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Resource allocation strategies for multicarrier radio systems
Marco MorettiInformation Engineering Department
Università di [email protected]
-2-
Summary
IntroductionConvex optimization
Single-cell resource allocation Rate adaptiveMargin adaptiveOptimal allocationMIMO
Multi-cell resource allocationDistributedCentral coordinationStatic
-3-
Introduction
-4-
OFDMA resource allocation schemes
Orthogonal Frequency Division Multiplexing (OFDM) is one of the most adopted modulation techniques by current air interface standards (e.g. 802.16, 3GPP Long Term Evolution)
OFDM is robust to the multi-path wireless propagation channelIn OFDMA systems it is possible to exploit channel frequency diversity by dynamically assigning the radio resources to the users.
-5-
System model
OFDMA synchronized system
Nc cellsN subcarriersK users per cell
Received signal for user kin cell i on channel n
X(i)k,n = H
(i)k,n,iS
(i)n + d
(i)k,n
d(i)k,n N (0, 2 + I
(i)k,n)
-6-
OFDMA signal model
The received signal-to-interference-plus- noise ratio (SINR) k,n(i) is (Gk,n,i(i)=|Hk,n,i(i)|2)
The power of the MAI Ik,n(i), affecting user k in cell i on the n channel, is given by
Accordingly, the capacity of user k in cell i on channel n
C(i)k,n = log2 1 +
(i)k,n
(i)k,n =
P (i)n G
(i)
k,n,i
2+I(i)
k,n
I(i)k,n =
Nc
j=1,j=i
P(j)n G
(j)k,n,i
-7-
Definition of convexity
f:D R is convex if D is a convex set and
for any x,y D and 0 1Example: function f(x) = maxixi
f ( x+ (1 )y) = maxi( xi + (1 )yi)
maxi
xi + (1 )maxi
yi
= f(x) + (1 )f(y)
f ( x+ (1 )y) f(x) + (1 )f(y)
-8-
Convex optimization
Standard form convex optimization problem
f0,f1, ,fm are convex; equality constraints are affine
0min ( ).
0 1, ,.
( )i
f xs t
f x i mbAx= �…
=
-9-
Convex optimization
Standard form problem (not necessarily convex)
Variable , domain D, optimal value p*=f0(x*)Lagrangian
Weighted sum of objective and constraint functions
0
0 1, ,1
min ( ). .
( )( , ,) 0
i
i
f xs t
f xx
i mh i p
=
= �…=
�…
nx
01 1
( , ) ( ) ) ), ( (i
pm
i ii i iL x f x f x h x
= =
= + +
-10-
Convex Optimization
Lagrangian dual function g
g is affine in and and therefore concave AND convexLower bound property: if 0, then g( , ) pProof: if x0 is feasible and 0, then
minimizing over all feasible x0 gives p g( , )
01 1
( ) ( ) ( ) (, inf ( , n ), ) i fpm
i i i ix i ixg f x f h xx xL
= =
= += +D D
f0(x0) L(x0, , ) infx D
L(x, , ) = g( , )
-11-
The dual problem
Lagrange dual problem (LDP)
d = max g( , ) finds best lower bound on p , obtained from Lagrange dual function
, , are dual feasible if 0 and , dom gConvex optimization problem: since the objective to be maximized is concave and the constraint is convex
max g( , )
s.t.
0
-12-
Weak and strong duality
Weak duality: d palways holds (for convex and non-convex problems)can be used to find nontrivial lower bounds for difficult problems
Strong duality: d = pdoes not hold in general(usually) holds for convex problemsconditions that guarantee strong duality in convex problems are called constraint qualifications
-13-
Slater's constraint qualification
Strong duality holds for a convex problem
if it is strictly feasible, i.e.,
It also guarantees that the dual optimum is attained when `d >- , i.e., there exists a dual feasible ( , ) with g( ; ) = d = p
min f0(x)
s.t.
fi(x) 0 i = 1, . . . , m
Ax = b
x D : fi(x) < 0(i = 1, · · · , m); Ax = b
-14-
Complementary slackness
When strong duality holds if x is primal optimal and ( , ) is dual optimal
x minimizes L(x, , )fi(x )=0 for i=1, ,m, which implies one of the two
Complementary: either i or fi(x) are zeroSlack: not binding
f0(x ) = g( , ) = infx D
f0(x) +
m
i=1
i fi(x) +
p
i=1
i hi(x)
f0(x ) +m
i=1
i fi(x ) +
p
i=1
i hi(x )
f0(x )
i > 0 fi(x ) = 0 fi(x ) < 0 i = 0
-15-
Karush-Kuhn-Tucker (KKT) conditions
We now assume that the functions f0, ,fm and h0, ,hp are differentiable and f0(x ) =g( , ). Then the following conditions hold
fi(x ) 0 i = 1, · · · , mhi(x ) = 0 i = 1, · · · , p
i 0 i = 1, · · · , m
i fi(x ) = 0 i = 1, · · · , m
f0(x ) +
m
i=1
i fi(x ) +
p
i=1
i hi(x ) = 0
-16-
KKT conditions for convex problems
Single-cell single-user power allocation: allocate the power over the N channels with the goal of maximizing the rate subject to a
The problem is convex since it satisfies the Slater’s qualifications
maxP
N
n=1
log2(1 +Pn|Hn|2
2 )
s.t.
