Rate balancing for multiuser MIMO systems

Rate Balancing for Multiuser MIMO Systems

Im`ene Ghamnia^1,2, Dirk Slock² and Yi Yuan-Wu¹

1Orange Labs, Chˆatillon, France,

2Communication Systems Department, EURECOM, France {imene.ghamnia, yi.yuan}@orange.com, dirk.slock@eurecom.fr

Abstract—We investigate and solve the rate balancing problem in the downlink for a multiuser Multiple-Input-Multiple-Output (MIMO) system. In particular, we adopt a transceiver structure to maximize the worst-case rate of the user while satisfying a total transmit power constraint. Most of the existing solutions either perform user Mean Squared Error (MSE) balancing or streamwise rate balancing, which is suboptimal in the MIMO case. The original rate balancing problem in the downlink is complicated due to the coupled structure of the transmit filters.

This optimization problem is here solved in an alternating manner by exploiting weighted MSE uplink/downlink duality with proven convergence to a local optimum. Simulation results are provided to validate the proposed algorithm and demonstrate its performance improvement over unweighted MSE balancing.

Index Terms—rate balancing, max-min fairness, MSE duality, tranceiver optimization, multiuser MIMO systems

I. INTRODUCTION

One important criterion in designing wireless networks is ensuring faireness requirements. Fairness is said to be achieved if some performance metric is equally reached by all users of the system, depending on their priority allocations. With respect to applications in communication networks, fairness is closely related to min-max or max-min optimization problems, also referred to asbalancing problems. Actually, balancing a given metric or a utility function among users implies that the system performances are limited by the weak users. At the optimum, the performance of the latter is brought to be improved [1].

However, most of balancing optimization problems are non- convex and can not be solved directly. Despite that, several works over the litterature have developped optimal solutions.

For instance, [2] solved the max-min problem by a sequence of Second Order Cone Programs (SOCP). Also, [3] showed that a semidefinite relaxation is tight for the problem, and the optimal solution can be constructed from the solution to a reformulated semidefinite program. In [4], the authors proposed an algorithm based on fixed-point that alternates between power update and beamformer updates, and the nonlinear Perron-Frobenius theory was applied to prove the convergence of the algorithm.

Another way to solve balancing optimization problems is to convert the problem from the downlink (DL) channel to its equivalent uplink (UL) channel, by exploiting the UL/DL duality. Doing so, the transformed problem has better mathe- matical structure and convexity in the UL, thus, the computa- tional complexity of the original problem can be reduced [5].

The UL/DL duality has been widely used to design optimal transmit and receive filters that ensure faireness requirements

(a)

(b)

Fig. 1. System model: (a) DL channel, (b) equivalent DL channel.

w.r.t. the Signal-to-Interference-plus-Noise Ratio (SINR), the Mean Square Error (MSE), and the user or stream rate.

With the objective being to equalize all user SINRs, the SINR balancing problem is of particular interest because it is directly related to common performance measures like system capacity and bit error rates. Maximizing the minimum user SINR in the UL can be done straightforwardly since the beamformers can be optimized individually and SINRs are only coupled by the users’ transmit powers. In contrast, DL optimization is generally a nontrivial task because the user SINRs depend on all optimization variables and have to be optimized jointly [5]–[10].

Another well-known duality is the stream-wise MSE duality where it has been shown that the same MSE values are achievable in the DL and the UL with the same transmit power constraint. This MSE duality has been exploited to solve various minimum MSE (MMSE) based optimization problems [11]–[13].

In this work, we focus on user rate balancing in a way to maximize the minimum (weighted) rate among all the users in the cell, in order to achieve cell-wide fairness. This balancing problem is studied in [14]. However, the authors do not provide an explicit precoder design. Here we provide a solution via the relation between user rate (summed over its streams) and a weighted sum MSE. But also another ingredient is required: the exploitation of a scale factor that can be freely chosen in the weights for the weighted rate balancing. User- wise rate balancing outperforms user-wise MSE balancing or streamwise rate (or MSE or SINR) balancing when the streams of any MIMO user are quite unbalanced.

II. SYSTEMMODEL

The considered network is a multiuser MIMO DL system, (see Figure 1). We focus on a Base Station (BS) ofM transmit antennas servingK users of eachN_k antennas, (k= 1, ..., K is the users’ index). The channel between thekth user and the

BS is denoted by H_k^H ∈C^M×Nk, andH^H = [H₁^H, ...,H_K^H] is the overall channel matrix.

