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An Efficient Heuristic Adaptive Power Controlfor Downlink Transmission in Infeasible Scenarios

Noriyuki TAKAHASHI Masahiro YUKAWA Isao YAMADA

Department of Communications and and Integrated Systems, Tokyo Institute of Technology

E-mail: {takahashi, masahiro, isao}@comm.ss.titech.ac.jp

Abstract This paper provides a novel heuristic approach tothe infeasible power control problems [i.e., there is no powerallocation that simultaneously achieves target quality of service(QoS) for multiple accessing users] in wireless communicationsystems. It is known that the existing power control techniquestend to diverge in such situations. We first present a newformulation, to deal with the infeasible situations, which coincideswith a typical formulation in feasible situations. Finding anoptimal power control in our formulation requires, in general,high computational complexity because of its combinatorialnature. We thus propose a simple but effective distributed powercontrol algorithm, which keep low computational complexity and

just requires local information. Simulation results demonstrate

that the proposed algorithm successfully increases the numberof accepted users and reduces the total amount of transmittedpower.

I. INTRODUCTION

This paper considers the infeasible power control problemsin wireless communication systems; i.e., there is no powerassignment that simultaneously achieves target quality of ser-vice (QoS) for multiple accessing users [1, 2]. The goal of thispaper is to attain a novel heuristic approach to this challengingproblem. We concentrate on downlink transmission withoutexchanging transmitted-power information among base sta-tions (BSs). In this case, each BS should control its trans-mitted power with only local measurements, which is calleddistributed (decentralized) power control [3]. To clarify ourmotivation more, let us start with a brief introduction of powercontrol systems.

Spread spectrum systems, such as code division multipleaccess (CDMA), are widely used in 3G cellular commu-nications because of its channel efficiency [4]. However, itis known that a conventional matched filter severely losesits efficacy when there exists multiple access interference(MAI) [5]. Hence, a great deal of effort has been devoted forsuppressing MAI (see [6]), at receiver sides, to provide highquality of service (QoS). To attain satisfactory performancein the MAI suppression, sufficiently high level of signal-to-interference-plus-nose ratio (SINR) is necessary at every

receiver. An increase of transmitted power improves SINRperformance, ironically, however, the increase of power forone user degrades SINR for the other users. Hence, the powersof all active users should be balanced and the power controlhas been considered to be one of the central burdens in thespread spectrum systems.

Centralized power control had been a major issue in wire-less communications, however, for the reasons of additionalinfrastructure, latency and network vulnerability, the inter-ests of most researchers have been making the transition todistributed power control [3]. The distributed power control

problem is mathematically formulated as a linear programmingwith lack of information about constraints because each BS hasonly local measurements. This defection creates very difficultsituations for the power control systems. In [3], a reasonabledistributed power control algorithm was proposed, and it hasbeen widely studied [7, 8]. This algorithm is effective whenthe problem is feasible and the system is not overloaded.Unfortunately, however, in infeasible cases, it is known that thealgorithm diverges, which degrades the SINR [1, 2]; infeasiblesituations happen when call admission control (CAC) is im-perfect. In such situations, the performance of overall systemsdeteriorates unbearably. A natural question will be what thebest strategy is for such infeasible situations.

In this paper, we propose an efficient distributed powercontrol technique, which exploits the technique in [3] butperforms effectively even in infeasible situations. First, wepresent a new formulation for infeasible power control prob-lem; maximize the number of users satisfying target SINRwhile minimizing the total power at all BSs. Especially,our formulation coincides with the conventional formulation.Finding an optimal power control in our formulation requires,in general, high computational complexity because of itscombinatorial nature. We then present the proposed algorithm,which requires almost the same computational complexity as

the technique in [3]. Simulation results verify the efficacy ofthe proposed algorithm.

I I . PRELIMINARIES

We consider a cellular wireless system with N base stations(BSs) and M cochannel users (see Fig. 1) [7]. Assume thatthe channels are stationary, and each BS updates its powerallocation synchronously. Also assume the ith user is assignedto the bith BS, where bi {1, . . . , N }, and BS assignmentsfor all users are fixed during considered period.

