Optimized Backhaul Compression for Uplink Cloud Radio

arX

iv:1

304.

7509

v3 [

cs.IT

] 17

Aug

201

41

Optimized Backhaul Compression for Uplink CloudRadio Access Network

Yuhan Zhou,Student Member, IEEEand Wei Yu,Fellow, IEEE

Abstract—This paper studies the uplink of a cloud radio accessnetwork (C-RAN) where the cell sites are connected to a cloud-computing-based central processor (CP) with noiseless backhaullinks with finite capacities. We employ a simple compress-and-forward scheme in which the base-stations (BSs) quantizethe received signals and send the quantized signals to theCP using either distributed Wyner-Ziv coding or single-usercompression. The CP decodes the quantization codewords first,then decodes the user messages as if the remote users and thecloud center form a virtual multiple-access channel (VMAC).This paper formulates the problem of optimizing the quantizationnoise levels for weighted sum rate maximization under a sumbackhaul capacity constraint. We propose an alternating convexoptimization approach to find a local optimum solution to theproblem efficiently, and more importantly, establish that settingthe quantization noise levels to be proportional to the backgroundnoise levels is near optimal for sum-rate maximization whenthesignal-to-quantization-noise ratio (SQNR) is high. In addition,with Wyner-Ziv coding, the approximate quantization noiselevelis shown to achieve the sum-capacity of the uplink C-RANmodel to within a constant gap. With single-user compression,a similar constant-gap result is obtained under a diagonaldominant channel condition. These results lead to an efficientalgorithm for allocating the backhaul capacities in C-RAN. Theperformance of the proposed scheme is evaluated for practicalmulticell and heterogeneous networks. It is shown that multicellprocessing with optimized quantization noise levels across theBSs can significantly improve the performance of wireless cellularnetworks.

Index Terms—Cloud radio access network, multicell process-ing, compress-and-forward, Wyner-Ziv compression, heteroge-neous network, network MIMO, coordinated multipoint (CoMP )

I. I NTRODUCTION

CLOUD Radio Access Network (C-RAN) is a futurewireless network architecture in which base-station (BS)

processing is uploaded to a cloud-computing based centralprocessor (CP). By taking advantage of the high-capacitybackhaul links between the BSs and the CP, the C-RANarchitecture enables joint encoding and decoding of messagesfrom multiple cells, and consequently, effective mitigation ofintercell interference. As future 5G wireless cellular networksare expected to be deployed with progressively smaller cellsizes in order to support higher data rate demands and as inter-cell interference increasingly becomes the main physical-layer

Manuscript received December 1, 2013; revised April 25, 2014; acceptedMay 25, 2014. This work was supported by Huawei Technologies, Canada.This paper was presented in part at the Canadian Workshop on InformationTheory, Toronto, ON, Canada, June 2013 [1], and at the IEEE InformationTheory Workshop, Seville, Spain, September 2013 [2].

The authors are with Edward S. Rogers Sr. Department of Electrical andComputer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada(e-mail: [email protected]; [email protected]).

….

….

Central

Processor

Y1 : Y1

Y2 : Y2

YL : YL

X1

X2

XL

Z1

Z2

ZL

h11

h21 h12

h1L hL1

h2L hL2

h22

hLL

Y1

Y2

YL

C

Fig. 1. The uplink of a cloud radio access network with a finitesum backhaul

bottleneck, the C-RAN architecture is seen as a path towardeffective implementation of coordinated multi-point (CoMP),also known as the network multiple-input multiple-output(network MIMO) system. It has the potential to significantlyimprove the overall throughput of the cellular network [3].

This paper deals with the capacity limits and system-level optimization of uplink C-RAN under practical finite-capacity backhaul constraints. The uplink of C-RAN model,as shown in Fig. 1, consists of multiple remote users sendingindependent messages while interfering with each other at theirrespective BSs. The BSs are connected to the CP via noiselessbackhaul links with a finite sum capacity constraintC. Theuser messages are eventually decoded at the CP. This uplinkC-RAN model can be thought of as avirtual multiple-accesschannel(VMAC) between the users and the CP, with the BSsacting asrelays. The antennas of multiple BSs essentiallybecome a virtual MIMO antenna array capable of spatiallymultiplexing multiple user terminals.

To explore the advantage of the C-RAN architecture, thispaper considers a compress-and-forward relay strategy inwhich the BSs send compressed version of their receivedsignals to the CP through the backhaul, and the CP eitherjointly or successively decodes all the user messages. De-pending on the different compression strategies used at BSs,either with Wyner-Ziv (WZ) coding or with single-user (SU)compression, the coding strategies in this paper are namedVMAC-WZ or VMAC-SU respectively. A key parameter inbackhaul compression design is the level of quantization noiseintroduced by the compression operation. The main objectiveof this paper is to identify efficient algorithms for the optimalsetting of quantization noise levels in uplink C-RAN withcapacity-limited backhaul.

A. Related Work

The achievable rates and the relay strategy of the uplinkC-RAN architecture have been studied previously in the infor-

http://arxiv.org/abs/1304.7509v3

2

mation theory literature. Under a Wyner model, the achievablerate of an uplink cellular network with BS cooperation is stud-ied in [4] assuming unlimited cooperation, then extended tothe limited cooperation case in [5], where the performancesofrelaying strategies such as decode-and-forward and compress-and-forward are evaluated.

The uplink C-RAN model considered in this paper isclosely related to that in [6], [7], [8], where the fundamentalachievable rates using the compress-and-forward strategyarecharacterized under individual backhaul capacity constraints.The achievable rates of [6], [7], [8] are derived assuming thatthe quantization codewords and the user messages are decodedjointly at the CP. However, such a joint decoding strategyis computationally complex. Further, the question of how tooptimally set the quantization noise level is left open.

The uplink C-RAN model can be thought of as a particularinstance of a general relay network with a single destinationfor which several recent works [9], [10], [11] have been ableto characterize the information theoretical capacity to within aconstant gap. The achievability schemes of [9], [10], [11] arestill based on joint decoding, but with the new insight that inorder to achieve to within a constant gap to the outer bound,the quantization noise level should be set at the backgroundnoise level.

This paper goes one step further in identifying relayingand decoding schemes that have lower complexity than jointdecoding, while maintaining certain optimality. Toward thisend, this paper shows that asuccessivedecoding strategy inwhich the CP first decodes the quantization codewords, thendecodes the user messages based on the quantized signalsfrom all BSs can achieve to within a constant gap to the sumcapacity of the network. We note that the proposed schemeis different and performs better than the per-BS successiveinterference cancellation (SIC) scheme of [12], where eachuser message is decoded based on the quantization codewordof its own BS only and the previously decoded messages.

A main focus of this paper is the optimization of thequantization noise levels at the BSs for the uplink C-RANmodel. In this direction, the present paper is related to theworkof [13], which uses a gradient approach to solve a quantizationnoise level optimization problem for a closely related problem.The present paper is also closely related to [14], where thequantization noise level optimization problem is solved onaper-BS basis (and the robustness of the optimization procedureis addressed in addition). In contrast, the algorithm proposed inthis paper involves a more direct optimization objective wherethe quantization noise levels of all BSs are optimized jointly.

As related work, we also mention [15] which investigatesthe effect of imperfect channel state information (CSI) foruplink C-RAN, and [16] which evaluates the performanceof compress-and-forward for a two-user C-RAN model underlimited individual backhaul assuming only receiver side CSI.Finally, we mention briefly that the compute-and-forwardrelaying scheme has been studied for the uplink C-RAN modelwith equal-capacity backhaul links in [17], [18], where theBSs compute a function of transmitted codewords and sendthe function value to the CP for joint decoding.

B. Main Contributions

From a theoretical capacity analysis perspective, this papershows that VMAC-WZ with successive decoding can achievethe sum capacity of the C-RAN model to within a constantgap, while VMAC-SU achieves the sum capacity to withina constant gap under a channel diagonal dominant condition.Since the VMAC schemes have the advantage of low decodingcomplexity and low decoding delay as compared to jointdecoding, the constant-gap results provide a strong motivationfor the possible implementation of the VMAC schemes inpractical C-RAN systems.