We assume zero-mean white Gaussian noise nk ∈C^Nk×1 with distributionCN(0, σ_n²I)at thekth user. We assume inde- pendent unity-power transmit symbols s= [s^T₁ . . .s^T_K]^T, i.e., E

ss^H

=I, wheresk∈C^dk×1is the data vector to be trans- mitted to the kth user, with dk being the number of streams allowed by userk. The latter are transmitted using the transmit filtering matrix G = GP^1/2 ∈ C^M×Nd, composed of the beamforming matrix G = [G₁. . .G_K] = [g₁. . .g_Nd] with normalized columnskgik2= 1and the diagonal non-negative DL power allocationP^1/2=blkdiag{P₁^1/2, . . . ,P_K^1/2}where diag(Pk) ∈ R^d+^k×1 contains the transmission powers and Nd = PK

k=1dk is the total number of streams. The total transmit power is limitted, i.e., tr P)≤Pmax.

Similarly, the receive filtering matrix for each user is defined as F^H_k = P_k^−1/2β_kF_k^H ∈ C^dk×Nk, composed of beamforming matrix F_k^H ∈ C^dk×Nk and the diago- nal matrices β_k contain scaling factors which ensure that the columns of F_k^H have unit norm. We define β = blkdiag{β₁, . . . ,β_K} = diag{[β₁. . . β_Nd]} and F = blkdiag{F₁, . . . ,F_K} = [f₁. . .f_Nd] with normalized per- stream receivers, i.e., kfik2= 1.

The MSE per streamε^DL_i between the decision variablesˆ_i and the transmit data symbols_i is defined as follows

ε^DLi =E n

|ˆsi−si|²o

=β²i/pifi^HH

N_d

X

j=1

pjgjg^Hj

H^Hfi

−2βiRe

fi^HHgi +σn²β²i/pi+ 1,∀i∈ {1, ..., Nd}. (1)

III. PROBLEMFORMULATION

In this work, we aim to solve the weighted user-rate max-min optimization problem under a total transmit power constraint, i.e., the user rate balancing problem expressed as follows

max

{G,P,F,β}min

k rk/r^◦_k

s.t. tr P)≤Pmax (2) wherer_k is thekth user-rate

rk= ln det

I+HkGkG^HkH_k^H σ_n²I+X

j6=k

HkGjG^HjH_k^H−1 (3) and r^◦_k is the rate scaling factor for user k. However, the problem presented in (2) is complex and can not be solved directly.

Lemma 1. The rate of userk in (3) can also be represented as

rk= max

W_k,Fk

ln det Wk

−tr WkE^DL_k +dk

. (4) where

E^DLk =E h

(ˆsk−sk)(ˆsk−sk)^Hi

= (I−F^HkHkGk)(I−F^HkHkGk)^H +X

j6=k

F^HkHjGjG^HjHj^HFk+σ²nF^HkFk (5)

is thekth-user DL MSE matrix between the decision variable ˆ

s_k and the transmit signal s_k, and W = {Wk}1≤k≤K are auxiliary weight matrix variables with optimal solutionW_k=

Fig. 2. Dual UL channel.

E^DL_k −1

and Fk = (σ_n²I+PK

j=1HkGjG^H_jH_k^H)⁻¹HkGk, [15].

Now considering both (2) and (4), and introducing t = minkrk/r^◦_k, we have ∀k

rk/(t r_k^◦)≥1 or rk/r^◦_k ≥t

⇐⇒(a) ln det Wk

+dk−tr WkE^DL_k

≥tr^◦_k (6)

⇐⇒ tr WkE^DL_k ln det Wk

+dk−tr^◦_k

(b)= ^DL_w,k ξk

≤1

where(a)follows from (4) (with optimal Wk) and (b) from ^DL_w,k= tr WkE^DL_k

, the matrix-weighted MSE (WMSE), and ξk = ln det Wk

+dk −r^M_k the WMSE requirement, with r^M_k = tr^◦_k the individual rate target, i.e. rk ≥ r^M_k. What we exploit here is a scale factor t that can be chosen freely in the rate weights r_k^o in (2), to transform the rate weights r_k^◦ into target ratesr_k^M=tr_k^◦, which at the same time allows to interpret the WMSE weightsξ_k as target WMSE values.

