uplink multicell processing with limited backhaul via per ...processing problem. toward this end,...

arX

iv:1

208.

3024

v2 [

cs.IT

] 25

May

201

3

Uplink Multicell Processing with Limited Backhaulvia Per-Base-Station Successive Interference

CancellationLei Zhou,Member, IEEEand Wei Yu,Senior Member, IEEE

Abstract—This paper studies an uplink multicell joint pro-cessing model in which the base-stations are connected to acentralized processing server via rate-limited digital backhaullinks. Unlike previous studies where the centralized processorjointly decodes all the source messages from all base-stations,this paper proposes a simple scheme which performs Wyner-Ziv compress-and-forward relaying on a per-base-station basisfollowed by successive interference cancellation (SIC) atthecentral processor. The proposed scheme significantly reduces theimplementation complexity of joint decoding while resulting in aneasily computable achievable rate region. Although suboptimalin general, this paper proves that the proposed per-base-stationprocessing SIC scheme can achieve the sum capacity of a specialcellular Wyner model to within a constant gap. Under theproposed SIC framework, this paper also studies the capacityscaling with limited backhaul. It is established that in orderto achieve to within constant gap to the maximum SIC ratewith infinite backhaul, the limited-backhaul system must havebackhaul link capacities that scale logarithmically with the signal-to-interference-and-noise ratios (SINR) at the base-stations. Thispaper further studies the optimal backhaul rate allocationproblem for an uplink multicell joint processing model witha total backhaul capacity constraint. The analysis revealsthatthe optimal rate allocation that maximizes the overall sum rateshould have backhaul link rates that also scale logarithmicallywith the SINR at each base-station. Finally, the proposed per-base-station SIC scheme is evaluated in a practical multicellorthogonal frequency division multiple access (OFDMA) networkto quantify the performance gain brought by the centralizedprocessor.

Index Terms—Coordinated multi-point (CoMP), interferencechannel, limited backhaul network multiple-input multipl e-output (MIMO), relay channel, successive interference cancel-lation, Wyner-Ziv coding

I. I NTRODUCTION

Traditional wireless cellular networks operate on a cell-by-cell basis where out-of-cell interference is treated as part ofthe noise. However, as cellular base-stations become more

Manuscript submitted to IEEE Journals on Selected Areas of Communica-tions on Oct 13, 2012, revised on March 20, 2013, accepted on May 2, 2013.This work was supported by the Natural Science and Engineering ResearchCouncil (NSERC) of Canada. The materials in this paper have been presentedin part at IEEE Global Communications Conference (Globecom), Anaheim,CA, U.S.A., Dec 2012.

L. Zhou was with the The Edward S. Rogers Sr. Department of Electricaland Computer Engineering, University of Toronto, Toronto,ON M5S 3G4Canada. He is now with Qualcomm Technologies Inc., Santa Clara, CA 95051USA (email: [email protected]).

W. Yu is with The Edward S. Rogers Sr. Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada(email: [email protected]). Phone: 416-946-8665. FAX: 416-978-4425. Kindly address correspondence to Wei Yu.

densely deployed and heterogeneous networks become com-monplace where small cells overlap heavily with macro-cells, modern cellular networks are becoming increasinglyinterference limited. Because of the intercell interference, theper-cell achievable rates in traditional cellular deployments aretypically much smaller than that of a single isolated cell.

Multicell joint processing is a promising technique that hasthe potential to significantly improve the cellular throughput bytaking advantage of its capability for intercell interference mit-igation. When base-stations share the transmitted and receivedsignals, the codebooks, and the channel state information(CSI) with each other through high-capacity backhaul, it istheoretically possible to perform joint transmission in thedownlink and joint reception in the uplink to eliminate out-of-cell interference completely. This paper deals with thein-formation theoretical capacity analysis of the joint-processingarchitecture in the uplink. In this architecture, antennasfrommultiple base-stations essentially become a virtual multiple-input multiple-output (MIMO) array capable of spatial multi-plexing multiple user terminals.

One way to implement a multicell joint processing systemis to deploy a centralized processing server which is connectedto all the base-stations via high-capacity backhaul links.Whenthe capacities of the backhaul links are sufficiently large,thejoint processing system implemented across the different cellsin the network can be modeled as a multiple-access channelin the uplink and a broadcast channel in the downlink, givingrise to the concept of network MIMO [1], [2].

The practical implementation of a network MIMO systemmust also consider the effect of finite capacity in the back-haul. While the capacity of the network MIMO system withinfinite backhaul is easy to compute, when finite backhaul isconsidered, the information theoretical analysis of multicellprocessing becomes significantly more complicated. This pa-per focuses on an uplink network MIMO model as shown inFig. 1. From an information theoretical perspective, the overallsystem can be thought of as a combination of a multiple-accesschannel (with remote terminals acting as the transmitters andthe centralized processor as the receiver) and a relay channel(with the base-stations acting as relays).

A. Related Work

The uplink network MIMO model considered in this paper,where the joint processing takes place in the central server, isrelated to but is different from the multicell joint processing

http://arxiv.org/abs/1208.3024v2

SUBMITTED TO IEEE JOURNALS ON SELECTED AREAS IN COMMUNICATIONS 2

….

….

Centralized

Processor

Y1 : Y1

Y2 : Y2

YL : YL

X1

X2

XL

Z1

Z2

ZL

h11

h21

hL1 h1L

hL2 h2L

h22

hLL

C1

C2

CL

Y1

Y2

YL

Fig. 1. Uplink multicell joint processing via a centralizedprocessor

model with finite-capacity uni- or bi-direction backhaul linksdeployed between the base-stations. In this latter setting, theuplink multicell model becomes an interference channel withpartial receiver cooperation. Although a complete characteri-zation of capacity for such a model is still an open problem,information theoretical studies have been carried out for thecase of two-user Gaussian interference channel with receivercooperation [3]–[7]. Likewise, the analysis of uplink cellularnetwork under this setup has been carried out in [8], [9],where strategies such as decode-and-forward and compress-and-forward are proposed, and analytical results under theWyner model are obtained. We mention briefly here that thedownlink counterpart of this model has been studied in [10].

For the uplink multicell joint processing model consideredin this paper, where the limited-capacity backhaul links aredeployed between the base-stations and a central processoras depicted in Fig. 1, although a complete characterizationof its information theoretical capacity region is also an openproblem, much work has been done on finding its achievablerate region and the gap to the outer bound [11]–[16]. The earlywork by Sanderovich et al. [11] considers a joint decodingscheme in which the base-stations quantize the received signalsand forward them to the centralized processor, which thenperforms joint decodingof both the source messages andquantized codewords. Instead of joint decoding, [14], [15]describe schemes whichsuccessivelydecode the quantizationcodewords first, then the source messages. A comparison ofsuccessive decoding vs. joint decoding is contained in [16].Alternatively, base-stations can alsodecodepart of the mes-sages of users in their own cell, then forward the decoded dataalong with the remaining part to the centralized processor [11],[12], [17]. The counterpart of these results for the downlinkwith finite backhaul has been described in [18]. A review ofsome of these results is available in [2].

The uplink multicell joint processing model is actually aspecial case of a general multiple multicast relay networkstudied in [19], [20], for which a generalization of quantize-and-forword can be shown to achieve to within a constant gapto the information theoretical capacity of the overall network.This is proved in [19] using a quantize-map-and-forward tech-nique, and in [20] using noisy network coding, both of whichrequire the joint decoding of the source messages and thequantized codewords. Thus in retrospect, the joint processingscheme of [11] already achieves the approximate capacity

of the uplink finite-backhaul multicell model, provided thatthe quantization noise level is appropriately chosen and jointdecoding is done at the central processor.

It should be emphasized that joint decoding is challengingto implement. Not only is the network-wide CSI required atthe central processor, the computational complexity of jointdecoding typically scales exponentially with the total numberof nodes in the network. Moreover, even a mere evaluationof the achievable rate can be computationally prohibitive.The achievable rate region of the joint decoding schemeas presented in [11, Proposition IV.1] involves2L − 1 rateconstraints, each requiring a minimization of2L terms, whereL is the number of users in the uplink multicell model. Toobtain analytical results, [11] and likewise [9], [21] assumecertain symmetry and obtain rate expressions under a Wynercellular model. But the rate expression for an arbitrary networkremains difficult to evaluate. On top of this, the achievablerateregion needs to be optimized over all choices ofL quantizationnoise levels, which is a difficult task, although recent progresshas been made in [14], [16], [22].