From an optimization perspective, this paper proposes analternating convex optimization algorithm for optimizingthequantization noise levels for weighted sum-rate maximizationfor the VMAC-WZ scheme, and proposes reformulation ofthe problem in term of optimizing backhaul capacities forthe VMAC-SU scheme. Further, this paper shows that inthe high signal-to-quantization-noise-ratio (SQNR) regime, thequantization noise level should be set to be proportional tothe background noise level, regardless of the transmit powerand the channel condition. Based on this observation, low-complexity algorithms are developed for the quantization noiselevel design in practical C-RAN scenarios.

Finally, this paper evaluates the performance of the proposedVMAC schemes in multicell networks and in heterogeneoustopologies where macro- and pico-cells may have significantlydifferent backhaul capacity constraints. Numerical simulationsshow that the C-RAN architecture can bring significant per-formance improvement, and that the proposed approximatequantization noise level setting can already realize much ofthe gains.

C. Paper Organization and Notation

The rest of the paper is organized as follows. Section IIintroduces the VMAC scheme with WZ compression andwith SU compression. Section III focuses on optimizing thequantization noise level for the VMAC-WZ scheme, where analternating convex optimization algorithm and an approxima-tion algorithm are proposed. It is shown that the VMAC-WZscheme achieves the sum capacity of the uplink C-RAN modelto within a constant gap. Section IV focuses on the optimiza-tion of quantization noise levels for the VMAC-SU scheme,and formulates an equivalent backhaul capacity allocationproblem. A constant-gap capacity result for the VMAC-SUscheme is demonstrated. The proposed VMAC schemes areevaluated numerically for practical multicell/picocell networksin Section V. Conclusions are drawn in Section VI.

The notations used in this paper are as follows. Lower-case letters denote scalars and upper-case letters denote scalarrandom variables. Boldface lower-case letters denote columnvectors. Boldface upper-case letters denote vector randomvariables or matrices, where context should make the dis-tinction clear. The superscripts(·)T , (·)H , and (·)−1 denotetranspose, Hermitian transpose, and matrix inverse operators;Tr(·) denotes the trace operation. We usediag(xi) to denote adiagonal matrix with diagonal elementsxi’s. The expectationoperation is denoted asE(·). Calligraphy letters are used to

3

denote sets. For a vectorX, X(S) denotes a vector withelements whose indices are elements ofS; for a matrixX, X(S) denotes a matrix whose columns are indexed byelements ofS.

II. PRELIMINARIES

A. System Model

This paper considers the uplink C-RAN, whereL single-antenna remote users send independent messages toL single-antenna BSs forming a fixed cluster, as shown in Fig. 1. TheBSs are connected to a CP through noiseless backhaul linksof capacitiesCi, i = 1, . . . , L. The user messages need to beeventually decoded at the CP. A key modelling assumption ofthis paper is that the backahul capacitiesCi can be adapted tothe channel condition and user traffic demand, subject to anoverall capacity constraint, i.e.

∑L

i=1 Ci ≤ C. For simplicity,both the remote users and the BSs are assumed to have asingle antenna each here, but most results of this paper can beextended to the MIMO case.

The sum backhaul capacity constraint considered in thispaper is particularly suited to model the scenario where thebackhaul is implemented in a wireless shared medium. Forexample, when the wireless backhaul links are implementedusing an orthogonal access scheme such as time/frequencydivision multiple access (TDMA or FDMA), and the totalnumber of time/frequency slots that can be utilized by differ-ent access points can be shared, the sum-capacity constraintcaptures the essential feature of the backhaul constraints.

The uplink C-RAN model can be thought of as anL × Linterference channel between the users and the BSs, followedby a noiseless multiple-access channel between the BSs andthe CP. Alternatively, it can also be thought of as a virtualmultiple-access channel between the users and the CP withthe BSs serving as relay nodes. LetXi denote the signaltransmitted by theith user. The signal received at theith BScan be expressed as

Yi =

L∑

j=1

hijXj + Zi for i = 1, 2, . . . , L,

whereZi ∼ CN (0, σ2i ) is the independent background noise,

and hij denotes the complex channel from thejth user tothe ith BS. In this paper, we assume that the user schedulingis fixed, and perfect CSI is available to all the BSs and tothe centralized processor. Further, it is assumed that eachuser transmits at a fixed power, i.e.,Xi’s are complex-valuedGaussian signals withE|Xi|2 = Pi, for i = 1, . . . , L.

This paper uses a compress-and-forward scheme in whichthe BSs quantize the received signalsY = [Y1, Y2, . . . , YL]

T

into Y = [Y1, Y2, . . . , YL]T using either Wyner-Ziv coding or

single-user compression and transmit the compressed bits tothe CP through noiseless backhaul links. A two-stage succes-sive decoding strategy is employed, where the CP first recoversthe quantized signalsY, and then decodes user messagesX = [X1, X2, . . . , XL]

T based on the quantized signalsY.The successive decoding nature of the proposed scheme over-comes the delay and high computational complexity associated

with joint decoding (e.g., [7], [8]). Letqi = E(Yi − Yi)2 be

the average squared-error distortion betweenYi andYi. In thispaper, the distortion levelqi is referred to as the quantizationnoise level.

B. The VMAC-WZ Scheme

Because of the mutual interference between the neighboringusers, the received signals at the different BSs are statisticallycorrelated. Consequently, Wyner-Ziv compression can be usedto achieve higher compression efficiency and to better utilizethe limited backhaul capacities than per-link single-usercom-pression.

Proposition 1: For the uplink C-RAN model with backhaulsum capacity constraintC as shown in Fig. 1, the rate tuples(R1, R2, . . . , RL) that satisfy the following set of constraintsare achievable using the VMAC-WZ scheme:

∑

i∈S

Ri ≤ log

∣∣H(S)KX(S)H(S)H + Λq + diag(σ2i )∣∣

|Λq + diag(σ2i )|

(1)

such that

log

∣∣HKXHH + Λq + diag(σ2i )∣∣

|Λq|≤ C (2)

for all S ⊆ 1, 2, . . . , L, whereKX(S) = E[X(S)X(S)H ]is the covariance matrix ofX(S), Λq = diag(q1, q2, . . . , qL)is the covariance matrix of the quantization noise, andH(S)denotes the channel matrix fromX(S) to Y.

Proof: This theorem is a generalization of [6, Theorem1], which treats the case of a single transmitter with multiplerelays under individual backhaul capacity constraints. In[6,Theorem 1], it has been shown thatR < I(X; Y) is achievablesubject to

I(Y(S); Y(S)|Y(Sc)) ≤∑

i∈S

Ci, ∀S ⊆ 1, 2, . . . , L (3)

under a product distributionp(y|y) = ΠLi=1p(yi|yi). Note

that under the sum backhaul constraint∑L

i=1 Ci ≤ C, theconstraint (3) simply becomesI(Y; Y) ≤ C. Now, withmultiple users and considering the sum rate over any subsetS, we likewise have

∑

i∈S

Ri ≤ I(X(S); Y|X(Sc)), ∀S ⊆ 1, 2, . . . , L (4)

subject to

I(Y; Y) ≤ C. (5)

Let p(yi|yi) be defined by the test channelYi = Yi + Qi,whereQi ∼ CN (0, qi) is the quantization noise independentof everything else, andqi is the quantization noise level. Theachievable rate region (1) subject to (2) can now be derivedby evaluating the mutual information expressions (4) and (5)assuming complex Gaussian distribution forXi.

4

C. The VMAC-SU Scheme

Although Wyner-Ziv coding represents a better utilizationof the backhaul, it is also complex to implement in practice.In this section, Wyner-Ziv coding is replaced by single-usercompression. We derive the achievable rate region when thecompression process does not take advantage of the statisticalcorrelations between the received signals at different BSs. Inthis case, each BS simply quantizes its received signals usinga vector quantizer.

