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648 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 2, FEBRUARY 2014

Dynamic MIMO Precoding forFemtocell Interference Mitigation

Ahmed R. Elsherif, Student Member, IEEE, Zhi Ding, Fellow, IEEE, and Xin Liu, Member, IEEE

Abstract—This paper studies interference mitigation in hetero-geneous cellular networks consisting of traditional macrocells andnewly envisioned femtocells. The mutual interference betweenmacrocells and femtocells arises as a result of decentralizedfemtocell deployment and backhaul delay. To mitigate downlinkinterference between the femtocell clients, known as Home UserEquipments (HUEs), and macrocell clients, known as MacrocellUser Equipments (MUEs), we present methods of dynamicdistributed beamforming that are fully compatible with MIMOprecoding mechanisms in existing LTE standard releases. Wedevelop three MIMO beamforming schemes for interferencemitigation that take into account the Quality of Service (QoS)requirement of both femtocell and macrocell clients. These newheterogeneous MIMO precoding strategies improve flexibilityin resource provisioning and signaling requirement while re-sponding to different QoS needs. We also present MUE meanthroughput analysis by applying order statistics to our proposedmethods. Moreover, we provide an approximate closed form forthe mean throughput in terms of basic transmitter, channel,and receiver parameters. Furthermore, we extend our proposedinterference control precoding schemes to spatial multiplexing forMIMO transmissions. Finally, we extend our solution to tacklethe more general case involving multiple MUEs, multiple HUEs,and multiple femtocells.

Index Terms—Femtocells, heterogeneous networks, home eNB,interference control, MIMO systems, precoding.

I. INTRODUCTION

RECENT studies have shown that over 60% of cellularvoice calls and over 90% of cellular data traffic origi-

nate from indoor subscribers [1]. Yet, historically, substantialactivities in wireless research and development have focusedon high velocity mobile users and on the resulting fast channelfading problems. In fact, current cellular technologies anddeployment tend to exhibit poor indoor coverage, especiallyfor high speed data networks where broadband service encoun-ters severe channel distortions and packet losses in complexindoor environments. One recent proposal for improving in-door wireless coverage introduces the promising concept of“heterogeneous” networking (HetNet) and more specificallythe deployment of femtocells [2]. A femtocell is an indoorcellular base station that connects subscribers at a high speedand low power by reusing the same cellular spectrum. These

Manuscript received January 22, 2013; revised July 5 and October 24,2013. The editor coordinating the review of this paper and approving it forpublication was O. Oyman.

This material is based upon work supported by the National ScienceFoundation under Grants CNS1147930, ECCS1307820, and CCF1321143.

A. R. Elsherif and Z. Ding are with the Department of Electrical andComputer Engineering, University of California, Davis, CA 95616 USA (e-mail: {arelsherif, zding}@ucdavis.edu).

X. Liu is with the Department of Computer Science, University of Cali-fornia, Davis, Davis, CA 95616 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2013.122913.130062

femtocell base stations are linked to the core service networkby utilizing common broadband connections such as DigitalSubscriber Line (DSL), cable modem, or RF as their backhaulchannel.

The concept of heterogeneous networking, and femtocellsin particular, has already been proposed in the standardizationprocess for next generation communication systems such as,LTE, LTE-A, and WiMAX [3]. Femtocell base stations arereferred to as Home eNB (HeNB) in the LTE standardization.From the network operator’s point of view, femtocells improveindoor coverage, and can offload traffic from the macro-cell. Therefore, femtocell deployment can improve macrocellthroughput and strengthen link reliability. Moreover, the costof a femtocell HeNB, including equipment and deployment,is much lower than that of a macrocell base station deployedby the operator.

The main challenge of femtocell-macrocell deployment liesin interference management required by spectrum sharing inheterogeneous frameworks. Because the backhaul connectionof HeNB to the core-network relies, in general, on commercialInternet, HeNB control channels and data traffic cannot befully coordinated by the mobile network controller. In fact,HeNB can receive control information of its neighboringmacrocell base-station (MBS). But such information under-goes Internet delays and cannot be depended upon for timelyHeNB resource assignment and interference control. Indeed,because of control signal delays and decentralized HeNBresource allocation, both femtocells and macrocells mustcarefully manage their mutual interference to avoid seriousdisruptions due to their shared use of time-frequency physicalresource blocks (PRBs).

Some recent studies have attempted to address the problemof interference management in HetNets. The authors in [4]gave an overview of the problem together with a list ofexpected challenges. In [5], the authors proposed an adaptivepower control technique to limit the transmission power offemtocells in an effort to maximize frame utilization. Onemajor drawback of this scheme is that it is oblivious to the QoSneed of macrocell user equipment (MUE) that experiencesHeNB interference. In [6], the authors investigated proba-bilistic assignment of PRBs by the HeNB during downlink.More specifically, to reduce mutual interference, the proposedresource assignment scheme allows home user equipment(HUE) to access those PRBs occupied by outdoor MUEswith higher access probability. To implement such schemes,however, HeNBs must acquire MBS control signals in atimely manner. Furthermore, HeNB and MBS would eachneed to correctly classify MUEs as indoor versus outdoor user

0090-6778/14$31.00 c© 2014 IEEE

ELSHERIF et al.: DYNAMIC MIMO PRECODING FOR FEMTOCELL INTERFERENCE MITIGATION 649

equipments (UEs). Additionally, application of this approachto cases involving multiple femtocells remains open. Anotherwork in [7] proposed an adaptation technique that combinesnull steering of antenna array and spectral band selection fromthe perspective of cognitive radio. The application of this ideain femtocells for interference control is presented in [8].

Among existing works, there are two related publications([9] and [10]) on beamforming techniques in multiple antennasystems for interference control. The authors of [9] addressedthe interference problem between two HeNBs: HeNB1 andHeNB2. If the channel quality between HeNB2 and its HUE ishigh enough, HeNB2 will choose the multi-input multi-output(MIMO) precoding matrix index that minimizes the mutualinterference to the HUE served by HeNB1. The obstacle ariseswhen the channel between HeNB2 and its HUE suffers fromdistortions and losses such as multipath fading. The authorsin [10] considered the interference between MUE and HeNB,as in our formulation here. The major difference is that theproposed solution in [10] requires a connection between MBSand HeNB through the backhaul. This channel may leadto significant delay in precoder restriction, thereby causingperformance degradation as the channel between MUE andHeNB varies rapidly. In our work, we overcome the delayproblem by exploiting MUE feedback signals decoded bythe femtocells as part of the heterogeneous network. Moreimportantly, the analysis in [10] relies on a strong assumptionthat the projection power of normalized channel vectors ontodifferent beamforming directions generates a set of indepen-dent random variables. This assumption is not necessarily true,particularly for the codebooks defined in LTE for differentantenna configurations.

Another possible approach for solving the interference man-agement problem in the literature is the concept of coordinatedmulti-point transmission/reception (CoMP) [11]. The basicidea is to utilize multiple transmitting BS nodes to covercell-edge UEs (user equipments) that have poor reception. Inthis context, there are a variety of CoMP techniques amongwhich the most popular techniques are coordinated schedul-ing/beamforming (CS/CB), joint transmission (JT), and dy-namic cell selection [12]. Although the concept of CoMPcan enhance cell coverage and increase system throughput,it has a number of challenges for practical implementation.These challenges include determining the collaborating BSs,synchronization among the collaborating BSs, channel estima-tion at the collaborating BSs, and more importantly, increasedtraffic on the backhaul.

Some recent works have studied applying the concept ofCoMP for interference management in heterogeneous net-works. In [13], the authors proposed a scheme for beam-forming codebook restriction at the MBS which allows MUEsand HUEs to select the best channel that is robust to cross-tier interference prior to transmission. Beamforming codebookrestriction is performed based on codeword correlation witha reference codeword vector generated according to a pre-determined pseudo-random sequence for each channel. Thisscheme, however, requires the coordinated BSs to know thechannel state information (CSI) of the interference channelprior to transmission which incurs a considerable amount offeedback signaling over the backhaul. The work in [14] studied

the impact of the backhaul channel (wired or wireless) on theperformance gain of CoMP in femtocell networks. The authorsformulated the problem as a cooperative game where HeNBsjointly decide on their cooperative partners and on whether touse wired or wireless backhaul for the CoMP operation. Thiswork considered joint transmission (JT) with user data sharingwhich mandates full knowledge of CSI at each transmitter.In [15], the authors proposed a CoMP scheme that selectsthe optimal antenna patterns combination of all HeNBs withthe objective of maximizing network capacity. The proposedsolution searches for the optimal antenna pattern combinationsof all HeNBs using simulated annealing. Both the works in[14] and [15] consider the interference among different HeNBsbut are not concerned about the cross-tier interference betweenHeNBs and MBSs as in our work.

The chief focus and contribution of this manuscript is thedevelopment of new and low complexity algorithms for down-link MIMO precoding cooperation between macrocells andfemtocells. More specifically, we take advantage of a specialLTE standard feature known as MIMO precoder restriction forinterference control between MUE and HeNB in downlink.Based on MUE feedback signals over-heard at the HeNB, aswell as a delayed information exchange between the MUEand the HeNB, each HeNB can define a restricted subset of itsprecoding codebook during HUE downlink that can effectivelyreduce the interference to the MUE which shares the samePRBs. Our second major contribution consists of practicaland novel approaches of defining precoder subset restrictionsfor downlink interference suppression in order to meet theQoS requirement of MUE and HUE. We also propose a novelscheme where no connection is required between MUE andHeNB which helps avoid link delay. Our third contribution isanalysis of the mean throughput of MUE under our proposedprecoding algorithms.

In our conference presentation [16], we introduced threedifferent schemes for interference control based on MUEs andHUEs requirements and priorities. In [16], we studied a simpli-fied Z-channel model where the interference from the MBSat the HUE is assumed negligible. We also presented meanthroughput analysis for the first scheme. In this manuscript,we consider the more general X-channel model, where boththe MBS-to-HUE and HeNB-to MUE interferences are con-sidered. Moreover, we present more comprehensive meanthroughput analysis for the first scheme as well as for theother two schemes. We elaborate more on the third scheme,where no connection is required between MUE and HeNB.Furthermore, we present a comparison of the analytical andsimulation results. Moreover, we extend our work in [16] tospatial multiplexing with multiple receive antennas. Finally,we extend our solution to the more general case involvingmultiple UEs and multiple femtocells.

The rest of the paper is organized as follows: the systemmodel that depicts HeNB downlink interference is presented inSection II. Three precoding restriction methods are proposedfor HeNB codebook restriction in Sections III, IV, and V,respectively, along with their performance analysis. Precodingstrategies for spatial multiplexing MIMO transmission inHeNB are further studied in Section VI. In Section VII, weextend our studies to the more general case of multiple MUEs,


Fig. 1: Downlink interference channel model.

multiple HUEs, and multiple HeNBs. Numerical results arethen provided in Section VIII for performance evaluation andverifications before the conclusions of Section IX.