Doing so, the initial rate balancing optimization problem (2) can be transformed into a matrix-weighted MSE balancing problem expressed as follows

min

{G,P,F,β}max

k ^DL_w,k/ξk

s.t. tr P

≤Pmax, (7) which needs to be complemented with an outer loop in which Wk = E^DL_k −1

,t = minkrk/r_k^◦,r^M_k =tr^◦_k andξk =dk+ rk−r^M_k get updated.

The problem in (7) is still difficult to be handled directly. In the next sections, we solve the problem via UL and DL MSE duality. To this aim, we model an equivalent UL-DL channel plus transceivers pair by separating the filters into two parts:

a matrix with unity-norm columns and a scaling matrix [16].

Then, the UL and DL are proved to share the same MSE by switching the role of the normalized filters in the UL and DL. Doing so, an algorithmic solution can be derived for the optimization problem (7).

IV. DUALUL CHANNEL

In the equivalent UL model represented in Figure 2, we switch between the role of the normalized transmit and receive filters. In fact, F_kQ^1/2_k is the kth transmit filter and Q^−1/2βG^H is a multiuser receive filter, where Q = blkdiag

Q₁, ...,Q_K with diag(Q_k)∈R^d+^k×1 being the UL power allocation.

Although the quantitiesH,G,F and β are the same, the UL power allocation q= [q₁. . . q_Nd]^T=diag(Q)may differ from the DL allocation p = [p₁. . . p_Nd]^T = diag(P), both verifying the same sum power constraint kpk1 = kqk1 ≤ P_max.

The corresponding UL per stream MSE ε^UL_i is given by ε^UL_i =β²_i/qig^H_iH^H

N_d

X

j=1

qjfjf_j^H

Hgi−2βiRe

g^H_iH^Hfi

+σ_n²β²_i/qi+ 1,∀i. (8)

V. MSE DUALITY

With the equivalent DL channel and its dual UL, it has been shown that the same per stream MSE values are achieved in both links, i.e., ε^UL/DL = diag

[ε^UL/DL₁ . . . ε^UL/DL_N

d ] =

diag

[ε₁. . . ε_Nd] =ε [16].

The UL and DL power allocation, obtained by solving the MSE expressions as in (8) for UL w.r.t. the powers, are given by q=σn²(ε−D−β²Ψ)⁻¹β²1N_d (9)

and _p=σ2

n(ε−D−β²Ψ^T)⁻¹β²1N_d (10) respectively, where the diagonal matrixD is defined as

[D]ii=β²_ig_i^HH^Hfif_i^HHgi−2βiRe{g^HiH^Hfi}+ 1 and

[Ψ]ij= (

g^HiH^Hfjfj^HHgi, i6=j

0, i=j.

In fact, the MSE duality allows to optimize the transceiver design by switching between the virtual UL and actual DL channels. The optimal receive filtering matrices in both UL and DL are MMSE filters and given by

GkβkQ^−1/2_k = H^HF QF^HH+σ²nI−1

Hk^HFkQ^1/2_k (11) and

FkβkP_k^−1/2= HkGP G^HHk^H+σn²I−1

HkGkP_k^1/2. (12) VI. THE MATRIX WEIGHTEDUSER-MSE OPTIMIZATION

In this section, the problem (7) with respect to the matrix weighted user-MSE is studied. First, we start by the UL power allocation strategies. Then, the joint optimization will follow given the MSE duality. In fact, the MSE duality opens up a way to obtain optimal MMSE receiver designs in (11) and (12). The DL matrix weighted user-MSE optimization problems can be solved by optimizing the weighted MSE values of the dual UL system. The latter can be formulated as

min

{G,F,W}max

k ^UL_w,k/ξk

s.t. tr Q

≤Pmax (13) where^UL_w,k= tr WkE^UL_k

, and

E^ULk = (I−Q^−1/2_k βkG^HkHk^HFkQ^1/2_k )

×(I−Q^−1/2_k βkG^H_kH_k^HFkQ^1/2_k )^H +X

j6=k

Q^−1/2_k βkG^H_kH_j^HFjQjF_j^HHjGkβkQ^−1/2_k

+σ²nQ^−1/2_k βkG^HkGkβkQ^−1/2_k . (14) Then, based on the equivalent UL/DL channel pair, we derive a general framework for joint DL MSE design. First, in theUL channel, we find the globally optimal powersQaccording to the optimization problem under consideration; then, we update the UL receivers as MMSE filters (11) and we compute the associated per stream MSE valuesε^UL_i , ∀i. Second, in theDL

channel, we find the DL power allocation P which achieves the same UL MSE values; and we update the DL receivers as MMSE filters (12). Finally, we updateWk.