Pn 0 n = 1, · · · , NN
n=1Pn = P0
-17-
KKT conditions for convex problems
In standard convex form( n= /|Hn|2)
The Lagrangian L(P, , ) is
The KKT conditions are
minP
N
n=1
log2(1 +Pn2n)
s.t.
Pn 0 n = 1, · · · , NN
n=1Pn = P0
L(P, , ) =N
n=1log2(1 +
Pn2n)
N
n=0nPn +
N
n=1Pn P0
Pn 0 n 0 nPn = 0 n = 1, · · · , NN
n=1Pn = P0
12n+Pn
1log 2 n + = 0 n = 1, · · · , N
-18-
KKT conditions for convex problems
The optimal value can be found using the complementary slackness condition
There are two possible solutions that depend on the value log2 …
…leading to the well-known waterfilling strategy
Pn log 2 12n+Pn
= 0 n = 1, · · · , N
* 2*
2
2
0 1/1/ ( log
log 21, ,
2) l 2 /o 1gn
n nnP n N= =
<
Pn = max 0, 1log 2
2
Gnn = 1, · · · , N
-19-
Uniform power allocation
Given any 0 and , it holds
and the duality gap (P, , )=f0(P)-g( , ) for any vector P is
after some manipulations one obtains
Pn +2n =
1n
1log 2 n = 1, · · · , N
=N
n=1
2n
Pn+ 2n
1log 2 + P0
Nlog 2
= 1log 2
N
n=1
Pnmink{Pk+ 2
k}Pn
Pn+ 2n
-20-
Uniform power allocation
If power is uniformly allocated on M channels it yields
The strategy that allocates uniform power P0/M over the M subchannels that would receive positive power in exact waterfilling is close to the optimum
=1
log 2
M
n=1
P0/M
P0/M +mink { 2k}
P0/M
P0/M + 2n
1
log 2
M
n=1
2n mink
2k
P0/M +mink { 2k}+ 2
n
-21-
Waterfilling dual: power minimization
The power necessary to achieve a certain rate rnover channel n is Pn = (2rn-1)· n
Minimize the power with rate constraints
minr
N
n=1(2rn 1) · 2
n
s.t.
rn 0 n = 1, · · · , NN
n=1rn = r0
-22-
Waterfilling dual: power minimization
The Lagrangian is
the optimal rate allocation is
and the power allocated on subcarrier n is
L(r, , ) =N
n=1(2rn 1) 2
n
N
n=1nrn
N
n=1rn r0
Pn = log 22n
+
2
2
** 2 2
2
log 21, ,
log l0 /
( / log 2) /og log 2n
nn
n
n Nr =<
=
-23-
Single-cell allocation
-24-
OFDMA resource allocation
The goal is to take advantage of the multi-user and frequency diversity of system to maximize the spectral efficiency and reduce the power consumptionRadio resources are:
Transmission powerTransmission formatsSubcarriers
As channels are statistically independent for each user, a channel that is “bad” for one user may be “good” for another
-25-
OFDMA resource allocation
-26-
Assumptions
We consider the downlink of a wireless multi-carrier communication system with K users and N subcarriersPerfect CSI at the base stationIdeal feedback channel to signal the assignment decision.
We introduce a binary allocation variable
,
1 if channel is assigned to user 0 otherwisek n
n k=
-27-
Resource allocation: constraints
Univocal assignment:Subcarriers are allocated univocally to the users: only one user at the time can occupy a given subcarrier
The presence of an integer assignment variable greatly complicates the allocation problem since it makes the problem NOT convex
,1
1 1,...,K
nk
k n N=
=
-28-
Resource allocation schemes
Rate adaptive Objective: maximize the overall rate rtot subject to a global power constraint nPn=P0
Margin adaptiveObjective: minimize the overall power Ptot subject to the different users’ rate constraints.
,1 1
n k n
K N
totk n
P P= =
=
,2
1 1, 2log 1 n k n
K N
kkot nt
nr
P G
= =
+=
-29-
Rate adaptive schemes
-30-
Sum-rate maximization
The sum-rate maximizationproblem is solved by1. assigning each sub-carrier to
the user that maximizes its gain
2. performing waterfilling over all the sub-carriers allocated.
Such a solution maximizes the cell throughput but is extremely unfair since it privileges the users closest to the BS and starves all the others.