Let pi [0, pmax] be the transmitted power for the ith user,where pmax is the maximum of the power allocation for eachuser, and hibj 0 the channel gain from the bj th BS to theith user. Then the received SINR at the ith user is defined as

follows [7]:

i(p) :=hibipiM

j=i hibjpj + 2i

=:hibipi

Ii(p), (1)

where p = (p1, . . . , pM)t is the power vector, 2i and Ii(p)

denote powers of additive noise and the interference-plus-noiseat the ith user, respectively. In a CDMA system, hibj can bewritten as [9]

hibj = wHj Ribj wj , (2)

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bith BS bj th BS

ith user jth user

hibihibjhjbi hjbj

Fig. 1. Model of downlink transmission. Solid and dotted arrows are targetand interference passes, respectively.

where wj is the normalized (i.e. wj = 1) beamformerweight vector for the jth user, and Ribj is the downlinkchannel correlation (DCC) matrix between ith user and bj thBS.

In order to suffice QoS requirements of all users, the SINRat the ith user must exceed at least a certain level i, which werefer to as target SINR. The objective of the power control isto minimize the total power of all users under such a condition.Then the power control problem is formulated as follows [7]:

minimizepi[0,pmax]

M

i=1

pi

subject to i(p) i, i = 1, . . . , M .

(3)

Let P = [0, pmax]M denote the set of all power vectors.

Then, we can rewrite the problem (3) in a matrix form asfollows [7]:

minimizepP

1tp

subject to (I H)p u,(4)

where 1 = (1, . . . , 1)t,

[H]ij =

0 if i = jihibj

hibi > 0 otherwise

(5)

and

[u]i =i

2i

hibi> 0. (6)

We assume that H is an irreducible matrix (see Appendix,Definition 2) [7]. Let (H) denote the spectral radius (i.e. themaximum absolute eigenvalue) ofH. Most existing studies [7,8] consider the case that (H) < 1. Hence matrix (I H) isinvertible and (I H)1 > O (see Appendix, Definition 1)[10]. In this case,

p = (I H)1u > O, (7)

and, ifp P, the total transmitted power is minimized by p,that is p is the optimal power vector. In order to calculate p,the following distributed power control algorithm is proposed[7].

Generate a sequence of the power vectors (p(n))nN as[N: the set of non-negative integers]

p(n+1)i =

i

hibi

M

j=i

hibjp(n)j +

2i

= i

hibiIi(p

(n)), (8)

where p(n)i is the transmitted power for the ith user at the nth

iteration. In a matrix form, (8) can be written as

p(n+1) = Hp(n) + u, n N. (9)

Eq. (8) suggests that there is no need to know all the channelgains and the power of additive noise. That is, at each iteration,each BS can update the power only using the interference-plus-noise power and the path gain from itself to the target user.

Starting from an arbitrary power vector p(0), this algorithmconverges to the optimal power vector p [3]. When theproblem (3) is infeasible (i.e. there does not exist p Psatisfying (I H)p u), the algorithm (9) has no guaranteeto perform appropriately; e.g. it diverges if (H) 1.

III . POWER CONTROL FOR INFEASIBLE CASES

First we reformulate the power control problem for infea-sible situations. Then we propose an efficient heuristic powercontrol algorithm to solve the reformulated problem.

A. A General Formulation of Power Control

Consider the case that there exist no power vector psatisfying i(p) i, for all i = 1, . . . , M , i.e. the problem(3) is infeasible. In this case, we need to find a compromisesolution. What we can do is to provide the target SINR to asmany users as possible at the expense of the minimum numberof users. It is natural to consider that a best solution to theinfeasible cases is to maximize the number of users satisfyingthe target SINR. Define

M(p) := {i M | i(p) i} , (10)

where M := {1, . . . , M }. Then we reformulate the problem(3) as follows:

minimize 1tp

subject to p arg maxpP

|M(p)| =: Q. (11)

It is trivial that, if the original problem (3) is feasible, then

maxpP

|M(p)| = M (12)

and

Q = {p P | (I H)p u}. (13)

Thus the problem (11) is a natural extension of the originalproblem (3).