Proposition 2: For the uplink C-RAN model withL BSsand sum backhaul capacityC shown in Fig. 1, the followingrate tuple(R1, R2, . . . , RL) is achievable using the VMAC-SU scheme:

∑

i∈S

Ri ≤ log

∣∣H(S)KX(S)H(S)H + Λq + diag(σ2i )∣∣

|Λq + diag(σ2i )|

(6)

such that

log

∣∣diag(HKXHH) + Λq + diag(σ2i )∣∣

|Λq|≤ C (7)

for all S ⊆ 1, 2, . . . , L, whereKX(S) = E[X(S)X(S)H ] isthe transmit signal covariance matrix,Λq = diag(q1, . . . , qL)is the covariance matrix of the quantization noise, andH(S)denotes the channel matrix fromX(S) to Y.

Proposition 2 is a straightforward extension of Proposition1, where the rate expression (6) is given by the achievablesum rateI(X(S); Y) and the constraint (7) follows from thebackhaul constraint

∑L

i=1 I(Yi; Yi) ≤ C. The rate expressionimplicitly assumes the successive decoding of the quantizationcodewords first, then the transmitted signals.

III. QUANTIZATION NOISE LEVEL OPTIMIZATION FOR

VMAC-WZ

The achievable rate regions for the VMAC schemes have anintuitive interpretation. The quantization process adds quanti-zation noise to the overall multiple-access channel. Finerquan-tization results in higher overall rate, but also leads to higherbackhaul capacity requirements. To characterize the tradeoffbetween the achievable rate and the backhaul constraint, thissection formulates a weighted sum rate maximization problemover the quantization noise levelsq1, . . . , qL under a sumbackhaul capacity constraint for VMAC-WZ.

A. Problem Formulation

Let µi be the weights representing the priorities associatedwith the mobile users typically determined from upper layerprotocols. Without loss of generality, letµL ≥ µL−1 ≥· · · ≥ µ1 ≥ 0. The boundary of the achievable rate regionfor VMAC-WZ can be attained using a successive decodingapproach with a decoding order from user1 to L. A weightedrate sum maximization problem that characterizes the VMAC-

WZ achievable rate region can be written as:

maxΛq

L∑

i=1

µi log

∣∣∣∑L

j=i PjhjhHj + diag(σ2

i ) + Λq

∣∣∣∣∣∣∑L

j>i PjhjhHj + diag(σ2

i ) + Λq

∣∣∣

s.t. log

∣∣∣∑L

j=1 PjhjhHj + diag(σ2

i ) + Λq

∣∣∣|Λq|

≤ C,

Λq(i, j) = 0, for i 6= j,

Λq(i, i) ≥ 0, (8)

whereΛq(i, j) is the(i, j)th entry of matrixΛq, and the opti-mization is over the quantization noise levelsΛq = diag(qi).

The objective function of (8) is a convex function ofΛq (in-stead of concave). Consequently, finding the global optimumsolution of (8) is challenging. In [13], an algorithm based onthe gradient projection method together with a bisection searchon the dual variable is proposed for a related problem, wherethe quantization noise levels are optimized one after anotherin a coordinated fashion. The above problem formulation isalso related to that in [14] where the quantization noise levelsat the BSs are optimized for sum-rate maximization on a per-BS basis. The advantage of the present formulation is that thequantization noise levels across the BSs are optimized jointly,resulting in better overall performance.

B. Alternating Convex Optimization Approach

This section proposes an alternating convex optimization(ACO) scheme capable of arriving at a stationary point ofthe problem (8). The key observation is that the objectivefunction of (8) is a difference of two concave functions. Theidea is to linearize the second concave function to obtain aconcave lower bound of the original objective function, thensuccessively approximate the optimal solution by optimizingthis lower bound. The ACO scheme is closely related tothe block successive minimization method [19] or minorize-maximization algorithm [20], which can be used to solve abroad class of optimization problems with nonconvex objectivefunctions over a convex set. These optimization techniqueshave also been previously applied for solving related problemsin wireless communications; see [21], [22].

Before presenting the proposed algorithm, we first state thefollowing lemma, which is a direct consequence of Fenchel’sinequality for concave functions.

Lemma 1:For positive definite Hermitian matricesΩ,Σ ∈CL×L,

log |Ω| ≤ log |Σ|+Tr(Σ−1Ω

)− L (9)

with equality if and only ifΩ = Σ.

Applying Lemma 1, we reformulate problem (8) as a double

5

maximization problem:

maxΛq,Σ0

L∑

i=1

(µi − µi−1) log

∣∣∣∣∣∣

L∑

j=i

PjhjhHj + diag(σ2

i ) + Λq

∣∣∣∣∣∣−µL

(log |Σ|+Tr

(Σ−1(diag(σ2

i ) + Λq)))

s.t. log

∣∣∣∑L

i=1 PihihHi + diag(σ2

i ) + Λq

∣∣∣|Λq|

≤ C

Λq(i, j) = 0, for i 6= j,

Λq(i, i) ≥ 0, (10)

whereµL ≥ µL−1 ≥ · · · ≥ µ1 > µ0 = 0.Although the maximization problem (10) is still nonconvex

with respect to(Λq,Σ), the advantage of the reformulation isthat fixing eitherΛq or Σ, problem (10) is a convex optimiza-tion with respect to the other variable. This coordinate-wiseconvexity property enables us to use an iterative coordinateascent algorithm. Specifically, whenΛq is fixed, we solve

minΣ0

log |Σ|+Tr(Σ−1(diag(σ2

i ) + Λq)). (11)

Following Lemma 1, problem (11) has the following closed-form solution:

Σ∗ = diag(σi) + Λq. (12)

If Σ is fixed, problem (10) becomes

maxΛq

L∑

i=1


∣∣∣∣∣∣

L∑

j=i

PjhjhHj + diag(σ2

i ) + Λq

∣∣∣∣∣∣−µLTr

(Σ−1(diag(σ2

i ) + Λq))

s.t. log

∣∣∣∑L

i=1 PihihHi + diag(σ2

i ) + Λq

∣∣∣|Λq|

≤ C,

Λq(i, j) = 0, for i 6= j,

Λq(i, i) ≥ 0. (13)

It is easy to verify that the above problem is a convexoptimization problem, as the objective function is now concavewith respected toΛq. So, it can be solved efficiently withpolynomial complexity. We summarize the ACO algorithmbelow:

Algorithm 1 Alternating Convex Optimization

1: Initialize Λ(0)q = Σ(0) = γI.

2: Fix Σ = Σ(i), solve the convex optimization problem (13)overΛq. SetΛ(i+1)

q to be the optimal point.3: UpdateΣ(i+1) = diag(σ2

i ) + Λ(i+1)q .

4: Repeat Steps 2 and 3, until convergence.

The ACO algorithm yields a nondecreasing sequence of ob-jective values for problem (10). So the algorithm is guaranteedto converge. Moreover, it converges to a stationary point oftheoptimization problem.

Theorem 1:From any initial point(Λ(0)q ,Σ(0)), the limit

point (Λ∗q ,Σ

∗) generated by the alternating convex optimiza-tion algorithm is a stationary point of the weighted sum-ratemaximization problem (8).

The proof of Theorem 1 is similar to that of [21, Proposition1] and is also closely related to the convergence proof ofsuccessive convex approximation algorithm [22]. First, basedon a result on block coordinate descent [23, Corollary 2], itcan be shown that the ACO algorithm converges to a stationarypoint of the double maximization problem (10). Now, supposethat (Λ∗

q ,Σ∗) is a stationary point of (10), we have

Tr(∇Λq

F(Λ∗q ,Σ

∗)H

, (Λq − Λ∗q))≤ 0, ∀ Λq ∈ W , (14)

whereF (Λq,Σ) denotes the objective function of (10). Usingthe same argument as the proof of [21, Proposition 1], we cansubstituteΣ∗ = diag(σi) +Λ∗

q into (14) and verify thatΛ∗q is

also a stationary point of (8).We mention here that although the ACO algorithm is stated

here for the SISO case, it is equally applicable to the MIMOcase, where the BSs are equipped with multiple antennas, andthe optimization is over quantization covariance matrices. Inthe following, we highlight the advantage of our approach ascompared to that of [13], [14].