B. Main Results

To fully overcome the aforementioned difficulty yet stilltake full advantage of the centralized processing is expectedto be challenging. This paper instead aims to design schemeswith simple receiver structures that result in computationallyfeasible achievable rate regions for the uplink multicell jointprocessing problem. Toward this end, this paper focuses onthe uplink multicell model with finite-capacity backhaul tothe centralized processor, and proposes suboptimal schemesbased onsuccessive interference cancellation(SIC). Insteadof performing joint decoding of the source messages andquantization codewords [11], or successive decoding of allthequantization codewords first, then the source messages [14],[15], this paper applies the Wyner-Ziv compress-and-forwardrelaying scheme on a per-base-station basis and performssingle-user decoding with SIC at the centralized processor.The Wyner-Ziv coding here also uses the previously decodedmessages as side information. The main advantage of thisproposed scheme is that it leads to a receiver architecture inwhich only the decoded transmit signals are shared betweenthe successive stages, so it is more amenable to implementa-tion. Further, the quantization noise levels can now be easilyoptimized, resulting in an achievable rate region computablewith complexity that scales linearly withL.

Although the proposed per-base-station SIC scheme is nolonger the best possible for a general uplink joint processingmodel, this paper shows that for a special Wyner soft-handoffmodel [23], in which the mobile terminals and the base-stations are placed along a line, the proposed per-base-stationSIC scheme achieves a sum rate that is at most a constant gapaway from capacity. This paper shows that the constant gapis a linear function of the number of transmitter-receiver pairsin the uplink multicell joint processing model.

Under the proposed per-base-station SIC framework forthe finite-backhaul uplink model, we also ask the followingquestion: How much backhaul capacity is needed to approach


the theoretical infinite-backhaul successive-decoding rate? Theresults of this paper show that the backhaul rates need to scalelogarithmically with the received signal-to-interference-and-noise ratio (SINR) in order to achieve to within12 bit of thesuccessive decoding rates attainable with unlimited backhaulcapacity.

Further, this paper addresses the question of how to allocatethe backhaul rates optimally across the different backhaullinks. Under a sum rate constraint on the total backhaulcapacity, this paper shows that in order to maximize thesum rate over the entire network, the allocation of individualbackhaul link capacity should again scale logarithmicallywiththe SINR under the proposed per-base-station SIC framework.

This paper focuses on capacity analysis with the assumptionthat the CSI for the entire network is available at the centralprocessor. This is reasonable as the base-stations may estimatethe channel in the uplink, then send CSI via the backhaul linksto the central processor. For convenience, this paper neglectsthe additional backhaul capacity needed to support the sharingof CSI. We mention that the capacity of network MIMO underchannel estimation error has been investigated in [8]. Study onthe use of processing techniques that are robust against channeluncertainty has been reported in [15].

Finally, under the perfect CSI assumption, this paper eval-uates the performance of the proposed per-base-station SICscheme in an OFDMA network. Numerical simulations showthat the proposed per-base-station SIC scheme can almostdouble the per-cell sum-rate over a baseline scheme whereno centralized processor is deployed and each base-stationsimply treats interference as noise. Further, with optimizedbackhaul capacity allocation, most of such gains can alreadybe obtained when the backhaul sum capacity is at about 1.5times the achieved per-cell sum rate.

C. Organization of the Paper

The rest of the paper is organized as follows. SectionII presents the multicell joint processing model. Section IIIintroduces the per-base-station SIC scheme and compares itwith other achievability schemes such as noisy network codingand joint-base-station processing. The optimality of the per-base-station SIC scheme for the Wyner model is provided inSection IV. The backhaul rate allocation problem with a totalbackhaul constraint is solved in Section V. Finally, SectionVI numerically evaluates the per-base-station SIC scheme fora two-user symmetric setting and for a realistic OFDMAnetwork.

II. CHANNEL MODEL

Consider the uplink of a multicell network with a centralprocessor. Assuming that there is only one user operating ineach time-frequency resource block in each cell, the multicellnetwork can be modelled byL users each sending a messageto their corresponding base-stations. Base-stations serve asintermediate relays for the centralized server, which eventuallydecodes all the transmit messages.

As depicted in Fig. 1, the uplink joint processing modelconsists of two parts. The left half is anL-user interference

channel withXi as the input signal from theith user,Yi asthe output signal,Zi as the additive white Gaussian noise(AWGN), and hij as the channel gain from useri to base-station j, where i, j = 1, 2, · · · , L. The right half can beseen as a noiseless multiple-access channel with capacitiesCi, i = 1, 2, · · · , L, modelling the backhaul links between thebase-stations and the central processor. The input signalXi ispower constrained, i.e.,E[|Xi|2] ≤ Pi; the receiver noises areassumed to be independent and identically distributed withZi ∼ N (0, N0), i = 1, 2, · · · , L. The signal-to-noise ratios(SNRs) and the interference-to-noise ratios (INRs) are definedas follows:

SNRi =h2iiPi

N0, INRi,j =

h2ijPi

N0, i, j = 1, · · · , L (1)

In addition, define the vectorsX = {X1, X2, · · · , XL} andY = {Y1, Y2, · · · , YL}.

III. M ULTICELL PROCESSINGSCHEMES

This paper focuses on uplink multicell processing schemesin which the base-stations quantize the received signals, thensubsequently transmit a function of the quantized signal tothe central processor via noiseless backhaul links. Quantize-and-forward is a natural strategy in the uplink setting. This isbecause in order to reap the benefit of multicell processing,theuser terminals must transmit at rates higher than that decodableat its own base-station alone. This prevents the use of thedecode-and-forward strategy. In Fig. 1, the received signal isdenoted asYi; its quantized version is denoted asYi. Thispaper also assumes that the base-stations either transmitYi

directly, or performs a binning operation onYi. The task atthe central processor is to decode{X1, X2, · · · , XL} based oneither{Y1, Y2, · · · , YL} or their bin indices. In the following,we examine the information theoretical achievable rates forvarious quantization and decoding strategies.

A. Joint Decoding

Consider a coding scheme in which each base-stationquantizes its received signal using a Gaussian codebook atcertain distortion or quantization noise levelqi, bins thequantized message, then forwards the bin index to the centralprocessor. In the joint decoding strategy, the central processor,upon receiving all the bin indices, decodes{X1, X2, · · · , XL}and {Y1, Y2, · · · , YL} jointly based on joint typicality. Thisstrategy is first proposed by Sanderovich et al. [11, PropositionIV.1]. (Note that there is no requirement that{Y1, Y2, · · · , YL}are decoded correctly; only{X1, X2, · · · , XL} need to.)

The key parameter here is the setting of the quantizationnoise levelsqi. For example,qi’s can be set to be such that{Y1, Y2, · · · , YL} can be correctly decoded based on the binindices. The values ofqi’s can then be determined based onthe capacities of the backhaul links(C1, C2, · · · , CL). But,such a setting ofqi may not be optimal. Indeed, as shownin [19] and also later in [20], for a much larger class ofgeneral relay networks, setting the quantization noise levelsto be at the background noise level, i.e.,qi = N0, leads to anachievable rate that is within at most a constant gap from the


capacity outer bound. This gap scales linearly with the numberof nodes in the network, but is independent of the channelgains and the backhaul capacities. Applying this result to theuplink multicell model in Fig. 1, we immediately have thefollowing joint decoding achievable rate. The achievable rateexpression can be derived based on the noisy-network-codingtheorem [20], and is equivalent to the expression in [11].

Lemma 1. For the multicell joint processing model as shownin Fig. 1, the following rate region is achievable

R(S) ≤ minT ⊆L

{

1

2log∣

∣I + Λ−1T c (N0 + q)HST cΛS(P )HT

ST c

∣

∣

+∑

i∈T

(

Ci −1

2log(1 +

N0

qi)

)

}

, ∀ S ⊆ L , {0, 1, · · · , L}

(2)

where R(S) =∑

i∈S Ri, qi is a positive real numberrepresenting the quantization noise level at base-stationi,HST c denotes for the transfer matrix from inputX(S) tooutputY(T c), Λ−1

T c(N0 + q) is a diagonal matrix of 1N0+qi

with i ∈ T c, and ΛS(P ) is a diagonal matrix ofPi withi ∈ S. This rate region is within a constant gap to thecapacity region of the multicell joint processing model if wesetqi = N0, i = 1, 2, · · · , L.

The above achievable rate region is valid for any choiceof qi, and is approximately optimal whenqi = N0. But, theevaluation of such an achievable rate region can be difficult.For the uplink joint processing model, the achievable rateregion as derived in Lemma 1 requires a minimization of2L

terms for each rate constraint, and there are2L − 1 differentrate constraints describing the rate region. Even when the sizeof the network is in a reasonable range, for example as in a19-cell topology, it is computationally prohibitive to minimizeover219 terms each involving219−1 different rate constraints.Further, the implementation of the joint decoding scheme itselftends to have exponential complexity, so the achievable rategiven in Lemma 1 is not practically implementable for areasonably-sized network.