II. SYSTEM MODEL

In this paper, we consider a heterogeneous network thatconsists of femtocells deployed within the coverage and thespectrum of a macrocell. Because of spectrum-sharing be-tween HUEs and MUEs, the downlink mutual interferencebetween femtocells and MUEs is, generally, captured byan X-channel model as shown in Fig. 1. The X-channelmodel means that the transmission from the HeNB to theHUE causes interference to MUE, and the MBS transmissioncauses interference to the downlink reception at the HUE.Since the MBS operates at a much higher transmit powercompared to the HeNB, we may not generally neglect theinterference caused by the MBS at the HUEs, especially ifthe HeNB is not very far from the MBS. Another reasonis that although HeNBs are mostly set up in locations ofweak MBS coverage, they can also be deployed for the mainpurpose of traffic offloading from the MBS rather than onlyfor improving indoor coverage (e.g. when considering multiplefemtocells/picocells for offloading traffic from the MBS at aconvention center, a stadium, or other popular events). In thiscase, the HeNB may be close enough to the MBS and, leadingto significant interference from MBS to HUEs.1 We would alsolike to highlight that for cases involving indoor MUEs and notauthorized to connect to the HeNB, the interference seen bythe MUE can be much stronger than that seen by the HUEowing to the extra penetration loss (typically 20 dB) of MBSsignals versus the loss of HeNB signals. In this case, the Z-channel model can provide sensible approximation as basis forperformance analysis. For sake of completeness, in the rest ofthe paper we will consider the more general X-channel model.

In this case, the HeNB needs to adapt its transmission basedon the channel quality between itself and the MUE, withMIMO channel matrix denoted by H10, to reduce interferenceon the MUE as well as to enhance the throughput achieved bythe HUE. H00 and H11 are the direct MIMO channel matricesfrom the MBS to the MUE and from the HeNB to the HUE,

1In our earlier work in [16], we neglected the interference from the MBSto the HUEs and, thus, the X-channel model was simplified into a Z-channelmodel.

respectively. Moreover, the matrix H01 represents the MIMOchannel matrix between the MBS and the HUE. All entries ofthe MIMO channel matrices are assumed to be Rayleigh, i.e.,complex Gaussian i.i.d. random variables.

We assume that HeNB, HUE, and MUE are equipped withmultiple antennas and MIMO transceivers. A connection isestablished between the MUE and the interfering HeNB. TheHeNB and the MUE can collaborate via control signals andfeedback signals to set up beamforming. Based on the LTEstandard, MIMO transmit beamforming is restricted to a code-book. The beamforming code index, namely the precodingmatrix index (PMI), is chosen via certain performance criteriaby the receiving unit before being passed onto the transmittingunit for precoder implementation. The chosen PMI shouldbe updated according to how fast the interference channelchanges.

One may be concerned about the feasibility of setting up aconnection between HeNB and MUE. In fact, this connectionis practical and well-defined. For example, in the LTE-A andUMTS standards, a UE can connect with more than one eNBat a time. The signaling overhead is insignificant becausethe MUE only needs to transmit the indices of the chosenbeamformers each time which consist of a few bits. Forexample, we need two bits in LTE-A for two transmit antennasand four bits for four transmit antennas. Nevertheless, we doconsider and propose a separate scheme without this signalingconnection in Section V for special applications.

In this paper, we use lowercase and uppercase boldfaceletters for vectors and matrices, respectively. During downlinkbeamforming, the received signal at the HUE is given by

rf =√G01H01Wm∗sm +

√G11H11Wi∗sf + nf , (1)

where sm is the MBS transmit vector and sf is the HeNBtransmit vector with E[sHmsm] = Pm and E[sHf sf ] = Pf ,where Pm and Pf are the MBS and HeNB transmission pow-ers, respectively. The matrix H01 is of dimension Nf

r ×Nmt

with Nfr HUE receive antennas and Nm

t MBS transmitantennas. Similarly, the matrix H11 is of dimension Nf

r ×Nft

where Nft is the number of HeNB transmit antennas. For ease

of analysis, we separate the small-scale and large-scale fadingcomponents where Hij and

√Gij represent, respectively, the

small-scale and large-scale fading between base station i anduser j. This makes the entries of the channel matrices Hij

have zero mean and unit variance. Without loss of generality,Gij is defined as the path power gain between base stationi and user j and is expressed in terms of the correspondingpath loss as Gij = 1

10(PLij/10), where PLij is the path loss

between base station i and user j in dB as defined in Table IIin terms of the distance between base station i and user j. Itis, thus, obvious that 0 < Gij ≤ 1 ∀i, j.

The matrices Wm∗ and Wi∗ are the precoding matricesused by the MBS and the HeNB, respectively, and nf isthe channel noise vector, which is assumed to be AWGN.In LTE-A, the beamforming matrix Wi∗ ∈ W , where W isa finite set of all possible beamforming weights. Each matrixWi∗ ∈ W is of length Nf

t and has a unit norm. The receiverselects the beamforming matrix by feeding its index back tothe transmitter.

Similarly, the interference from the HeNB on the MUE is


TABLE I: Comparison of the proposed schemes.

Scheme Selection Decision Priority SignalingMRS MUE HeNB MUE Yes

HAPMI HeNB MUE HUE YesHRS HeNB HeNB MUE No

characterized by H10Wi∗ . Thus, the received signal at theMUE is given by

rm =√G00H00Wm∗sm +

√G10H10Wi∗sf + nm, (2)

where nm is the channel noise assumed to be AWGN. H00 isthe small-scale fading MIMO channel matrix between MBSand MUE with dimension Nm

r ×Nmt where Nm

r is the numberof MUE receive antennas.

In the next four sections, we first consider a single receiveantenna at both the HUE and MUE (Nf

r = Nmr = 1), thereby

dealing with vectors wi∗ and wm∗ and scalar received signalsrf and rm, respectively. Later in Section VI, we will considerspatial multiplexing with Nf

r > 1 and Nmr > 1. In Section VI,

we will deal with precoder matrices Wi∗ and Wm∗ instead ofthe precoding vectors wi∗ and wm∗ as well as signal vectorsrf and rm instead of the scalars rf and rm, respectively.

Our proposed solution is to be implemented at the HeNB(since it is the new element to be added to the cellular systemand, thus, it makes more sense from a practical point of viewto have the changes in the newly added modules). Therefore,we focus on precoder selection at the HeNB rather than at theMBS.

In subsequent sections, we describe possible criteria forselecting the precoder wi∗ among W that satisfies the perfor-mance requirement of both MUE and HUE. Different selec-tions of the precoder or beamformer wi∗ can lead to differentlevels of MUE interference during downlink. Since MUE andHUE have different QoS requirements, the optimum selectionof the precoder wi∗ is often a tradeoff between MUE andHUE needs. We propose three different schemes to determinethe most suitable PMI based on the UE requirements andpriorities. The three schemes are based on the availabilityof a restricted subset codebook which is supported in LTE[17]. The first scheme is “MUE Restricted Subset (MRS)Codebook”. This method gives priority to MUEs, and is, there-fore, suitable for heavily loaded macro networks and/or highpriority macrocell traffic. The second is “HUE AugmentedPMI (HAPMI) Feedback”. This approach targets the HUEQoS provisioning, which makes it more suitable for lightlyloaded macro networks. The third scheme is “HeNB RestrictedSubset (HRS) Codebook”, which is a modification of the firstscheme where HeNB can choose the PMI without establishinga connection with the MUE. The following sections discussthe three schemes in detail and then present the performanceanalysis of MUE mean throughput under these three schemes.

Table I compares the proposed schemes in terms of userpriority and the need for connection between MUE and HeNB.

Before proceeding, we introduce the following notations tobe used throughout the paper:

• Pf : constant HeNB transmission power.• Pm: constant MBS transmission power.

Fig. 2: MUE restricted subset codebook scheme.

• Gij : Path power gain between base station i and user j.• W : set of all precoders with cardinality K .• W1/W2/W3: set of restricted precoders for

MRS/HAPMI/HRS schemes, respectively, withcardinality K1/K2/K3, that satisfy the requiredcriteria out of W .

• ε: MUE’s interference power threshold, decreasing εimproves MUE’s QoS.

• ηsinr: HUE’s desired SINR threshold, increasing ηsinrimproves HUE’s QoS.

III. MUE RESTRICTED SUBSET (MRS) CODEBOOK

PRECODING

A. Precoder Selection

In our first heterogeneous cooperative precoding scheme,illustrated in Fig. 2, a connection is established between theMUE and the interfering HeNB. This connection is usedto exchange information between the HeNB and the MUE.First, MUE estimates the cross channel response H10 basedon HeNB downlink reference signals (pilots). Once H10 isdetermined, the MUE informs the HeNB a subset of in-dices from the precoder codebook. The subset of precodersmust satisfy a predetermined MUE interference constraintPfG10 ‖H10wi‖2 ≤ ε. In other words, the MUE restrictedsubset (MRS) of precoders can be denoted as

W1 ={

wi : PfG10 ‖H10wi‖2 ≤ ε}. (3)

Note that (3) characterizes the interference from the HeNBon the MUE and the choice of W1 that satisfies the MUEinterference constraint. The QoS concern expressed by theMUE as the maximum HeNB interference level is definedby ε. The smaller the value of ε, the higher the MUE priority.For real time traffic and in heavily loaded networks, ε canbe very small so as to force a more conservative HeNB. Afterreceiving the set of permissible indices W1 from the MUE, theHeNB subsequently chooses the optimum one that maximizesthe performance metric for its HUE (defined as the throughputof the HUE) as follows:

maxwi∈W1

PfG11 ‖H11wi‖2PmG01 ‖H01wm∗‖2 +N0Bm

, (4)


where wm∗ is the precoder used by the MBS to maximize‖H00wm‖2, i.e.

wm∗ = arg maxwm∈W

‖H00wm‖2. (5)

By noting that the maximization in (4) is on the subset ofprecoders at the HeNB and is, consequently, independent ofthe precoder chosen by the MBS, wm∗ , we can rewrite thecriterion for selecting the precoder that maximizes the HUE’sperformance as follows:

maxwi∈W1

PfG11 ‖H11wi‖2. (6)

B. Throughput Analysis

To analyze the mean throughput, we need to derive theprobability density function (PDF) of the MUE’s SINR. First,the MUE observes the following SINR

SINR =PmG00‖H00wm∗‖2

PfG10‖H10wi∗‖2 +N0Bm, (7)

where wi∗ is the precoding vector that satisfies both (3) and(6). The instantaneous capacity for MUE can thus be writtenas

Cm = Bm log2

(1 +

PmG00 ‖H00wm∗‖2PfG10 ‖H10wi∗‖2 +N0Bm

), (8)

where Bm is the bandwidth assigned to MUE. Similarly, theinstantaneous capacity for the HUE, Cf , can be written asfollows:

Cf = Bf log2

(1 +

PfG11 ‖H11wi∗‖2PmG01 ‖H01wm∗‖2 +N0Bf

), (9)

where Bf is the bandwidth assigned to the HUE.We first define xi = ‖H10wi‖2 ∀i ∈ {1 · · ·K} . To

derive the distribution of xi, we note that for the case of Nmr =

1 and single layer transmission, H10 has dimension 1 × Nft

and wi has dimension Nft ×1. The matrix H10 can be written

as,

H10 =[h(1)c h

(2)c · · · h

(Nft )

c

]+ j

[h(1)s h

(2)s · · · h

(Nft )

s

],

where h(m)c and h

(m)s are the real and imaginary components,

respectively, of the m-th entry in H10. Since each element ofH10 is an i.i.d. complex Gaussian with zero mean and unitvariance, h(k)

c and h(k)s are both i.i.d. N (0, 1/2). We assume

that the beamformer vector is normalized to ‖wi‖2 = 1.Consequently, xi = ‖H10wi‖2 is an exponential randomvariable with a unit rate.