In [13], a gradient projection method together with a bisec-tion search on the dual variable is used to solve the weightedsum-rate maximization for a related problem. Although thegradient projection approach also converges to a stationarypoint of the problem, it is slower than the proposed ACOalgorithm. This is because the algorithm of [13] relies on per-BS block coordinate gradient descent, which has sublinearconvergence [24], rather than joint optimization across allthe BSs. The gradient-type approach used in [13] is alsotypically much slower than optimization techniques which usesecond-order Hessian information (e.g. Newton’s method) thatcan be applied to convex problems. In [14], the optimizationof the quantization noise covariance matrices for sum-ratemaximization is solved on a per-BS basis in a greedy fashion,one BS at a time. This approach in general does not convergeto a local optimal solution, (as has already been pointedout in [14]). It cannot be applied to the weighted sum-ratemaximization problem considered in this paper. In contrast,the ACO algorithm presented here is capable of solving theoptimal quantization noise covariance matrices across alltheBSs jointly, and the convergence to the stationary point isguaranteed.

C. Optimal Quantization Noise Level at High SQNR

Although locally optimal quantization noise level can beeffectively found using the proposed ACO algorithm for anyfixed user schedule, user priority, and channel condition, theimplementation of ACO in practical systems can be compu-tationally intensive, especially in a fast-fading environment orwhen the scheduled users in the time-frequency slots changefrequently. In this section, we aim to understand the structureof the optimal solution by deriving the optimal quantizationnoise level in the high SQNR regime. The main result ofthis section is that setting the quantization noise level tobeproportional to the background noise level is approximatelyoptimal for maximizing the overall sum rate. This leads toan efficient way for setting the quantization noise levels inpractice.

6

Consider the sum-rate maximization problem:

max log

∣∣HKXHH + diag(σ2i ) + Λq

∣∣|diag(σ2

i ) + Λq|

s.t. log

∣∣HKXHH + diag(σ2i ) + Λq

∣∣|Λq|

≤ C

Λq(i, j) = 0, for i 6= j

Λq(i, i) ≥ 0. (15)

This optimization problem is nonconvex, but its Karush-Kuhn-Tucker (KKT) condition still gives a necessary condition foroptimality. To derive the KKT condition, form the Lagrangian

L(Λq, λ,Ψ) = (1− λ) log∣∣HKXHH + diag(σ2

i ) + Λq

∣∣− log

∣∣diag(σ2i ) + Λq

∣∣+ λ log |Λq|+ Tr(ΨΛq) (16)

whereΨ is a matrix whose diagonal entries are zeros andthe off-diagonal entries are the dual variables associatedtheconstraintΛq(i, j) = 0 for i 6= j, andλ is the Lagrangian dualvariable associated with the backhaul sum-capacity constraint.

Setting∂L/∂Λq to zero, we obtain the optimality condition

(1−λ)(HKXHH+diag(σ2i )+Λq)

−1−(diag(σ2i )+Λq)

−1

+ λΛ−1q +Ψ = 0 (17)

Recall thatΨ has zeros on the diagonal, but can have arbitraryoff-diagonal entries. Thus, the above optimality condition canbe simplified as

(1− λ)diag(HKXHH + diag(σ2i ) + Λq)

−1

− (diag(σ2i ) + Λq)

−1 + λΛ−1q = 0 (18)

First, it is easy to verify that the optimality condition canonly be satisfied if0 ≤ λ < 1. Second, sinceΛq+diag(σ2

i ) isthe combined quantization and background noise, if the overallsystem is to operate at reasonably high spectral efficiency,wemust have1 diag(HKXHH) ≫ diag(σ2

i ) + Λq. Under thishigh SQNR condition, we have

diag(HKXHH + diag(σi) + Λq)−1 ≪ (diag(σ2

i ) + Λq)−1

in which case the optimality condition becomes

qi ≈λ

1− λσ2i (19)

where λ ∈ [0, 1) is chosen to satisfy the backhaul sum-capacity constraint. Thus we see that under high SQNR, theoptimal quantization noise level should be proportional tothebackground noise level. Note thatλ = 0 corresponds to theinfinite backhaul case whereqi = 0. As λ increases, the sumbackhaul capacity becomes increasingly constrained, and theoptimal quantization noise levelqi also increases accordingly.

1 Here, “≫” denotes component-wise comparison on the diagonal entries.

D. Sum Capacity to Within a Constant Gap

We now further justify the setting of the quantization noiselevel to be proportional to the background noise level byshowing that this choice in fact achieves the sum capacityof the uplink C-RAN model with sum backhaul capacityconstraint to within a constant gap. The gap depends on thenumber of BSs in the network but is independent of thechannel matrix and the signal-to-noise ratios (SNRs).

Theorem 2:For the uplink C-RAN model with a sumbackhaul capacityC as shown in Fig. 1, the VMAC-WZscheme with the quantization noise levels set to be proportionalto the background noise levels achieves a sum capacity towithin one bit per BS per channel use.

Proof: See Appendix A.The proof of above theorem depends on a comparison of

achievable rate with a cut-set outer bound. The basic idea istoset the quantization noise levels to be at the background noise

levels ifC is large, (specifically,C ≥ log|HKXH

H+2diag(σ2

i )||diag(σ2

i)|

as in the proof), resulting in at most1 bit gap per channel useper BS. WhenC is small, scaling the quantization noise levelby a constant turns out to maintain the constant-gap optimality.

This result is reminiscent of the more general constant-gapresult for arbitrary multicast relay network [9], [10], butthisresult is both more specific, as it only applies to the sum-capacity constrained backhaul case, and also more practicallyuseful, as it assumes successive decoding of quantizationcodeword first then user messages, rather than joint decoding.

A similar constant-gap result can be obtained in the casewhere both transmitters and receivers are equipped with mul-tiple antennas. For example, considering the scenario whereGusers withM transmit antennas each send independent mes-sages toL BSs withN receive antennas each. It can be shownthat the constant gap for sum capacity isminGM,NL bitsper channel use. In particular, whenG = NL, i.e., when thedegree of freedom in the system is fully utilized, the constant-gap result becomes one bit per BS antenna per channel use.

E. Efficient Algorithm for Setting Quantization Noise Level

The main observation in the previous section is that settingthe quantization noise levels at different BSs to be proportionalto the background noise levels is near sum-rate optimal underhigh SQNR and from a constant-gap-to-capacity perspective.This holds regardless of the transmit power, the channel ma-trix, and the user schedule, which is especially advantageousfor practical implementation as no adaptation to the channelcondition is needed.

In the following, we propose a simple algorithm for settingthe quantization noise level to beqi = ασ2

i for some appropri-ateα. Note that with this setting ofqi, the backhaul constraintbecomes:

CWZ(α) , log

∣∣∣∑L

j=1 PjhjhHj + (1 + α)diag(σ2

i )∣∣∣

|αdiag(σ2i )|

≤ C.

(20)Since the backhaul constraint should be satisfied with equalityand sinceCWZ (α) is monotonic inα, a simple bisectionsearch can be used to find the suitableα. The algorithm

7

is summarized below as Algorithm 2. As simulation resultslater in the paper show, Algorithm 2 performs very closeto the optimized scheme (Algorithm 1) for practical channelscenarios.

Algorithm 2 Approximate Algorithm for VMAC-WZ1: Setα = 1.2: While CWZ (α) > C, Setα = 2α; End3: Setαmax = α andαmin = 0.4: Use bisection in[αmin, αmax] to solveCWZ(α) = C.5: Setqi = ασi.

IV. OPTIMAL BACKHAUL ALLOCATION FOR VMAC-SU

A. Problem Formulation

We now turn to the VMAC-SU scheme and consider theweighted sum-rate maximization problem under a sum back-haul constraint for the more practical single-user compressionscheme. The optimization problem can be stated as follows:

maxΛq

L∑

i=1

µi log

∣∣∣∑L

j=i PjhjhHj + diag(σ2

i ) + Λq

∣∣∣∣∣∣∑L

j>i PjhjhHj + diag(σ2

i ) + Λq

∣∣∣

s.t.