In order to derive more tractable performance analysis of themulticell joint processing scheme, [11] resorts to a modifiedWyner model (see [23]), where each transmitter-receiver pairinterferes only with one other neighboring transmitter-receiverpair, and is subject to interference from only one neighboringtransmitter-receiver pair. Further, certain symmetry is intro-duced so that all the direct channels are identical, and so areall interfering channels. With this symmetric cyclic structure,computation of the sum rate becomes tractable under jointdecoding [11].

B. Per-Base-Station SIC

This paper focuses on the general multicell model (insteadof the symmetric Wyner model) and proposes suboptimalschemes based on the successive decoding of source messages.Based on the observation that the exponential complexity ofthe rate expression in Lemma 1 is due to the joint decoding

step at the destination, this paper proposes a per-base-stationSIC approach (in contrast to joint-base-station SIC to be de-scribed in detail later) at the centralized processor that signifi-cantly reduces the complexity of computation of the achievablerate region. In addition, the proposed scheme involves theuse of compress-and-forward relaying technique [24] at eachbase-station independently, which also significantly reducesthe complexity of its eventual implementation in a practicalsystem.

Specifically, we use an SIC approach assuming afixedorder of decoding firstX1, thenX2, X3, · · · , XL. A centralfeature of the proposed scheme is that only the decodedsignals are used as side information in subsequent stages. Thissimplifies the sharing of information among the base-stationsand facilitates practical implementation.

At the kth stage, the base-stationk, upon receivingYk,quantizesYk into Yk using the compress-and-forward tech-nique and sends bin index ofYk to the destination via thenoiseless link of capacityCk. Note that the quantizationprocess at the base-stationk includes interference from allother users. The quantization noise level at each base-stationk is chosen so as to fully utilizeCk such thatYk can beimmediately recovered at the central processor. This quan-tization process can be done either with Wyner-Ziv codingthat takes advantage of the already-decoded user messages(X1, · · · , Xk−1) as side information, or without Wyner-Zivcoding for ease of implementation.

At the central processor, to decode userk’s messageXk thecentral processor first decodes the quantization messageYk

upon receiving its description from the digital linkCk. It thendecodes the message of userk using joint typicality decodingbetweenXk and the quantized messageYk. The decoding ofXk also takes advantage of the knowledge of previously de-coded messages(X1, · · · , Xk−1) at the centralized processor.In this way, the impact of interference fromX1, · · · , Xk−1

eventually disappears and the effective interference is only dueto users not yet decoded, i.e.,Xj , for j > k. After decodingXk, the central processor moves to the next decoding stage,addingXk to the set of known side information.

1) Per-Base-Station SIC with Wyner-Ziv:The followingtheorem gives the achievable rate of the proposed per-base-station SIC scheme with Wyner-Ziv compress-and-forwardrelaying.

Theorem 1. For the uplink multicell joint processing channeldepicted in Fig. 1, the following rate is achievable usingWyner-Ziv compress-and-forward relaying at the base-stationsfollowed by SIC at the centralized processor with a fixeddecoding order:

Rk =1

2log

1 + SINRk

1 + 2−2CkSINRk

, (3)

where

SINRk =SNRk

1 +∑

j>k INRj,k

. (4)

Proof: In thekth stage of the SIC decoder,X1, · · · , Xk−1

decoded in the previous decoding stages serve as side informa-tion for stagek. The equivalent channel of userk is depicted


Centralized

Processor

Xk

hkkCk

Zk +∑

j 6=khjkXj

Yk : Yk Yk

X1, X2, · · ·Xk−1

Fig. 2. Equivalent channel of userk in the kth decoding stage

in Fig. 2. This is a three-node relay channel without thedirect source-destination link. Specifically, the source signalXk is sent from the transmitter to the relay, which receivesYk, quantizes it intoYk and forwards its description to thecentralized processor via the noiseless digital link of capacityCk. At the centralized processor,X1, · · · , Xk−1 serve asside information and facilitate the decoding ofYk and Xk.According to [24, Theorem 6], the achievable rate of userkusing Wyner-Ziv compress-and-forward can be written as

Rk = I(Xk; Yk|X1, · · · , Xk−1) (5)

subject to the constraint

I(Yk; Yk|X1, · · · , Xk−1) ≤ Ck. (6)

We constrain ourselves to Gaussian input signalsXk ∼N (0, Pk), and the Gaussian quantization scheme1, i.e.,

Yk = Yk + ek, (7)

where ek ∼ N (0, qk) is the Gaussian quantization noiseindependent of everything else. To fully utilize the digital link,it is natural to set

I(Yk; Yk|X1, · · · , Xk−1) = Ck. (8)

Now, substitutingYk =∑L

j=1 hjkXj +Zk and Yk = Yk + ekinto (8), we have

Ck = I(Yk; Yk|X1, · · · , Xk−1)

= h(Yk|X1, · · · , Xk−1)− h(ek)

=1

2log

(

1 +N0 +

∑

j≥k h2jkPj

qk

)

, (9)

which results in the following quantization noise level thatfully utilizes the digital linksCk:

q∗k =N0 +

∑

j≥k h2jkPj

22Ck − 1. (10)

With the aboveq∗k, the achievable rate of userk can becalculated as

Rk = I(Xk; Yk|X1, · · · , Xk−1)

= h(Yk|X1, · · · , Xk−1)− h(Yk|X1, · · · , Xk)

=1

2log

q∗k +N0 +∑

j≥k h2jkPj

q∗k +N0 +∑

j>k h2jkPj

=1

2log

N0 +∑

j>k h2jkPj + h2

kkPk

N0 +∑

j>k h2jkPj + 2−2Ckh2

kkPk

=1

2log

1 + SINRk

1 + 2−2CkSINRk

. (11)

1Gaussian input and Gaussian quantization are used for convenience andfor simplicity only. They are not necessarily optimal; see [12] for an example.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

Ck

Rk

12

log(1 + SINRk)

12

log(1 + SINRk), 12

log(1 + SINRk)2

1 + 2SINRk

≤ 12

Fig. 3. Achievable rate of userk versus the backhaul capacityCk

Finally, note thatSINRk =h2kkPk

N0+∑

j>k h2jk

Pj,which is equivalent

to (4).

The rate expression (3) shows how the achievable rates areaffected by the limited capacities of the digital backhaul linksunder the proposed per-base-station SIC decoding framework.Fig. 3 plots the achievable rate ofRk as a function of thebackhaul link capacityCk with SINRk equal to20dB. WhenCk is small,Rk grows almost linearly withCk, which meansthat each bit of the backhaul link provides approximately onebit increase in the achievable rate for userk. The digitalbackhaul is efficiently exploited in this regime. However, asCk

grows larger, each bit of the backhaul link gives increasinglyless achievable rate improvement. In the limit of unboundedbackhaul capacity, i.e.,Ck = ∞, Rk saturates and approaches12 log(1 + SINRk)

△= Rk, which is the upper limit for the rate

of userk when the SIC decoder is employed.Since backhaul link capacity can be costly in practical

implementations, it is natural to ask how largeCk needs tobe in order to achieve a rateRk that is close to the maximumSIC rate with unlimited backhaul. It is easy to see that whenCk = 1

2 log(1+SINRk), Rk−Rk is upper bounded by one-halfbit, i.e.,

Rk −Rk =1

2log(1 + SINRk)

−1

2log

1 + SINRk

1 + 2−2CkSINRk

∣

∣

∣

∣

Ck=12 log(1+SINRk)

=1

2log

(

1 +SINRk

1 + SINRk

)

≤1

2. (12)

Therefore, when the backhaul link has capacityCk =12 log(1 + SINRk), the achievable rate is half a bit away fromthe SIC upper limit. This is also the point under which theutilization of Ck is the most efficient, as shown in Fig. 3.

It is worth noting that the above result, which suggests thatCk should scale aslog(SINRk), is analogous to the conclusion


of [11, Corollary IV.6], which states that the backhaul rateshould scale aslog(P ) in order to approach the capacity ofthe infinite backhaul case for the symmetric Wyner model. Themain difference is that this paper uses SIC, so interferencenotyet cancelled still appears in the SINR expression.