,2 ,2,
,
0
,
max log 1
. .1
{0,1} ,
n k nk nP nk
kk n
nn
k n
P G
s
P
tn
P
k n
+
-31-
Max-min rate allocation
Fairness is introduced by allocating resources with the goal of maximizing the minimum capacity offered to each user, thus introducing fairness among the users.
In general, fairness comes at the cost of a reduction of the overall throughput of the cell.
,2 ,2,
,
,
0
max min log 1
. .1
{0,1} ,
n
k
n k nk nkP
k n
nn
k n
P G
s tn
P
k n
P
+
-32-
Max-min rate allocation
Problem is NOT convexA heuristic solution is implemented:1. Uniform power
allocation on all sub-channels P=P0/N
2. A greedy assignment strategy that iteratively allocates the sub-carriers to the user with the smallest rate
-33-
Sum-rate maximization with proportional rate constraints
Different users may require different data rates. In this case, a fair solution is to allocate radio resources proportionally to the users’ different rate constraints.
,2 ,2,
,
1 2 1 2
0
,
max log 1
. .1
: : ... : : : ... :{0,1} ,
n
n k nk nP k
k n
nn
K K
k n
k
PG
s tn
P
r rk n
P
r
+
=
-34-
Sum rate maximization with proportional rate constraints
The optimization problem is a mixed binary integer programming problem and as such is not convex and in general very hard to solve. We follow an heuristic approach:
Subcarrier allocation phase: assuming an uniform power distribution, the subcarriers are allocated complying as much as possible with the proportional rate constraints.Power allocation phase: once the subcarrier are allocated to the users, the power is distributed so that the proportional rate constraints are exactly met.
-35-
Sum rate maximization with proportional rate constraints: subcarrier allocation
At each iteration, the user with the lowest proportional capacity has the option to pick the best sub-channel. The sub-channel allocation algorithm is suboptimal, because it is greedy and assumes a uniform distribution of power.
-36-
Sum rate maximization with proportional rate constraints: power allocation
The proportional rate constraints are enforced by allocating the power to the users in two steps1. Find for each user the expression of Pk(tot) the total power
allocated to each user k.2. Distribute the power among users in such a way that the
proportional rate constraints are met and the overall power doesnot exceed P0
-37-
Sum rate maximization with proportional rate constraints: power allocation
The Lagrangian is (Zk,n=Gk,n/ ) :
Leading to the power per user Pk(tot)
The optimal distribution of the Pk(tot) is found iteratively with the Newton-Raphson method.
L(P, ) =K
k=1 n k
log2 (1 + Pk,nZk,n) + 1
K
k=1 n k
Pk,n P0
+
K
k=2
k
n 1
log2 (1 + P1,nZ1,n)1
kn k
log2 (1 + Pk,nZk,n)
P totk = NkPk,1 +
Nk
n=1
Zk,n Zk,1Zk,nZk,1
-38-
Rate adaptive results
Simulation parametersNumber of cells: 1Maximum BS transmission power: 1 WCell radius: 500 mMT speed: staticCarrier frequency: 2 GHzNumber of sub-carriers: 192Sub-carrier bandwidth 15 kHzPath loss exponent: 4Log-normal shadowing standard dev. 8 dBSmall-scale fading Typical Urban (TU)
-39-
Rate adaptive results
Algorithms simulatedSum rate maximizationMax-minSum rate maximization with proportional rate constraints
Equal rate constraintsRate constraints proportional to the pathloss
1 2 K= =�…=
-40-
Rate adaptive results
4 6 8 10 12 14 161.5e7
2.0e7
2.5e7
3e7
3.5e7
4e7
Number of users
Thro
ughp
ut
Sum rateProp rate 2Prop rate 1Max min
-41-
Rate adaptive results
4 6 8 10 12 14 160
0.2
0.4
0.6
0.8
1
Number of users
Fairn
ess
Sum rateMax minProp rate 2Prop rate 1
-42-
Margin adaptive schemes
-43-
Resource allocation: constraints
Each user has to meet a certain target rate, which constraints the task of resource allocation
The power necessary for user k to transmit with rate rk,n on subcarrier n is
1,( ) 1,...,
N
k nn
r k r k K=
= =
( ),, ,2 1 /k nr
k n k nP Z=
-44-
WCLM: formulation
The allocator solves the optimization problem by assigning the subcarriers and the rate on each subcarrier.Address the fairness issue but there are no explicit limits to the transmitted power.Problem is NOT convex
( ), ,
,,
,
, ,
,
min 2 1
. .1
( )
{0,1} ,
k nr k n
r n k n
k n
k n k
k
k
k
nn
n
Zs t
n
r kr
k n
-45-
WCLM: convexification
1. Relax the integer allocation variable k,n.
Another way to interpret the optimization is to consider k,n as the time-sharing factor for the kuser of subcarrier n.