L∑

i=1

log

(1 +

∑L

j=1 Pj |hij |2 + σ2i

qi

)≤ C

Λq(i, j) = 0, for i 6= j,

Λq(i, i) ≥ 0. (21)

As mentioned earlier, the objective function in the aboveis convex in qi (instead of concave), which is not easy tomaximize. But the ACO algorithm proposed earlier can stillbe used here to find locally optimalqi’s. However for VMAC-SU, because the compression at each BS is independent, it ispossible to re-parameterize the problem in term of the ratesallocated to the backhaul links. It is instructive to work withsuch a reformulation in order to obtain system design insight.Introduce the new variables

Ci = log

(1 +

∑Lj=1 Pj |hij |2 + σ2

i

qi

). (22)

Let γi be the combined quantization and background noise,i.e., γi = σ2

i + qi. Then,

γi =

∑Lj=1 Pj |hij |2 + σ2

i 2Ci

2Ci − 1. (23)

Further, defineΥ = diag(1/γi). By a variable substitution, itis straightforward to establish that the optimization problem(21) is equivalent to the following:

max

L∑

i=1


∣∣∣∣∣∣Υ

L∑

j=i

PjhjhHj + I

∣∣∣∣∣∣

s.t.L∑

i=1

Ci ≤ C, Ci ≥ 0, i = 1, . . . , L (24)

where, without loss of generality, it has been assumedµL ≥· · · ≥ µ1 > µ0 = 0. The above problem is easier to

solve than (21), because the feasible set of the problem isa polyhedron with only linear constraints. For example, it ispossible to dualize with respect to the sum backhaul constraint,then numerically find a local optimum of the Lagrangian. Abisection on the dual variable can then be used in an outerloop to solve (24).

B. Optimal Quantization Noise Level at High SQNR

For the VMAC-WZ scheme under high SQNR assumption,settingqi = ασ2

i is approximately optimal for maximizing theoverall sum rate. This section establishes a similar resultforthe VMAC-SU case. We first introduce Lagrange multipliersνi ≥ 0 for the positivity constraintsCi ≥ 0, andβ ≥ 0 for thebackhaul sum-capacity constraint

∑L

i=1 Ci ≤ C, we obtainthe following KKT condition

Tr

[HKXHH

(ΥHKXHH + I

)−1 ∂Υ

∂Ci

]−β+νi = 0. (25)

Note that γi is the combined quantization and backgroundnoise. So, under the high SQNR assumption, where SNR≫ 1and Ci ≫ 1, we must havediag(HKXHH) ≫ diag (γi).Thus HKXHH

(ΥHKXHH + I

)−1≈ Υ−1. After some

manipulations, the optimality condition now becomes∑L

j=1 Pj |hij |2 + σ2

i∑Lj=1 Pj |hij |2 + σ2

i 2Ci

− β + νi ≈ 0 (26)

where we also use the approximation2Ci − 1 ≈ 2Ci . Notethat νi = 0 wheneverCi > 0. Solving (26) together with∑L

i=1 Ci = C yields the following approximately optimalbackhaul rate allocation:

Ci ≈ log

(1− β

βSNRi +

1

β

)(27)

whereSNRi = (∑L

j=1 Pj |hij |2)/σ2i andβ is chosen such that∑L

i=1 Ci = C. The corresponding quantization noise level isgiven by

qi ≈β

1− βσ2i . (28)

We point out here that the same result can also be derivedfrom the KKT condition of (21).

The above result shows that setting the quantization noiselevel to be proportional to the background noise level is nearoptimal for maximizing the sum rate for VMAC-SU at highSQNR. This is similar to the VMAC-WZ case. Intuitively, inthe VMAC schemes the intercell interference is completelynulled by multicell decoding. The achievable sum rate is onlylimited by the combined quantization noise and backgroundnoise. Thus, it is reasonable that the optimal quantizationnoiselevels only depend on the background noise levels.

C. Sum Capacity of Diagonally Dominant Channels

This section provides further justification for choosing thequantization noise level to be proportional to the backgroundnoise level by showing that doing so achieves the sum ca-pacity of the VMAC model to within a constant gap whenthe received signal covariance matrix satisfies a diagonally

8

dominant channel criterion. The received signal covariancematrix is defined asE[YYH ] = HKXHH + diag(σ2

i ). It isoften diagonally dominant, because the path losses from theremote users to the BSs are distance dependent, and typicallyeach user is associated with its strongest BS. In the following,we define a diagonally dominant condition for matrices, andstate a constant-gap result for sum capacity for the VMAC-SUscheme under a sum backhaul constraint.

Definition 1: For a fixed constantκ > 1, an×n matrix Ψis said to beκ-strictly diagonally dominant if

|Ψ(i, i)| ≥ κ

n∑

j 6=i

|Ψ(i, j)| for all i = 1, . . . , n,

whereΨ(i, j) is the(i, j)-th entry of matrixΨ.Theorem 3:For the uplink C-RAN model with a sum back-

haul capacityC as shown in Fig. 1, if the received covariancematrixHKXHH+diag(σ2

i ) is κ-strictly diagonally dominantfor κ > 1, then the VMAC-SU scheme achieves the sumcapacity of the uplink CRAN model to within

(1 + log κ

κ−1

)

bits per BS per channel use.Proof: See Appendix B.

We note that the above result can be further strengthenedwhenC is large. In this case, setting the quantization noiselevels to be at the background noise levels results in at most1bit gap per channel use per BS to sum capacity. It is not hard tofurther verify that, in this case, the VMAC-SU scheme is actu-ally approximately optimal for the entire capacity region of theuplink C-RAN model. Analogous to Wyner-Ziv compression,a similar constant-gap result for single-user compressioncanalso be obtained in the case where both users and BSs areequipped with multiple antennas.

D. Backhaul Allocation for Heterogeneous Networks

The fact that setting the quantization noise levels to beproportional to the background noise levels is approximatelyoptimal gives rise to an efficient algorithm for allocatingcapacities across the backhaul links. This section describesan approach similar to the corresponding algorithm for theVMAC-WZ case. In addition, we further generalize to the caseof heterogeneous network with multiple tiers of BSs.

Consider a multi-tier heterogeneous network consisting ofnot only macro BSs, but also pico-BSs, coordinated togetherin a C-RAN architecture. The macro- and pico-BSs typicallyhave very different backhaul capacities, so they may be subjectto different backhaul constraints. LetCm be the sum backhaulcapacity constraint across the macro-BSs, andCp be thebackhaul constraint for pico-BSs. Assuming a VMAC-SUimplementation, the backhaul constraints can be expressedas:

∑i∈Sm

log

(1 +

∑Lj=1

Pj |hij|2+σ2

i

qi

)≤ Cm (29)

∑i∈Sp

log

(1 +

∑Lj=1

Pj |hij |2+σ2

i

qi

)≤ Cp (30)

where Sm and Sm are the sets of macro- and pico-BSs,respectively.

It can be shown that for multi-tier networks, it is also nearoptimal to set the quantization noise levels to be proportional

to the background noise levels under high SQNR. However,different tiers may have different proportionality constants.Since the quantization noise level (or equivalently the backhaulcapacity) for each BS may be set independently without af-fecting other BSs for VMAC-SU, a simple bisection algorithmcan be used to optimize the quantization noise level (orequivalently the backhaul capacity) in each tier independently.

Let

CSU (β) =∑

i∈S

log

(1− β

βSNRi +

1

β

)(31)

be the sum backhaul capacity across a particular tier (whereScan beSm for macro-BSs orSp for pico-BSs). The bisectionalgorithm described in Algorithm 3 can run simultaneously ineach tier.

Algorithm 3 Approximate Algorithm for VMAC-SU1: Setβmin = 0, βmax = 1.2: Use bisection in[βmin, βmax] to solveCSU (β) = C.

3: Setqi =β

1−βσ2i , andCi = log

(1−ββ

SNRi +1β

).

We point out here that practical heterogeneous networkmay have other types of backhaul structure. For instance, inpractical implementation the pico-BSs may not have directbackhaul links to the CP, but may connect to the macro-BSsfirst then to the CP. In this case, the backhaul constraints canbe formulated as

∑i∈Sm

log

(1 +

∑Lj=1

Pj |hij |2+σ2

i

qi

)≤ Cm

∑i∈Sp

log

(1 +

∑Lj=1

Pj |hij |2+σ2

i

qi

)≤ Cp

Cm + Cp ≤ C, Cp ≤ Cp

(32)

where the optimization variables areqi, Cm and Cp. HereCp is the sum-capacity constraint for the backhaul linksconnecting pico-BSs to the macro-BSs, andC is the total sumbackhaul constraint for both pico-BSs and maco-BSs. In thiscase, the backhaul constraints for maco-BSs and pico-BSs arecoupled together. However, Algorithm 3 can be still be helpfulin finding the approximately optimal quantization noise levels.Specifically, for each fixed pair ofCm and Cp, Algorithm 3can be used to find theqi’s for the macro-BSs and the pico-BSs respectively. The problem is now simplified to finding theoptimal partition ofC betweenCm and Cp.