2) Per-Base-Station SIC Without Wyner-Ziv:Implementingthe Wyner-Ziv compression may not be easy in practice. Themain difficulty lies in the binning operation that is necessaryin order to take advantage of the side information. To makethe proposed per-base-station SIC scheme more amenableto practical implementation, the Wyner-Ziv quantizer can bereplaced by a simple vector quantizer that does not take sideinformation (i.e., previously decoded messages in the SICframework) into account. In this way, the following rate canbe achieved for userk:

Rk = I(Xk; Yk|X1, · · · , Xk−1) (13)

subject to the constraint

I(Yk; Yk) ≤ Ck, (14)

for k = 1, 2, · · · , L. Following the same lines of the proof ofTheorem 1, it is easy to see that by replacing (9) withCk =

12 log

(

1 +N0+

∑

jh2jkPj

qk

)

, which givesq∗k =N0+

∑

jh2jkPj

22Ck−1,

we obtain

Rk =1

2log

q∗k +N0 +∑

j≥k h2jkPj

q∗k +N0 +∑

j>k h2jkPj

=1

2log

N0 +∑

j≥k h2jkPj + 2−2Ck

∑

j<k h2jkPj

N0 +∑

j>k h2jkPj + 2−2Ck

∑

j≥k h2jkPj

. (15)

Define

SINR′k =

h2kkPk

N0 +∑

j>k h2jkPj + 2−2Ck

∑

j<k h2jkPj

. (16)

We have just proved that the following rate is achievableusing per-base-station SIC without Wyner-Ziv compress-and-forward.

Theorem 2. For the uplink multicell joint processing channeldepicted in Fig. 1, the following rate is achievable usingvector quantization at the base-stations followed by SIC atthe centralized processor with a fixed decoding order:

Rk =1

2log

1 + SINR′k

1 + 2−2CkSINR′k

, (17)

where

SINR′k =

SNRk

1 +∑

j>k SIRj,k + 2−2Ck∑

j<k SIRj,k

. (18)

Comparing Theorem 2 with Theorem 1, it is easy to see thatachievable rates in (3) and (17) share the same structure. How-ever, without Wyner-Ziv compression, the effective SINR (18)becomes smaller due to the extra terms2−2Ck

∑

j<k SIRj,k.This means that the impact of previously decoded signalsX1, · · · , Xk−1 is not fully cancelled at stagek. We also notethat asCk → ∞, we haveSINR

′k → SINRk. Thus, Wyner-Ziv

coding is most useful only when the backhaul link capacityis small. For reasonably largeCk, the benefit of Wyner-Zivcoding is expected to be marginal.

C. Joint Base-Station SIC Schemes

The per-base-station SIC scheme takes advantage of mul-ticell processing only in so far as to the use of the decodedmessages as side information. It is possible to further improveupon these schemes by joint decoding across the base-stations.

Note that in the per-base-station SIC scheme, the decodingof source messages and quantized messages follow the orderY1 → X1 → Y2 → X2,→ · · · ,→ YL → XL, where inthe kth compression and decoding stage, previously decodedsource signalsX1, · · · , Xk−1 are used as side information.However, note that in the decoding process the quantizationcodewordsY1, · · · , Yk−1 are also decoded along the way, andthus are also available for possible joint processing at stagek, which can lead to better performance. In particular, we canincorporateY1, · · · , Yk−1 in the expressions forCk andRk,leading to the following achievable rate:

Rk = I(Xk; Y1, · · · , Yk|X1, · · · , Xk−1), (19)

subject to

Ck ≥ I(Yk : Yk|X1, · · · , Xk−1, Y1, · · · , Yk−1), (20)

for k = 1, 2, · · · , L.Alternatively, we can also proceed in a two-stage successive

process of decoding all of{Yk}Lk=1 first, then{Xk}Lk=1 [14],[15]. Further, each of these two stages can be accomplishedin an SIC fashion, resulting in rate expressions

Rk = I(Xk; Y1, · · · , YL|X1, · · · , Xk−1), (21)

subject toI(Yk; Yk|Y1, · · · , Yk−1) ≤ Ck, (22)

for k = 1, 2, · · · , L. We note here that the above rate expres-sions (19) and (21) can outperform the rates (3) in Theorem 1and (17) in Theorem 2, because in the above expressions eachXk is decoded based on the quantized observations of allbase-stations (or that of all previous decoded base-stations),rather than just thekth base-station in the per-base-station SICscheme. However, for this same reason, the implementationsofthe above joint-base-station SIC schemes are also expectedtobe somewhat more complicated. For the rest of this paper, weonly focus on the per-base-station SIC schemes of Theorem 1and Theorem 2, and leave the joint-base-station SIC schemesas subject for future studies.

IV. A PPROXIMATE OPTIMALITY OF PER-BASE-STATION

SIC IN A WYNER MODEL

The proposed per-base-station SIC scheme is in general sub-optimal. However, in certain asymmetric settings, the per-base-station SIC scheme can be shown to be approximately optimalin the sense of achieving the sum capacity to within a constantgap. This section studies a class of such asymmetric channelsin which each transmitter-receiver pair interferes only with oneneighbor and gets interfered by only one other neighbor asshown in Fig. 4. This model is called the soft-handoff Wynermodel in the literature [23]. The Wyner model is an abstractionof the cellular network to a one-dimensional setting. It can,for example, model a communication scenario where base-stations are placed along a highway and mobile terminals


….

…. Centralized

Processor

Y1 : Y1

Y2 : Y2

YL : YL

X1

X2

XL

Z1

Z2

ZL

h11

h21

h32

h22

hLL

C1

C2

CL

Y1

Y2

YL

Fig. 4. Wyner model where each transmitter-receiver only interferes withone neighbor.

travel between the base-stations. This section proves thattheper-base-station SIC scheme achieves to within a constant gapto capacity for the Wyner model. This result holds either withor without Wyner-Ziv coding.

Consider first the per-base-station SIC with Wyner-Zivcoding. Intuitively, decoding order plays an important role inthis asymmetric setting. Since receiverL sees a non-interferedversion of the source signalXL, decoding should take placestarting from userL. After XL is decoded, it can serve as sideinformation, which facilitates the decoding of userL−1, thenL − 2, etc. With this decoding order and applying the resultof Theorem 1 to the channel setting of Fig. 4, we arrive at thefollowing achievable rate region:

{

Ri = I(Xi; Yi|Xi+1), i = 1, 2, · · · , L− 1

RL = I(XL; YL)(23)

subject to{

I(Yi; Yi|Xi+1) ≤ Ci, i = 1, 2, · · · , L− 1

I(YL; YL) ≤ CL

(24)

which, when specialized to Gaussian inputsXi ∼ N (0, Pi)and the Gaussian quantization scheme:Yi = Yi + ei, whereei is the quantization noise independent of everything elseand follows Gaussian distributionsei ∼ N (0, qi), gives thefollowing achievable rate

Ri =1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)

, i = 1, 2, · · · , L. (25)

The achievable sum rate for the Wyner model with decodingorder from userL to user1 is then

Rsum =

L∑

i=1

1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)

. (26)

The above sum rate can be slightly improved by noticing thatthe Lth base-station can use decode-and-forward instead ofcompress-and-forward, which leads to

RL = min

{

1

2log(1 + SNRL), CL

}

. (27)

The main result of this section is that under a mild weakinterference condition, the sum rate achieved by the per-base-station SIC scheme is at most a constant number of bits

away from the sum capacity for the Wyner model, so it isapproximately optimal in the high SNR regime.

Theorem 3. For a multicell processing Wyner model shownin Fig. 4, in the weak-interference regime ofINRi+1,i ≤SNRi, i = 1, 2, · · · , L − 1, the per-base-station SIC schemewith Wyner-Ziv coding achieves a sum rate that is withinL− 1

2bits of the sum capacity.

Proof: To show that the difference between the achievablesum rate in (26) and the sum capacity is bounded by aconstant gap, we first write down the cut-set bound [25,Theorem 14.10.1] for the Wyner channel model. LetL ={1, 2, · · · , L} represent the index set for the base-stations.We partitionL into two sets,S and Sc, and only considercuts for which all theXi’s and a selected subset ofYi’s,with i ∈ S, are on one side, and the rest of theYi’s, withi ∈ Sc, and the central processor are on the other side. DenoteX(L) = {Xi | i ∈ L} andY(Sc) = {Yi | i ∈ Sc}. We havean upper bound to the cut-set bound as follows:

Rcut−setsum ≤ max

p(X(L))minS⊆L

{

I (X(L);Y(Sc)) +∑

i∈S

Ci

}

≤ minS⊆L

{

maxp(X(L))

I (X(L);Y(Sc)) +∑

i∈S

Ci

}

,

(28)

where the maximization is taken over all the possible jointdistributions ofX(L), and the minimization is taken over2L

different choices of cuts that separate theL transmitters andthe centralized processor.

Evaluate the mutual-information term in (28) as follows:

I (X(L);Y(Sc))

= h (Y(Sc))−∑

i∈Sc

h(Zi)

≤∑

i∈Sc

(h(Yi)− h(Zi))

(a)

≤∑

i∈Sc

1

2log(1 + SNRi + INRi+1,i)

(b)

≤

{

∑

i∈Sc12 log(1 + SNRi) +

|Sc|2 , if L ∈ S

∑

i∈Sc12 log(1 + SNRi) +

|Sc|−12 , if L ∈ Sc

(29)

where (a) holds because Gaussian distribution maximizes theentropyh(Yi) and (b) is due to the fact thatINRi+1,i ≤ SNRi

thus 12 log(1+SNRi+INRi+1,i) ≤

12 log(1+SNRi)+

12 for i =

1, · · · , L− 1. For i = L, h(YL)− h(ZL) =12 log(1+ SNRL).