2. Introduce a new rate variable sk,n=rk,n k,n so that the objective function becomes convex (positive semidefinite Hessian )
mins,
N
n=1
K
k=1k,n 2
sk,n
k,n 1 1Zk,n
s.t.N
n=1sk,n = r(k)
K
k=1k,n = 1
k,n [0, 1]
3. Once the problem is convex, it can be solved in the dual domain
-46-
WCLM dual function
Lagrangian of the problem is
Given the Lagrangian multipliers yields
While for k,n holds
,
,, , ,
1 11 11, 1
1( , ) 2 1 ( ), , 1k n
k n
sN K N K
n
K N
k n k k n n k nk nnk n kk
L s s r kZ
µ µ= = == = =
=
( ),
,, 2 , ,
log 2
/ log 2 log 2
0 /
log /k k n
k nk n k k n k k nZ
sZ
Z=
>
, ,2
, ,
11 if arg min 1 loglog 2 log 2
0 else
k k n k k nk
k n k n
Zk
ZZ
==
-47-
WCLM: algorithm flow chart
The algorithm is iterativeStart from some arbitrary values of and computes
k,n and sk,n.If the users’ rate constrains are not satisfied iteratively increase the values of until all the rate constraints are met. This procedure requires the inversion of non –linear functions to converge.
-48-
WCLM algorithm
Complexity is a major issue: due to the nature of the allocation problem, the solution proposed is iterative: depending on system parameters, convergence may be extremely slow.Solution admits non-integer values of the allocation variable.
It is necessary to implement a heuristic (multi-user adaptive OFDM, MAO) to set the vector of allocation variables to integer values
-49-
Linear programming
Resource allocation can be formulated as a linear programming problem:
Linear objective functionLinear constraints
-50-
All rate requests are expressed as a multiple integer of a certain fixed rate corresponding to a spectral efficiency
The power necessary for user k to transmit the rate b(b=0, ,B) on the subcarrier n is a fixed cost
( )2 ,,log 1 k nk nP Z= +
(,
), (2 1) /b b
k n k nP Z=
Linear programming: multiple tx formats
-51-
Linear programming: multiple tx formats
( ) ( ), ,
( ),
( ),
( ),
min
. .1 1,...,
( ) 1,.
{0,1} , ,
..,
b bk n k n
k n b
bk n
k b
k
bk n
bk n
b
P
stn N
b r k k K
b k n
=
= =
After linearization of the objective function and of the constraints, resource allocation can be formulated as a linear integer programming (LIP) problemCombinatorial problem with exponential complexity in N, K, and B
-52-
Provided that there is enough multi-user diversity, it is possible adopt only one transmission format (B=1) with very limited performance loss. The rate requirements r(k) are translated into a minimum number n(k) of subcarriers to be allocated per userBy relaxing the integer constraint, RRA turns into a standard LP problem
, ,
,
,
,
min
. .1 1,...,
( ) 1,.
{0, }
,
1
..
k n k nk n
k nk
kk n
k n
P
s tn N
n k k K
=
==
Linear programming: single tx format
-53-
The relaxed LP RRA problem has the characteristic that it can be modeled as a a network flow problem.
The network simplex method (NSM) is the most efficient solver for min-cost-max-flow network problems and outperforms other existing techniquesBecause of its topology, the solution of the relaxed LP RRA is integral and thus, regardless of the relaxation, always a combination of 0 and 1
The single format choice allows a great simplification of the solution of the RRA problem at the cost of only a modest worsening of system performance. Dynamical assignment of subcarriers already provides a great deal of diversity!
Linear programming: single tx format
-54-
Linear programming: power allocation
After having solved the LP single-format resource allocation, power can be further reduced by solving a single-user waterfilling problem for each user on the assigned subcarriers.For user k, who is allocated the set k of subcarriers, the problem is formulated as
( ),
,
,
1min 2 1
. .( )
k n
k
k
r
rn
kn
k
n
n
r
Zs t
r k=
-55-
Optimal allocation
The solution of an optimization problem can be bounded by resorting to the Lagrange dual Duality gap is the difference between the solution of the primal problem and the solution of the dual problemQualification conditions. It has been showed that in multi-carrier applications, even if the original RRA problem is non-convex, the duality gap tends to zero as the number of tones goes to infinity.
-56-
Optimal rate allocation: primal
The primal problem is formulated as a minimization problem with standard rate and exclusive allocation constraintsThe problem is combinatorial (i.e. all possible allocations should be evaluated!) and its complexity grows exponentially with K and N
( ),
1 1
1
,
,,
, ,
, , ,1
min 2 1
. .