B. An Efficient Heuristic Power Control Scheme

We present an efficient heuristic algorithm to solve the prob-lem (11). First, we simply modify the conventional algorithm

so as not to diverge as follows:

p(n+1)i = min

i

hibiIi(p

(n)), pmax

, n N. (14)

Henceforth, we call (14) the conventional algorithm instead of(8).

Let us give an essential idea of the proposed algorithm. Toresolve the infeasibility, it is necessary to eliminate some usersbecause the system cannot provide the target SINR to all users.The key point is to select the users to be eliminated so thatthe problem becomes feasible at the expense of the minimum

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number of users. A user allocated pmax is a factor deterioratingthe SINRs of other users. Hence, if this user does not achievethe target SINR, the user should be eliminated. We call sucha user bottleneck user. In the presence of bottleneck users,it is a natural strategy to eliminate those users by priority.Actually, the step 2 of the proposed algorithm realizes theabove strategy. The proposed algorithm is given as follows.

Algorithm 1: Let 0 be the tolerance for convergence,

and 0 < 1 the tolerance for admission. Starting from anarbitrary power vector p(0) P, for all pi at the nth (n N)iteration, do the following:

Step 1 Minimize the power. Update pi by (14).Step 2 Eliminate possible bottleneck users to resolve the

infeasibility.

If

p(n)i = pmaxp(n)i p(n1)i

i(p(n)) i

p()i = 0, n

,

(15)

otherwise do nothing.

Step 3 If, for all i M,p(n)i p(n1)i but (15)

is not executed, then the algorithm exits. Otherwise,n := n + 1 and return to step 1.

Referring to the statement after (13), it is obvious that theproposed algorithm generate the same power vector as theconventional one when the problem is feasible. In step 2, ifthe power is pmax and sufficiently converges, but the SINRdoes not achieve the target SINR, then the user is consideredas a bottleneck, thus eliminated from the system. Finally, afterenough number of bottleneck users are eliminated, the systembecomes feasible. Therefore the remaining users can achievethe target SINR. It is clear that the proposed algorithm keepsa computational complexity of the conventional one.

To clarify the advantage of the proposed algorithm, let usgive a simple example of an infeasible case.

Example 1: Consider the case with M = 3 and the follow-ing system parameters:

pmax = 1, 1 = 2 = 3 = 2, 21 =

22 =

23 = 0.01,

h1b1 = 0.8, h1b2 = 0.2, h1b3 = 0.2,

h2b1 = 0.2, h2b2 = 0.8, h2b3 = 0.2,

h3b1 = 0.2, h3b2 = 0.2, h3b3 = 0.5.

In this case, we have

H =

0.0 0.5 0.50.5 0.0 0.50.8 0.8 0.0

, u =

0.0250.0250.040

,

(H) 1.18( 1),

(I H)1u = (1.5, 1.5, 2.0)t,

which implies that the problem is infeasible. The proposedand conventional algorithms respectively update the powervectors as shown in Table 1. At 13th iteration, the proposedalgorithm sets p3 to 0 by Step 2, because the 3rd user isthe bottleneck user. Finally, the proposed algorithm achieves1(p

(25)) = 2 = 1, 2(p(25)) = 2 = 2, 3(p

(25)) =0. On the other hand, the conventional algorithm achieves

TABLE I

COMPARISON OF POWER ALLOCATIONS OF THE PROPOSED AND

CONVENTIONAL ALGORITHMS.

Proposed Conventionaln p1 p2 p3 p1 p2 p30 0 0 0 0 0 0

.

.

....

.

.