V. SIMULATIONS

A. Multicell Network

In this section, the performances of the VMAC-WZ andVMAC-SU schemes with different quantization noise leveloptimization strategies are evaluated in a wireless cellularnetwork setup with19 cells wrapped around,3 sectors per cell,and 20 users randomly located in each sector. The central7BSs (i.e.,21 sectors) form a C-RAN cooperation cluster, whereeach BS is connected to the CP with noiseless backhaul linkwith a sum capacity constraint across the7 BSs. The users areassociated with the sector with the strongest channel. Round-robin user scheduling is used on a per-sector basis. Perfect

9

TABLE IMULTICELL NETWORK SYSTEM PARAMETERS

Cellular Layout Hexagonal,19-cell, 3 sectors/cellBS-to-BS Distance 500 mFrequency Reuse 1

Channel Bandwidth 10 MHzNumber of Users per Sector 20

Total Number of Users 420User Transmit Power 23 dBm

Antenna Gain 14 dBiBackground Noise −169 dBm/Hz

Noise Figure 7 dBTx/Rx Antenna No. 1

Distance-dependent Path Loss 128.1 + 37.6 log10(d)Log-normal Shadowing 8 dB standard deviation

Shadow Fading Correlation 0.5Cluster Size 7 cells (21 sectors)

Scheduling Strategy Round-robin

channel estimation is assumed, and the CSI is made availableto all BSs and to the CP. In the simulation, fixed transmitpower of 23dBm is used at all the mobile users. Variousalgorithms are run on fixed set of channels. Detailed systemparameters are outlined in Table I.

In the simulation, weighted rate-sum maximization is per-formed over the quantization noise levels, with weights equalto the reciprocal of the exponentially updated long-term aver-age rate. In the implementation of VMAC schemes, successiveinterference cancelation (SIC) decoding is used at the CP. Thedecoding order of the users is determined by their weights, i.e.,the user with high weight is decoded last. The baseline systemis the conventional cellular networks without joint multicellprocessing at the CP. Cumulative distribution function (CDF)of the user rates is plotted in order to visualize the performanceof various schemes.

Fig. 2 compares the performance of the baseline system withthe VMAC-WZ scheme under the sum backhaul capacities of120Mbps per macro-cell (40Mbps per sector) and270Mbpsper cell (90Mbps per sector). The VMAC-WZ scheme isimplemented with two choices of quantization noise levels:theapproximately optimalqi proportional to the background noiselevel as given by Algorithm 2 (labeled as “appro. opt. q”) andthe optimalqi given by Algorithm 1 (labeled as “optimizedq”). It is shown that the VMAC-WZ schemes significantlyoutperform the baseline system. The figure also shows thatsetting qi to be proportional to the background noise levelis indeed approximately optimal, especially whenC is large.This confirms our earlier theoretical analysis on the approxi-mately optimalqi.

The VMAC schemes considered in this paper is superiorto the per-BS SIC scheme considered in [12]. To illustratethis point, Fig. 3 compares the performance of the VMAC-WZ scheme under the approximately optimalqi with the per-BS SIC scheme of [12] (labeled as “Per-BS SIC”). For faircomparison, we run the simulation over the users in the7-cellcluster only, and ignore the out-of-cluster interference,whichis the case considered in [12]. The figure shows that significantgain can be obtained by the VMAC-WZ scheme over the per-BS successive cancellation scheme.

Fig. 4 shows the CDF curves of user rates for the VMAC-

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu

mu

lative

Dis

trib

utio

n F

un

ctio

n (

CD

F)

Uplink User Rates (Mbps)

Baseline: No central processor

Approx. opt. q, C=120 Mbps

Optimized q, C=120 Mbps



Fig. 2. Cumulative distribution of user rates with the VMAC-WZ scheme

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu

mu

lative

Dis

trib

utio

n F

un

ctio

n (

CD

F)



Per-BS SIC with opt. q, C=180 Mbps

VMAC-WZ with approx. q, C=180 Mbps

Per-BS SIC with opt. q, C=270 Mbps

VMAC-WZ with approx. q, C=270 Mbps

Fig. 3. Performance comparison of the VMAC-WZ scheme with the per-BSinterference cancellation scheme of [12].

SU scheme with three choices of quantization noise levels: thequantization noise levels given by allocating the backhaulca-pacity equally across the BSs (labeled as “uniform backhaul”),the approximately optimalqi proportional to the backgroundnoise as given by Algorithm 3 (labeled as “approx. opt. q”),and the optimalqi derived from the backhaul capacity allo-cation formulation of the problem (labeled as “optimized q”).It can be seen that VMAC with single-user compression alsosignificantly improves the performance of baseline system andthat the approximately optimalqi is near optimal, especiallywhenC is large. The figure also shows that allocating backhaulcapacity uniformly across the BSs is strictly suboptimal.

To further compare the performance of the VMAC-SUscheme with different choices of quantization noise levels,Fig. 5 plots the average per-cell sum rate of the baselineand the VMAC-SU schemes as a function of the backhaulcapacity. The figure clearly shows the advantage of optimizingthe quantization noise levels (or equivalently the allocation ofbackhaul capacities). For example, to achieve80Mbps per-

10

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0C

um

ula

tive

Dis

trib

utio

n F

un

ctio

n (

CD

F)



Uniform backhaul, C=120 Mbps



Uniform backhaul, C=270 Mbps



Fig. 4. Cumulative distribution of user rates with the VMAC-SU scheme

100 150 200 250 300 350 40030

40

50

60

70

80

90

100

110

120


VMAC-SU with uniform backhaul

VMAC-SU with approx. opt. q

VMAC-SU with optimized q Pe

r-ce

ll S

um

Ra

te (

Mb

ps)

Average Per-cell Backhaul (Mbps)

Fig. 5. Per-cell sum rate vs. average per-cell backhaul capacity of the VMAC-SU scheme.

cell sum rate, we need200Mbps sum backhaul if backhaulcapacities are allocated uniformly,170Mbps sum backhaulif qi is chosen to be proportional to the background noise,and 150Mbps sum backhaul ifqi is optimized. Thus, theoptimization of the quantization noise level can save up to25% in backhaul capacity.

Further, it can be seen from Fig. 5 that under infinitesum backhaul, the achieved per-cell sum rate saturates andapproaches about115Mbps for this cellular setting. But whenthe quantization noise level is optimized, a finite sum backhaulcapacity at about200Mbps is already sufficient to achieveabout 100Mbps user sum rate, which is90% of the fullbenefit of uplink network MIMO. Note that the performancegap between the approximately optimalqi and the optimalqi becomes smaller as the sum backhaul capacity increases,confirming the approximate optimality ofqi = ασ2

i in the highSQNR regime.

Fig. 6 compares the performance of Wyner-Ziv coding andsingle-user compression for the VMAC scheme. It is observed

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


SU with optimized q, C=120 Mbps

WZ with optimized q, C=120 Mbps





Cu

mu

lative

Dis

trib

utio

n F

un

ctio

n (

CD

F)


Fig. 6. Comparison of the VMAC-SU and VMAC-WZ schemes

TABLE IIHETEROGENEOUSNETWORK CHANNEL PARAMETERS

Cellular Layout Hexagonal, wrapped aroundBS-to-BS Distance 500 m

Number of Macro Cells 7 cells, 3 sectors/cellNumber of Pico Cells 3 pico cells per macro sector

Frequency Reuse 1Channel Bandwidth 10 MHz

Number of Users perMacro Sector 20

User Transmit Power 23 dBmAntenna Gain 14 dBi

Background Noise −169 dBm/HzNoise Figure 7 dB

Pico BS Antenna Pattern Omni-directionalTx/Rx Antenna No. 1

Path Loss Macro to User 128.1 + 37.6 log10(d)Path Loss Pico to User 140.7 + 36.7 log10(d)

8 dB standard deviationLog-normal Shadowing for macro-user link;

4 dB for pico-user linkShadow Fading Correlation 0.5

Cluster Size 1 macro cell and9 pico cellsMin. Dist. between BSs 75 m

Scheduling Strategy Round-robin

that Wyner-Ziv coding is superior to single-use compression.However, as the sum backhaul capacity becomes larger, thegain due to Wyner-Ziv coding diminishes.