(
; {0,1}
)
| 1
k nr k n
k n
k n k n
k n k n k
N K
n k
N
n
K
kn
Zs t
r r k k
= =
=
=
= =
r
-57-
Optimal allocation: Lagrange dual
The Lagrangian is
defined over the set R and all the positive rates rk,n
The Lagrangian dual function is
( ),
1 1
,, ,
1 1,
( , ) 2 1 (, )k nN K N N
n
r k nk k n k n
nnk k nL r r r k
Z= = = =
=
( ), ,, ,,
,
,
1 1 1
1
( ) min 2 1 ( )
. .
1
k nr k nk k n k n
k n
N K N
n k n
kk
K
n
g r r kZ
n
s t= = =
=
=
=
r
-58-
Optimal allocation: Lagrange dual
The Lagrange dual of the RRA can be written as the sum of N reduced-complexity minimization problems
Solving the per-carrier problem still involves an exhaustive search over the whole set of users.
( ), ,,,
,
,
1
1
( ) min 2 1
. .
1
k nr k nn k k n
K
r k
K
k
k n
k n
g
n
rZ
s t=
=
=
=
-59-
Optimal allocation: dual update method
The solution to the Lagrange dual problem is found following an iterative process:
Given the multiplier vector , we find g( ) and the rates allocated for each userThe rate results for the different users contribute to the subgradient
The subgradient is employed to update the multiplier vector (Ellipsoid method)
,1
( ) ( ) ( 1,) ,N
nk nd kk k r Kr
=
==
-60-
Optimal allocation: ellipsoid method
It is the multi-dimenionalextension of the bisection methodThe idea is to localize the set of candidate s within some closed and bounded set. Then, by evaluating the subgradient of g( ) at an appropriately chosen center of such a region, roughly half of the region may be eliminated from the candidate set. The iterations continue as the size of the candidate set diminishes until it converges to an optimal
An ellipsoid with a center z and a shape defined by positive semidefinite matrix A is defined as
The update rule is the following
E(A, z) = x|(x z)TA(x z) 1
-61-
Optimal power allocation
The same approach can be used to solve the maximization of the sum of weighted rates
( )2 , ,,
0
1
1
,
1
1, ,
log
; {
max 1
. .
| 1 0,1}
k n k n k nP
n
k n k n k
K N
k n
N
n
K
kn
w P Z
s t
P P
R
= =
=
=
+
= =
-62-
Simulation setup
Number of cells = 1Radius of each cell R = 500 mTotal available bandwidth W = 5MHzCenter frequency = 2 GHzNumber of subcarriers N = 64Number of users K = 8
-63-
Power vs. spectral efficiency
0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
12
14
Spectral efficiency (bit/s/Hz)
Powe
r [W
]
OptimalWCLMLP
-64-
Power vs. number of users
2 4 8 162
3
4
5
6
7
8
Number of users
Aver
age
tx p
ower
[W]
OptimalWCLMLP
-65-
MIMO resource allocation
-66-
MIMO system
We are considering a NT×NR MIMO system with NT>NR so that at least Q = NT/NR users can transmit on the same frequency channel.Users signals are separated by the implementation of linear precoding and receving filtersSignal model
, , , , , , , , , , ,, n
H H Hk n k n k n k n k n k n k n k n k n j n k n
j k j= = + +z W y W H B s W H x
U
Desired signalMultiple access
interference and noise
-67-
MMO optimal allocation
Margin adaptive optimization problem:Optimize linear precoder, transmit power distribution and channel allocation to minimize the overall transmit powerProblem is NOT convex and prohibitively complex
( )
( )
,
, ,
, 1
1, 2 , ,
1
min
log det ( ) 1
1
tr
. .
k nn
R k n k n
N
n k
NH
k n N k n k nn
n
r k k K
s t
Q n N
=
=
+
xx
x i
R
I H R H R
U
U
-68-
Block diagonalization (BD) approach
To simplify the problem, we first decide the precoding strategy and then allocate remaining resources (channels and power).By projecting each user’s MIMO channel on the interference null space, users’ channels are decoupled so that the users transmit on the same channel do not interfere with each otherAllocation task is greatly simplified in absence of interference
-69-
BD-based RA
Problem is still not convex but as the number of subcarriers increase the duality gap tends to zero and can be solved in the dual domain.
( )
,, 1
, ,1
min
. .
, ( ) 1
1
n
N
k nn k
N
k n k nn
n
P
s t
r r k k K
Q n N
=
=
p
P
U
U
( )
,
,
( ), ,
1
( ) ( ), ,
, 2 21 0
, log 1
k n
k n
lk n k n
l
l lk n k n
k nl
P P
P gr
=
=
=
= +P
-70-
BD-based RA – Dual domain
On each subcarrier the dual function must be evaluated over all the possible combinations of Q usersFor each combination the precoding and receive linear filters must be evaluated!!
( ) ( ), ,,min ,
. .
k k kk
g P r
s tQ
µ=P
µ PU
U
1. Exhaustively compute all user combination
2. For each user combination evaluate
3. Choose the combination that minimizes the metric
( )
2( ) 0, ( )
,log 2l k
k n lk n
Pg
µ+
=
-71-
Successive channel assignment
To reduce allocation complexity, we first group the users on the base of their channel quality and then sequentially solve the RA problem.The implementation of a sequential allocation strategy forces a change in the design of the linear precoder.