Recall that the MUE chooses a precoder subset W1 thatsatisfies the condition of (3) rewritten as

W1 =

{wi : xi = ‖H10wi‖2 ≤ ε′ =

ε

PfG10

}. (10)

The resulting subset W1 has K1 precoders and associatedprecoder index set, M1. This subset is sent to the HeNBwhich will search for the precoder that satisfies the objec-tive function (6). Thus, we define a new random variable

yi = ‖H11wi‖2 , i ∈ M1. Given this notation, we find that

i∗ = arg maxi∈M1

yi, (11)

which selects the precoder index i∗ and the correspondingprecoder denoted wi∗ used in (7).

Now, we define the random variable in the denominator of(7) as v = ‖H10w∗

i ‖2 = xi∗ where v is chosen according tothe maximization in (6) over the subset W1 that satisfies (3).Consequently, by ordering the random variables xi ascend-ingly, v will be one of the leading K1 ordered values. Fornotation simplicity, we assume that the random variables xi

come from a random variable x with the same distribution.The PDF of v can be obtained from the PDF of x usingorder statistics as in [27]. To simplify the analysis, we assumethat {xi} are i.i.d. Strictly speaking, this i.i.d. assumptionholds if and only if all codeword vectors are orthogonal,i.e., wH

i wj = δ[i − j]. When the codebook contains vectorsthat are not pairwise orthogonal, our analysis serves as anapproximation to the true system throughput.

Based on order statistics, the conditional PDF of v giventhe size of W1 is

fv(v|K1 = L) =

L∑�=1

Pr(M1(�) = i∗)gx(�)(v), L > 0,

(12)where gx(�)

(v) is the PDF of the �-th smallest order statisticsof x and is given by

gx(�)(v) = αfx(v)Fx(v)

�−1(1− Fx(v))K−�, (13)

where α = K!(K−�)!(�−1)! , fx(v) and Fx(v) are the PDF and the

cumulative distribution function (CDF) of the random variablex, respectively. Because xi and yi are independent, as H10

and H11 are independent, Pr(M1(�) = i∗) = 1L . Thus, the

conditional PDF in (12) can be rewritten as

fv(v|K1 = L) =1

L

L∑�=1

gx(�)(v), L > 0. (14)

The PDF of v can be obtained by considering all possiblevalues of K1. We note that (14) assumes that the set W1

is non-empty. When W1 is empty, i.e. the condition in (3)does not hold for any precoder, we can have two differentscenarios that will slightly change the derivation of the PDFof the variable v. Here we discuss them separately.

1) Best-of-the-worst Scenario: In this scenario, the MUEhas to send at least one precoder to the HeNB, i.e. the casethat K1 = 0 is not allowed. In this case, if the interferencecondition in (3) is not satisfied for any precoder in the setW , the MUE will pick the one that generates the minimuminterference, i.e. a best-of-the-worst scenario. This scenariois compliant with LTE standard releases where a UE caninform its serving base station of a preferred precoder orset of precoders. Hence, this scenario does not require anymodification to existing LTE standards.

In this case, the interference condition is not satisfied for anycodeword, i.e. xi > ε′ ∀i ∈ {1 · · ·K}. Thus, the conditionalPDF of v given that the precoder subset is empty can beobtained as fv(v|K1 = 0) = gx(1)

(v − ε′)U(v − ε′), wheregx(1)

(v) is the PDF of the smallest x that can be calculated


by substituting � = 1 in (13) with step function U(·).2) Empty Subset Scenario: This scenario allows the set W1

to be empty when no precoder satisfies the MUE interferenceconstraint. This scenario proposes that the MUE can send abit to an interfering HeNB warning it not to transmit becauseit will be causing intolerable interference to the MUE. In thisscenario, the conditional PDF of v given that the precodersubset is empty can be written as fv(v|K1 = 0) = δ(v),where δ(v) is the Dirac Delta function.

Consequently, the PDF of v can be written as follows:

fv(v) =

K∑L=1

fv(v|K1 = L)Pr(K1 = L)

+ fv(v|K1 = 0)Pr(K1 = 0). (15)

To find Pr(K1 = L), we first define pi(ε′) as the probability

that codeword wi satisfies the condition in (3). Thus, pi(ε′)can be written as

pi(ε′) = Pr(xi ≤ ε′) = 1− exp(−ε′), (16)

where we used the fact that xi is a unit rate exponentialrandom variable as proven earlier. Based on our earlier as-sumption that the random variables xi = ‖H10wi‖2 are i.i.d,pi(ε

′) will be the same for all codewords. Therefore, wedrop the index i for simplicity and define p(ε′) = pi(ε

′)∀i ∈ {1 · · ·K}. Therefore, Pr(K1 = L) for L > 0 becomes

Pr(K1 = L) =

(K

L

)(p(ε′))L(1− p(ε′))K−L. (17)

To calculate Pr(K1 = 0), we need to calculate the probabilitythat none of the variables xi satisfy the condition in (3). Thiscan be calculated as

Pr(K1 = 0) = (Pr(xi > ε′))K

= (1− Pr(xi ≤ ε′))K

= exp(−ε′K). (18)

By substituting (17) and (18) into (15), we complete the PDFof the random variable v.

Next, we define a real-valued random variable u =‖H00wm∗‖2. The random variable u can be written as u =maxwm∈W ‖H00wm‖2 = maxwm∈W b(wm). The PDF ofthe random variable b = ‖H00wm‖2 can be shown to beexponential using the same derivation steps used for the PDFof xi. The PDF of u can thus be obtained using the maximumorder statistic of b as follows [27] :

fu(u) = Kfb(u)Fb(u)K−1, (19)

where fb(u) and Fb(u) are the PDF and CDF of the expo-nential random variable b, respectively.

Given PDFs of both u and v plus the fact that they areindependent, the mean throughput can be determined as

E(Cm) = Bm

∫u

∫v

log2(1 + SINR)fv(v)fu(u)dvdu. (20)

Though the mean throughput analysis does not exhibit a closedform, it can be numerically evaluated via standard integrationor importance sampling techniques [28], as shown in SectionVIII.

C. Mean throughput closed form

In this section, we derive an approximate closed form forthe mean throughput in (20). Using Jensen’s Inequality [33],the following inequality applies to the mean throughput

E(Cm)

= Bm

∫u

∫v

log2

(1 +

PmG00u

PfG10v +N0Bm

)· fv(v)fu(u)dvdu

≤ Bm log2

(1 +

∫u

∫v

PmG00ufv(v)fu(u)dvdu

PfG10v +N0Bm

)= Cm, (21)

where Cm represents an upper bound for the mean throughput.Since u = ‖H00wm∗‖2 and v = ‖H10w∗

i ‖2 are independent,Cm can be written as

Cm = Bm log2

(1 +

PmG00

PfG10u

∫v

1

v + N0Bm

PfG10

fv(v)dv

),

(22)where u is the expected value of the random variable u whosePDF is given in (19). We define the integral on v as J where

J =

∫v

1

v + N0Bm

PfG10

fv(v)dv. (23)

The integral J can be expressed in terms of the first inversemoment of v [34], where the first inverse moment of a randomvariable x is defined as

∫x

1xfx(x)dx. The calculation of

inverse moments is difficult in general. Some literature workshave provided different conditions for approximating the in-verse moment by the inverse of the moment [29] - [31]. Ingeneral, for any positive random variable, Schwarz’s inequalityleads to E[1/X ] ≥ 1/E(X) [29]. More specifically, in [31], itwas shown that, under some conditions, the following holdstrue for a random variable X

E[(a+X)−α] ∼ [a+ E(X)]−α, (24)

where α > 0. Applying the result in (24), to the integral J in(23), we get

J ≈ 1

v + N0Bm

PfG10

. (25)

Substituting (22)- (25) into (21), we conclude that

E(Cm) ≈ Bm log2

(1 +

PmG00

PfG10

u

v + N0Bm

PfG10

). (26)

It is worth mentioning that the approximation of J in (25)holds, more accurately, with a greater than or equal sign,i.e., on a real axis, J approaches the value 1

v+N0BmPfG10

from

the right. This can be seen from Schwarz’s inequality or byapplying Jensen’s Inequality on the concave function 1

v+N0BmPfG10

in (23). Consequently, since E(Cm) ≤ Cm but J ≥ 1

v+N0BmPfG10

,

the expression in (26) represents an approximation for themean throughput rather than an upper bound on the meanthroughput. We also note that in general, the integral J , canbe evaluated using numerical evaluation as in (20).


Next, we need to get expressions for u and v. We recall thatboth u and v are order statistics of exponential random vari-ables, and, thus, are themselves exponential random variables[32]. According to [32], the mean of the �-th order statisticof an exponential random variable X with a sample size Kis given by

E(X(�)) =1

λ

K∑n=K−�+1

1

n, (27)

where λ is the rate parameter of the exponential randomvariable. According to (19), u is the maximum order statisticof b = ‖H00wm‖2, therefore

u =

K∑n=1

1

n. (28)

The mean of the random variable v can be obtained usingconditional expectation on K1 as follows

v =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∑KL=1 Pr(K1 = L)

∑L�=1

1LE(X(�))

+Pr(K1 = 0)E(X(1)) Best-of-the-worst,∑KL=1 Pr(K1 = L)

∑L�=1

1LE(X(�))

Empty subset.(29)

The significance of the expression in (26) is that it gives asimple approximate closed form for the mean throughput as afunction of basic transmitter, channel, and receiver parameters.The mean of the random variable u depends merely on thenumber of precoders K (28), whereas the mean of v dependson both K and the normalized MUE’s interference powerthreshold ε′ (29). Consequently, the MUE’s mean through-put in (26) is expressed in terms of K , ε′, MBS transmitpower Pm, HeNB transmit power Pf , path gain between theMBS/HeNB and the MUE G00/G10 (which in turn dependson the distance between the MBS/HeNB and the MUE), andnoise power at the MUE N0Bm.

IV. HUE AUGMENTED PMI (HAPMI) FEEDBACK


We now consider the scenario where we need to givehigher priority to the HUE service quality. As in the MRSscheme, we consider that a connection exists between theHeNB and the MUE. The precoder set Wf that guaranteesacceptable performance for the HUE is first determined beforetransmitting to the MUE. This subset can be characterizedaccording to the SINR at the HUE as

W2 =

{wi :

PfG11 ‖H11wi‖2E(PmG01 ‖H01wm∗‖2 +N0Bf )

≥ ηsinr

},

(30)

where ηsinr is the HUE’s desired SINR threshold. Fortractability of the derivation, we consider the average interfer-ence plus noise power, where the expectation is over the chan-nel matrix H01. We define a random variable u′ such that u′ =‖H01wm∗‖2 = maxwm∈W ‖H01wm‖2 = maxwm∈W b′(wm),where the PDF of the random variable b′ = ‖H01wm‖2 isexponential using the same derivation steps used for the PDF

Fig. 3: HUE augmented PMI feedback scheme.

of xi. Similar to the PDF of u = ‖H00wm∗‖2, the PDF ofu′ can be obtained using the maximum order statistic of b′ asfollows :

fu′(u′) = Kfb′(u′)Fb′(u

′)K−1, (31)

where fb′(u′) and Fb′ (u

′) are the PDF and CDF of theexponential random variable b′, respectively. Therefore, theset W2 can be redefined as

W2 ={

wi : PfG11 ‖H11wi‖2 ≥ η},

η = ηsinr

∫u′(PmG01u

′ +N0Bf ) fu′(u′)du′, (32)

where η represents HUE’s desired received signal powerthreshold.

