ISIT2007, Nice, France, June 24 - June 29, 2007

Uplink Macro Diversity with Limited Backhaul

CapacityAmichai Sanderovich*, Oren Somekht, and Shlomo Shamai (Shitz)*

* Department of Electrical Engineering, Technion, Haifa 32000, Israelt Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA

Email: amichi@tx.technion.ac.il, orens @princeton.edu, sshlomo @ee.technion.ac.il

Abstract- In this contribution we present new achievablerates, for the non-fading uplink channel of a cellular network,with joint cell-site processing, where unlike previous results,the error-free backhaul network has finite capacity per-cell.Namely, the cell-sites are linked to the central joint processorvia lossless links with finite capacity. The cellular network ismodeled by the circular Wyner model, which yields closed formexpressions for the achievable rates. For this idealistic model,we present achievable rates for cell-sites that use compress-andforward scheme, combined with local decoding, and inter-celltime-sharing. These rates are then demonstrated to be ratherclose to the optimal unlimited backhaul joint processing rates,already for modest backhaul capacities, supporting the potentialgain offered by the joint cell-site processing approach.

I. INTRODUCTION

The growing demand for ubiquitous access to high-data rateservices, has produced a huge amount of research analyzingthe performance of wireless communications systems. Cellularsystems are of major interest as the most common method forproviding continuous services to mobile users, in both indoorand outdoor environments. In particular, the use of joint multi-cell processing (MCP) has been identified as a key tool forenhancing system performance (see [1] and references thereinfor recent results on multi-cell processing).

Since its introduction in [2], the Wyner cellular model hasprovided a framework for many works dealing with multi-cell processing. Albeit its simplicity, this model captures theessential structure of a cellular system and facilitates analyticaltreatment. The uplink channel of the Wyner linear and planarmodels are analyzed in [2] for optimal and linear minimummean square error (MMSE) MCP receivers, and Gaussianchannels. In [3], the Wyner model is extended to includefading channels and the performance of single and two cell-site processing under various setups is addressed. In [4] theresults of [2] are extended to include fading channels.

All the above cited studies assume that the backhaul networkconnecting the cell-sites to the remote central processor (RCP)is error-free with infinite capacity. In this work we relax thisassumption and allow each cell-site to connect to the RCPvia a reliable error-free connection, but with limited capacity.Such a model suits cellular networks, where joint decoding canimprove the overall network performance, with the underlyingassumption that the received signals are forwarded to onelocation to be jointly processed. Since network resourcesare finite, in particular when the cell-sites are in fact "hot

spots" with limited complexity, the inclusion of finite backhaulresources facilitates better prediction of the ultimate theoreticalperformance of MCP.

Recently, the common problem of nomadic terminals send-ing information to a remote destination via agents with losslessconnections has been investigated in [5] and extended in [6]to include multiple-input multiple-output (MIMO) channels.These works focus on the nomadic regime, where the nomadicterminals use codebooks which are unknown to the agents, butfully known to the remote destination. Such setting suits theuplink channel of the limited backhaul cellular system withMCP, where the oblivious cell-sites play the role of the agents.

Using the techniques developed in [5][6], we assess theimpact of limited backhaul capacity on the performance ofvarious transmission schemes. Throughout this work we usea circular variant of the linear Wyner cellular setup [2],which provides an homogenous framework with respect to themobile users and cell-sites. In particular we are interested inthe asymptotic scenario of infinitely many nodes, where thecircular and linear Wyner models are equivalent.The rest of the paper is organized as follow. In Section II we

define the system model. In Section III we prove an achievablerate for oblivious cell-sites, while in Section IV we considerthe case where cell-sites perform partial local decoding andpractice time-sharing. Numerical examples are presented inSection V, which demonstrate the effect of limited backhaulcapacity on MCP performance. Several proofs are relegated tothe Appendix.

