[IEEE 2007 IEEE 18th International Symposium on Personal, Indoor and Mobile Radio Communications -...

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

COORDINATING BASE STATIONS FOR GREATER UPLINK SPECTRAL

EFFICIENCY: PROPORTIONALLY FAIR USER RATES

Sivarama Venkatesan

Bell Laboratories, Alcatel-Lucent

Holmdel, New Jersey, U.S.A.

ABSTRACT

To overcome the spectral efficiency limits imposed by cochan-

nel interference on the uplink of a cellular network, we propose

coordinating each base station with several of its neighbors in

the reception of user signals, and decoding each user at one

such cluster of base stations. We evaluate the impact on the

user rate distribution due to such coordination when there is

1 user per base station antenna in the network, each user has 1

transmitting antenna, all users transmit at maximum power, and

the data rates of the users assigned to each cluster are subject

to a proportional fairness criterion. We highlight the depen-

dence of the user rate distribution on the number of rings of

neighbors with which each base station is coordinated, as well

as the underlying signal-to-noise ratio (SNR) distribution in the

network. Our results point to the possibility of increasing cell-

edge user rates almost tenfold, while simultaneously doubling

the overall system throughput, in high-SNR conditions.

I INTRODUCTION

Interference between cochannel users places a fundamental

limit on the spectral efficiency achievable in today’s cellular

networks. Within the limits imposed by cochannel interfer-

ence, link performance is already close to optimal, thanks to

the use of sophisticated error correcting codes, adaptive mod-

ulation, incremental redundancy, etc. Therefore, novel strate-

gies for mitigating cochannel interference are likely to be cru-

cial in meeting the spectral efficiency requirements of future-

generation cellular networks.

In [1], one such strategy was proposed for the uplink of a cel-

lular network, viz., coordinating several base stations in the re-ception of users within their coverage area. The premise in [1]

is that the network has several “coordination clusters”, each

consisting of a base station and one or more rings of its neigh-

bors, and that the antennas of all the base stations in each clus-

ter can act as a single coherent antenna array. Results therein

show that the interference affecting each user in the network

can then be suppressed quite effectively by means of coherent

linear beamforming at an appropriately chosen cluster.

The potential gain in uplink spectral efficiency from such co-

ordination was evaluated in [1] by simulation, with 1 user per

base station antenna in the network and 1 transmitting antenna

per user, and the requirement that all users (but for a small

fraction in outage) be served at equal data rates. Further, the

receiver architecture was restricted to be based on linear min-

imum mean squared error (MMSE) beamforming. The equal-

rate requirement was met by controlling the powers at which

the users transmitted.

In this paper, we investigate the theoretical potential of base

station coordination on the uplink from a different and comple-

mentary perspective. We relax the requirement of equal user

rates, and instead assume that all users always transmit at max-

imum power. Each user is assigned to the coordination cluster

at which it attains the highest signal-to-interference-plus-noise

ratio (SINR). Each cluster then jointly decodes all the users

assigned to it, treating all users assigned to other clusters as in-

terference (i.e., without attempting to decode them). The rates

of the users decoded at a cluster are chosen to be proportionallyfair [2] within the capacity region of the multiple access chan-

nel corresponding to that cluster. Apart from the restriction

of not attempting to decode users assigned to other clusters,

the receiver architecture at each cluster is not constrained in

any way. Interestingly, the proportional fairness criterion max-

imizes the sum as well as the product of the user rates within

each cluster. As in [1], we gloss over channel estimation issues

and simply assume the availability of perfect channel state in-

formation wherever needed.

Within the above framework, we compare the user rate distri-

butions achievable with coordination clusters of different sizes,

with the signal-to-noise ratio (SNR) distribution over the net-

work as a parameter. Our results show that base station coordi-

nation has the potential to dramatically increase cell-edge user

rates, while simultaneously improving the mean user rate sig-

nificantly. Thus, much greater fairness between users can be

achieved without sacrificing overall system spectral efficiency.

The rest of the paper is organized as follows. We describe

the cellular network model assumed in this study in Sec. II,

and the methodology used to evaluation the coordination gain

with proportionally fair user rates in Sec. III. The results from

the simulations are in Sec. IV, and the conclusions in Sec. V.

II NETWORK MODEL

We focus our study on an idealized cellular network of 127

regular hexagonal cells (center cell plus 6 rings of neighbors),

with a base station at the center of each cell. We assume that

the network is wrapped around at its edges, so that every cell

can effectively be regarded as the center cell of the network.