The MUE receives the indices of the precoder set W2 andestimates the cross channel with the HeNB using the pilotsymbols in the same transmission that provided the indices ofthe precoder set W2. Based on the complex channel gains, theMUE chooses the minimum interference beamformer

wi∗ = arg minwi∈W2

PfG10 ‖H10wi‖2. (33)

The MUE sends the index of the optimum precoder wi∗ backto the HeNB. Fig. 3 summarizes this scheme.


For performance analysis of this scheme, the HeNB firstselects the restricted subset according to (32). The restrictedsubset comprises of K2 precoders with indices M2. We definexi = ‖H11wi‖2 which can be shown to be exponentiallydistributed following the same argument as in the previoussection.

Out of the restricted subset, the MUE chooses the precoderthat minimizes the interference according to (33). We defineanother random variable yi = ‖H10wi‖2, i ∈ M2. Similar tothe MRS scheme, we define v = ‖H10w∗

i ‖2 as in the denom-inator of (7). Therefore, we see that v = yi∗ = mini∈M2 yi.Because yi has only a predefined subset selection that isdictated by the HeNB and is independent of xi, i.e. not afunction of H11, it becomes clear that yi is exponentiallydistributed with unit rate.

We can also assume that {yi} are i.i.d. Therefore, theconditional PDF of v given a certain size of W2 can be given


by minimum order statistic of y as follows [27] :

fv(v|K2 = L) = Lfy(v)(1 − Fy(v))L−1, L > 0. (34)

The PDF of v can, consequently, be obtained from theconditional PDF as follows :

fv(v) =K∑

L=1

fv(v|K2 = L)Pr(K2 = L)

+ fv(v|K2 = 0)Pr(K2 = 0). (35)

Similar to the MRS scheme, the set W2 can be empty,i.e. none of the precoders satisfied the HUE throughputrequirement in (32). Thus, if the HeNB finds no precodersatisfying its HUE requirement, it can pick the precoder thatmaximizes its HUE throughput, even though it is below itsdesired requirement. This choice is considered a best-of-the-worst case and can be suitable if the HUE traffic does notrequire Guaranteed Bit Rate (GBR), i.e. a best-effort trafficsuch as TCP-based traffic. However, if the traffic has a GBRrequirement, e.g. video conferencing, and the HeNB is not ableto meet its rate requirement, it makes sense for the HeNBto suppress its transmission, thereby reducing unnecessaryinterference to the MUE. This is because any achieved SINRbelow a predefined QoS is useless. This is denoted by theempty-subset scenario.

Therefore, for the Best-of-the-worst scenario, when K2 = 0,the HeNB will select the precoder that gives maximum xi

among the K precoders. Denoting the index of the chosenprecoder in this case as i′, we can write fv(x|K2 = 0) as

fv(v|K2 = 0) =

K∑�=1

Pr(� = i′)fyi(v)

= exp(−v)U(v) Best-of-the-worst, (36)

where, from the MUE’s point of view, the probability that theHeNB chooses any precoder when K2 = 0 is the same, i.e.Pr(� = i′) = K−1. For the Empty-subset scenario, however,since the HeNB suppresses transmission, fv(x|K2 = 0) isgiven by

fv(v|K2 = 0) = δ(v) Empty-subset. (37)

Similarly, by defining the probability that a codeword wi

satisfies the condition in (32), we see that

p(η′) = Pr(xi = ‖H11wi‖2 ≥ η

PfG11= η′)

= exp(−η′)U(η′), (38)

where we also used the fact that xi = ‖H11wi‖2 is unit-rateexponential. Therefore, Pr(K2 = L) is given by

Pr(K2 = L) =

(K

L

)(p(η′))L(1− p(η′))K−L. (39)

The probability Pr(K2 = 0) is the probability that allvariables xi do not satisfy the condition in (32). This canbe calculated as

Pr(K2 = 0) = (Pr(xi < η′))K = [1− exp(−η′)]K . (40)

By substituting (39) and (40) into (35), we get the PDF ofv. We also define the random variable u = ‖H00wm∗‖2 with

Fig. 4: HeNB restricted scheme.

the same distribution as in (19). Finally, with the PDF of uand v, we obtain the mean throughput for the HAPMI schemeby substituting in (20).

V. HENB RESTRICTED SUBSET (HRS) CODEBOOK


This scheme can be considered as a modification of theMRS scheme where the HeNB estimates the channel betweenitself and the MUE. This can be done by listening to uplinkfeedback signals, and the reference signals therein, at theHeNB to get an estimate H10 of the actual downlink channelseen by the MUE, H10. Based on channel reciprocity, we canapply (3) on the reciprocally generated channel, H10, betweenthe MUE and the HeNB to determine a codebook subset W3,from which the PMI is obtained by applying (6) on the setW3. This operation is illustrated in Fig. 4.

The concept of channel reciprocity has been studied in [18]and references therein. The reciprocally generated channel canbe written as H10 = H10 + Hδ, where Hδ is a zero meancomplex Gaussian random variable with a variance of ρ. Therandom variable Hδ accounts for the error between the esti-mated downlink channel obtained through channel reciprocity,H10, and the actual downlink channel, H10. For Time DivisionDuplex (TDD) operation, channel reciprocity generally holdsbecause both uplink and downlink transmission occur onthe same frequency. However, the Radio Frequency (RF)circuitry in the transmitter and the receiver are, generally, notsymmetric. One way to handle this non-symmetry is throughcareful RF circuitry calibration as in [19], producing smallreciprocity error which we simply model by Hδ. On the otherhand, for Frequency Division Duplex (FDD) systems, channelreciprocity is not straightforward over frequency selectivechannels as uplink and downlink transmission use differentfrequency bands. Studies showed that channel spatial informa-tion in macro and micro cellular environment, characterizedby Direction of Arrival (DOA), in both uplink and downlinkare correlated [20]. Frequency correction algorithms based onlong-term statistical channel characteristics, such as channelcovariance matrix and DOA were analyzed and compared in[21]. However, in femtocell environments, the antenna heightof HeNB would be lower than that of MBS. Also, sincethe HeNB is in indoor environment, the propagation betweenHeNB and MUE is completely without LOS. In such case,


the angular spreading would be much wider than that of microcellular environment and the channel reciprocity may not hold.In the context of our work, we can still model inaccuracyin channel reciprocity for FDD by Hδ, probably with largervariance, ρ, compared to the case of TDD channels.

The advantage of this approach is that the decision is solelymade at the HeNB. No signaling is needed between the MUEand the HeNB. However, this simplicity is at the expense ofpotential performance degradation when channel reciprocitydoes not hold accurately. In short, the difference Hδ betweenthe the actual and estimated channels can lead to performancedegradation. Thus, there is a trade-off between the overhead ofsignaling and performance loss due to inaccuracy of channelestimation Hδ.

It is worth noting that although we defined Hδ as thechannel reciprocity error at the HeNB for the HRS scheme, itcan also be considered as a channel estimation error of H10 atthe MUE for the MRS scheme. In Section VIII, we evaluatethe performance of the HRS scheme for different values of ρ.This also corresponds to the evaluation of the MRS schemewhen considering a channel estimation error at the MUE forinterference channel H10. This provides good insight abouthow our proposed solution would perform when consideringpractical channel estimation errors.


To derive the mean throughput for this scheme, the actualchannel H10 should be replaced with the reciprocally gener-ated channel H10, which is equal to the actual channel plus areciprocity error Hδ. Thus, the codebook subset is defined asfollows :

W3 ={

wi : PfG10 ‖(H10 + Hδ)wi‖2 ≤ ε}, (41)

where, as defined before, Hδ is ∼ CN (0, ρ). We define themeasurement xi = ‖(H10 + Hδ)wi‖2, ∀i ∈ {1 · · ·K}. Onceagain, it is clear that xi is exponentially distributed, albeitwith rate = 1

1+ρ . After obtaining the restricted subset W3 withprecoder indices M3, the HeNB will pick the precoder thatmaximizes the throughput of its HUE as in (4) and (6). Similarto the MRS scheme, the maximization of the HUE throughputover the subset of precoders at the HeNB is independent ofthe precoder chosen by the MBS. To model this, we defineyi = ‖H11wi‖2, i ∈ M3. The chosen precoder index will,similarly, be obtained according to i∗ = argmaxi∈M3 yi.

Because SINR calculation should use the actual channelseen by the MUE, H10, the variable v in (7) should still bedefined using the actual channel H10 as v = ‖H10w∗

i ‖2. Now,because of the presence of the error Hδ, v cannot be directlyobtained from order statistics of xi, because Hδ might alter theorder. To handle this problem, we first define a new randomvariable z as the order statistics of xi. The PDF of z can bewritten as

fz(z) =

K3∑L=1

fz(z|K3 = L)Pr(K3 = L)

+ fz(z|K3 = 0)Pr(K3 = 0). (42)

Note that, as in the MRS scheme, we have

fz(z|K3 = L) =1

L

L∑�=1

gx(�)(z), L > 0, (43)

where gx(�)(z) is the �-th order statistics of x as given in (13),

and,

fz(z|K3 = 0)

=

{gx(1)

(z − ε′)U(z − ε′) Best-of-the-worst,

δ(z) Empty subset.(44)

To find Pr(K3 = L), we define pi(ε′) as the probability

that codeword wi satisfies the condition in (41). Thus, pi(ε′)can be written as

pi(ε′) = Pr(xi ≤ ε′) = 1− exp(− ε′

1 + ρ), (45)

where xi is exponential with rate(1+ρ)−1. Similarly, becausethe random variables xi = ‖H10wi‖2 are i.i.d, pi(ε′) will bethe same for all codewords. Therefore, Pr(K3 = L) is givenas follows :

Pr(K3 = L) =

(K

L

)(p(ε′))L(1− p(ε′))K−L. (46)

The probability Pr(K3 = 0) is the probability that noneof the variables xi satisfy the condition in (41). This can becalculated as

Pr(K3 = 0) = (Pr(xi > ε′))K = exp(− ε′K1 + ρ

). (47)

To find the PDF of v, we will try to relate it to z. We write(H10 + Hδ)wi∗ = zr + jzi. Thus, z can be written as z =z2r + z2i . Similarly, we write Hδwi∗ = δr + jδi, where δr andδi are ∼ N (0, ρ/2). Thus,

v = (zr − δr)2 + (zi − δi)

2 ≈ z − 2(δrzi + δizr)

= z − 2 < [δr δi], [zi zr] >= z − 2δaza cos(θ), (48)

where we have neglected the second order terms in the approx-imate equality. This approximation is reasonable as long as thereciprocity error Hδ is small enough. The notation < m,n >refers to the inner product of two vectors m and n. The randomvariable δa =

√δ2r + δ2i and za =

√z2r + z2i =

√z. Because

δr and δi are ∼ N (0, ρ/2), the random variable δa becomesRayleigh distributed with parameter ρ/2. To have a tractablesolution, we assume that δa and za are independent and,thus, the angle θ is uniformly distributed. We note that thisassumption is common in the field of stochastic approximation[22],[23]. A similar and well-known assumption was used inthe seminal work by Widrow et al. in [24] for analyzing theconvergence of the least mean squares (LMS) algorithm andin [25],[26]. In section VIII, we show the combined effectof all approximations in the analytical expression where wecompare analysis with numerical evaluation.