The matrix weighted per user MSE can be expressed as follows

^UL_w,k= tr WkE_k^UL

(15)

= tr Wk

+ tr WkQ^−1/2_k βkG^H_kH_k^HFkQkF_k^HHkGkβkQ^−1/2_k

−2Ren

tr Q^1/2_k WkQ^−1/2_k βkG^HkHk^HFk

o

+σn²tr WkQ^−1/2_k βkG^HkGkβkQ^−1/2_k +X

j6=k

tr WkQ^−1/2_k βkG^H_kH_j^HFjQjF_j^HHjGkβkQ^−1/2_k ,∀k.

We define Qk = ˜qkQ¯k where tr ¯Qk

= 1 and q˜k is the individual power of thekth user. Then, the transmit covariance matrix R_k =F_kQ_kF_k^H can be written as R_k = ˜q_kR¯_k with tr ¯R_k

= 1. Thus, the matrix weighted MSE_w,k becomes a function of q˜= [˜q1, ...,q˜K]^T

^ULw,k=ak+ ˜q_k⁻¹X

j6=k

˜

qjbkj+ ˜q_k⁻¹ckσn²,∀k (16) where

ak= tr Wk

+ tr WkQ¯^−1/2_k βkG^H_kH_k^HR¯kHkGkβkQ¯^−1/2_k

−2Ren

tr Q^1/2_k WkQ^−1/2_k βkG^HkHk^HFko , bkj= tr WkQ¯^−1/2_k βkG^HkHj^HR¯jHjGkβkQ¯^−1/2_k andck = tr WkQ¯^−1/2_k βkG^H_kGkβkQ¯^−1/2_k

.

Actually, problem (13) always has a global minimizerq˜^opt characterized by the following equations:

∆^UL= ^UL_w,k(q^opt) ξk

, ∀k, (17)

kq^optk1=Pmax (18) where∆^UL is the minimum balanced matrix-weighted user MSE.

We aim to form an eigensystem by combining (17) and (18).

For that, we rewrite (16) as

^UL_w q˜=Aq˜+σ²_nC1K (19) where^UL_w =diag{^UL_w,1, . . . , ^UL_w,K},C =diag

c1, . . . , cK

and

[A]kj= (

bkj, k6=j ak, k=j.

Now, we defineξ=diag

[ξ1. . . ξK] and multiply both sides byξ⁻¹to have

ξ^−1UL_w q˜=ξ⁻¹Aq˜+σ²_nξ⁻¹C1K. (20) From (17), we have ξ^−1UL_w ( ˜q^opt) = ∆^ULI. Thus, (20) becomes

∆^ULq˜=ξ⁻¹Aq˜+σn²ξ⁻¹C1K. (21) From (18), we can reparameterize q˜= ^P_1T^max

Kq⁰q⁰ whereq⁰ is unconstrained. This allows to rewrite (21) as [7]

Λq˜⁰ = ∆^ULq˜⁰, Λ=ξ⁻¹A+ σ²_n Pmax

ξ⁻¹C1K1^T_K (22)

TABLE I

PSEUDO CODE OF THE PROPOSED ALGORITHM

1. initialize:F_k^H(0,0) = (Id_k:0),Q¯^(0,0) = ^P_N^max

d I,m=n= 0 andnmax, mmaxand fixr_k^◦(0)

2. compute UL receive filterG^(0,0)andβ^(0,0)with (11) 3. setW_k⁽⁰⁾=I andξ⁽⁰⁾_k =dk

4. find optimal user power allocation q˜^(0,0) by solving (22) and computeQ^(0,0)_k = ˜q^(0,0)_k Q¯^(0,0)_k

5. repeat 5.1 repeat

n←n+ 1 UL channel:

• updateG^(n,m−1)andβ^(tmp,tmp)with (11)

• compute the MSE valuesε^UL,(n) with (8) DL channel:

• computeP^(n,m−1)with (10)

• updateF^(n,m−1)andβ^(tmp,tmp)with (12)

• compute the MSE valuesε^DL(n)with (1) UL channel:

• compute Q^(tmp,tmp) with (9) and Q¯^(n,m−1)_k = Q^(tmp,tmp)_k /trQ^(tmp,tmp)_k

• find optimal user power allocationq˜^(n,m−1)by solving (22) and computeQ^(n,m−1)_k = ˜q^(n,m−1)_k Q¯^(n,m−1)_k

5.2 untilrequired accuracy is reached orn≥nmax

5.3 m←m+ 1

5.4 update W_k^(m) = (E^UL(m)_k )⁻¹, r^(m)_k = ln det(W_k^(m)), t = mink

r_k^(m)

r^◦(m−1)_k , r^◦(m)_k = t r^◦(m−1)_k , and ξ_k^(m) = dk+r^(m)_k −r_k^◦(m)

5.5 don← 0and set(.)^(nmax,m−1) →(.)^(0,m) in order to re-enter the inner loop

6. untilrequired accuracy is reached orm≥mmax

It can be observed that ∆^UL is an eigenvalue of the non- negative extended coupling matrixΛ. However, not all eigen- values represent physically meaningful values. In particular,

˜

q^opt>0 and∆^UL>0 must be fulfilled.

It is known that for any non-negative irreducible real matrix X with spectral radius ρ(X), there exists a unique vector q > 0 andλmax(X) = ρ(X) such that Xq = λmax(X)q.

The uniqueness of λmax(Λ) also follows from immediately from the function ∆^UL(Pmax) being strictly monotonically decreasing in Pmax. This rules out the existence of two different balanced levels with the same sum power. Hence, the balanced level is given by

∆^UL,opt=λmax(Λ). (23) Therefore, the optimal power allocation q˜⁰ is the principal eigenvector of the matrixΛin (22). As noted in [5], we have in fact

λmax(Λ) = min

p˜ max

q˜

p˜^HΛq˜

˜

p^Hq˜ = max

p˜ min

q˜

p˜^HΛq˜

˜

p^Hq˜ (24) where in [5] p˜ was said to have no particular meaning but actually can be shown to relate to the DL powers. So, the proposed algorithm provides in the inner loop an alternating optimization of (24) w.r.t.p,˜ q,˜ F,G[5], [16]. If we take for p˜theK standard basis vectors, then we get

λ_max(Λ) = min

˜ q max

k

Λq˜

k

q˜k

(25) which from (17), (20), (22) can be seen to be exactly the WMSE balancing problem we want to solve.

VII. ALGORITHMICSOLUTION ANDSIMULATIONS

A. Algorithm

The proposed optimization framework is summarized here- after in Table I. Superscripts(.)⁽ⁿ⁾and(.)^(tmp)denote thenth iteration and a temporary value, respectively. This algorithm is based on a double loop. The inner loop solves the WMSE balancing problem in (7) whereas the outer loop iteratively transforms the WMSE balancing problem into the original rate balancing problem in (2).

B. Proof of Convergence

In case the rate weightsr^◦_k would not satisfyr_k ≥r^◦_k, this issue will be rectified by the scale factort after one iteration (of the outer loop). It can be shown thatt= mink

r^(m)_k r^◦(m−1)_k ≥1.

By contradiction, if this was not the case, it can be shown to lead to ^{tr W}

(m−1) k E^(m)_k

ξ_k^(m−1) >1, ∀k and hence ∆^(m) > 1. But we have

∆^(m)=^{tr W}

(m−1) k E^(m)_k

ξ_k^(m−1) , ∀k,= maxk

tr W_k^(m−1)E^(m)_k

ξ_k^(m−1) (a)

<maxk

tr W_k^(m−1)E^(m−1)_k

ξ^(m−1)_k = maxk d_k

ξ^(m−1)_k (b)

< 1. (26) LetE={Ek, k= 1, ..., K} and

f^(m)(E) = max_k^{tr W}

(m−1) k E_k

ξ_k^(m−1) . Then (a)is due to the fact that the algorithm in fact performs alternating minimization of f^(m)(E)w.r.t.G,F,q˜and hence will lead tof^(m)(E^(m))<

f^(m)(E^(m−1)). On the other hand, (b) is due to ξ_k^(m−1) = dk+r_k^(m−1)−r^◦(m−1)_k > dk, for m≥3.