.11 0.647 0.647 0.872 0.647 0.647 0.87212 0.784 0.784 1.000 0.784 0.784 1.000

13 0.917 0.917 0.000 0.917 0.917 1.00014 0.484 0.484 0.000 0.984 0.984 1.00015 0.267 0.267 1.000 1.000 1.000 1.000

.

.

....

.

.

.25 0.050 0.050 0.000 1.000 1.000 1.000

TABLE II

SOME COMPARISONS OF THE PROPOSED AND CONVENTIONAL

ALGORITHMS FOR = 5.

Convent ional Proposed

Averaged power 0.8840 9.97 104

Maximum power 1.0000 3.86 103

Minimum power 0.0594 0.0 (rejected)Averaged SINR 1.775 dB 1.990 dB

Maximum SINR 5.000 dB 5.000 dBMinimum SINR 4.395 dB (rejected)

1(p(25)) = 1.95 < 1, 2(p

(25)) = 1.95 < 2, 3(p(25)) =

1.22 < 3. Therefore, the proposed algorithm succeeds toresolve the infeasibility.

IV. SIMULATION RESULTS

We consider a CDMA wireless system with N = 7 BSsillustrated in Fig. 2. Each BS is equipped with a uniformcircular array (UCA) of eight omnidirectional sensors. Thechannel gain is assumed to be proportional to r4ib

j

, where ribjis the distance between the ith user and the bj th base station.All users are randomly distributed in the area shown in Fig. 2with uniform distribution, and they are assigned to the nearestBS. We set to pmax = 1. The beamformer weight vector wi iscomputed by [11]

P

Mi=j

Ri,bj

1

Ribi

, (16)

where P{} denotes the principal eigenvector. We assumethat the target SINR is common to all users. The noise poweris randomly distributed between 0 and 0.01 with uniform

distribution. For the proposed algorithm, we set = 109

and = 0.8. All the results are averaged over 1000 timesindependent runs.

Fig. 3 shows the averaged number of accepted users. Thetarget SINR is ranges between 1 dB and 8 dB every 1 dB. Wecan see that the proposed algorithm always provides acceptedusers more than the conventional one. Especially, for = 5,4.8 times more users are accepted by the proposed algorithmcompared to the conventional one. For 2 6 (: targetSINR), the proposed algorithm achieves more than twice asmany accepted users as the conventional one.

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-2

-1

0

1

2

-2 -1 0 1 2

x

y

Fig. 2. Base stations and users distribution (N = 7 and M = 16). Basestations and users are pointed out by + and , respectively.

0

2

4

6

8

10

12

14

16

-1 0 1 2 3 4 5 6 7 8

conventionalproposed

Target SINR (dB)

Numberofuserssatisfying

Fig. 3. Comparison of the averaged number of accepted users.

Fig. 4 exemplifies how amount of the total power is re-duced thanks to the proposed algorithm. We observe that theproposed algorithm attains about 30dB gain. Detailed data aregiven in Table II.

V. CONCLUSIONS

We considered infeasible power control problems and pre-sented the new problem formulation. To solve this problem,we also proposed the heuristic distributed power control al-gorithm. The proposed algorithm increased the number ofusers satisfying the target SINR in infeasible cases withoutincreasing the computational complexity. We also confirmedthe advantage of the proposed algorithm by simulations.

APPENDIX

Definition 1: An m n matrix A with real elements iscalled non-negative (positive) if all the elements are non-negative (positive), and notated A O (> O).

-40

-30

-20

-10

0

0 20 40 60 80 100

Averagedpower(dB)

Number of iteration

conventionalproposed

Fig. 4. An averaged power of users;P

M

i=1pi/M for = 5 dB.

Definition 2: An n n square matrix A is called reducibleif there exists a permutation matrix P that puts A into the

formP APt =

B OC D

,

where B and D are square matrices. Otherwise A is calledirreducible.

Acknowledgement: The authors would like to express theirdeep gratitude to Prof. K. Sakaniwa of Tokyo Institute ofTechnology for fruitful discussions. This work was supportedin part by JSPS grants-in-Aid (178440).

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