B. Multi-Tier Heterogeneous Network

The performance of the VMAC-SU scheme is further eval-uated for a two-tier heterogeneous network with7 macro-cellswrapped around,3 sectors per cell,3 pico BSs randomly lo-cated in each sector, and20 mobile users per macro-cell sector.The cellular topology is shown in Fig. 7. Each user establishesconnection with the macro/pico BS with the highest receivedSNR. Note that the number of users in each pico/macro-cell isnot fixed. On average there are8 users per macro-cell sectorand 4 users per pico-cell. In this network, every macro-cellforms a C-RAN cluster, consisting of3 macro-sectors and9 pico-cells. The macro BSs and pico BSs are subject todifferent sum backhaul capacity constraints. Specifically, the

11

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Macro Base Station

Macro Mobile User

Pico Base Station

Pico Mobile User

Fig. 7. A picocell network topology with7 cells, 3 sectors per cell, and3pico base-stations per sector placed randomly.

0 2 4 6 8 10 12 14 160.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu

mu

lative

Dis

trib

utio

n F

un

ctio

n (

CD

F)


Picocell network: No central processor

VMAV-SU with uniform backhaul

VMAV-SU with approx. opt. q

VMAV-SU with optimized q

Fig. 8. Cumulative distribution of user rates in the picocell network where the3 macro-BSs and 9 pico-BSs within each 3-sector macrocell form a cluster.The VMAC-SU scheme is applied and the sum backhaul constraints for macroand pico BSs are189Mpbs and81Mpbs per cluster, respectively.

sum backhaul capacity is set to be189Mbps for the3 macro-BSs and81Mbps for the9 pico BSs. Perfect CSI is madeavailable to all the BSs and to the CP. System parameters areoutlined in Table II.

Fig. 8 shows the CDF plots of user rates achieved by thebaseline scheme and the VMAC-SU scheme. It is clear that theC-RAN architecture significantly improves upon the baseline,more than doubling the50-percentile rate. The optimization ofthe quantization noise level is important, as a naive uniformbackhaul allocation only achieves half of the potential gainfor C-RAN. Finally, setting the quantization noise level tobeproportional to the background noise is indeed approximatelyoptimal. In this multi-tier heterogeneous network case, theproportionality constant is set independently for each tier usingAlgorithm 3.

VI. CONCLUSION

This paper studies an uplink C-RAN model where theBSs within a cooperation cluster are connected to a cloud-

computing based CP through noiseless backhaul links oflimited sum capacity. We employ two VMAC schemes wherethe BSs use either Wyner-Ziv compression or single-usercompression to quantize the received signals and send thecompressed bits to the CP. At the CP, quantization codewordsare first decoded; subsequently the user messages are decodedas if the users form a virtual multiple-access channel.

The main findings of the paper are concerned with efficientoptimization of the quantization noise levels for both VMAC-WZ and VMAC-SU. We propose an alternating optimizationalgorithm for VMAC-WZ and a backhaul capacity allocationformulation for VMAC-SU. More importantly, it is observedthat setting the quantization noise levels to be proportional tothe background noise levels is approximately optimal. Thisleads to efficient algorithms for optimizing the quantizationnoise levels, or equivalently, for allocating the backhaulca-pacities.

From an analytic point of view, this paper shows that settingquantization noise levels to be proportional to the backgroundnoise levels is near optimal for maximizing the sum rate whenthe system operates in the high SQNR regime. With such achoice of quantization noise levels, the VMAC-WZ schemecan achieve the sum capacity of the uplink C-RAN model towithin a constant gap. A similar constant-gap result is alsoobtained for VMAC-SU under a diagonally dominant channelcondition. From a numerical perspective, simulation resultsconfirm that the proposed VMAC schemes can significantlyimprove the performance of wireless cellular systems. Theimprovement is maximized with optimized quantization noiselevels or equivalently optimized backhaul capacity allocations.The near optimal choice of quantization noise levels indeedperforms very close to the optimal one over the SQNR regionof practical interest.

APPENDIX APROOF OFTHEOREM 2

The idea is to chooseqi = ασ2i , i = 1, 2, . . . , L where

α > 0 is an appropriately chosen constant, then compare theachievable rate of VMAC-WZ with the following cut-set likesum-capacity upper bound [7]

C = min

log

∣∣HKXHH + diag(σ2i )∣∣

|diag(σ2i )|

, C

(33)

where the first term is the cut from the users to the BSs, andthe second term is the cut across the backhaul links.

We choose the quantization noise levelα depending onC

as follows: WhenC ≥ log|HKXH

H+2diag(σ2

i )||diag(σ2

i)|

, we choose

α = 1, i.e., the quantization noise levels are set to be at thebackground noise levels. Sinceα = 1, it can be verified that

I(Y; Y) = log

∣∣HKXHH + 2diag(σ2i )∣∣

|diag(σ2i )|

. (34)

Thus, we haveC ≥ I(Y; Y). This implies that the sumbackhaul constraint (2) is satisfied. Therefore, the sum rate

Rsum = I(X; Y) = log


|2diag(σ2i )|

(35)

12

is achievable. In this case, the gap betweenC andRsum canbe bounded by

C −Rsum ≤ log


|diag(σ2i )|

− log


|2diag(σ2i )|

< L.

WhenC < log|HKXH

H+2diag(σ2

i )||diag(σ2

i)|

, we chooseα such that

I(Y; Y) = C. First, note that for such a choice ofα the sumrateRsum = I(X; Y) is achievable. Next, observe that

I(Y; Y) = log

∣∣HKXHH + diag(σ2i ) + αdiag(σ2

i )∣∣

|αdiag(σ2i )|

(36)

is a monotonically decreasing function ofα. Since C =

I(Y; Y) < log|HKXH

H+2diag(σ2

i )||diag(σ2

i)|

, we haveα > 1. Now,

we useC = I(Y; Y) as an upper bound. The gap betweenCandRsum can be bounded by

C −Rsum ≤ I(Y; Y)− I(X; Y)

= log

∣∣HKXHH + (1 + α)diag(σ2i )∣∣

|αdiag(σ2i )|

− log


|(1 + α)diag(σ2i )|

= L log

(1 +

1

α

)< L

where the last inequality follows from the fact thatα > 1.Combining the two cases, we see that the gap to the

sum capacity for the VMAC-WZ scheme with appropriatelychosen quantization noise levels (which are proportional to thebackground noise levels) is always less than1 bit per BS perchannel use.

APPENDIX BPROOF OFTHEOREM 3

Lemma 2:For fixedκ > 1, suppose that an× n matrix Ψis κ-strictly diagonally dominant, then

|Ψ| ≥

(1−

1

κ

)n n∏

i=1

|Ψ(i, i)|. (37)

Proof: The proof follows from the lower bound given in[25], which shows that ifΨ is strictly diagonally dominant, i.e.

|Ψ(i, i)| >n∑

j 6=i

|Ψ(i, j)| for i = 1, . . . , n, then the determinant

of Ψ can be bounded from below as follows,

|Ψ| ≥n∏

i=1

|Ψ(i, i)| −

n∑

j 6=i

|Ψ(i, j)|

. (38)

Under the condition thatΨ is κ-strictly diagonally dominant,i.e.∑n

j 6=i |Ψ(i, j)| ≤ |Ψ(i,i)|κ

we further bound|Ψ| by

|Ψ| ≥n∏

i=1

(|Ψ(i, i)| −

|Ψ(i, i)|

κ

)

=

(1−

1

κ

)n n∏

i=1

|Ψ(i, i)|, (39)

which completes the proof.We now prove Theorem 3. The proof uses the same tech-

nique as in that of Theorem 2. We first choose the quantizationnoise levelsqi = ασ2

i , i = 1, 2, . . . , L, whereα > 0 is aconstant depending onC, then compare the achievable rate ofthe VMAC-SU scheme with the following cut-set like upperbound [7]

C = min

log


|diag(σ2i )|

, C

. (40)

We consider two different cases as follows: whenC ≥

log|diag(HKXH

H)+2diag(σ2

i )||diag(σ2

i)|

, i.e. the sum backhaul capacity

is large enough to support the choice ofqi = σ2i , we choose

α = 1. In this case, the gap betweenC and Rsum can bebounded by

C −Rsum ≤ log


|diag(σ2i )|

− log


|2diag(σ2i )|

< L.

