Transmission Probability of an AHAP based on Stochas-

4.2 Downlink MU-MIMO LTE-LAA for Coexistence with Asymmetric Hidden

4.2.4 Proposed MU-MIMO LTE-LAA System

4.2.4.4 Transmission Probability of an AHAP based on Stochas-

E[eα]to compute ˜γl,u in (72).

Letx_i andy₀ be a point for APiand the eNB in an XY-plane, respectively. The large-scale fading is denoted as l(d), where d is the distance between a transmitter and a receiver. The small-scale fading from the eNB to AP iis denoted by G^LW_eNB,i, and the small-scale fading from AP j to AP i is denoted by G^W_j,i. In our coexistence scenario, the eNB is placed at the origin o= (0,0). In addition,Φ denotes a set including all the locations of AHAPs.

Similarly to the work [78], we define the medium access indicator for AP iinA_AH as e_i = Y

xj∈Φ\{x_i}

1_tW

j ≥t^W_i + 1_tW

j <t^W_i 1_GW

j,i/l(kx_j−x_ik)≤Γcs/PW

, (74)

where1_A>B is an indicator function whereAandB are integer numbers. If the conditionA > B is true, then 1A>B = 1. Otherwise,1A>B = 0;t^W_j is the random backoff period of thej-th AP, which is uniformly distributed on[0,1];Γ_cs is the carrier sensing threshold of Wi-Fi; andP_W is the transmit power of an AP.

The meaning of (74) is that APican start a new transmission if its own backoff period (t^W_i ) is shorter than the backoff periods (t^W_j ,∀j6=i) of all other APs or if the signal from each AP is less thanΓcs even thought^W_i > t^W_j .

The goal is to computeE[e_i], which is the average medium access probability of each AHAP during eNB’s transmission.

Lemma 1 When the channel information between the eNB and each AHAP is given, the cor- responding medium access probability E[ei]is obtained as:

E[e_i] = Z 1

0

"

Z

B(o,reNB)

f_X,Y

(a_i, b_i)|kh^WL_i,eNBk², r≤r_eNB

×

Y

j∈A_AH,j6=i

Z

B(o,reNB)

fX,Y

(aj, bj)|kh^WL_j,eNBk², r≤reNB

×

(

1−texp

−µΓcs

P_Wl(k(a_j, bj)−(ai, bi)k) )

dajdbj



daidbi





dt,

where h^WL_i,eNB = [(h^(1,1)_i,eNB)^T, . . . ,(h^(N

W sub,N_sub^L )

i,eNB )^T]^T. Here, h^(w,l)_i,eNB ∈ C^NT×1 is the channel vector from subcarrier w of Wi-Fi to subcarrier l of LTE-LAA.

Proof 1 By the law of total expectation,E[ei] is rewritten as

Et



Exi

"

EΦ\{xi}

E

h

e_i|Φ, t^W_i =ti

|x_i, t^W_i =t

|t^W_i =t

#

.

Definingξ(x_j)asξ(x_j) = 1_tW

j ≥t+1_tW j <t1_GW

j,i/l(kx_j−x_ik)≤^Γcs

PW

, we first derive the part,E h

e_i|Φ, t^W_i =ti as follows:

E h

e_i|Φ, t^W_i =ti_(a)

= E

"

Y

xj∈Φ\{xi}

ξ(x_j)

#

=P





 Y

xj∈Φ\{xi}

ξ(xj)







= 1

!

= Y

xj∈Φ\{xi}

P ξ(xj) = 1

(75)

(b)= Y

xj∈Φ\{xi}

P

1_t^W

j ≥t= 1

+P

1_t^W

j <t= 1

×

P

1_GW

j,i/l(kx_j−x_ik)≤^Γcs

PW

= 1 )

(c)= Y

xj∈Φ\{x_i}

(

1−texp

−µΓ_cs PW

l kx_j−x_ik )

, (76)

where (a) can be derived by inserting the conditions (Φ, t^W_i = t) to (74). Note that ‘Φ is conditioned’ means all the locations of AHAPs are fixed. The step (b) follows from the fact that (t^W_j ≥t^W_i ) and (t^W_j < t^W_i ) are the opposite cases and that (t^W_j < t^W_i ) and (G^W_j,i/l(kx_j −x_ik)≤ Γcs/PW) are independent cases. Finally, (c) follows from the fact that G^W_j,i is assumed to be Rayleigh fading and that t^W_j is uniformly distributed on [0,1].