We further define a random variable q = δa cos(θ), whichis, thus, ∼ N (0, ρ/2). Rewriting the expression of v as afunction of q and za, we get

v = z2a − 2qza. (49)


To derive the PDF of v, we define an auxiliary variable s = zaso that the transformation from (q, za) to (v, s) is one-to-one.The Jacobian of the transformation can be found as J(za, q) =2za = 2s. Thus the joint PDF of v and s is given by

fvs(v, s) =1

2sfzaq(s, (s

2 − v)/(2s)), (50)

where fzaq(za, q) is the joint PDF of za and q. Based on theearlier approximation that δa and za are independent, we cansimilarly approximate za and q to be independent, and thusfzaq(za, q) is given by

fzaq(za, q) = fza(za)fq(q) = 2zafz(z2a)

1√πρ

exp(−q2

ρ),

(51)where the PDF of za is obtained from the PDF of z using thefact that za =

√z, thus, fza(za) = 2zafz(z

2a) where fz(z

2a)

is obtained by substituting z = z2a in (42). Consequently, thePDF of v can be obtained from (50) by integrating over s :

fv(v) =

∫ ∞

0

1√πρ

fz(s2) exp(− (s2 − v)2

4s2ρ)ds. (52)

Finally, using the PDFs of both u and v, we get the meanthroughput for the HRS scheme according to (20), which canbe evaluated numerically.

VI. PRECODING FOR SPATIAL MULTIPLEXING

In this section, we generalize our schemes to precodingstrategies in spatial multiplexing MIMO transmissions. Westart by considering the case of two transmission layers andtwo receive (Rx) antennas. In other words, there are twostreams of data being transmitted simultaneously and the twoRx antennas allow the receiver to recover both signal streams.We then consider four Rx antennas and study the cases of twoand four signal layers. When there are more than one receiveantenna, we must revise the capacity definitions in (8) and (9)to account for the direction of the interference relative to thedirection of the received signal.

The capacity of the MIMO interference channel (MIMOIC) is still an open problem, in general. The widely studiedIC model is the two-user single-input-single-output (SISO)Gaussian IC. The capacity region for the SISO Gaussian ICis obtained for the strong and very strong interference caseswhere it was shown in [36] that strong interference does notreduce capacity since it can be decoded and subtracted at eachreceiver. For the general case including weak interference,the best inner bound was given by Han and Kobayashi [37]and shown to be within one bit of the capacity region in[38]. For the MIMO IC, the Han and Kobayashi’s region wasshown to be within one bit per receive antenna of the capacityregion [39]. Few recent works have provided expressions forthe capacity regions and sum-rate capacities as in [40] and[41]. However, to the best of the knowledge of the authors,no existing literature has studied the capacity of the MIMOIC with constrained precoder selection. For the purpose of ourwork, we use the sum-rate capacity presented in [41] as an un-constrained sum-rate capacity where no constraint is imposedon precoder selection and compare it with constrained sum-rate for the proposed precoder selection schemes. The basic

idea is that in the low interference regime, the interferencecan be considered as noise.

We first define Wm as a MBS precoder matrix, Q0 =PmWmWH

m as the MBS covariance matrix when using pre-coder Wm, and Q1 = PfWiWH

i as the HeNB input covari-ance matrix when using precoder Wi. Applying the resultsobtained in [41] to our problem, the unconstrained sum-ratecapacity for the MIMO IC defined in (1) and (2) can be givenby

Cs = Bm log2 det(

I +G00H00Q0HH00

· (N0BmI +G10H10Q1HH10)

−1)

+Bf log2 det(

I +G11H11Q1HH11

· (N0Bf I +G01H01Q0HH01)

−1),

(53)

where the operator det(·) is the matrix determinant operatorand Q0 and Q1 are the capacity achieving covariance matricesfor the MBS and HeNB, respectively, i.e. the covariancematrices Q0 and Q1 that correspond to the precoders thatmaximize the sum-rate capacity out of the set of all possibleprecoders W .

The expression for the sum-rate capacity in (53) mandatesthe following two conditions to be satisfied:

1) Invertibility condition : Both H00 and H11 are left-invertible.

2) Low interference condition :

r(Φi) ≤ 0.5, i = 0, 1, (54)

where r(X) is the numerical radius [42] defined as

r(X) = maxαHα≤1

(abs(αHXα)), (55)

where α is a complex vector and abs(·) is the absolutevalue operator. The matrices Φ0 and Φ1 are defined asfollows

Φ0 =(

I − AH0 A0 − A1AH

1

)−0.5

AH0 AH

1

·(

I − AH0 A0 − A1AH

1

)−0.5

,

Φ1 =(

I − A0AH0 − AH

1 A1

)−0.5

AH1 AH

0

·(

I − A0AH0 − AH

1 A1

)−0.5

,

(56)

where

A0 =(I +G10H10Q1HH

10

)√G00H00

· (G00HH00H00

)−1(√

G01HH01 − BH

0

),

A1 =(I +G01H01Q0HH

01

)√G11H11

· (G11HH11H11

)−1(√

G10HH10 − BH

1

),

(57)


such that B0 ∈ B0 and B1 ∈ B1, where

Bi =

{B|all columns of BH are in the null space of Qi},i = 0, 1. (58)

For the MRS scheme, the criteria for precoder selection, i.e.selecting Wi∗ , are given by

maxWi∈W1

det(G11H11Q1HH

11

), (59)

where the precoder subset W1 is given by

W1 ={

Wi : det(G10H10Q1HH

10

) ≤ ε}. (60)

The criteria for the HAPMI and the HRS schemes can bederived similarly.

In Section VIII, we compare the unconstrained sum-ratecapacity obtained in (53) with the constrained sum-rate for theproposed precoder selection schemes. The constrained sum-rate for each scheme corresponds to the sum-rate obtainedwhen using the chosen precoder according to the criteria ofthe corresponding scheme. This corresponds to the sum-rateachieved when replacing Q0 and Q1 by Q∗

0 = PmWm∗WHm∗

and Q∗1 = PfWi∗WH

i∗ , respectively.

A. Two Rx antennas

For transmission on two antennas with two layers and twoRx antennas, H11 and H10 are 2× 2 matrices. LTE-A definesthree codes of size 2 × 2 [43]. Through simple derivations,we can see that regardless of the channel matrix H11, thevalue of det(G11H11QHH

11) will always be the same ∀ w ∈W . The same is true for det(G10H10QHH

10). This means that,regardless of the chosen precoder, the throughput performanceat the receiver will be the same. In this case, the receiver willchoose the precoder that gives the Minimum Mean SquareError (MMSE) which does not benefit from our work here.However, our proposed schemes can still be applied in thesame manner as before for the special case when the set W1,W2, and W3 in (3), (32), and (41) will be either the entireprecoder set W or the empty set.

B. Four Rx antennas

For four transmit and four receive antennas, the precodingmatrix, W , is selected from a 4×4 matrix, Wn, obtained viahouseholder transformation of specific 4×1 vectors [43]. Forsingle layer signal transmissions, W is chosen as the firstcolumn of the corresponding Wn. For more than one layer,W is given by certain column permutations of Wn with thenumber of columns equal to the number of signal layers.Simulation results are shown later in Section VIII.

VII. GENERALIZATION TO MULTIPLE UES AND MULTIPLE

HENBS

In this section, we extend the previous scenario to includethe more general case of multiple MUEs, multiple HUEs, andmultiple HeNBs. We first consider the case of multiple MUEsand a single HeNB with a single HUE, then consider the caseof multiple HeNBs, each with a single HUE, and a single

MUE. Finally, we show the extension of all the previousscenarios to the case when a HeNB serves multiple HUEs.For simplicity of presentation, we limit the discussion to thecase of single transmission layer while the extension to spatialmultiplexing follows immediately.

A. Multiple MUEs and Single HeNB

In this case, we consider Nm neighboring MUEs in thevicinity of an HeNB. Because the neighboring MUEs arein the coverage range of the HeNB, they are very likelyto be in the coverage of the same MBS and thus havingorthogonal channel assignment from the serving MBS (exceptin the very special case when the HeNB is at the MBS celledge and some of the neighboring MUEs are served by theMBS while the other MUEs are served by a neighboringMBS). Since in OFDMA, orthogonal channels are assigned todifferent users within a cell, we assume that the MUEs haveorthogonal frequency allocation. This mandates that precoderselection be done in frequency domain for each neighboringMUE frequency band. Thus, precoder selection should bedone using the channel frequency responses for each MUEband. This means that for each MUE band, the HeNB mightuse a different precoder that satisfies the requirements ofthe corresponding MUE. Consequently, the solution of thisproblem can be seen as solving Nm independent single-MUE-single-HeNB problems as before.

Moreover, instead of using a single precoder per MUEband, better performance can be obtained by consideringmore precoders per band. For example, a single precoder persubband (8 PRBs for 20 MHz bandwidth) or per multiple sub-bands can be used for better precoder selection resolution but,of course, at the expense of more information exchange. Wenow characterize the precoder selection resolution by the term“precoding channel unit” (PCU), where the PCU for an MUEn can be the whole frequency band assigned to MUE n, onesubband, or multiple sub-bands. The performance improve-ment of using multiple precoders per MUE frequency banddepends on the channel frequency selectivity (i.e., coherencebandwidth) relative to the resolution of precoder selection. Forbrevity, we show the details for the MRS scheme but similarapproach can be applied for the HAPMI and the HRS schemes.

For the MRS scheme, the HeNB selects a precoder (ormultiple precoders) for each neighboring MUE n and for eachPCU a as follows

W(n,a)1

=

⎧⎨⎩wi :

∑k∈Ωn,a

Pf (k)G10‖H10(n, k)wi‖2 ≤ ε(n,a)

⎫⎬⎭ ,

n = 1, · · · , Nm, a = 1, · · · , An, (61)

where W(n,a)1 is the set of precoders that satisfy the maximum

interference requirement for MUE n in PCU a. The parameterAn is the number of PCUs for MUE n. For example, if asingle precoder per MUE band is chosen, An will be 1. If,instead, one precoder per subband is chosen for MUE n, An

will be equal to the number of subbands in the frequency bandof MUE n. The set Ωn,a represents the subcarrier indices ofPCU a for MUE n. The index k is the subcarrier index and


Pf (k) is the per-subcarrier transmission power over subcarrierk. H10(n, k) is the channel frequency response between theHeNB and MUE n over subcarrier k. The parameter ε(n,a)

represents the interference requirement for MUE n and forPCU a. This can be set as ε(n)/An where ε(n) is the maximumtolerable interference for MUE n.

Each MUE n informs the interfering HeNB with the set ofindices W(n,a)

1 that result in tolerable interference to that MUEaccording to (61). Consequently, the HeNB chooses a precoder(or multiple precoders) for each MUE frequency band so asto maximize the received signal power at the HUE in eachcorresponding MUE frequency band as follows

maxwi∈W(n,a)

1

∑k∈Ωn,a

Pf (k)G11‖H11(k)wi‖2, (62)

where H11(k) is the channel frequency response between theHeNB and the HUE over subcarrier k.