II. SYSTEM MODEL

We consider the circular variant of the Wyner model [2],which includes an array of N cell-sites, indexed by j =

O, ... ,N -1, arranged on a circle. Each cell contains asingle user (j user for the jth cell), which transmits X. tothe channel, incorporating intra-cell time division multipleaccess (TDMA) transmission scheme. Each cell-site receivesthe transmission of the single user with independent interfer-ences from the users of the adjacent cells and additive whitecircularly-symmetric complex Gaussian noise. The receivedsignal at the jth cell-site for an arbitrary time index reads

Y Xj (X[j-l]N + X[j+l]N) +Zj (1)

where [j]N A mod N, and the inter-cell interference factoris 0 < a < 1. The additive noise Zj is circularly-symmetric

1-4244-1429-6/07/$25.00 ©2007 IEEE 11

ISIT2007, Nice, France, June 24 - June 29, 2007

complex Gaussian, with EBZA 2 = 1 and the transmissionpower is ElXj 2 = P, where the users use circularly-symmetric complex Gaussian codebooks. Using vector rep-resentation, expression (1) can be rewritten as

Yg = HXAr + ZAr

where =V {O,...,N -1}, and YAr {YO,...,YN_1}.The matrix H is the N x N circulant channel transfer matrix,with first line (1, a ,O,... , O, a). Each cell-site is connectedto the RCP through a lossless link, with bandwidth of Cjbits per channel use. The RCP jointly processes the signalsand decodes the messages sent by all the users of the cellularsystem, where the code rate in bits per channel use of the jthuser is Rj.

Using similar argumentation as in [2], it is easy to verify thatintra-cell TDMA protocol is optimal in terms of the achievablethroughput, for the non-fading homogenous setup considered.

III. OBLIVIOUS CELL-SITES

In this section we consider cell-sites which are obliviousto the users' codebooks and can not perform local decoding.Instead, each cell-site forwards a compressed version of Yj,namely Uj, to the RCP, through the lossless link of bandwidthCj. The RCP then receives the compressed {Uj} and decodesthe messages sent by all the users. We begin by stating thefollowing achievable rate-region for the multi access channel(MAC).

Proposition 1: An achievable rate region for a general Nuser MAC with oblivious N cell-sites, connected by error-freelimited capacities {Cj} links to the RCP is given by

VLC {, ...,N -1}:

1:R < min :c T

t(EL - jCs

where

PXN, UAg,YN (XA, UA, Yg)

N

j=l

rj] + I(X,; Usc XLc) }, (2)

N

= H Pxj (Xi)j=l

N

PYj IXN (Yj XAg) Hl Pu, Yx (Uj byi)j=

and rj = I(Yj; Uj XAr).

The proof outline appears in part A of the Appendix, isbased on [5], and is given for Rj = R (a single point onthe achievable rate region boundaries) and equal capacitiesCj = C, for the sake of brevity. The extension of the proof tothe complete achievable rate region under any links' capacitiesis straightforward.

For the Gaussian channel, we use {Xj, Uj } that are complexGaussian, so that the joint probability (3) is Gaussian. It isnoted that the Gaussian statistics is used due to the simplicity

and relevance of the reported results, with no claim of optimal-ity. For the Gaussian channel, the mutual information includedin (2) boils down to [5],[6]

I(XL; USc XLc)=1ug2 det(I + Pdiag(1-2-rj)jgscHscLHscL)I (4)

where HscL is the transfer matrix between the output vectorYsc and the input vector XL, and rj are positive parametersthat are subjected to optimization over 0 < rj < Cj. Focusingon the setup at hand, with Hi,j which is zero for N -1 >li-j' > 1, equation (2) becomes

S:R min 5:[Cj -rjl+I(XLs;USC~XLsc)tEEL SC[L+1]NU[L-1]NUL jEL

where [L: 1]NA {j: j = (i± 1) mod N, i C L}. Let us

define Hs = HsA, which is the transfer matrix between XVand Ys.