The users of a group do not interfere with the users already allocated but do generate interference versus the sets of user allocated successively. MAI is treated as spatially colored noise
-72-
Successive channel assignment
Taking into account the colored interference the allocation problem can be formulated as the solution of Q successive problems
( )
( )
,
, ,
, 1
( ) ( ) 1, ,, 2
1
,
min
. .
log det ( )
1 1
trk n
q
R k n k n
q
N
k n
N q q Hk n k nk n N q
n
k nk
s t
r k k
n N
=
=
+
xx
x i
R
I H R H R
K
K
K
-73-
Successive channel assignment
Problem becomes more tractable by whitheningthe colored noise at the receiver multiplying the received signal by ( )
,, 1
, ,1
,
min
. .
, ( )
1 1
q
q
N
k nk n
N
k n k n qn
k nk
s t
r r k k
n N
=
=
p aP
P
K
K
K,
1/2k niR
-74-
LP-based channel assignment
Supposing that each user transmits with a fixed spectral efficiency on all his channels:
power needed becomes a cost rate requirements r(k) translates into requesting a certain number of channels n(k)
Each successive problem can be formulated as a LP problem
, ,1
,1
,
min
( )
1 1, ,
q
q
N
k n k nn k
N
k n qn
k nk
P
n k k
n N
=
=
=
= �…
K
K
K
-75-
Simulation setup
Number of cells = 1Radius of each cell R = 500 mTotal available bandwidth W = 5MHzCenter frequency = 2 GHzNumber of subcarriers N = 64Number of users K = 8Two scenarios: 4x2 and 2x1
-76-
Computational complexity
2 4 8 16 32104
105
106
107
108
109
Number of users
Com
plex
ity
BDRAASCAALPSCA
-77-
Power vs number of users
2 4 8 16 321
1.5
2
2.5
3
3.5
4
Number of users
Powe
r [W
]
BDRASCAALPSCALPOA
2 4 8 16 321
1.5
2
2.5
3
3.5
4
Number of usersPo
wer [
W]
BDRASCAALPSCALPOA
-78-
Power vs. spectral efficiency
0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6
7
8
9
10
Average spectral efficiency [bit/s/Hz]
Powe
r [W
]
BDRASCAALPSCALPOA
0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6
7
8
9
10
Average spectral efficiency [bit/s/Hz]Po
wer
[W]
BDRASCAALPSCALPOA
-79-
Multi-cell algorithms
-80-
Multi-cellular RRA schemes
We consider a downlink communication in an OFDMA-based multi-cell network with full reuseof the frequency spectrum among cellsThe most limiting factor for this systems is represented by multiple access interference (MAI), caused by users in adjacent cells that share the same spectrum
-81-
Multi-cell RRA: interference
With respect to the conventional single-cell scenario, the main problem is the feasibility of the allocation.
Given a certain traffic configuration, there might be no solution that satisfies the rate requirements of all users in the system.It is equivalent to the problem of power control for single-carrier cellular networks.
RRA schemes need to enforce strategies designed to control the users’ requirements in order to meet a feasible solution.
-82-
Multi-cell RRA: interference
MAI depends on the users allocated on the same channel in other cells. Interference power is computed as
The required transmitted power to achieve a certain target k,n(i) is:
I(i)k,n =
Ncells
j=1,j=i
P(j)n G
(j)k,n,i
P(i)k,n =
(i)k,n
2+I(i)
k,n
G(i)
k,n,i
-83-
Multi-cell power control
Let us focus on subcarrier nSuppose that in cell i it is allocated to user k(i). Let , and . Power control consists in solving a set of linear equations in P(1) (1) 2 ( )
1,1,1
( ) (
2
1
1
) 2 ( ),
,
1
1
c
cc c
c
c c
jN
j
N
j
j
N N jN j
N N
G
G
P G P
P G P
=
=
= +
= +
( ), ( ), ,
jk ii nj iG G= )( () ii
nP P=
( ) =I uG P
( )
2,
,
1
;
diag ,,c
l jii l j
i l
N
GG
uG
=
=
=
�…
G
( ) ( ),
i ik n=
-84-
Multi-cell power control
The matrix has non-negative elements and it is by its nature irreducible. Invoking the Perron-Frobenius theorem, these three statements are equivalent
It exist a power vectormax eigenvalue of
limk +
�˜Gk
= 0
G
( ) ( )1,M M<G G G( ) 1* : =P uGP I
-85-
Multi-cellular RRA schemes
In a multi-cell system, RRA algorithms can be classified as distributed and centralized
In a distributed scheme, resource allocation is performed locally by each base station, exploiting the knowledge of channel conditions of only the users in the cellIn a centralized approach, a radio network controller (god), that ideally knows the channel state information of all users in the system, assigns the radio resources aiming at a global optimum
-86-
Distributed schemes
Attractive because of the limited (!!) amount of feedback and computational complexity.Each cell has its own controller: RRA is performed on the base of the information available in the cell.Hybrid schemes allow a certain amount of
information exchanged on the network backbone.The lack of centralized information is partially compensated by the implementation of iterative algorithms.