The per-subcarrier HeNB transmission power Pf (k) can beset equal to the total HeNB transmission power divided bythe total number of sub-carriers used by the HeNB. However,a better solution is to consider per-subcarrier HeNB powerallocation rather than using a constant per-subcarrier power.Therefore, precoder selection and power allocation should beoptimized jointly. Nevertheless, instead of solving a difficultjoint optimization problem for precoder selection and powerallocation, we propose a more practical sub-optimal iterativeprecoder selection and power allocation approach. The pro-posed approach applies the precoder selection step as in (61)and (62) for given per-subcarrier powers obtained using thewater-filling concept [44] as follows

Nm∑n=1

An∑a=1

∑k∈Ωn,a

1

λ− N0Bk

G11‖H11(k)wi(n, a)∗‖2 = Pf,max,

(63)where 1/λ represents the water level, Bk is the bandwidthof subcarrier k, wi(n, a)

∗ is the chosen precoder for MUEn for PCU a according to (61) and (62), and Pf,max is themaximum HeNB transmission power. Accordingly, the per-subcarrier power is given by

P ∗f (k) =

(1

λ− N0Bk

G11‖H11(k)wi(n, a)∗‖2)+

, ∀k ∈ Ωn,a,

a = 1, · · · , An, n = 1, · · · , Nm, (64)

where (x)+ = max(0, x). The obtained power through (63)and (64) can then be used in the precoder selection step[(61) and (62)] and so on. The algorithm terminates whena pre-determined stopping criterion is met. The stoppingcriterion can be either an iteration limit or stagnation betweensuccessive iterations. The initial value of the per-subcarrierspower P (0)

f (k) is given by

P(0)f (k) =

Pf,max∑Nm

n=1

∑An

a=1 |Ωn,a|, (65)

where |Ωn,a| is the cardinality of the set Ωn,a representing thenumber of sub-carriers in the set Ωn,a. In fact, the value of∑Nm

n=1

∑An

a=1 |Ωn,a| represents the total number of subcarriersused by the Nm MUEs under consideration.

It is worth noting that although the previous analysis con-

sidered the extension of single MUE to multiple MUEs, thesame analysis can be applied as an extension from single MUEwith flat fading, as in earlier sections, to single MUE withfrequency selective fading. For frequency selective fading,precoder selection has to be performed in the frequencydomain by considering the channel response in each frequencyband. This is, in fact, a special case of precoder selectionin (61) and (62) with Nm = 1. Moreover, it is clear andobvious that the most general case of multiple MUEs eachwith frequency selective channel is already handled in theprecoder selection in (61) and (62).

B. Multiple HeNBs and Single MUE

In this case, we consider one MUE in the common coveragerange of Nh HeNBs. We note that this is not a typical situationbecause the coverage range of HeNBs is small due to its lowtransmission power which means that it is not common that anMUE happens to be in the common coverage range of multipleHeNBs. Therefore, we only briefly discuss the modification ofthe proposed schemes.

For the MRS scheme, the MUE shall try all possiblecombinations of all the Nh HeNB precoders and choose theones that satisfy its requirement. The number of combinationshere is exponential in the number of HeNBs of interest, Nh.If more than one combination satisfies the MUE requirement,the MUE should send each HeNB its restricted subset of pre-coders. However, this requires coordination between HeNBsto ensure that they collectively choose one of the MUE’sproposed combinations. To avoid such coordination, the MUEcan choose a single combination and ask each HeNB touse a specific precoder. This will be a special case of theoriginal MRS scheme where, here, the MUE maximizes itsown interest only without caring about HeNBs performance.

For the HRS scheme, if no coordination is allowed betweenHeNBs, then each HeNB will independently select a precoderthat satisfies MUE’s maximum interference requirement dueto this HeNB’s interference. However, this individual selectionof HeNB precoders might not collectively satisfy the MUE’smaximum interference requirement.

For the HAPMI algorithm, each HeNB sends the MUE arestricted subset of codewords that satisfy their HUE require-ments. The MUE should try all possible combinations of theset of restricted precoders it receives. Thereafter, the MUEselects the combination that minimizes its interference andsends an individual precoder request to each HeNB.

C. Multiple HUEs

This section describes the extension to the case when anHeNB is serving Nf HUEs instead of one HUE as before. Inthis case, the HeNB has to assign non-overlapping frequencydomain resources to each HUE. Therefore, precoder selectionshould be done in frequency domain. This extension can beeasily done on top of any of the previous scenarios consideringthe frequency band of each HUE. For example, for the MRSscheme with single MUE, single HeNB and multiple HUEs,precoder selection should be done in the frequency bandfor each HUE (similar to the approach in Section VII-Afor multiple MUEs). For multiple MUEs, single HeNB and


TABLE II: Simulation parameters.

Parameter ValueMBS/HeNB Tx Power 47dBm/20dBm

MBS to MUE Path Loss 131.1 + 42.8 · log10(d00/1000)(PL00) d00: distance between MBS & MUE

MBS to HUE Path Loss 151.1 + 42.8 · log10(d01/1000)(PL01) d01 : distance between MBS & HUE

HeNB to MUE Path Loss 147 + 30 · log10(d10/1000)(PL10) d10: distance between HeNB & MUE

HeNB to HUE Path Loss 127 + 30 · log10(d11/1000)(PL11) d11: distance between HeNB & HUE

multiple HUEs, the HeNB should consider the two steps ofprecoder selection in the frequency bands for each MUE andeach HUE. For the HAPMI case, the HeNB has to satisfy therequirements for each HUE ηm, m = 1, · · · , Nf .

VIII. PERFORMANCE EVALUATION

In this section, we evaluate the performance of our proposedschemes. Two performance metrics are considered, namely, the5%-outage capacity and the mean throughput. The 5%-outagecapacity is defined as C0 for which Pr(C(H) < C0) = 0.05,where C(H) is the instantaneous capacity under a randomchannel realization H. The instantaneous capacity for MUEsand HUEs are given, respectively, in (8) and (9). On theother hand, the mean throughput is the ensemble average ofinstantaneous capacity over different channel realizations. Fornormalization of the results, we use the capacity expressionsin bps/Hz (i.e. Cm/Bm and Cf/Bf ). Unless otherwise men-tioned, we set ε = −70dBm and ηsinr = 18dB and considerthe Best-of-the-worst scenario. For the HRS scheme, standarddeviation of 0.1 is tested for the reciprocity error Hδ (i.e. 10%channel reciprocity error).

The general simulation parameters used in the rest of thepaper are summarized in Table II. The performance met-rics are evaluated and shown against the distance betweenthe MUE and the interfering HeNB, d10, since it directlydetermines the path loss from the HeNB to the MUE and,thus, the interference signal power at the MUE. We com-pare the performance of the three schemes described earlierwith two other competing schemes that can characterize theperformance bounds. The first competing scheme, labeled“min interference”, simply uses the PMI that minimizes theinterference at the MUE. It does not consider the HUEthroughput maximization (same as proposed in [9]). Hence,the performance of “min interference” constitutes an upperbound on the throughput of the MUE. The second competingscheme, labeled “max throughput”, uses the PMI that directlymaximizes the throughput of the HUE while ignoring theinterference seen by the MUE. This results in an upper boundon the throughput of the HUE.

A. Single layer transmission

For the case of two transmit antennas, we use 2 bits toencode the PMI which is consistent with the LTE-A standard.The codebook for precoding is given in [43]. We also tested 4-bit indexing for which the simulation exhibited only marginalimprovement. Channel gains are assumed to be complexGaussian i.i.d. random variables.

10 20 30 40 50 60 70 80 90 1006

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18

Distance between HeNB and MUE (d10

) [m]

Mean

th

rou

gh

pu

t [b

ps/H

z]

2 Tx, 1 Rx

MUE, MRSHUE, MRSMUE, min interferenceHUE, min interferenceMUE, HAPMIHUE, HAPMIMUE, max throughputHUE, max throughputMUE, HRS, ρ = 0.1HUE, HRS, ρ = 0.1

MUE

HUE

(a) Mean throughput vs d10

10 20 30 40 50 60 70 80 90 1002

4

6

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14

16


) [m]

5%

ou

tag

e c

ap

acit

y [

bp

s/H

z]

2 Tx, 1 Rx


MUE

HUE

(b) 5%-outage capacity vs d10

Fig. 5: Performance comparison for 2 Tx antennas and 1 Rxantenna vs d10.

Fig. 5a and 5b show the mean throughput and the 5%-outagecapacity, respectively, seen by both the MUE and the HUEgiven two Tx antennas vs the distance between HeNB andMUE, d10. As shown in Fig. 5a and 5b, the system has theflexibility to assign priority to MUE interference mitigationin heavily loaded networks (by using the MRS scheme) orto HUE throughput maximization in lightly loaded networks(by using the HAPMI scheme). With respect to the MUEthroughput, the “min interference” performance representsthe upper bound whereas the “max throughput” performancerepresents the lower bound. On the other hand, the oppositeis true with respect to the HUE performance.

In Fig. 5a and 5b, for the MRS and HRS schemes, we noticethat as the HeNB-to-MUE distance d10 increases, the HUE’sthroughput grows. The reason for this is that as d10 increases,


the interference power at the MUE drops and, therefore, itbecomes easier to satisfy the maximum interference conditionsin (3) and (41). Consequently, the size of precoder subsets W1

(for MRS) and W3 (for HRS) becomes larger which givesmore flexibility for HUE’s throughout maximization. Thus, asd10 increases the performance of the MRS and HRS schemesapproaches that of the “max throughput” bound for both theHUE and the MUE. We also observe that for very small valuesof d10, the performance of the MRS scheme for the MUEis the same as that of the “min interference” bound. This isbecause, at such high level of interference, the precoder subsetW1 becomes empty. Therefore, since we are considering theBest-of-the-worst scenario, the precoder resulting in minimuminterference at the MUE will be chosen, similar to the “mininterference” scheme.

On the other hand, for the HAPMI scheme, the HUE’sperformance is independent of d10. The HUE’s performancefor the HAPMI scheme is affected only by the distancebetween the HeNB and the HUE, d11, since precoder subsetrestriction for the HAPMI scheme is based on the HUE’sthroughput (as in (30)), instead of the interference power at theMUE as in the MRS and HRS schemes. Fig. 6a and 6b showthe mean throughput and the 5%-outage capacity, respectively,as HeNB-to-HUE distance d11 varies. In this simulation,we set d10 to 40m. Intuitively, as d11 increases, the HUE’sthroughput decreases. For large values of d11, the precodersubset W2 for the HAPMI scheme becomes empty because thetarget minimum throughput cannot be achieved. Consequently,the precoder that achieves the maximum throughput will bechosen, giving the same performance as the “max throughput”bound. Moreover, since the size of the precoder subset W2

decreases with increasing d11, the MUE’s performance of theHAPMI scheme decreases as d11 grows. However, the MUE’sperformance of the MRS and HRS schemes is independent ofd11 since d11 does not affect the precoder subsets of theseschemes (namely, W1 and W3, respectively).

Fig. 7a and 7b show, respectively, the mean throughput and5%-outage capacity seen by both the MUE and the HUE, nowgiven four transmit antennas. From the results, we observesimilar trends as in the case of two antennas but with a largergap in performance among various schemes. This is expectedas the codebook for four antennas has more codewords and,thus, the gain of properly selecting the best precoder becomesmore obvious.

In Fig. 8a and 8b, we study the impact of the thresholdsε and ηsinr on the 5%-outage capacity for 4 Tx antennasand 1 Rx antenna for MRS and HAPMI, respectively. ForMRS, increasing ε means that the MUE is more tolerant tolarger levels of interference. This, intuitively, will decreasethe achieved 5%-outage capacity for the MUE while givingmore flexibility for precoder selection for the HUE since moreprecoders will satisfy the interference condition at the MUE.This, in turn, will increase the 5%-outage capacity for theHUE as shown in Fig. 8a. Similarly, for HAPMI, increasingηsinr means that the HUE is requesting higher average SINRand, in turn, better performance. Accordingly, the precodersubset W2 will get smaller as ηsinr increases, leading to lessflexibility in minimizing the interference at the MUE and,thus, the achieved 5%-outage capacity for the MUE degrades.