Each cell has 3 sectors, with main lobe directions as indi-

cated by the arrows in Figure 1. There are N receiving anten-

nas per sector, each having an idealized antenna beam pattern

given by

A(θ) = min

{12

(θ

Θ

)2

, Am

}, −π ≤ θ < π. (1)

Here, A(θ) represents the beam attenuation in dB along a di-

rection making an angle of θ radians with the main lobe di-

1-4244-1144-0/07/$25.00 c©2007 IEEE


D

s

Figure 1: Sector orientation

rection. The parameters Θ and Am are respectively the 3-dB

beamwidth (in radians) and the maximum beam attenuation (in

dB). We set Θ = 7π/18 and Am = 20.

A Channel model

In the interest of simplicity, we assume that all user-to-sector

links in the network are flat-fading and time-invariant, and that

there is perfect symbol synchronization between all users at

each sector. Further, we assume that each user in the network

has a single omnidirectional transmitting antenna. Accord-

ingly, we model the complex baseband signal vector ys(t) ∈C

N received at the N antennas of sector s during symbol pe-

riod t as

ys(t) =U∑

u=1

hs,uxu(t) + zs(t). (2)

Here, U is the total number of users in the network; xu(t) ∈ C

is the complex baseband signal transmitted by user u during

symbol period t; hs,u ∈ CN is the vector representing the

channel from user u to sector s; and zs(t) ∈ CN is a circularly

symmetric complex Gaussian vector representing additive re-

ceiver noise, with E [zs(t)] = 0 and E [zs(t)z∗s(t)] = I. We

subject each user to a transmitted power constraint of 1, i.e.,

E |xu(t)|2 ≤ 1.

Each channel vector hs,u has a position-dependent power

loss component, a lognormal shadow fading component, and a

complex Gaussian multipath fading component. Specifically,

hs,u =√

η

(ds,u/D)α 10A(θs,u)/1010γs,u/10gs,u. (3)

In (3), ds,u is the distance between user u and sector s, and

D is half the distance between neighboring base stations (see

Figure 1); α is the path loss exponent, taken to be 3.8; θs,u ∈[−π, π) is the angle in radians that the position vector of user urelative to sector s makes with the main lobe direction of sec-

tor s; A(·) is the sector antenna beam pattern, as in (1); γs,u is

a real Gaussian random variable of mean 0 and standard devi-

ation 8, representing the effects of large-scale shadow fading;

and gs,u is an N -dimensional circularly symmetric complex

Gaussian vector of mean 0 and covariance I, representing the

effects of small-scale multipath fading.

For each user u, we assume that the shadow fading ran-

dom variables γs,u corresponding to different sectors s are in

fact jointly Gaussian, with 100% correlation between sectors

of the same cell and 50% correlation between sectors of differ-

ent cells. But for these constraints, the random variables γs,u,

gs,u, and zs(t) in (2) and (3) for different s, u, and t are all

statistically independent.

Under all the above assumptions, we can interpret the param-

eter η in (3) as the average SNR at sector s of a user u located

at half the distance to the neighboring base station along the

main lobe direction (for the sector s in Figure 1, this location is

indicated by a cross), when the shadow fading random variable

γs,u is at its mean value of 0 (or, equivalently, when there is no

shadow fading). The SNR distribution over the network is then

determined by η and the path loss exponent α (equal to 3.8).

Note that the value of D is immaterial (i.e., we have scale in-

variance). We will henceforth refer to η simply as the referenceSNR at the cell edge.

As η is increased from a very small value to a very large

value, the network goes from being limited primarily by re-

ceiver noise to being limited primarily by interference between

users. We can therefore expect the mitigation of interference

through coordinated reception at multiple base stations to be

more beneficial at higher η values. In the simulations, we will

determine the coordination gain as a function of η, varying the

latter over a wide range of values.

B Coordination clusters

We define a coordination cluster to be a subset of base stations

that jointly and coherently process the received signals at the

antennas of all their sectors. We suppose that the network has

a predefined set of coordination clusters, and that each user

can be assigned to any one of these clusters. Each cluster then

jointly decodes all the users assigned to it, in the presence of

interference from all other users in the network.