-87-
Centralized schemes
Centralized solutions aim at optimizing the system performance globally:
Controller possesses full information about all users in the system
Unfortunately, they are practically unfeasible due to the large amount of signaling they require their complexity, which grows exponentially with the number of users in the system (scale with the number of cells)
-88-
Multi-cell algorithms: distributed schemes
-89-
Distributed schemes: the PB algorithm
This distributed algorithm addresses the problem by dividing it in three steps.1. Set max SINR bounds per subcarrier per user per
cell2. Solve the allocation problem 3. Implement an admission control strategy
-90-
PB algorithm: SINR bounds
The spectral radius of matrix is lower than any sub-multiplicative matrix norm of
By imposing , we can set a bound for the max target SINR per user per subcarrier, i.e.
( )( ) ( ), ,
1
,,( ), ,
maxc
i jk n k n ij j i
M ii Nk n iG
G=G G
( )M GG
( ) 1M <G
( ) ( ) ( ), , ,, , ,( ) ( )
, ,( ) ( ), , , ,,
1i j i
k n k n ij j i k n ii ik n k ni j
k n i k n ij j i
GG
GE
G< < =
-91-
PB algorithm: the allocation problem
PB propose an iterative scheme that 1. implements a heuristic that
allocates the subcarriers to users2. solves a convex problem in the
SINR variable designed to minimize the transmit power, having assumed that the interference power is fixed
3. performs power control so that each user meets its target SINR
Algorithm is iterated until convergence
( ) 2,( )
, ( ), ,
( )2 ,
1( ) ( ), ,
. .
log
min
(1 ) ( )
,
ik ni
k n ik n i
Ni
k nn
i ik n k n
IG
s t
k
E k
r k
n=
+
+
-92-
PB algorithm: the allocation problem
The admission control strategy consists in switching off those users that:
due the max SINR bounds, do not reach their target ratesexceed a certain predetermined power limit
There is a fairness problem!!
-93-
Distributed schemes: a LP approach
As in the single-cell scenario, we have formulated a linear programming approach with just a single transmission format for all users.The distributed approach leads to an iterative procedure: at the beginning of each new iteration the resources’ costs in each cell are updated taking into account the interference levels of the previous iteration.Due to interference, allocation in one cell perturbs the allocation in all neighboring cells
-94-
LP algorithm: load control
Allocation convergence is not guaranteed and the algorithm needs to be modified to reach a stable allocation.We implement a load control mechanism that progressively reduce the total amount of resources allocated in each cell until a stable allocation is achievedLP formulation still maintains the network flow topology
-95-
LP algorithm: formulation
In each cell the LP problem is formulated so that a certain number N(i)
of subcarriers has to be allocated.Each user k can get at most n(i)(k) resources.
( ),
1 1
( ),
1
( ),
( )
( )
( ),
1
( ),
1 1
min
. .
1
( )
K Ni
k nk n
K
ik n
i
ik n
kN
ik n
nK N
ik n
k n
i
s
n
N
P
t
n k k
= =
=
=
= =
=
-96-
LP algorithm: packet scheduler
By assigning a different number of subcarriers to users, the LP RRA sets the actual rate offered by the system to each user.Exploiting multi-user diversity, it tends to assign
most of the resources to users with the best channels.In order to compensate the displacement of resources due to the RRA, in each cell we implement a Packet Scheduler (PS) that aims at maximizing fairness among users by setting the max number of resources for each user
-97-
LP algorithm: architecture
At the beginning of each frame, in each cell the PS sets the maximum rate per userGiven the requirements dictated by the PS, the RRA LP iterates until it finds a stable allocation in each cellIf, after a certain number of iterations, a stable allocation has not been found, the load of the cells is progressively reduced until allocation converges.The allocation results are fed-back to the PS so that it updates the requirements to enforce fairness
-98-
LP algorithm: packet scheduler
Packet Scheduler
ResourceAllocator
Feedback on last resource allocation
( ),i
k n
( )ikn
Feedback on max. rate constraints
-99-
Multi-cell algorithms: centralized schemes
-100-
Centralized approach
The maximum achievable performance of multi-cell resource allocation is currently unknown. In analogy with the bound developed for the single-cell scenario, we develop a bound on the performance of centralized resource allocation in the dual domain.