Now, we are going to choose a particular cut to further upperbound the cut-set bound. We compare1

2 log(SNRi) at eachbase-station with the backhaul capacityCi, and let the cut gothrough either the user-base-station link or the backhaul link,whichever corresponds to the smaller of the two quantitiesabove, as shown in Fig. 5. More precisely, define

S =

{

i

∣

∣

∣

∣

1

2log SNRi ≥ Ci

}

. (30)

With this particular cut, we can now bound the differencebetween the cut-set bound and the achievable sum rate. First,


Y1 : Y1

Y2 : Y2

YL : YL

X1

X2

XL

Z1

Z2

ZL

h11

h21

h32

h22

hLL

C1

C2

CL

Y1

Y2

YL

S Sc

Fig. 5. Choice of cut-set for the Wyner model. The cut goes through theaccess link or the backhaul link depending on whether1

2log(SNRi) is greater

or smaller thanCi.

consider the case thatL ∈ Sc, we use the achievable sum rateas given in (26):

Rcut−setsum −Rsum

≤∑

i∈Sc

{

1

2log(1 + SNRi)−

1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)}

+∑

i∈S

{

Ci −1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)}

+|Sc| − 1

2

=∑

i∈Sc

1

2log(

1 + 2−2CiSNRi

)

+∑

i∈S

1

2log

(

22Ci + SNRi

1 + SNRi

)

+|Sc| − 1

2

(a)

≤|Sc|

2+

|S|

2+

|Sc| − 1

2

≤ L−1

2(31)

where in (a) we used the definition of the cut (30), i.e.,SNRi ≤22Ci for i ∈ Sc, andSNRi ≥ 22Ci for i ∈ S, and the lastinequality is due to the fact that|Sc|+ |S| = L and|Sc| ≤ L.

Now, consider the case thatL ∈ S. We tighten the sum rateexpression (26) by noticing that since by the definition ofSwe have1

2 log SNRL ≥ CL, so by (27) we haveRL = CL. Inthis case,

Rcut−setsum −Rsum

≤∑

i∈Sc

{

1

2log(1 + SNRi)−

1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)}

+∑

i∈S\{L}

{

Ci −1

2log

(

1 + SNRi

1 + 2−2CiSNRi

)}

+|Sc|

2

≤|Sc|

2+

|S| − 1

2+

|Sc|

2≤ L−

1

2(32)

This completes the proof.

It turns out that even without Wyner-Ziv compress-and-forward, it is still possible to achieve the sum capacity towithin a constant, albeit larger, gap under the same weak-interference condition.

Theorem 4. For a multicell processing Wyner model shownin Fig. 4, in the weak-interference regime ofINRi+1,i ≤SNRi, i = 1, 2, · · · , L − 1, the per-base-station SIC schemewithout Wyner-Ziv coding achieves a sum rate that is within12 (1 + log(3))L − 1

2 bits of the sum capacity.

Proof: Applying Theorem 2 to the Wyner model of Fig. 4with the decoding order from userL to user1, it is easy tosee that the achievable rate of userL remains the same whilethe rates of other users become

Ri =1

2log

1 + 2−2Ci INRi+1,i + SNRi

1 + 2−2CiINRi+1,i + 2−2CiSNRi

. (33)

Under the weak-interference conditionINRi+1,i ≤ SNRi,when i ∈ Sc, we have0 ≤ 2−2Ci INRi+1,i ≤ 1, and whenwhen i ∈ S, we have0 ≤ 2−2Ci INRi+1,i ≤ 2−2CiSNRi.Replace theRi expressions in (31) and (32) by (33) whilenoting the inequalities above, we obtain the1

2 (1+log(3))L− 12

bound.Note that a looser bound can be obtained by noting that

under the weak-interference conditionINRi+1,i ≤ SNRi, theachievable rate without Wyner-Ziv coding (33) is alreadywithin 1

2 bits of the rate with Wyner-Ziv coding as in (25). Asa result, a looser sum-rate gap to the cut-set upper bound canimmediately be obtained asL− 1

2 + L−12 = 3

2L− 1 bits2.

V. OPTIMAL RATE ALLOCATION WITH A TOTAL

BACKHAUL CAPACITY CONSTRAINT

A practical system may have a constraint on the sumcapacity of all digital backhaul links, for example, when thebackhaul links are implemented in a wireless medium withshared bandwidth. So, it may be of interest to optimize theallocation of backhaul capabilities among the base-stations inorder to achieve an overall maximum sum rate under a totalbackhaul capacity constraint. The structure of the solutionof such an optimization can also yield useful insight. Theoptimization problem can be formulated as the following:

maximize

L∑

k=1

Rk (34)

subject to Ck ≥ 0, k = 1, 2, · · · , LL∑

k=1

Ck ≤ C

whereRk, k = 1, 2, · · · , L are functions ofCk as derivedin Theorem 3, andC > 0 is the total available backhaulcapacity. The following theorem provides an optimal solutionto the above optimization problem for the per-base-stationSICscheme with Wyner-Ziv coding.

Theorem 5. For the uplink multicell joint processing modelshown in Fig. 1, with the per-base-station SIC at the centralprocessor with Wyner-Ziv coding, the optimal allocation ofbackhaul link capacities subject to a total backhaul capacityconstraintC is given by

C∗k = max

{

1

2log(SINRk)− α, 0

}

, (35)

2The authors wish to thank Yuhan Zhou to point out the tighter bound inTheorem 4.


whereα is chosen such that∑L

k=1 C∗k = C.

Proof: Substituting the rate expression (19) forRk intothe optimization problem (34), we obtain the following equiv-alent minimization problem:

minimize

L∑

k=1

1

2log(

1 + 2−2CkSINRk

)

(36)

subject to Ck ≥ 0, k = 1, 2, · · · , L.L∑

k=1

Ck ≤ C

It can be easily seen that (36) is a convex optimizationproblem, as the constraints are linear and the objective functionis the sum of convex functions, as can be verified by takingtheir second derivatives.

Now introducing Lagrange multipliersν ∈ RL+ for the

positivity constraintsCk ≥ 0, k = 1, 2, · · · , L, andλ ∈ R+

for the backhaul sum-capacity constraint∑L

k=1 Ck ≤ C, weform the Lagrangian

L(Ck,ν, λ) =

L∑

k=1

1

2log(

1 + 2−2CkSINRk

)

−L∑

k=1

νkCk + λ

(

L∑

k=1

Ck − C

)

(37)

Taking the derivative of the above with respect toCk, weobtain the following Karush-Kuhn-Tucker (KKT) condition

−2−2C∗

kSINRk

1 + 2−2C∗

kSINRk

− νk + λ = 0, (38)

for the optimalC∗k , wherek = 1, 2, · · · , L. Note thatνk = 0

wheneverCk > 0. Now, the optimalC∗k must satisfy the back-

haul sum-capacity constraint∑L

k=1 C∗k ≤ C with equality,

because the objective of the minimizationRk monotonicallyincreases withCk. Solving the condition (38) together withthe fact that

∑Lk=1 C

∗k = C gives the following optimalC∗

k :

C∗k = max

{

1

2log

SINRk

β, 0

}

, (39)

whereβ is chosen such that∑L

k=1 C∗k = C. This is equivalent

to (35).

An interpretation of (39) is that whenever the SINR of userk is above a thresholdβ, 1

2 logSINRk

βbits of the backhaul

link should be allocated to userk. Otherwise, this user shouldnot be active as far as maximizing the uplink sum rate isconcerned. This optimal rate allocation is quite similar totheclassic water-filling solution for the sum-capacity maximiza-tion problem for a parallel set of Gaussian channels, in whichmore power (backhaul capacity in this case) is assigned tousers with a better channel.

When written as (35), the optimal backhaul capacity allo-cation can be interpreted as follows:Ck = 1

2 log(SINRk) canbe thought of as the nominal optimal backhaul link capacity.If the total backhaul capacity is above (or below) the nominal∑

k12 log(SINRk), then the extra capacity must be distributed

(or taken away) from each base-station equally. In other words,

all base-station should nominally operate at the point 1/2 bitsaway from the SIC limit (as shown in Fig. 3). If more (orless) backhaul capacity is available than the nominal value,all base-stations should move above (or below) that operatingpoint in the same manner.