To highlight the dependence of the spectral efficiency gain

on the number of rings of neighbors with which each base sta-

tion is coordinated, we will be interested in coordination clus-

ters having a specific structure. For any integer r ≥ 0, we de-

fine an r-ring coordination cluster to consist of any base station

and the first r rings of its neighboring base stations (accounting

for wraparound), and Cr to be the set of all r-ring coordination

clusters in the network. Figure 2 illustrates 0-ring, 1-ring, 2-

ring, and 4-ring coordination clusters.

Note that each base station is at the center of a unique cluster

in Cr. As a result, all cells are equally favored from the point

of view of coordination (this spatial homogeneity is the reason

we consider overlapping clusters, instead of disjoint ones). To

ensure that coordination is truly limited to the first r rings of

neighbors, we will disallow any exchange of information be-

tween clusters in Cr through base stations that they may have

in common.

With C0 as the set of coordination clusters in the network,

there is in fact no coordination between base stations. This

case will serve as the benchmark in estimating the spectral effi-

ciency gain achievable with sets of larger coordination clusters.

Specifically, we will compare C1, C2, and C4 with C0.

With some abuse of notation, we will denote by hC,u ∈C

3N |C| the channel from user u to the antennas of all the base

stations in the coordination cluster C (here |C| denotes the

number of base stations in C). Then, with each user transmit-

ting at its maximum power of 1, the SINR attained by user u


(a) r = 0 (b) r = 1

(c) r = 2 (d) r = 4

Figure 2: r-ring coordination clusters

at cluster C is h∗C,u

(I +

∑v �=u hC,vh∗

C,v

)−1

hC,u. Note that

this expression assumes perfect knowledge at cluster C of the

channel vector hC,u and the composite interference covariance∑v �=u hC,vh∗

C,v .

III SIMULATION METHODOLOGY

The objective of the simulations is to compare the user rate

distributions achievable with the coordination cluster sets C0,

C1, C2, and C4 (as defined above), when all users transmit at

maximum power and the rates of the users decoded at each

cluster are chosen to satisfy a proportional fairness criterion.

Note that all users are served, though some users may achieve

much lower rates than others.

The rate distribution achievable with a given set of coordina-

tion clusters is determined from several independent trials, in

each of which we do the following:

1. Populate the network with users.

2. Assign each user to a coordination cluster.

3. Find the proportionally fair rate vector in the capacity re-

gion of each cluster.

Finally, we compute the distribution of all the user rates in all

the trials. Details are in the following subsections.

A Populating the network with users

We assume that the network is loaded randomly and uniformly

with N users per sector, N being the number of receiving an-

tennas in each sector. In other words, we allow only 1 user

per sector antenna. The justification for this assumption is that

a larger pool of users can be split between orthogonal dimen-

sions, e.g., time slots or frequency bands, so that there are only

N users per sector in each dimension (note that we would have

to assume such dimensions to be orthogonal over the entire net-

work, and not just within each sector).

We choose the N users associated with sector s to have

higher average path gain to sector s than to any other sector,

1/4 1/2 1 2 4 80.1

0.3

0.5

0.7

0.92 antennas/sector, 2 users/sector, 12 dB ref. SNR

User rate (bits/sym)

CD

F of

use

r rat

e

No coord.

1−ring coord.

2−ring coord.

4−ring coord.

Figure 3: Rate distributions: N = 2, η = 12 dB

while ensuring that users are equally likely to be situated at all

locations in the network. The average path gain from user u to

sector s is the quantity under the square root sign in (3).

B Assigning users to clusters

Suppose that the set of coordination clusters in the network is

Cr for some r ≥ 0. We assign user u to the coordination cluster

C(u) � arg maxC∈Cr

h∗C,u

I +

∑v �=u

hC,vh∗C,v

−1

hC,u, (4)

i.e., the cluster in Cr at which it attains the highest SINR (with

a linear MMSE receiver) in the presence of interference from

all other users in the network.

For each cluster C ∈ Cr, let U(C) denote the subset of users

assigned to C as above. These users are jointly decoded based

on the received signals at the antennas of all the base stations in

C. In doing so, all users not in U(C) are treated as interferers,

i.e., no attempt is made to decode them at C. Thus, we have

a multiple access channel corresponding to each coordination

cluster C, defined by

yC(t) =∑

u∈U(C)

hC,uxu(t) + wC(t). (5)

Here yC(t) ∈ C3N |C| is the vector of received signals at the

antennas of all the base stations in C, and wC(t) ∈ C3N |C|

is the sum of the noise and the interference from users not in

U(C) at those antennas. We will require all users to transmit

signals with a circularly symmetric complex Gaussian distri-

bution of mean 0 and variance 1. Therefore, wC(t) is itself

circularly symmetric complex Gaussian, with mean 0 and co-

variance QC � I +∑

u/∈U(C) hC,uh∗C,u.