10 15 20 25 30 35 40 45 50 55 600

5

10

15

Distance between HeNB and HUE (d11

) [m]

Mean

th

rou

gh

pu

t [b

ps/H

z]

2 Tx, 1 Rx


MUE

HUE


10 15 20 25 30 35 40 45 50 55 600

2

4

6

8

10

12

Distance between HeNB and HUE (d11

) [m]

5%

ou

tag

e c

ap

acit

y [

bp

s/H

z]

2 Tx, 1 Rx


MUE

HUE



In conclusion, we see that the threshold values ε and ηsinrcan determine how throughputs of the MRS, HAPMI, or theHRS schemes may edge toward one of the bounds dependingon user priorities. It is also worth noting that the relativeperformance of MRS and HAPMI depends merely on thevalues of the thresholds ε (for MRS scheme) and ηsinr (forHAPMI). Thus, neither the MRS scheme nor the HAPMI canbe claimed to provide a better performance for all values ofthreshold pairs (ε,ηsinr).

Fig. 9 shows the mean throughput for the HRS scheme fordifferent values of ρ, the standard deviation of the reciprocityerror Hδ , in comparison with the MRS scheme. It is clearthat the performance of the HRS scheme is very close to thatof the MRS when channel reciprocity holds with reasonable


10 20 30 40 50 60 70 80 90 1006

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20


) [m]

Mean

th

rou

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pu

t [b

ps/H

z]

4 Tx, 1 Rx


MUE

HUE


10 20 30 40 50 60 70 80 90 1002

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) [m]

5%

ou

tag

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acit

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bp

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z]

4 Tx, 1 Rx


MUE

HUE



error variances. Thus, the HRS scheme is capable of deliveringcomparable performance to the MRS scheme without the needfor any connection between the HeNB and the MUE. On theother hand, as described earlier, Hδ can be considered as achannel estimation error of H10 for the MRS scheme. Thisshows that the proposed solution is robust against channelestimation errors.

In Fig. 10, we provide numerical results using importancesampling on the mean throughput for the case of two antennasfor the three schemes. We observe a gap between the analyticalmean throughput in (20) and simulation results for the en-semble average of instantaneous capacity in (8) for differentchannel realizations. This gap is the direct consequence ofnon-orthogonal precoders in LTE-A. We also observe that the

10 20 30 40 50 60 70 80 90 1002

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) [m]

5%

ou

tag

e c

ap

acit

y [

bp

s/H

z]

4 Tx, 1 Rx, ηsinr

= 18dB

MUE, min interferenceHUE, min interferenceMUE, max throughputHUE, max throughput

ε = −85 dBm

ε = −85 dBm

MUE

HUE

ε = −60 dBm ε increases in steps of 5 dB

ε = −60 dBm

ε increases in steps of 5 dB

(a) Impact of ε on MRS scheme

10 20 30 40 50 60 70 80 90 1002

4

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) [m]

5%

ou

tag

e c

ap

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bp

s/H

z]

4 Tx, 1 Rx, ε = −70 dBm

MUE, min interferenceHUE, min interferenceMUE, max throughputHUE, max throughput η

sinr = 21 dB

ηsinr

= 9 dB

HUE

MUE

ηsinr

increases in steps of 3 dB

ηsinr

increases in steps of 3 dB

ηsinr

= 21 dB

ηsinr

= 9 dB

(b) Impact of ηsinr on HAMPI scheme

Fig. 8: Impact of ε and ηsinr on MRS and HAPMI schemes,respectively.

gap for the HRS scheme is slightly larger than that for theMRS scheme due to the added approximation that δa and zaare independent. We also plot the approximate closed formderived in (26) for the MRS scheme and show that it gives agood approximation to the mean throughput.

B. Precoding for spatial multiplexing

In our setup we assume that the number of Rx antennas isequal to the number of layers. Fig. 11 shows the performancefor four Tx antennas and two Rx antennas. In Fig. 11a, weshow the sum-rate for each of the proposed precoder selec-tion schemes in comparison with the unconstrained sum-ratecapacity defined in (53) (obtained when there is no restriction


10 20 30 40 50 60 70 80 90 1006

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) [m]

Mean

th

rou

gh

pu

t [b

ps/H

z]

2 Tx, 1 Rx

MUE, MRSHUE, MRSMUE, HRS, ρ = 0.01HUE, HRS, ρ = 0.01MUE, HRS, ρ = 0.1HUE, HRS, ρ = 0.1MUE, HRS, ρ = 0.2HUE, HRS, ρ = 0.2MUE, HRS, ρ = 0.3HUE, HRS, ρ = 0.3MUE, HRS, ρ = 0.4HUE, HRS, ρ = 0.4MUE, HRS, ρ = 0.5HUE, HRS, ρ = 0.5

40 45 50

13.4

13.6

13.8

14

14.2

MUE

HUE

Fig. 9: Impact of ρ on HRS scheme.

10 20 30 40 50 60 70 80 90 1008

9

10

11

12

13

14

15

16

17


) [m]

Me

an

th

rou

gh

pu

t [b

ps

/Hz]

2 Tx, 1 Rx, ε = −60dBm, ηsinr

= 18, ρ = 0.1

MRS, simulationMRS, analyticalMRS, approx. closed formHAPMI, simulationHAPMI, analyticalHRS, simulationHRS, analytical

Fig. 10: Simulation versus analytical MUE performance.

on precoder selection so that any of the predefined precodersin the set W can be used). It is to be noted that the differencebetween the achieved sum-rate for the proposed schemes andthe unconstrained sum-rate capacity should not be viewed asa performance gap. The reason is that the objective of theproposed schemes is not to maximize the achieved sum-rateof the MUE and the HUE. Instead, each of the scheme hasa specific QoS objective for either the MUE or the HUE thatis described in the precoder selection criteria. In Fig. 11b,we examine the conditions for the validity of the sum-ratecapacities in MIMO IC. Namely, we plot the average valuesof r(Φ0) and r(Φ1) over different channel realizations andshow that they are both below 0.5 for the different schemesand, thus, the low interference condition, defined in (54),is satisfied for most channel realizations. We also plot theprobability of meeting the low interference condition as well

10 20 30 40 50 60 70 80 90 10030

32

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36

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) [m]

Su

m r

ate

cap

acit

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bp

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z]

4 Tx, 2 Rx

MRSmin interferenceHAPMImax throughputHRS, ρ = 0.1Unconstrained sum−rate

(a) Sum-rate capacity

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5


) [m]

Rad

ius(φ

i)

4 Tx, 2 Rx r(Φ

0), MRS

r(Φ1), MRS

r(Φ0), min interference


r(Φ0), HAPMI

r(Φ1), HAPMI

r(Φ0), max throughput


r(Φ0), HRS, ρ = 0.1

r(Φ1), HRS, ρ = 0.1

10 20 30 40 50 60 70 80 90 1000.95

0.96

0.97

0.98

0.99

1


) [m]

Pro

b. o

f L

ow

in

terf

. / In

vert

ib. co

nd

.

MRSInvertibility cond.min interferenceHAPMImax throughputHRS, ρ = 0.1

(b) Validity condition

Fig. 11: Performance comparison for 4 Tx antennas and 2layers.

as that of meeting the invertibility condition of H00 and H11.Both probabilities are shown to be very close to 1.

In Fig. 12, we show the same performance metrics for thecase of four Tx antennas and four Rx antennas. Similarly, wesee that both the low interference and invertibility conditionscan be satisfied with very high probabilities. Comparing resultsin Fig. 12, Fig. 11, and Fig. 7a, we see that the gap between thelower and upper bounds narrows as the number of signal layersincreases. This is because as the number of layers increases,the precoders in the codebook become less orthogonal as morecolumns are selected from the matrix Wn.

C. Multiple MUEs and single HeNB

In this section, we consider Nm MUEs in the coverageof a single HeNB. For simplicity, we consider single layer


10 20 30 40 50 60 70 80 90 10055

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MRSmin interferenceHAPMImax throughputHRS, ρ = 0.1Unconstrained sum−rate

(a) Sum-rate capacity

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5


) [m]

Rad

ius(φ

i)

4 Tx, 4 Rx

r(Φ

0), MRS

r(Φ1), MRS



r(Φ0), HAPMI

r(Φ1), HAPMI



r(Φ0), HRS, ρ = 0.1

r(Φ1), HRS, ρ = 0.1

10 20 30 40 50 60 70 80 90 1000.95

0.96

0.97

0.98

0.99

1


) [m]Pro

b. o

f L

ow

in

terf.

/ In

verti

b. co

nd

.

MRSInvertibility cond.min interferenceHAPMImax throughputHRS, ρ = 0.1

(b) Validity condition

Fig. 12: Performance comparison for 4 Tx antennas and 4layers.

transmission. The simulation parameters for this case aresummarized in Table III. Fig. 13 shows the sum of the5% outage capacity of the MUEs for the MRS scheme.We compare the performance for different cases of precoderselection, namely single precoder per MUE frequency band,precoder per subband, and precoder per Physical ResourceBlock (PRB). Furthermore, we consider random precoderselection at the HeNB with optimal precoder selection at theMBS as well as random precoder selection at both HeNBand MBS. Fig. 13 shows that under the simulation parametersin III, using multiple precoders per MUE frequency bandgives an improvement in the sum of 5% outage capacityfor the MUEs. The performance gap between using singleprecoder versus multiple precoders increases as the channel

TABLE III: Simulation parameters for multiple MUEs andsingle HeNB.

Parameter ValueNumber of MUEs (Nm) 3

Total bandwidth (MBS/HeNB) 20MHzNumber of usable PRBs 100

Number of PRBs per MUE [25 33 42]Maximum delay spread per MUE [3.7 2.3 0.2] μsec

Doppler Frequency 100HzNumber of Tx Antennas at HeNB/MBS 4Number of Rx Antennas at MUE/HUE 1

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MUE, Single precoderHUE, Single precoderMUE, Uniform, Precoder / subbandHUE, Uniform, Precoder / subbandMUE, Uniform, Precoder / PRBHUE, Uniform, Precoder / PRBMUE, Uniform, Random HeNBHUE, Uniform, Random HeNBMUE, Uniform, Random HeNB & MBSHUE, Uniform, Random HeNB & MBSMUE, Joint power & precoder / subbandHUE, Joint power & precoder / subbandMUE, Joint power & precoder / PRBHUE, Joint power & precoder / PRB

MUE

HUE

Fig. 13: Sum 5% throughput, multiple MUEs and singleHeNB.

becomes more selective in the frequency domain, i.e. asthe channel’s maximum delay spreads increase. As expected,using a precoder per PRB gives an even better perfomanceat the expense of increased signalling. We also notice thatrandom precoder selection gives worse performance for bothMUEs and HUEs. It is also clear from Fig. 13 that selectingrandom precoders at both the HeNB and MBS gives the worstperformance for both MUEs and HUEs.

All the aforementioned schemes use equal per-subcarrierpower allocation (uniform power allocation). We also show theperformance using our proposed iterative precoder selectionand power allocation scheme in the same figure (Fig. 13).We notice that for the chosen multipath fading channels,power allocation gives a very slight improvement comparedto the corresponding case with uniform power allocation.The reason for this is that for multiple antenna, the gain ofantenna diversity becomes more dominant compared to thegain of power allocation. In other words, for the gain ofpower allocation to be significant, simultaneous deep fadesin H11 should be encountered, which is unlikely when havingmultiple antennas.