Hereafter, we limit our attention to the symmetric caseof Ci = C for all cell-sites, and Rt = R for all users.By symmetry and concavity, this limits the optimal rj to beinvariant with respect to j: rj = r, and the sum-rate inequality(L {O,... ,N -1}) to be the dominant inequality in (2).Consequently we get the following.

Corollary 1: An achievable rate for the circular Wynermodel with equal capacity links C, equal rate users andoblivious cell-sites is given by

Robl max< min < S [C -r]N O<r ScAV

+log2 det (I + P(1- 2r)HscHc)}} (5)

This rate is achieved with complex Gaussian {U, Xi}.Next, we need to calculate the logarithm of the determinant

of (5). In the case where no inter-cell interference is presence(a = 0), it is easily verified that Hsc H3, is an SeC identitymatrix, and in this case the rate equals the rate achieved by anequivalent single-user single-agent channel [5], which is givenby log2 (1 + 2C+P

For a > 0, we focus on the case where the number of cellsN is large. An achievable rate for this asymptotic scenario isgiven by the following proposition, which is the main resultof this work.

Proposition 2: An achievable rate for the circular Wynermodel with equal limited capacities, oblivious cell-sites andinfinite number of cells (N --> oc), is given by

Robl = F(r*),

where r* is the solution of

F(r*) = C -r*,

and

F(r)

(6)

(7)

J1

1092 (1 + P(1- 2-r) (I + 2a8 cos 270 )2)dO. (8)

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ISIT2007, Nice, France, June 24 - June 29, 2007

Notice that when C -> oc, then also r* > oc, and (6) boilsdown to the per-cell sum-rate capacity of the Wyner modelwith optimal joint processing and unlimited backhaul capacity[2] 1

Robl = Fx(o) log2 (1 + P(1 + 2a cos 270O)2)dO.

For finite C, the implicit equation (7) is easily solvednumerically, since F(r) is monotonic. The following lemma,which is proved in part B of the Appendix, is required forthe proof of Proposition 2.

Lemma 1: Any subset S such that IS f(N)(f :R+ H- R+, limN, f (N) = A, O < A < 1),which minimizes equation (5), when N -> oc, includes onlyconsecutive indices (considering also modulo operation).

Denote a subset which contains only consecutive indices by

Proof of Proposition 2 (outline): First, note that applyingSzego's theorem [2], to log2 det(I + P(1 -2-r)HS(,)HS(,))when IS(C) X->oo, we get the following simple explicitexpression

lim 102(c) log2det(I + P(1- 2r)Hs()HS) F(r).

Let us define S(c) s, so that

log2 det(I + P(1- 2-r)Hs(,)HSH ) sF(r) + e(s), (9)

where lim,,, E(s)/s = 0.Secondly, from Lemma 1, when N - oc, the minimum in

equation (5) is within the subspace of subsets that contain onlyconsecutive indices {f(c)}. Combining (9), when N -> oc,equation (5) becomes

Robl = lim max minN--Coo O<r O<s<N

{ N [C -r]+ NF(r)+ N)}}}

max {min {(1 -A)[C-r] + AF(r)}} (10)

Since F(r) is monotonic increasing, (10) is maximized by r*,which is defined by F(r*) = C r. A

IV. CELL-SITES WITH DECODING

In order to better utilize the bandwidth between the cell-sites and the RCP, we may consider using local decoding atthe cell-sites. In this case the cell-sites should be aware of theassociated codebooks, and thus do not operate in the nomadicregime [5]. It is noted that in general, decoding decreases thenoise uncertainty, thus increasing the efficiency of backhaulusage.