Next, we derive EΦ\{xi}

E

h

e_i|Φ, t^W_i =ti

|x_i, t^W_i =t

as

EΦ\{xi}





 Y

xj∈Φ\{x_i}

(

1−texp

−µΓ_cs

P_Wl kx_j −x_ik )







(a)= Y

xj∈Φ\{x_i}

Exj

"

1−texp

−µΓcs

P_Wl kx_j−xik #

(b)= Y

xj∈Φ\{x_i}

Z

B(o,reNB)

f_X,Y

a_j, b_j|kh^WL_j,eNBk², r≤r_eNB

× (

1−texp

−µΓcs

P_Wl k(a_j, bj)−xik )

dajdbj, (77)

whereB(x, r) is a closed ball with centerx and radiusr, r_eNB is the coverage of the eNB, andµ is the parameter of Rayleigh fading channel. Finally, fX,Y

(aj, bj)|kh^WL_j,eNBk², r≤reNB

is the probability density function (pdf ) of probability that AP j is placed at x_j = (a_j, b_j) for given kh^WL_j,eNBk² and the condition r ≤r_eNB. Note that r ≤r_eNB means AP j should be in the eNB’s

coverage. The step (a) can be derived by using the law of total expectation, while (b) follows from the definition of expectation.

By using (77) and the definition of expectation, we have

Exi

"

EΦ\{xi}

E

h

e_i|Φ, t^W_i =ti

|x_i, t^W_i =t

|t^W_i =t

#

= Z

B(o,reNB)

f_X,Y

ai, bi|kh^WL_i,eNBk², r≤r_eNB

×

Y

xj∈Φ\{x_i}

(Z

B(o,reNB)

f_X,Y

a_j, b_j|kh^WL_j,eNBk², r≤r_eNB

×

(

1−texp

−µΓcs

P_Wl kx_j−xik )

dajdbj







daidbi. (78)

Recall that t^W_i is uniformly distributed over [0,1]. Thus,E[e_i]can be obtained as the expectation of (78)with respect to t, which proves the lemma.

Lemma 2 The truncated distribution for the probability that APiis at(ai, bi),f_X,Y

ai, bi|kh^WL_i,eNBk², r≤r_eNB

, is given by:

f_X,Y

a_i, b_i|kh^WL_i,eNBk², r≤r_eNB

= (

q

a²_i +b²_i)⁻¹µ^NT

kh^WL_i,eNBk²·l(r)NT−1

·e^{µkhWLi,eNBk2·l(r)} Γ (N_T)R

B(o,reNB)f_X,Y

a_i, b_i| kh^WL_i,eNBk² dxdy

, (79)

where r= q

a²_i +b²_i.

Proof 2 The eNB acquires h^WL_i,eNB by listening to APi’s beacon signals, and hence can compute kh^WL_i,eNBk². Note that both the long-term and short-term fading effect, denoted as l(R) and G, respectively, is taken into account within kh^WL_i,eNBk²; that is kh^WL_i,eNBk² = G/l(R), where R is a random variable representing the distance between AP iand the eNB.

At this point, we assume that each element of h^WL_i,eNB is an independent and identically dis- tributed (i.i.d.) complex Gaussian random variable, or equivalently assume that the absolute value of each element of h^WL_i,eNB is a Rayleigh random variable with the parameter µ. Then, the long-term fading effect l(R)is a function of R for givenR, and the short-term fading effectGis given as the sum ofNT i.i.d. exponential random variables with the parameterµ. Consequently, G is distributed according to the Gamma distribution with the parametersα=N_T andβ =µ.

Note again that kh^WL_i,eNBk² is known by the eNB. Then, for given kh^WL_i,eNBk² = G/l(R), we

have

FG

kh^WL_i,eNBk²·l(r) | kh^WL_i,eNBk²

(80)

=P

G≤ kh^WL_i,eNBk²·l(r) | kh^WL_i,eNBk²

=P

kh^WL_i,eNBk²·l(R)≤ kh^WL_i,eNBk²·l(r) | kh^WL_i,eNBk²

=P

l(R)≤l(r) | kh^WL_i,eNBk²

=F_R

r | kh^WL_i,eNBk²

, (81)

whereF_G(·) 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.

f_R

r|kh^WL_i,eNBk²

= µ^NT

kh^LW_i,eNBk²·l(r)NT−1

·e^{µkhLWi,eNBk2·l(r)}

Γ (N_T) .

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

f_X,Y

x, y| kh^WL_i,eNBk²

=

f_Rp

x²+y²| kh^WL_i,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 f_X,Y. For the truncated distribution, we can derive

f_X,Y

x, y| kh^WL_i,eNBk², r≤r_eNB

=

fX,Y

x, y| kh^WL_i,eNBk² R

B(o,reNB)fX,Y

x, y| kh^WL_i,eNBk² dxdy

= (p

x²+y²)⁻¹µ^NT

kh^WL_i,eNBk²·l(r)NT−1

·e^{µkhWLi,eNBk2·l(r)} Γ (N_T)R

B(o,reNB)f_X,Y

x, y| kh^WL_i,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.