As is well known [3], the capacity regionR(C) ⊆ RU(C)+ of

the above multiple access channel is the convex (and compact)

polytope defined by the system of linear inequalities

∑u∈S

ru ≤ log

∣∣∣∣∣I + Q−1C

∑u∈S

hC,uh∗C,u

∣∣∣∣∣ ∀S ⊆ U(C),

ru ≥ 0 ∀u ∈ U(C). (6)


0 6 12 18 241

1.5

2

2.51 antenna/sector, 1 user/sector

Reference SNR at cell edge (dB)

Gai

n in

mea

n us

er ra

te

1−ring coord.

2−ring coord.

4−ring coord.

Figure 4: Coordination gain (mean): N = 1

0 6 12 18 241

1.5

2

2.52 antennas/sector, 2 users/sector


Gai

n in

mea

n us

er ra

te

1−ring coord.

2−ring coord.

4−ring coord.


The vertices of R(C) are achievable by successive decoding

and cancellation of users, with a linear MMSE beamformer

being employed for each user to optimally suppress the noise

and interference from users yet to be decoded. Each vertex of

the capacity region corresponds to a distinct decoding order.

Other rate vectors on the sum-rate hyperplane∑

u∈U(C) ru =

log∣∣∣I + Q−1

C

∑u∈U(C) hC,uh∗

C,u

∣∣∣ can be achieved by time-

sharing between different decoding orders.

C Proportional fairness

Once the assignment of users to clusters is fixed as above, for

each cluster, we must determine the rates at which the users as-

signed to it are served, i.e., we must choose an operating point

within the capacity region of each cluster. In general, choosing

such an operating point involves making a tradeoff between

overall system throughput and fairness among users. For our

purposes, we assume that the choice of operating point is made

according to a proportional fairness criterion [2].

The rate vector r̃ is proportionally fair w.r.t. the capacity re-

gionR(C) of cluster C, if r̃ ∈ R(C) and∑u∈U(C)

ru − r̃u

r̃u≤ 0 for any r ∈ R(C). (7)

In words, r̃ is proportionally fair if the sum of the fractional

increases in user rates in moving from r̃ to any other rate vector

r cannot be positive.

0 6 12 18 241

1.5

2

2.54 antennas/sector, 4 users/sector


Gai

n in

mea

n us

er ra

te

1−ring coord.

2−ring coord.

4−ring coord.


It can be shown that there exists a unique proportionally fair

rate vector inR(C), which is the unique maximizer overR(C)of the function f(r) �

∑u∈U(C) log ru. So the proportionally

fair rate vector maximizes the product of the user rates. We

now sketch the proof of the above claim. Since f is strictly

concave andR(C) is compact and convex, f has a unique max-

imizer over R(C). Further, r̃ ∈ R(C) is this maximizer if

and only if ∇f(r̃)T (r − r̃) ≤ 0 for all r ∈ R(C) [4]. But

∇f(r̃)T (r− r̃) =∑

u(ru − r̃u)/r̃u, so such an r̃ must be the

unique proportionally fair rate vector inR(C).Another observation to be made here is that the proportion-

ally fair rate vector in R(C) must lie on the sum-rate hy-

perplane∑

u∈U(C) ru = log∣∣∣I + Q−1

C

∑u∈U(C) hC,uh∗

C,u

∣∣∣.The reason is simply that, for any r ∈ R(C), we can find an

r̃ on the sum-rate hyperplane such that r̃ ≥ r (componentwise

inequality), implying that f(r̃) ≥ f(r). As a corollary, the pro-portionally fair rate vector inR(C) must maximize the sum aswell as the product of the rates of users decoded at cluster C.

We can find the proportionally fair rate vector in R(C) by

maximizing∑

u∈U(C) log ru over R(C). This is a standard

convex program, and as such can be solved by any generic con-

vex optimization algorithm. For our purposes, we use the sim-

ple conditional gradient algorithm [4], which reduces a convex

program to a sequence of linear programs and one-dimensional

optimizations. The structure ofR(C) (specifically, the fact that

it is a polymatroid) makes the linear programs particularly sim-

ple to solve. We sketch the overall algorithm below:

1. Let r(0) be any vertex ofR(C), corresponding to an arbi-

trary decoding order.