In this section we present an intuitive, simple scheme whichprovides an achievable rate. According to this scheme, eachuser employs rate splitting and divides its message into two

parts: one that is decoded at the RCP and another that isdecoded at the local cell-site. In this case the message which isintended for the RCP to decode, interferes with the decodingof the relevant message at the cell-site. Let the power used forthe former be 13P and the latter (1 -3)P, where 0 <Q < 1.There are two strategies for the cell-site to execute: to decodeonly its local user's message, or to decode also the interferingusers' messages, emerging from the two neighboring cells (see[3] Section III.D). Such approach allows decoding of messageswith rate of

Rd= max {1og2 (1 + (I +j2)p)

min { 1og2 (1 + (1 - /3)2a2P1 + /3(1+ 2a2)P'

(1 1)

Forwarding the decoded information through the losslesslinks reduces the bandwidth available for the compression,so the achievable rate is R,d (C) (sd stands for separatedecoding)

R,d (C) =max {F(rd) + Rd()} (12)

where Rd (/3) = min{Rd (3), C}, rd is the solution of

FB(rd) = C -Rd(3) -rd,

and Fa(r) is defined as F(r) from (8) while replacing P with13P

Fa (r) j10o2(1 + P(1- 2-r)(1 + 2a cos 2w0)2)dO.For a 0 this scheme is optimal, since there is no inter-

cell interference and each cell-site can decode messages at thesame rate as the RCP can.

Note that the rate Rsd(C) is not concave in C in general,thus time-sharing may improve the achievable rate, whichleads to the following (ch stands for the convex-hull).

Proposition 3: An achievable rate of the rate-splittingscheme deployed in the infinite circular Wyner model withlimited equal capacities C, is given by

Rsdch, 1 =

max {ARsd(Cl) + (1 -A)Rsd(C2)}A,C1,C2: ACl+(1-A)C2<C

(13)In fact, numerical calculations reveal that a good strategy

is to do time-sharing between the two approaches: usingdecoding at the cell-sites, with no decoding at the RCP, anddoing decoding only at the RCP (10), rather than using themixed approach of (12). Thus, defining t = Rd(O), the rateRsdch,l of (13) can be written as

Rdec(P, C) = max {t + (Cr>r*

13

11092 1 +(I + 2a2) (I /3)p

3 1 + /3(l + 2a2)p

t) F(r) t

- (14)F(r) + r t

ISIT2007, Nice, France, June 24 - June 29, 2007

and r* is calculated by (7). The derivation of (14) is based ontime-sharing between the point (t, t) and the concave curve(F(r) + r, F(r)). It is noted that the maximizing r must belarger than r*, thus limiting the optimization range.

It is expected that decoding at the cell-site will be benefi-cial when a is small (low inter-cell interference), or whenC is small, so that decoding before transmission saves onbandwidth, which otherwise would have been wasted on noisequantization.

A. Adding Inter-Cell Time-Sharing

Another improvement that can be made to the overallperformance is by allocating a fraction of the transmissioninterval in which adjacent users are not active simultaneously.Therefore allowing the cell-sites to decode their messageswithout interference during this fraction of time (see [3]Section IILD). For the circular Wyner model with N even,it means that odd and even cells are active alternately intime, hence the acronym inter-cell time-sharing (ICTS). In thefollowing we restrict our attention to even N. Redivide thetime frames, such that every user is active for (1 -0) of thetime (O < 0 < 1), indicating that the time fraction used fortransmitting in the interference-free phase is 0. The time usedfor the previous rate-splitting scheme is thus 1 -20. Sincethe users do not transmit all the time, they can scale theirtransmission power by a factor of 1 1 0. The resulting rate is

RICTS = max { Rd,2+

(1 -20)Rdec(~0 (C Rd,2)) }, (15)

where Rd,2 is the rate of the message sent in the interference-free phase, which equals

Rd,2 = min {01o2 (1 + P 0)C

Finally, using (14) we get the following.Proposition 4: An achievable rate of the ICTS with cell-site

decoding, deployed in the infinite circular Wyner model withlimited equal capacities, is given by

RICTS = max {Rd,2+

(1 -20) max t + ( ,2r>r* I1 20

t)F(r)-

F(r) + r

where t = min{C -Rd,2, Rd 3,= 0PI and r*5 is calculatedby

Fa(rts) 3= 110 C-Rd,2-r*tsV. NUMERICAL RESULTS

Achievable rates of the considered three schemes, are plot-ted in Figure 1 for SNR P = 10 [dB], and several backhaulcapacity values, as functions of the inter-cell interferencefactor a. Examining the curves, it is observed that in order

: ~~~~~~~~~dec:0~~~~~~~~~~~~~~~~~01

a _ _ ~~~~~~~C=8, Rdec {

,, ,,

CZ .......+ . C=8, RIT

C,,~~~~~~~~~~~~IT

(D 3.1

l... ,.........