Hence, t ≥ 1. Of course, during the convergence t > 1.

The increasing rate targetsr^◦(m)_k constantly catch up with the increasing rates r_k^(m). Now, the rates are upper bounded by the single user MIMO rates (using all power), and hence the rates will converge and the sequencetwill converge to 1. That means that for at least one userk,r_k^(∞)=r_k^◦(∞). The question is whether this will be the case for all users, as is required for rate balancing. Now, the WMSE balancing leads at every outer iterationm to ^{tr W}

(m−1) k E^(m)_k

ξ^(m−1)_k = ∆^(m),∀k. At convergence, this becomes ^dk

ξ_k^(∞) = ∆^(∞)whereξ_k^(∞)=d_k+r^(∞)_k −r_k^◦(∞). Hence, if we have convergence because for one user k∞ we arrive at r^(∞)_k

∞ = r^◦(∞)_k

∞ , then this implies ∆^(∞) = 1 which impliesr^(∞)_k =r^◦(∞)_k ,∀k. Hence, the rates will be maximized and balanced.

Remark 1. In fact, the algorithm also converges withn_max= 1, i.e., with only a single loop.

C. Simulation results

In this section, we numerically illustrate the performance of the proposed algorithm. The simulations are obtained under a channel modeled as follows :H_k^H=BkUkAk whereBk,Ak

are of dimensions(M×Nk)and(Nk×Nk)respectively, and have i.i.d. elements distributed asCN(0,1);Uk=µUk, with the normalization parameterµ= (trace U_k)−1/2

andU_k = diag

1, α, α², . . . , α^Nk−1 (α∈Rbeing a scalar parameter).

This model allows to control the rank profile of the MIMO

Fig. 3. Minimum rate in the system VS SNR:K= 3.

Fig. 4. Rate distribution among users:K= 3, SNR=10dB,M= 6, Nk= dk= 2.

channels. For all simulations, we fix α= 0.3 and take 1000 channel realisations andnmax= 20. The algorithm converges after 4-5 (or 13-15) iterations of m at SNR = ^P_σ^max2

n =10dB (or 30dB).

Figure 3 plots the minimum achieved per user rate using i) our max-min user rate approach with equal user priorities and ii) the user MSE balancing approach [16], as a func- tion of the Signal to Noise Ratio (SNR). We observe that our approach outperforms significantly the unweighted MSE balancing optimization, and the gap gets larger with more streams. Note that we observe the same behavior with the classical i.i.d. channel H_k^H = Bk , but with a smaller gap (e.g., for 15dB, ^minkrk(weighted-MSE)

min_kr_k(unweighted-MSE) = 1.05 instead of 1.18 withM = 6, N_k =d_k = 2in Figure 3).

In Figure 4, we illustrate how rate is distributed among users according to their priorities represented by the rate targetsr^◦_k. We can see that, using the min-max weighted MSE approach, the rate is equally distributed between the users with equal user priorities, i.e., r_k^◦ = r^◦₁ ∀k, whereas with different user priorities, the rate differs from one user to another accordingly.

Furthermore, the Sum Rate (SR) reaches its maximum when user priorities are equal, as the channel statistics are identical for each user.

VIII. CONCLUSIONS

In this work, we addressed the multiple streams per user case (MIMO links) for which we considered user rate bal- ancing, not stream rate balancing. Actually, we optimized the rate distribution over the streams of a user, within the rate balancing of the users. In this regard, we proposed an iterative algorithm to balance the rate between the users in a MIMO system. The latter was derived by transforming the max-min rate optimization problem into a min-max weighted MSE optimization problem to enable MSE duality. We also provided numerical comparisons between the proposed weighted rate balancing approach and unweighted MSE balancing.

ACKNOWLEDGMENTS

EURECOM’s research is partially supported by its industrial members: ORANGE, BMW, Symantec, SAP, Monaco Tele- com, iABG, and by the projects DUPLEX (French ANR) and MASS-START (French FUI). The research of Orange Labs is partially supported by the EU H2020 project One5G.

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