Finally, we remark that the decoding order at the centralizedprocessor plays an important role in the optimal rate allocation.Different decoding orders result in different rate expressionsin Theorem 1 and thus different rate allocations in Theorem 5,and as a consequence different achievable sum rates. In orderto determine the decoding order that results in the largest sumrate (or the maximum network utility), we need to exhaustivelysearch overK! different decoding orders. This is a nontrivialproblem that is also encountered in other contexts involvingsuccessive decoding such as in V-BLAST [26].

VI. N UMERICAL EXAMPLES

A. Two-User Symmetric Scenario

To obtain numerical insight on the per-base-station SIC-based schemes proposed in this paper, the achievable rateregions of Theorem 1 and Theorem 2 are compared with thatobtained by two other schemes: 1) single-user decoding with-out joint processing; 2) joint base-station processing as definedin (21) in Section III-C, for a two-user symmetric scenariowhereL = 2, P1 = P2 = N0 = 1, SNR = h2

11 = h222,

INR = h212 = h2

21, andC = C1 = C2. Under the symmetricsetting, both Theorem 1 and Theorem 2 give two symmetricachievable rate pairs depending on the decoding order. Time-sharing of the two achievable rate pairs gives a pentagonshaped achievable rate region.

Single-user decoding without joint processing is consideredas a baseline, in which each receiver decodes its own signalwhile treating the other user’s signal as noise. This gives thefollowing achievable rate pair

R1 = R2 = min

{

1

2log

(

1 +SNR

1 + INR

)

, C

}

(40)

which in the symmetric setting results in a square shapedachievable rate region with (R1, R2) as the top-right corner.

We also compare with the joint-base-station processing SICscheme of (21). We restrict ourselves to symmetric quantiza-tion noise levels here. The quantization noise levelq is chosensuch that the resulting average backhaul link capacity isC, i.e.,

I(Y1; Y1) + I(Y2; Y1|Y1) = 2C. (41)

This condition gives the following analytic expression forthequantization noise level:

q =a+

√

4b+ 24C (a2 − 4b)

24C − 1, (42)

wherea = 1 + SNR + INR, andb = SNR · INR. Plots of thejoint-base-station SIC are then obtained using (21) with thisquantization noise level.

The achievable rate regions obtained above are comparedfor the following channel settings:

• SNR = 30dB, INR = 20dB, C = 5 bits;• SNR = 30dB, INR = 5dB, C = 5 bits;


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

R1

R2

No Joint ProcessingPer−Base−Station SIC W/ Wyner−ZivPer−Base−Station SIC W/O Wyner−Ziv Joint Base−Station SIC

(a) SNR = 30dB, INR = 20dB, C1 = C2 = 5 bits

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

R1

R2


(b) SNR = 30dB, INR = 5dB, C = 5 bits

0 1 2 3 4 5 60

1

2

3

4

5

6

R1

R2


(c) SNR = 30dB, INR = 20dB, C = 10 bits

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

R1

R2


(d) SNR = 30dB, INR = 20dB, C = 2 bits

Fig. 6. Comparison of the proposed achievability scheme andanother two schemes

• SNR = 30dB, INR = 20dB, C = 10 bits;• SNR = 30dB, INR = 20dB, C = 2 bits.

Fig. 6(a) shows that at relatively strong interference levelINR = 20dB, the proposed per-base-station SIC schemes(both with and without Wyner-Ziv compression) expand thebaseline achievable rate region by about2.8 bits on either ofthe individual rates and on the sum rate. The joint base-stationSIC regions further outperforms the proposed scheme in sumrate by about2.5 bits due to the benefits of joint decoding.

However, when the interfering links are weak, as shownin Fig. 6(b) where INR = 5dB, all four achievable rateregions are close to each other. This is the regime wheretreating interference as noise is close to optimal, so multicellprocessing does not provide significant benefits.

In the above two examples, the capacities of the backhaullinks are already quite abundant, since they are set to be therate supported by the direct links:12 log(1 + SNR) ≈ 5 bits.In Fig. 6(c), we further increase the backhaul capacity to 10bits, and show that doing so does not significantly improve theachievable rate region for either proposed SIC-based schemes

or the joint base-station processing scheme.Lastly, we decrease the backhaul capacity from 5 bits to

2 bits. Interestingly, this is a situation in which the base-line scheme can outperform per-base-station SIC as shownin Fig. 6(d). Therefore, the proposed per-base-station schemescan be inefficient in term of sum rate, when the backhaulrates are very limited. Observe that the largest sum rate is stillobtained with joint base-station processing.

B. Multicell OFDMA Network

To further understand the performance of the proposed per-base-station SIC scheme in practical systems, in this section,the achievable rates of the two variations of the per-base-station SIC, i.e., with and without Wyner-Ziv coding, areevaluated for a wireless cellular network setup with19 cells,3sectors per cell, and10 users per sector, where an orthogonalfrequency-division multiple-access (OFDMA) scheme with64tones over a fixed10Mbps bandwidth is employed. The cellu-lar topology is shown in Fig. 7. A 19-cell wrap-around layoutis used to ensure uniform interference statistics throughout the


TABLE IWIRELESSCELLULAR MODEL PARAMETERS

Cellular Layout Hexagonal, 19 cells, 3 sectors/cellBS-to-BS Distance 600 mFrequency Reuse 1

Number of User per Sector 10Channel Bandwidth 10 MHzMS Transmit PSD −27 dBm/Hz

Antenna Gain 15 dBiBackground Noise −169 dBm/Hz

Noise Figure 7 dBMultipath Time Delay Profile ITU-R M.1225 PedADistance-dependent Path Loss 128.1 + 37.6 log10(d)

Number of Tones 64

network. The assignments of frequency tones to users withineach cell are non-overlapping. As a result, users experienceonly intercell interference and no intracell interference. Boththe base-stations and the mobile users are equipped with asingle antenna each. Each of the19 base-stations is connectedto the centralized processor via a rate-limited backhaul link.Perfect channel estimation is assumed, and the CSI is madeavailable to all base-stations and to the centralized processor.In the simulation, uniform power allocation of−27dBm/Hzis assumed at all the mobile users. For convenience, a round-robin scheduler is used for user assignment. The base-station-to-base-station distance is set to600m corresponding to atypical urban deployment. Detailed system parameters areoutlined in Table I.

In the first part of the simulation, the capacities of thebackhaul links are fixed per base-station and uniformly dis-tributed across the frequency tones, e.g., if the capacity ofa backhaul link is64Mbps, each frequency tone is assumedto have a backhaul of64Mbps/64 = 1Mbps. Cumulativedistribution function (CDF) of the user rates is plotted in orderto visualize the performance gain of the proposed schemesover a baseline system, in which base-stations decode the usermessages without joint processing at the centralized processor.For the proposed per-base-station SIC scheme, to account forthe fairness among users, the decoding order across the cellsis chosen to be in decreasing order of the user SINRs (prior toSIC). This decoding order is adopted on each of the OFDMtones independently.

Fig. 8 shows the CDF plots of user rates with a backhaulcapacity at each base-station of180Mbps (i.e.,60Mbps persector). It is seen that the two per-base-station SIC schemes(with or without Wyner-Ziv coding) both significantly outper-form the baseline system. The user rate at50th-percentile isaround0.8Mbps for the baseline,2.1Mbps for the per-base-station SIC scheme without Wyner-Ziv coding, and2.8Mbpsfor the per-base-station SIC scheme with Wyner-Ziv coding.Compared with the baseline curve, the per-base-station SICcurves also have a better distribution of user rates in termsoffairness. There is a noticeable performance gap between theSIC curve with Wyner-Ziv compression and without Wyner-Ziv-compression. This gap is due to the compression gainbrought by side information.

When the capacity of the backhaul per base-station increasesto 270Mbps, the proposed per-base-station SIC schemes pro-

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Base StationMobile User

Fig. 7. A cellular topology with19 cells,3 sectors per cell, and10 users persector placed randomly. The distance between the neighboring base-stationsis set to600m.

duce a further performance gain as compared to the180Mbpscase. As can be seen in Fig. 9, the50-percentile rate becomes2.6Mbps for the per-base-station SIC scheme without Wyner-Ziv compression, and3.05Mbps for the per-base-station SICscheme with Wyner-Ziv compression. However, the gap be-tween the two curves becomes smaller. As the capacity ofthe backhaul per base-station further increases to360Mbps,Fig. 10 shows that the per-base-station SIC schemes now onlyperform marginally better than the270Mbps case. This is whenthe benefit of multicell SIC starts to saturate.

Further, as shown in Fig. 10, the CDF curves for the SICschemes with and without Wyner-Ziv compression are veryclose to each other for the360Mbps backhaul case. Thus,the benefit of performing Wyner-Ziv compress-and-forwardrelaying at the base-stations becomes negligible when thecapacities of backhaul links are high, thus confirming ourearlier theoretical analysis.