WhenC < log|diag(HKXH

H )+2diag(σ2

i )||diag(σ2

i)|

, we chooseα so that∑L

i=1 I(Yi; Yi) = C. First, notice that

L∑

i=1

I(Yi; Yi) = log

∣∣diag(HKXHH) + (1 + α)diag(σ2i )∣∣

|αdiag(σ2i )|

is a monotonically decreasing function ofα. Since C =∑L

i=1 I(Yi; Yi) < log|diag(HKXH

H )+2diag(σ2

i )||diag(σ2

i)|

, we haveα >

1. Now, we useC =∑L

i=1 I(Yi; Yi) as an upper bound. LetΩ = HKXHH + (1 + α)diag(σ2

i ) and note thatΩ(i, i) ≥ 0.The gap betweenC andRsum is bounded by

C −Rsum ≤L∑

i=1

I(Yi; Yi)− I(X; Y)

= log

∣∣diag(HKXHH) + (1 + α)diag(σ2i )∣∣

|αdiag(σ2i )|

− log


|(1 + α)diag(σ2i )|

= log

(1 +

1

α

)L

L∏i=1

Ω(i, i)

|Ω|

.

Since matrixHKXHH + diag(σ2i ) is κ-strictly diagonally

dominant,Ω is alsoκ-strictly diagonally dominant. Followingthe result of Lemma 2, we further bound the gap as follows,

C −Rsum ≤ L log

(1 +

1

α

)+

L∑

i=1

logκ

κ− 1

< L

(1 + log

κ

κ− 1

),

where the last inequality follows from the fact thatα > 1.Combining the two cases, we see that the gap to sum

capacity for the VMAC-SU scheme with quantization noise

13

levels proportional to the background noise levels is alwaysless than1 + log κ

κ−1 per BS per channel use.

ACKNOWLEDGMENT

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

REFERENCES

[1] Y. Zhou and W. Yu, “Approximate bounds for limited backhaul uplinkmulticell processing with single-user compression,” inProc. IEEECanadian Workshop on Inf. Theory (CWIT), Jun. 2013, pp. 113–116.

[2] Y. Zhou, W. Yu, and D. Toumpakaris, “Uplink multi-cell processing:Approximate sum capacity under a sum backhaul constraint,”in Proc.IEEE Inf. Theory Workshop (ITW), Sep. 2013, pp. 569–573.

[3] 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.

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

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

[6] 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.

[7] 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.

[8] A. Sanderovich, S. Shamai, and Y. Steinberg, “Distributed MIMOreceiver–Achievable rates and upper bounds,”IEEE Trans. Inf. Theory,vol. 55, no. 10, pp. 4419–4438, Oct. 2009.

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

[10] 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.

[11] M. H. Yassaee and M. R. Aref, “Slepian–Wolf coding over cooperativerelay networks,”IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3462–3482,Jun. 2011.

[12] L. Zhou and W. Yu, “Uplink multicell processing with limited backhaulvia per-base-station successive interference cancellation,” IEEE J. Sel.Areas Commun., vol. 31, no. 10, pp. 1981–1993, Oct. 2013.

[13] 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.

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

[15] 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.

[16] R. Karasik, O. Simeone, and S. Shamai, “Robust uplinkcommunications over fading channels with variable backhaulconnectivity,” IEEE Trans. Wireless Commun., 2013. [Online]. Available:http://arxiv.org/abs/1301.6938

[17] B. Nazer, A. Sanderovich, M. Gastpar, and S. Shamai, “Structuredsuperposition for backhaul constrained cellular uplink,”in Proc. IEEEInt. Symp. Inf. Theory (ISIT), Jun. 2009, pp. 1530–1534.

[18] S.-N. Hong and G. Caire, “Compute-and-forward strategies for coop-erative distributed antenna systems,”IEEE Trans. Inf. Theory, vol. 59,no. 9, pp. 5227–5243, Sep. 2013.

[19] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergenceanalysis of block successive minimization methods for nonsmoothoptimization,”SIAM J. Optim., vol. 23, no. 2, pp. 1126–1153, Feb. 2013.

[20] D. R. Hunter and K. Lange, “A tutorial on MM algorithms,”Am. Stat.,vol. 58, no. 1, pp. 30–37, Jan. 2004.

[21] Q. Li, M. Hong, H.-T. Wai, Y.-F. Liu, W.-K. Ma, and Z.-Q. Luo,“Transmit solutions for MIMO wiretap channels using alternating opti-mization,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1714–1727,2013.

[22] M. Hong, Q. Li, and Y.-F. Liu, “Decomposition by successive convexapproximation: A unifying approach for linear transceiverdesign ininterfering heterogeneous networks,”submitted to IEEE Trans. SignalProcess., 2012. [Online]. Available: http://arxiv.org/abs/1210.1507

[23] L. Grippo and M. Sciandrone, “On the convergence of the blocknonlinear Gauss–Seidel method under convex constraints,”Oper. Res.Lett., vol. 26, no. 3, pp. 127–136, Mar. 2000.

[24] A. Beck and L. Tetruashvili, “On the convergence of block coordinatedescent type methods,”SIAM J. Optim., vol. 23, no. 4, pp. 2037–2060,Apr. 2013.

[25] A. M. Ostrowski, “Note on bounds for determinants with dominantprincipal diagonal,”Proc. Amer. Math. Soc., vol. 3, no. 1, pp. 26–30,1952.

Yuhan Zhou (S’08) received the B.E. degree inElectronic and Information Engineering from JilinUniversity, Chuangchun, Jilin, China, in 2005 andM.A.Sc. degree in Electrical and Computer Engi-neering from the University of Waterloo, Waterloo,Ontario, Canada, in 2009. He is currently workingtowards the Ph.D. degree with the Electrical andComputer Engineering Department at the Universityof Toronto, Toronto, Ontario, Canada. His researchinterests include wireless communications, networkinformation theory, and convex optimization.

Wei Yu (S’97-M’02-SM’08-F’14) received theB.A.Sc. degree in Computer Engineering and Math-ematics from the University of Waterloo, Waterloo,Ontario, 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 (Tier1) in Information Theory and Wireless Communica-

tions. His main research interests include information theory, optimization,wireless communications and broadband access networks.

Prof. Wei Yu served as an Associate Editor for IEEE Transactions onInformation Theory (2010-2013), as an Editor for IEEE Transactions onCommunications (2009-2011), as an Editor for IEEE Transactions on WirelessCommunications (2004-2007), and as a Guest Editor for a number of specialissues for the IEEE Journal on Selected Areas in Communications and theEURASIP Journal on Applied Signal Processing. He was a Technical ProgramCommittee (TPC) co-chair of the Communication Theory Symposium at theIEEE International Conference on Communications (ICC) in 2012, and aTPC co-chair of the IEEE Communication Theory Workshop in 2014. Hewas a member of the Signal Processing for Communications andNetworkingTechnical Committee of the IEEE Signal Processing Society (2008-2013).Prof. Wei Yu received an IEEE ICC Best Paper Award in 2013, an IEEESignal Processing Society Best Paper Award in 2008, the McCharles Prizefor Early Career Research Distinction in 2008, the Early Career TeachingAward from the Faculty of Applied Science and Engineering, University ofToronto in 2007, and an Early Researcher Award from Ontario in 2006. Prof.Wei Yu was named as a Highly Cited Researcher by Thomson Reuters in2014. He is a registered Professional Engineer in Ontario.

http://arxiv.org/abs/1301.6938

http://arxiv.org/abs/1210.1507

Documents

Optimized Backhaul Compression for Uplink Cloud Radio