.8 E- -- ....

2.5 - F0 0.2 0.4 0.6 0.8 1

Fig. 1. The achievable rates Robl, Rdec and RICTS are plotted as functionsof the inter-cell interference a for C = 4, 8, oc and user SNR P = 10 [dB].

to approach the rates achieved with C = oc across the wholeregion 0 < a < 1, we need an overhead with approximatelytwice as much bandwidth for the backhaul network (C = 2R).The gain of the cell-site decoding scheme is prominent when

the inter-cell interference is low, but also noticeable even forrather high interference levels, when C is low.

Interestingly, the ICTS scheme provides only a slight im-provement over the cell-site decoding scheme. A possibleexplanation for this phenomenon is that for 0 = (nooverlapping between transmissions of adjacent users), theachievable rate reads RICTS = 0.510g2(1 + 2P), whichfor P = 10 roughly equals 2.2 [bits/channel use], which issignificantly lower than all the considered schemes.

Another observation is that for high backhaul capacities, therates of the three schemes approach the optimal performance.As expected for a = 0, the cell-site decoding and ICTSschemes rates are optimal independently of C, since no inter-cell interferences are present.

VI. CONCLUDING REMARKS

In this paper we considered the circular Wyner model, withlimited backhaul capacity. Achievable rates were presented forthe case of cell-sites which use signal processing alone, andalso combined with decoding. Both considered no networkplanning, so inter-cell interferences dominate. Achievable rateswere derived also for time-sharing based, optimized networkplanning. Numerical calculations reveal that unlimited optimaljoint processing performance can be closely approached withrather limited backhaul capacity, and that network planningdoes not add significant performance gains.The faded Wyner model with limited backhaul capacity is

currently under study.APPENDIX

A. Proof Outline of Proposition ]

Assuming transmission over n channel uses, we generatethe random codebooks for the users by {X3 }, and the randomquantization codebooks (each of size 2nR) by {Uj}. Each

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ISIT2007, Nice, France, June 24 - June 29, 2007

quantization codebook is randomly partitioned into 2'c bins,so that each bin consists of 2n[R-C] codewords.Each cell-site uses the joint typicality criteria to compress

the received channel output Yjn into Uj, and then sends theresulting bin index (instead of the of the index of Uj, in orderto save bandwidth) to the RCP through the lossless link. Anerror in this stage occurs with arbitrary low probability oferror, as long as R > I(Yj; Uj).The RCP receives the bins' indices, and looks for a jointly

typical {Xj, Uj}, where the {Uj} are taken from the binsindicated by the cell-sites.The probability that no set {Xj,NUj}7 1 is found to be

jointly typical is arbitrary small with increasing n, due to thegeneralized Markov lemma [5]. The probability of a set witherroneous XL and Us (where S, L C JV) to be jointly typicalis upper bounded by

Pe < 21nILIR±SI ( ^C)±h(XA,UA)]2-n[h(XLC,Usc)+h(XL)+±jGs h(Uj)] (16)

Hence, the probability of error is arbitrary small as long as

LJR < S (C -I(U; Y)) -h(XL, Us XLc, Usc)+ JL:h(X) + ISlh(U), (17)

where the cell index j is omitted due to symmetric argumen-tation.