2. Given r(n), let r̂(n) be the vertex of R(C) corresponding

to decoding users in decreasing order of rates in r(n).

3. Let r(n+1) be the point along the line joining r(n) and r̂(n)

that maximizes∑

u∈U(C) log ru (by binary search).

4. Iterate until convergence.

IV SIMULATION RESULTS

For the simulation results, we consider 3 different values of

N , the number of antennas per sector as well as the number of

users per sector, viz., N = 1, 2, 4.


0 6 12 18 241

4

7

10

131 antenna/sector, 1 user/sector


Gai

n in

10th−p

ct. u

ser r

ate

1−ring coord.

2−ring coord.

4−ring coord.

Figure 7: Coord. gain (10th percentile): N = 1

0 6 12 18 241

4

7

10

132 antennas/sector, 2 users/sector


Gai

n in

10th−p

ct. u

ser r

ate

1−ring coord.2−ring coord.

4−ring coord.


Figure 3 shows the user rate distributions achievable with

coordination clusters of different sizes, when N = 2 and the

reference SNR η at the cell edge is 12 dB (the x-axis is on a

logarithmic scale). Note that, as the cluster size is increased,

the rate distribution becomes much steeper and also shifts to

the right, indicating greater fairness between users as well as

higher system throughput.

Figures 4, 5, and 6 illustrate the gain in mean user rate

achievable with different coordination cluster sizes, for the

three values of N under consideration. Specifically, each fig-

ure shows the ratio of the mean user rate achievable with C1(1-ring coordination), C2 (2-ring coordination), and C4 (4-ring

coordination) to that achievable with C0 (no coordination), for

a different value of N . Figures 7, 8, and 9 similarly show the

gain in the 10th-percentile user rate. The gains are all plotted

as functions of the reference SNR η at the cell edge.

Note that the coordination gain increases with the reference

SNR η in each case, because interference mitigation becomes

more helpful as the level of interference between users goes up

relative to receiver noise. Also, at the low end of the η range,

a large fraction of the gain comes just from 1-ring coordina-

tion. This is because most of the interferers that are significant

relative to receiver noise are within range of the first ring of

surrounding base stations. However, as η is increased, inter-

ferers that are further away start to become significant relative

to receiver noise, and therefore it pays to increase the coordi-

0 6 12 18 241

4

7

10

134 antennas/sector, 4 users/sector


Gai

n in

10th−p

ct. u

ser r

ate

1−ring coord.2−ring coord.

4−ring coord.


nation cluster size correspondingly. Finally, the gains in 10th-

percentile rate are much higher than those in mean rate, i.e.,

cell-edge users are helped the most by base station coordina-

tion, as might be expected.

The results indicate that, in high-SNR conditions, 4-ring

coordination can increase the 10th-percentile user rate al-

most tenfold, while simultaneously doubling the overall system

throughput.

V CONCLUSIONS

In this paper, we have investigated base station coordination as

an approach towards overcoming the uplink spectral efficiency

limits imposed by cochannel interference in a cellular network.

Specifically, we have studied the user rate distributions achiev-

able in theory with coordination clusters of different sizes, un-

der a proportional fairness criterion on the user rates (with all

users transmitting at maximum power). This criterion maxi-

mizes both the sum and the product of the user rates within each

cluster. We have also described a simple algorithm to compute

the proportionally fair rate vector within the capacity region of

each cluster.

We have quantified the dependence of the gain in mean and

10th-percentile user rates on the number of rings of neighbors

with which each base station is coordinated, as well as the SNR

distribution in the network. Our results suggest that, in the-

ory, 4-ring coordination can increase the 10th-percentile user

rate almost tenfold in high-SNR conditions, while simultane-

ously doubling the overall system throughput. So base station

coordination can lead to much greater fairness between users

without sacrificing overall system spectral efficiency. However

several technical challenges will have to be overcome before

these gains can be realized in practice.

REFERENCES

[1] S. Venkatesan, “Coordinating base stations for greater uplink spectral

efficiency in a cellular network”, Proc. IEEE PIMRC, 2007.

[2] F. Kelly and A. Maulloo and D. Tan, “Rate control for communication

networks: shadow prices, proportional fairness and stability”, J. Opera-tional Research Society, pp. 237-252, 1998.

[3] T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley,

1991.

[4] D. Bertsekas, Nonlinear Programming, Athena Scientific, 1995.

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