In order to quantitatively evaluate the performance gainbrought by the centralized processor, Table II shows theaverage per-cell sum rate obtained by different schemes.The baseline scheme gives an average per-cell sum rate of55.5Mbps. By utilizing a high-capacity backhaul with Wyner-Ziv compression, up to 94% sum rate improvement can beobtained. Note that although the rate improvement withoutWyner-Ziv is lower than that with Wyner-Ziv, considerableperformance gains in the range of 37% to 84% can still beobtained without Wyner-Ziv coding. As noted before, the gaindue to the backhaul saturates at around270Mbps.

The above simulation is performed assuming that thebackhaul capacity is the same for each base-station and isuniformly allocated across the frequencies. It is possibletofurther optimize the backhaul capacity allocation using (35)of Theorem 5. In the next set of simulations, we chooseαin (35) to satisfy an average backhaul constraint across thecells, and present the resulting performance with optimizedbackhaul allocation in Table II and Fig. 11. It can be seen that120−150Mbps optimized backhaul already achieves about thesame performance as that of180Mbps uniform backhaul. Like-wise, 180Mbps optimized backhaul already achieves about


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate (Mbps)

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

(CD

F)

Baseline: No Centralized ProcessorPer−BS SIC W/ Wyner−Ziv CompressionPer−BS SIC W/O Wyner−Ziv Compression

Fig. 8. CDF of user rates with 180Mbps backhaul per base-station uniformlyallocated across the frequencies

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate (Mbps)

Cum

ulat

ive

Dis

trib

utio

n F

unct

ion

(CD

F)



the same performance as that of270Mbps uniform backhaul.Thus, the optimization of the backhaul is quite beneficial.

Further, it can be seen from Fig. 11 that under infinitebackhaul, the achieved per-cell sum rate is about110Mbpsfor this cellular setting. But when optimized, a finite backhaulcapacity at about 1.5 times of the user sum rate (i.e., at about150Mbps) is already sufficient to achieve about100Mbps usersum rate, which is 90% of the full benefit of uplink networkMIMO. Note that the gain in per-cell sum rate due to theoptimization of the backhaul becomes smaller as the backhaulcapacity increases, due to the fact that increasing the backhaulcapacity eventually offers diminishing return.

Finally, we mention that the performance gain presentedhere is idealistic because the achievable rates are computed us-ing information theoretical expressions assuming ideal coding,modulation, and perfect CSI. In addition, all users in the 19-cell cluster are assumed to participate in cooperative multicellprocessing, and no out-of-cluster interference is accounted for.The results in this paper nevertheless serve as upper bound to

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Fig. 11. Per-cell sum rate vs. average per-cell backhaul capacity using per-base-station SIC with uniform and optimized backhaul capacity allocation

what is achievable in an uplink network MIMO system.

VII. C ONCLUSION

This paper presents an information-theoretical study ofa novel uplink multicell processing scheme employing thecompress-and-forward technique with the per-base-station SICreceiver structure for the uplink of a network MIMO system,in which the base-stations are connected to a centralizedprocessor with finite capacity backhaul links. The main ad-vantage of the proposed schemes is that it achieves significantperformance gain over the conventional scenario where nocentralized processor is deployed, while having an achievablerate region which is easily computable, and that it leads to anarchitecture that is more amendable to practical implementa-tions than the joint decoding scheme. Furthermore, theoreticalanalysis shows that the proposed per-base-station SIC schemeis within a constant gap to the sum capacity for a class ofWyner channel models.

The results of this paper also show that when employing theproposed per-base-station SIC scheme, the capacities of the


TABLE IIIMPROVEMENT IN PER-CELL SUM RATE OVER19 CELLS, 3 SECTORS PER CELL, 10 USERS PER SECTOR. CELL DIAMETER IS 600M . BASELINE PER-CELL

SUM RATE IS55.5MBPS

Per-cell Average Backhaul Capacity Per-base-station SIC Improvement Per-base-station SIC ImprovementBackhaul (Mbps) Allocation without WZ (Mbps) over Baseline(%) with WZ (Mbps) over Baseline (%)

180 uniform 75.3 37% 92.7 67%270 uniform 93.2 68% 104.2 88%360 uniform 102.3 84% 107.6 94%120 optimized 75.1 35% 89.1 60%180 optimized 83.3 50% 102.9 85%

backhaul links should scale with the logarithm of the SINR ateach base-station, both from a point of view of approachingthe theoretical maximum SIC rate with unlimited backhaul, aswell as for maximizing the overall sum rate subject to a totalbackhaul rate constraint. Numerical simulations reveal thatsignificant sum-rate gain can be obtained by the proposed SIC-based schemes with modest backhaul capacity requirement.

ACKNOWLEDGEMENTS

The authors would like to thank Dimitris Toumpakaris forhelpful discussions and valuable comments.

REFERENCES

[1] S. Venkatesan, A. Lozano, and R. Valenzuela, “Network MIMO: Over-coming intercell interference in indoor wireless systems,” in Conf.Record Forty-First Asilomar Conf. Signals, Systems and Computers,Nov. 2007, pp. 83–87.

[2] D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, andW. Yu,“Multi-cell MIMO cooperative networks: A new look at interference,”IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp. 1380–1408, Dec. 2010.

[3] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limitedreceiver cooperation,”IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 2913–2940, May 2011.

[4] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” IEEETrans. Inf. Theory, vol. 52, no. 4, pp. 1522–1544, Apr. 2006.

[5] C. T. K. Ng, N. Jindal, A. J. Goldsmith, and U. Mitra, “Capacity gainfrom two-transmitter and two-receiver cooperation,”IEEE Trans. Inf.Theory, vol. 53, no. 10, pp. 3822–3827, Oct. 2007.

[6] V. M. Prabhakaran and P. Viswanath, “Interference channels withdestination cooperation,”IEEE Trans. Inf. Theory, vol. 57, no. 1, pp.187–209, Jan. 2011.

[7] H. Do, T. Oechtering, and M. Skoglund, “Noisy network coding ap-proach to the interference channel with receiver cooperation,” in Proc.Forty-Ninth Annual Allerton Conf. Commun, Control and Computing,Sep. 2011, pp. 839–846.

[8] P. Marsch and G. Fettweis, “Uplink CoMP under a constrained backhauland imperfect channel knowledge,”IEEE Trans. Wireless Commun.,vol. 10, no. 6, pp. 1730–1742, Jun. 2011.

[9] O. Simeone, O. Somekh, H. Poor, and S. Shamai, “Local basestationcooperation via finite-capacity links for the uplink of linear cellularnetworks,” IEEE Trans. Inf. Theory, vol. 55, no. 1, pp. 190–204, Jan.2009.

[10] A. Chowdhery, W. Yu, and J. Cioffi, “Cooperative wireless multicellOFDMA network with backhaul capacity constraints,” inProc. IEEEInt. Conf. Commun. (ICC), Jun. 2011.

[11] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai, “Uplink macrodiversity of limited backhaul cellular network,”IEEE Trans. Inf. Theory,vol. 55, no. 8, pp. 3457–3478, Aug. 2009.

[12] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer,“Communi-cation via decentralized processing,”IEEE Trans. Inf. Theory, vol. 54,no. 7, pp. 3008–3023, Jul. 2008.

[13] O. Somekh, B. Zaidel, and S. Shamai, “Sum rate characterization ofjoint multiple cell-site processing,”IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 4473–4497, Dec. 2007.

[14] A. del Coso and S. Simoens, “Distributed compression for MIMOcoordinated networks with a backhaul constraint,”IEEE Trans. WirelessCommun., vol. 8, no. 9, pp. 4698–4709, Sep. 2009.

[15] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, “Robustand efficientdistributed compression for cloud radio access network,”IEEE Trans.Veh. Technol., vol. 62, no. 2, pp. 692–703, Feb. 2013.

[16] ——, “Joint decompression and decoding for cloud radio access net-works,” IEEE Signal Process. Lett., vol. 20, no. 5, pp. 503–506, May2013.

[17] A. Sanderovich, M. Peleg, and S. Shamai, “Scaling laws and techniquesin decentralised processing of interfered Gaussian channels.” EuropeanTrans. Telecommun., vol. 22, no. 5, pp. 240–253, 2011.

[18] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Downlink multi-cell processing with limited-backhaul capacity,”EURASIP J. AdvancesSignal Process., 2009.

[19] S. Avestimehr, S. Diggavi, and D. N. C. Tse, “Wireless networkinformation flow: A deterministic approach,”IEEE Trans. Inf. Theory,vol. 57, no. 4, pp. 1872–1905, Apr. 2011.

[20] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisynetworkcoding,” IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 3132–3152, May2011.