Noting that Yi- X -Yj and {Xg, Ui}Yj -Uj areMarkov chains for i t j, and that users cooperation is notallowed, it is easy to verify that the following equalities hold

h(Uj Yj) = h(Uj Yj,XA)h(XL) = h(XL XLC)

h(XL,UsXLc, Usc) = h(XL XLc, Usc) + lS h(UXA).Applying these to (17) yields

lL R < lSl (C- I(U; Y Xg)) + I(XL; USc XLc)).Noting that the last condition should hold for any S and Lcompletes the proof. C

B. Proof of Lemma ]We prove that S which minimizes

lim II(Xr; Us) = mN log2 det(I + P'HsH3),N-*~ocN N-*~ocN

when |S| = f (N), (f : R+ i-- R+, limN f (N) = A,N A0 < A < 1), is composed of only consecutive indices.

Following the method used in [8] to derive a lower bound onthe capacity of the Gaussian erasure channel, the proof hereuses an analogy between the multi-cell setup and an inter-symbol interference (ISI) channel, combined with a recentlyreported relation between the MMSE and the mutual informa-tion [7].

Proof: Denote by Ei, the MMSE of estimating Xi fromUg. Further, denote by Ei(S), the MMSE of estimating Xifrom Us. Naturally

VS CA, iArC A: Ei(S) > Ei. (18)

Next, we use the following relation between the MMSE andthe mutual information [7], to write

dpI(XAr; Us)

From (18), we can writeN-1

Z Ei(S)i=O

N-1

E Ei(S).i=O

(19)

(20)f(N) ZT1> N E Eii=o

Combining (19) and (20) yields

I(XAr; Us) > f N) j E EidPt=O

f(N)I(XAr; Ug).N

(21)On the other hand, in the asymptotic regime, for consecutiveindices set S(c), where limN,o IS A=N

lim N ZEi(Sw(c))N--soc

IN-1A lim -E: Ei.

t=o(22)

This is because the equivalent ISI channel is stationary, andsince the right hand side of (22) exists (attainable fromapplying Szeg&'s theorem to the mutual information and thendifferentiating with respect to P'). By integrating both flanksof equation (22) we get that

rim II(Xg; Us(,)) A rim I(Xg; U).N-Eaon N N-(oow N

Equation (23) together with (21) proves the lemma.

(23)

U

ACKNOWLEDGMENT

The research was supported by the REMON consortiumfor wireless communication and a Marie Curie OutgoingInternational Fellowship within the 6th European CommunityProgramme.

REFERENCES

[1] S. Shamai, 0. Somekh, and B. M. Zaidel, "Multi-cell communications:An information theoretic perspective," in Proceedings of the Joint Work-shop on Communications and Coding (JWCC'04), Donnini, Florence,Italy, Oct. 2004.

[2] A. Wyner, "Shannon theoretic approach to a Gaussian cellular multipleaccess channel," IEEE Trans. Inform. Theory, vol. 40, no. 6, pp. 1713-1727, Nov 1994.

[3] S. Shamai and A. Wyner, "Information-theoretic considerations for sym-metric cellular, multiple-access fading channels - part I," IEEE Trans.Inform. Theory, vol. 43, no. 6, pp. 1877-1894, Nov 1997.

[4] 0. Somekh and S. Shamai, "Shannon-theoretic approach to a Gaussiancellular multi-access channel with fading," IEEE Trans. Inform. Theory,vol. 46, no. 4, pp. 1401-1425, July 2000.

[5] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, "Communi-cation via decentralized processing," Submitted to IEEE Trans. Inform.Theory.

[6] A. Sanderovich, S. Shamai, Y. Steinberg, and M. Peleg, "Decentralizedreceiver in a MIMO system," in Proc. of IEEE Int. Symp. Info. Theory(ISIT'06), Seattle, WA, July 2006, pp. 6-10.

[7] D. Guo, S. Shamai, and S. Verdu, "Mutual information and minimummean-square error in Gaussian channels," IEEE Trans. Inform. Theory,vol. 51, no. 4, pp. 1261-1282, April 2005.

[8] A. M. Tulino, S. Verdu, G. Caire, and S. Shamai, "Capacity of theGaussian erasure channel," Submitted to IEEE Trans. Inform. Theory.

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