Pros: Analytically sound, useful bound to compare other algorithms’ performanceCons: Exponential complexity, requires the knowledge of all the users channel gains at the central controller
-101-
Optimal bound: primal
The primal is formulated as a global minimization problem The problem is combinatorial and its complexity grows exponentially with K and N and the number of cellsTo partially reduce the complexity we admit only a small set of possible transmission formats
( )( ) ( ) ( ), , ,,
( ) ( ) ( )
1
, ,
( ) ( ) ( ), , ,
1
1
1
min
.
,
, ; {
.
(
}
)
| 0,1 1
N K
i n k
N
i i ik n k n k np
i i ik n k n
i i ik n k n
n
K
kk n
r
i k
i n
f
s t
r r k
R
= =
=
=
= =
-102-
Optimal bound: Lagrange dual
The Lagrange dual of the RRA can be written as the sum of N reduced-complexity minimization problems
Due to interference, the solution of the dual optimization problem needs an iterative strategy (even in the centralized approach!)
( )( )( ) ( )
1 1 1
( ) ( ) ( ) ( ), , ,
( )
1,
min ( )
.
1 ,
.
i i i i iN K K
rn i k i k
K
k
ik n k n k k n k
ik n
r
n i
f r r k
s t= = =
=
+
=
-103-
Optimal bound: Lagrange dual
The main idea is to locally optimize via coordinate descent.For each , we first find the optimal user and transmission format for cell #1 while keeping fixed the allocation in all other cells, then we find the optimal user and transmission format for cell #2 keeping all other fixed, and so on.
Note that, during each iteration, only a small finite number of power levels need to be searched over.Such an iterative process is guaranteed to converge, because each iteration strictly decreases the objective function. The convergence point is guaranteed to be at least a local minimum.
-104-
FFR-A and FFR-B
Static channel allocation according the pattern in figure
-105-
FFR-A and FFR-B
-106-
Simulation setup
Number of cells = 7Radius of each cell R = 500 mTotal available bandwidth W = 5MHzCenter frequency = 2 GHzNumber of subcarriers N = 16Number of users K = 8
-107-
Measured spectral efficiency
0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
(bit/s/Hz)
m (b
it/s/
Hz)
LPRAPBRAFFR-AFFR-B
-108-
Power vs. spectral efficiency
0.5 1 1.5 2 2.5 3 3.5 40
5
10
15
20
25
30
m (bit/s/Hz)
P m (W
)
LPRAPBRAFFR-AFFR-B
-109-
Power vs. Number of users
2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
9
10
K
P m
LPRACRA M = 2CRA M = 4
-110-
References
1. J. Jang, K.B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Select. Areas Comm., pp. 171- 178, no.2, 2003.
2. W. Rhee, J. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” Proc. IEEE VTC 2000 Spring, Tokyo, Japan, May 2000.
3. Z. Shen and J. G. Andrews and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. on Wireless Comm., vol. 4, no. 6, pp. 2726–2737, 2005.
4. C. Wong, R. Cheng, K. Letaief, and R. Murch, “Multiuser OFDM with adaptive subcarrier, bit and power allocation,” IEEE J. Select. Areas Comm., vol. 17, no. 10, pp. 1747–1758, October 1999.
5. I. Kim, I. Park, and Y. Lee, “Use of linear programming for dynamic subcarrier and bit allocation in multiuser OFDM,” IEEE Trans. Veh. Technol., vol. 55, no. 4, pp. 1195–1207, July 2006.
6. W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Comm.,vol. 54, no. 7, pp. pp. 1310–1322, July 2006.
7. W. W. L. Ho and Y.-C. Liang, “Optimal resource allocation for multiuser MIMO-OFDM systems with user rate constraints,”IEEE Trans. Veh. Technol., vol. 58, no. 3, pp. 1190–1203, 2009.
8. N. Bambos, S. Chen, and G. Pottie, “Channel access algorithms with active link protection for wireless communication networks with power control,” IEEE/ACM Trans. Netw., vol. 8, no. 5, pp. 583–597, Oct 2000.
9. M. Pischella and J.-C. Belfiore, “Distributed resource allocation for rate-constrained users in multi-cell OFDMA networks,” IEEE Commun. Lett., vol. 12, no. 4, pp. 250–252, April 2008.
10. M. Moretti and A. Todini, “A reduced complexity cross-layer radio resource allocator for OFDMA systems,” IEEE Trans.Wireless Commun., vol. 6, pp. 2807–2812, 2007.
11. M. Moretti, A. Todini, A. Baiocchi, and G. Dainelli, “A layered architecture for fair resource allocation in multicellular multicarrier systems,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1788–1798, 2011.
12. R. Y. Chang, Z. Tao, J. Zhang, and C.-C. J. Kuo, “A graph approach to dynamic fractional frequency reuse (FFR) in multi-cell OFDMA networks,” in Proc. IEEE Int. Conf. Communications ICC ’09, 2009, pp. 1–6.