[21] O. Somekh, B. M. Zaidel, and S. Shamai, “Sum rate characterizationof joint multiple cell-site processing,”IEEE Trans. Inf. Theory, vol. 53,no. 7, pp. 4473–4497, Dec. 2007.

[22] Y. Zhou, W. Yu, and D. Toumpakaris, “Uplink multi-cell processing:Approximate sum capacity under a sum backhaul constraint,”Apr. 2013,[Online] Available: http://arxiv.org/abs/1304.7509.

[23] A. D. Wyner, “Shannon-theoretic approach to a Gaussiancellularmultiple-access channel,”IEEE Trans. Inf. Theory, vol. 40, no. 6, pp.1713–1727, Nov. 1994.

[24] T. M. Cover and A. El Gamal, “Capacity theorems for the relaychannel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sep.1979.

[25] T. M. Cover and J. A. Thomas,Elements of Information Theory, 1st ed.Wiley, 1991.

[26] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela,“V-BLAST: An architecture for realizing very high data rates over therich-scattering wireless channel,” inProc. URSI Int. Symp. Signals, Syst.and Electron. (ISSSE), Sep. 1998, pp. 295–300.

REFERENCES

[1] S. Venkatesan, A. Lozano, and R. Valenzuela, “Network MIMO: Over-coming intercell interference in indoor wireless systems,” in Conf.Record Forty-First Asilomar Conf. Signals, Systems and Computers,Nov. 2007, pp. 83–87.

[2] D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, andW. Yu,“Multi-cell MIMO cooperative networks: A new look at interference,”IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp. 1380–1408, Dec. 2010.

[3] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limitedreceiver cooperation,”IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 2913–2940, May 2011.

[4] A. Høst-Madsen, “Capacity bounds for cooperative diversity,” IEEETrans. Inf. Theory, vol. 52, no. 4, pp. 1522–1544, Apr. 2006.

[5] C. T. K. Ng, N. Jindal, A. J. Goldsmith, and U. Mitra, “Capacity gainfrom two-transmitter and two-receiver cooperation,”IEEE Trans. Inf.Theory, vol. 53, no. 10, pp. 3822–3827, Oct. 2007.

[6] V. M. Prabhakaran and P. Viswanath, “Interference channels withdestination cooperation,”IEEE Trans. Inf. Theory, vol. 57, no. 1, pp.187–209, Jan. 2011.

[7] H. Do, T. Oechtering, and M. Skoglund, “Noisy network coding ap-proach to the interference channel with receiver cooperation,” in Proc.Forty-Ninth Annual Allerton Conf. Commun, Control and Computing,Sep. 2011, pp. 839–846.


[8] P. Marsch and G. Fettweis, “Uplink CoMP under a constrained backhauland imperfect channel knowledge,”IEEE Trans. Wireless Commun.,vol. 10, no. 6, pp. 1730–1742, Jun. 2011.

[9] O. Simeone, O. Somekh, H. Poor, and S. Shamai, “Local basestationcooperation via finite-capacity links for the uplink of linear cellularnetworks,” IEEE Trans. Inf. Theory, vol. 55, no. 1, pp. 190–204, Jan.2009.

[10] A. Chowdhery, W. Yu, and J. Cioffi, “Cooperative wireless multicellOFDMA network with backhaul capacity constraints,” inProc. IEEEInt. Conf. Commun. (ICC), Jun. 2011.

[11] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai, “Uplink macrodiversity of limited backhaul cellular network,”IEEE Trans. Inf. Theory,vol. 55, no. 8, pp. 3457–3478, Aug. 2009.

[12] A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer,“Communi-cation via decentralized processing,”IEEE Trans. Inf. Theory, vol. 54,no. 7, pp. 3008–3023, Jul. 2008.

[13] O. Somekh, B. Zaidel, and S. Shamai, “Sum rate characterization ofjoint multiple cell-site processing,”IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 4473–4497, Dec. 2007.

[14] A. del Coso and S. Simoens, “Distributed compression for MIMOcoordinated networks with a backhaul constraint,”IEEE Trans. WirelessCommun., vol. 8, no. 9, pp. 4698–4709, Sep. 2009.

[15] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, “Robustand efficientdistributed compression for cloud radio access network,”IEEE Trans.Veh. Technol., vol. 62, no. 2, pp. 692–703, Feb. 2013.

[16] ——, “Joint decompression and decoding for cloud radio access net-works,” IEEE Signal Process. Lett., vol. 20, no. 5, pp. 503–506, May2013.

[17] A. Sanderovich, M. Peleg, and S. Shamai, “Scaling laws and techniquesin decentralised processing of interfered Gaussian channels.” EuropeanTrans. Telecommun., vol. 22, no. 5, pp. 240–253, 2011.

[18] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Downlink multi-cell processing with limited-backhaul capacity,”EURASIP J. AdvancesSignal Process., 2009.

[19] S. Avestimehr, S. Diggavi, and D. N. C. Tse, “Wireless networkinformation flow: A deterministic approach,”IEEE Trans. Inf. Theory,vol. 57, no. 4, pp. 1872–1905, Apr. 2011.

[20] S. H. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisynetworkcoding,” IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 3132–3152, May2011.

[21] O. Somekh, B. M. Zaidel, and S. Shamai, “Sum rate characterizationof joint multiple cell-site processing,”IEEE Trans. Inf. Theory, vol. 53,no. 7, pp. 4473–4497, Dec. 2007.

[22] Y. Zhou, W. Yu, and D. Toumpakaris, “Uplink multi-cell processing:Approximate sum capacity under a sum backhaul constraint,”Apr. 2013,[Online] Available: http://arxiv.org/abs/1304.7509.

[23] A. D. Wyner, “Shannon-theoretic approach to a Gaussiancellularmultiple-access channel,”IEEE Trans. Inf. Theory, vol. 40, no. 6, pp.1713–1727, Nov. 1994.

[24] T. M. Cover and A. El Gamal, “Capacity theorems for the relaychannel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sep.1979.

[25] T. M. Cover and J. A. Thomas,Elements of Information Theory, 1st ed.Wiley, 1991.

[26] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela,“V-BLAST: An architecture for realizing very high data rates over therich-scattering wireless channel,” inProc. URSI Int. Symp. Signals, Syst.and Electron. (ISSSE), Sep. 1998, pp. 295–300.

PLACEPHOTOHERE

Lei Zhou (S’05-M’12) received the B.E. degree inElectronics Engineering from Tsinghua University,Beijing, China, in 2003 and M.A.Sc. and Ph.D. de-grees in Electrical and Computer Engineering fromthe University of Toronto, ON, Canada, in 2008and 2012 respectively. During 2008-2009, he waswith Nortel Networks, Ottawa, ON, Canada. Since2012, he has been with Qualcomm TechnologiesInc., Santa Clara, CA. His research interests includemultiterminal information theory, wireless commu-nications, and signal processing.

He is a recipient of the Shahid U.H. Qureshi Memorial Scholarship in 2011,the Alexander Graham Bell Canada Graduate Scholarship in 2011, and theChinese government award for outstanding self-financed students abroad in2012.

PLACEPHOTOHERE

Wei Yu (S’97-M’02-SM’08) received the B.A.Sc.degree in Computer Engineering and Mathematicsfrom the University of Waterloo, Waterloo, On-tario, Canada in 1997 and M.S. and Ph.D. degreesin Electrical Engineering from Stanford University,Stanford, CA, in 1998 and 2002, respectively. Since2002, he has been with the Electrical and Com-puter Engineering Department at the University ofToronto, Toronto, Ontario, Canada, where he is nowProfessor and holds a Canada Research Chair inInformation Theory and Wireless Communications.

His main research interests include information theory, optimization, wirelesscommunications and broadband access networks.

Prof. Wei Yu currently serves as an Associate Editor for IEEETRANSAC-TIONS ON INFORMATION THEORY. He was an Editor for IEEE TRANSAC-TIONS ON COMMUNICATIONS (2009-2011), an Editor for IEEE TRANSAC-TIONS ON WIRELESSCOMMUNICATIONS (2004-2007), and a Guest Editorfor a number of special issues for the IEEE JOURNAL ON SELECTEDAREAS

IN COMMUNICATIONS and the EURASIP JOURNAL ON APPLIED SIGNAL

PROCESSING. He is member of the Signal Processing for Communicationsand Networking Technical Committee of the IEEE Signal Processing Society.He received the IEEE Signal Processing Society Best Paper Award in 2008,the McCharles Prize for Early Career Research Distinction in 2008, the EarlyCareer Teaching Award from the Faculty of Applied Science and Engineering,University of Toronto in 2007, and the Early Researcher Award from Ontarioin 2006.

http://arxiv.org/abs/1304.7509

uplink multicell processing with limited backhaul via per ...processing problem. toward this end,...

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