H and by the linearity property of the trace operation, we obtain
tr(W) =
rank(W)
X
i=1
tr
λiqi(qi)H
=tr(Wf∗) +
rank(W)
X
i=2
tr
λiqi(qi)H
=tr(Wf∗) +
rank(W)
X
i=2
λikqik2>tr(Wf∗), (87) where the inequality in (87)follows from the fact that rank(W)≥2and that non-zero Eigenvalues of a positive semidefinite matrix are positive. Then, we have
tr(W) =X
u∈U NsubL
X
l=1
tr(Eel,uW)>tr(Wf∗), (88)
where the inequality follows from (87). Since W is the solution satisfying (86c), Eq. (88) can be rewritten as
tr(Wf∗)<|U | ·NsubL , (89) where|U | denotes the number of users. IfWf∗ satisfies (86c), then we have tr(Wf∗) =|U | ·NsubL , which contradicts (89). Therefore, Wf∗ =λ1q1(q1)H cannot satisfy (86c).
The failure to satisfy the constraint (86c) implies that beamforming vectors wl,u,∀l, u have not been properly constructed as intended. This issue is more clearly explained in the sequel.
We start with the definition Wl,u = El,uW(El,u)H such that Wl,u ∈ CNT×NT. In addition, λ(l,u),i and q(l,u),i ∈ CNT×1 denote the i-th Eigenvalue and the corresponding Eigenvector of Wl,u, respectively. The following lemma provides insights on the problem of the conventional rank-1 approximation.
Lemma 4 For any given l∈[1, NsubL ]and u∈ U, let us construct the vector qˆj as ˆ
qj= [ 0, . . . ,0
| {z }
(l−1)NUENT+(u−1)NT
,(q(l,u),i)T, 0, . . . ,0
| {z }
NT{NsubL NUE−(l−1)NUE−u}+1
]T. (90)
Then, qˆj becomes an Eigenvector ofW, and the corresponding Eigenvalue of W is λ(l,u),i. Proof 4 From (90), q(l,u),i =El,uqj or qj = (El,u)Hq(l,u),i. Sinceq(l,u),i is the Eigenvector of Wl,u, we get
Wl,uq(l,u),i=λ(l,u),iq(l,u),i. (91) At this point, note that all the constraints in (86)are calculated only by the block diagonal terms of W. Hence, without loss of generality, the solution of the problemf (86)without (86f), i.e.,W, can be formed as a block diagonal matrix as
W =bldg
W1,1, . . . ,W1,NUE, . . . ,Wl,u, . . . , WNL
sub,1, . . . ,WNL
sub,NUE , (92)
where bldg{X1,X2, . . .} denotes the block diagonal matrix, the block diagonal terms of which are {X1,X2, . . .}. Then, we further have
Wˆqj = [ 0, . . . ,0
| {z }
(l−1)NUENT+(u−1)NT
,(Wl,uq(l,u),i)T, 0, . . . ,0
| {z }
NT{NsubL NUE−(l−1)NUE−u}+1
]T
= [0, . . . ,0,(λ(l,u),iq(l,u),i)T,0, . . . ,0]T (93)
=λ(l,u),iˆqj, (94)
where (93)follows from(91). Therefore, by (94),qˆj is an Eigenvector ofWwith the Eigenvalue of λ(l,u),i, which proves the lemma.
In fact, it can be shown by numerical simulations that q1, the Eigenvector of W associated with the largest Eigenvalue, is indeed obtained in the form of (90) in most of the simulations.
Thus, choosingq1 as the final solution gives us most of the beamforming vectors having all zero elements, thus failing to meet (86c).
4.2.4.5.3 Proposed Rank-1 Approximation The goal is to obtain a non-zero ap- proximated solution for the beamforming vector for each subcarrier at each UE satisfying (86c). Notice that W obtained by solving the problem (86) is a Hermitian matrix, and hence Wl,u=El,uW(El,u)H is also a Hermitian matrix from
(Wl,u)H= (El,uW(El,u)H)H=El,uW(El,u)H=Wl,u.
Instead of obtaining the augmented beamforming vector for all the subcarriers and UEs, we propose to determine each beamforming vector we∗l,u for given l and u such that wel,u∗ (we∗l,u)H is close to Wl,u. Therefore, we choose
we∗l,u=q(l,u),1, (95)
whereq(l,u),1 is the Eigenvector ofWl,u associated with the largest Eigenvalueλ(l,u),1.
With this choice, we havekwe∗l,uk2 = 1,∀l, usincekq(l,u),ik2 = 1, thus satisfying the constraint (86c).
Based on the approximation we∗l,u =q(l,u),1, the final step is to find the maximum possible θ in the problem (86), which results in a feasible solution of we∗. This calculation can be easily done by the bisection line search.
4.2.4.5.4 Feasibility Check To find Bsuch that PB1 is feasible, we conduct feasibility check by solving the following problem:
PFeasibility of B1:Find Wf (96a)
s.t tr
Eel,uWf
= 1,∀l, u (96b)
tr
eJαWf
≥ΓED,∀α∈ AE∪ B, (96c)
Wf 0, (96d)
rank
Wf
= 1. (96e)
This feasibility problem (96) without (96e) is also a convex problem with respect to the variable W, which can be evaluated by the SDP. Iff PFeasibility of B1can be solved, then the problem (86) can be feasible with the chosen B, and hence we find the maximum possible θ as discussed in Section 4.2.4.5.3. If PFeasibility of B1 cannot be solved, then the problem (86) is infeasible with the chosenB, and hence we need to choose different B.
To systematically determine B, the eNB constructs a list of all possible AHAPs’ combina- tions, and then performsPFeasibility of B1for each case of the list. Specifically, the list is composed of as follows:
• An element of the list consists of table index t, AHAP-index set Bt, and the sum of average interference from AHAPs in Bt to UEs associated with the eNB.
• The elements are sorted in descending order by the sum interference. If there are elements with the same average interference value, those elements are additionally sorted by the number
of AHAPs inBt in descending order.
To find a feasible Bt, the eNB increases the table index t from 1 and checks PFeasibility of B1
until the problem can be solved. That is, we propose to change AHAPs, generating high average sum-interference to the UEs, into EAPs so that those APs defer their transmissions, which results in improved SINRs of the UEs. If the generating sum-interference from the AHAPs is the same for multiple AHAPs sets, we propose to choose the set including more AHAPs. Such a choice will protect eNB’s transmission to the UEs more reliably, given that the APs’ behaviors are in fact random.
If PFeasibility of B1 can be solved for given Bt, the eNB sets cα = 1, ∀α ∈ Bt and cα0 = 0,
∀α0 ∈ AAH\ Bt, and then solves PB2 to find we∗. If there is no feasible case for all possible AHAPs sets, the eNB finds the beamforming vectors from
PDefault of BF: max min
wl,u,∀l,uγ˜l,u (97a)
s.t kwl,uk2 = 1, ∀l∈[1, NsubL ],∀u∈ U. (97b) The PDefault of BF does not have the constraint (83c), where only the performance of LTE-LAA is considered. The solution of PDefault of BF can be readily obtained in the same way for PB1.