IX. CONCLUSION

In this work, we propose three practical schemes for inter-ference control and mitigation in heterogeneous LTE cellular


network environments. We focus on spectrum sharing and het-erogeneous coverage between HeNB and MBS. In particular,we focus on the downlink interference control from the HeNBto nearby MUEs in LTE cellular deployment. The decisionon which scheme to use depends on the QoS requirementof MUEs and HUEs as well as their QoS priorities. Toavoid the need for control channel between the HeNB andneighboring MUEs, one of our proposed schemes requiresthe HeNB to estimate the channel between itself and theMUE by listening to feedback signals and pilots. The HeNBrestricts its precoding codebook set by taking into accountthe cross interference channel and the required QoS. Wealso generalize our findings to include precoding strategiesfor spatial multiplexing MIMO transmissions. Moreover, weaddress the general case of multiple MUEs, multiple HUEs,and multiple HeNBs. Simulation results show that the systemcan adapt between MUE interference mitigation in densenetworks and HUE throughput maximization in lightly loadednetworks. Moreover, we present performance analysis on theMUE mean throughput for the three proposed schemes byapplying order statistics theory and we obtain an approximateclosed form for the mean throughput as a function of basictransmitter, channel and receiver parameters. The analyticalresults are supported by our simulation tests.

ACKNOWLEDGMENT

The authors would like to thank Ahmed Ahmedin at Uni-versity of California, Davis and Prof. Jyri Hamalainen at AaltoUniversity, Finland for their useful comments and discussions.

REFERENCES

[1] D. Lopez-Perez, G. de la Roche, A. Valcarce, A. Juttner, and J. Zhang,“Interference avoidance and dynamic frequency planning for WiMAXfemtocells networks,” in Proc. 2008 IEEE Int. Conf. Commun. Syst., pp.1579-1584.

[2] V. Chandrasekhar, J. Andrews, and A. Gatherer, “Femtocell networks:a survey,” IEEE Commun. Mag., vol. 46, no. 9, pp. 59–67, Sep. 2008.

[3] J. Zhang and G. de la Roche, Femtocells: Technologies and Deployment.Wiley, 2010.

[4] P. Kulkarni, W. Chin, and T. Farnham, “Radio resource managementconsiderations for LTE femto cell,” ACM SIGCOMM Comput. Commun.Rev., vol. 40, no. 1, pp. 26–30, Jan. 2010.

[5] P. Mach and Z. Becvar, “Dynamic power control mechanism forfemtocells based on the frame utilization,” in Proc. 2010 Int. Conf.Wireless and Mobile Commun., pp. 498-503.

[6] Z. Zheng, J. Hamalainen, and Y. Yang, “On uplink power controloptimization and distributed resource allocation in femtocell networks,”in Proc. 2011 IEEE Veh. Tech. Conf. – Spring, pp. 1–5.

[7] J. Mitola, “Cognitive radio for flexible mobile multimedia communi-cations,” in Proc. 1999 IEEE Int. Workshop on Mobile MultimediaCommun., pp. 3–10.

[8] M. Husso, J. Hamalainen, R. Jantti, and A. M. Wyglinski, “Adaptive an-tennas and dynamic spectrum management for femtocellular networks:a case study,” in Proc. 2008 IEEE Symp. New Frontiers in DynamicSpectrum Access Networks, pp. 699–703.

[9] M. Husso, Z. Zheng, J. Hamalainen, and E. Mutafungwa, “Dominantinterferer mitigation in closed femtocell deployment,” in Proc. 2010IEEE Int. Symp. Personal Indoor and Mobile Radio Commun., pp. 169–174.

[10] J. Zhu and H. C. Yang, “Interference control with beamforming coor-dination for two-tier femtocell networks and its performance analysis,”in Proc. 2011 IEEE Int. Conf. Commun., pp. 1–5.

[11] R. Irmer, et al., “Coordinated multipoint: concepts, performance, andfield trial results,” IEEE Commun. Mag., vol. 49, no. 2, pp. 102–111,Feb. 2011.

[12] M. Sawahashi, Y. Kishiyama, A. Morimoto, D. Nishikawa, and M.Tanno, “Coordinated multipoint transmission/reception techniques forLTE-advanced [coordinated and distributed MIMO],” IEEE WirelessCommun. Mag., vol. 17, no. 3, pp. 26–34, Jun. 2010.

[13] S. Park, W. Seo, S. Choi, and D. Hong, “A beamforming codebookrestriction for cross-tier interference coordination in two-tier femtocellnetworks,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1651–1663,May 2011.

[14] F. Pantisano, M. Bennis, W. Saad, M. Debbah, and M. Latva-aho,“On the impact of heterogeneous backhauls on coordinated multipointtransmission in femtocell networks,” in Proc. 2012 IEEE Int. Conf.Commun., pp. 5064–5069.

[15] Y. Li, Z. Feng, D. Xu, Q. Zhang, and H. Tian, “Optimisation approachfor femtocell networks using coordinated multipoint transmission tech-nique,” Electron. Lett., vol. 47, no. 24, pp. 1348–1349, Nov. 2011.

[16] A. R. Elsherif, A. Ahmedin, Z. Ding, and X. Liu, “Adaptive precodingfor femtocell interference mitigation,” in Proc. 2012 IEEE Int. Conf.Commun., pp. 4315–4320.

[17] “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical LayerProcedures (Release 10),” 3GPP TSG RAN TS 36.213, v10.3.0, Sep.2011. Available: http://www.3gpp.org/ftp/.

[18] G. S. Smith, “A direct derivation of a single-antenna reciprocity relationfor the time domain,” IEEE Trans. Antennas Propag., vol. 52, no. 6, pp.1568–1577, Jun. 2004.

[19] F. Kaltenberger, H. Jiang, M. Guillaud, and R. Knopp, “Relative channelreciprocity calibration in MIMO/TDD systems,” in Proc. 2010 FutureNetwork and Mobile Summit, pp. 1–10.

[20] K. Hugl, K. Kalliola, and J. K. Laurila, “Spatial reciprocity of uplinkand downlink radio channels in FDD systems,” in Proc. COST 273, TD(02) 066, Finland, May 2002.

[21] Y. Han, J. Ni, and G. Du, “The potential approaches to achieve channelreciprocity in FDD system with frequency correction algorithms,” inProc. 2010 Int. ICST Conf. Commun. and Networking in China, pp.1–5.

[22] H. Robbins and S. Monro, “A stochastic approximation method,” TheAnn. of Math. Stat., vol. 52, no. 6, pp. 400–407, Sep. 1951.

[23] A. Dvoretzky, “On stochastic approximation,” in Proc. 1956 BerkeleySymp. Math. Stat. and Probability, vol. 1, pp. 39–55.

[24] B. Widrow, J. M. McCool, M. G. Larimore, and J. C. R. Johnson, “Sta-tionary and nonstationary learning characteristics of the LMS adaptivefilter,” Proc. IEEE, vol. 64, no. 8, pp. 1151–1162, Aug. 1976.

[25] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Prentice Hall,1985.

[26] S. Haykin and B. Widrow, Least-Mean-Square Adaptive Filters. Wiley,2003.

[27] H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed. Wiley, 2003.[28] P. W. Glynn and D. L. Iglehart, “Importance sampling for stochastic

simulations,” Manage. Sci., vol. 35, no. 11, pp. 1367-1392, Nov. 1989.[29] N. Garcia and J. Palaciosb, “On inverse moments of nonnegative random

variables,” Stat. & Probability Lett., vol. 53, no. 3, pp. 235-239, Jun.2001.

[30] S. Sung, “On inverse moments for a class of nonnegative randomvariables,” J. Inequalities and Applicat., vol. 2010, Article ID 823767,2010.

[31] X. Shi, Y. Wu, and Y. Liu, “A note on asymptotic approximationsof inverse moments of nonnegative random variables,” Statistics &Probability Lett., vol. 80, no. 15, pp. 1260–1264, Aug. 2010.

[32] N. Balakrishnan and S. Gupta, “Higher order moments of order statis-tics from exponential and righttruncated exponential distributions andapplications to lifetesting problems,” Purdue University, Indiana, Tech.Rep. 92-07C, Jan. 1992.

[33] J. Jensen, “Sur les fonctions convexes et les ingalits entre les valeursmoyennes,” Acta Mathematica vol. 30, no. 1, pp. 175-193, 1906.

[34] Z. Govindarajulu, “Theory of Inverse Moments,” No. CSL-1066. CASEInstitute of Technology, Cleveland, OH, Sep. 1962.

[35] M. D. Springer and W. E. Thompson, “The distribution of products ofindependent random variables,” SIAM J. on Appl. Math., vol. 14, pp.511–526, May 1966.

[36] A. B. Carleial, “A case where interference does not reduce capacity,”IEEE Trans. Inf. Theory, vol. 21, no. 5, pp. 569-570, Sep. 1975.

[37] T. S. Han and K. Kobayashi, “A new achievable rate region for theinterference channel,” IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 49-60, Jan. 1981.

[38] R. H. Etkin, D. Tse, and H. Wang, “Gaussian interference channelcapacity to within one bit,” IEEE Trans. Inf. Theory, vol. 54, no. 12,pp. 5534-5562, Dec. 2008.


[39] E. Telatar and D. Tse, “Bounds on the capacity region of a class ofinterference channels,” in Proc. 2007 IEEE Int. Symp. Inf. Theory, pp.2871-2874.

[40] V. S. Annapureddy and V. V. Veeravalli. “Sum capacity of MIMOinterference channels in the low interference regime,” IEEE Trans. Inf.Theory, vol. 57, no. 5, pp. 2565–2581, May 2011.

[41] X. Shang, B. Chen, G. Kramer, and H. V. Poor, “Capacity regionsand sum-rate capacities of vector Gaussian interference channels,” IEEETrans. Inf. Theory, vol. 56, no. 10, pp. 5030-5044, Oct. 2010.

[42] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge UniversityPress, 1985.

[43] “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Chan-nels and Modulation (Release 10),” 3GPP TSG RAN TS 36.211,v10.3.0, Sep. 2011. Available: http://www.3gpp.org/ftp/

[44] D. Tse and P. Vishwanath, Fundamentals of Wireless Communications.Cambridge University Press, 2005.

Ahmed R. Elsherif (S’08) received the B.Sc. andM.Sc. degrees in Electronics and CommunicationsEngineering from Cairo University, Egypt in 2006and 2010, respectively, and is currently a Ph.D. can-didate in the Electrical and Computer EngineeringDepartment at University of Davis. His industrialexperience includes full time position in NewportMedia Inc. as a senior systems engineer (2006 -2010), internship in Fujitsu Laboratories of America,Inc. (2012 - 2013), and internship in BroadcomCorporation (2013 - present). He holds 7 issued

and pending US patents in different topics in wireless communications. Hisresearch interest is on wireless communication networks and signal processingwith a focus on interference management, resource allocation, heterogeneousnetworks, design and analysis of PHY and MAC layers algorithms, andhardware implementation of communication systems.

Zhi Ding (S’88-M’90-SM’95-F’03) is the ChildFamily Endowed Professor of Engineering and En-trepreneurship at the University of California, Davis.He also holds a joint appointment as a thousand-talent professorship at Southeast University in Nan-jing, China. He received his Ph.D. degree in Elec-trical Engineering from Cornell University in 1990.

Dr. Ding is a Fellow of IEEE and has beenan active member of IEEE, serving on technicalprograms of several workshops and conferences. Hewas associate editor for IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 1994-1997, 2001-2004, and associate editor ofIEEE SIGNAL PROCESSING LETTERS 2002-2005. He was a member oftechnical committee on Statistical Signal and Array Processing and member oftechnical committee on Signal Processing for Communications (1994-2003).Dr. Ding was the Technical Program Chair of the 2006 IEEE Globecom. He isalso an IEEE Distinguished Lecturer (Circuits and Systems Society, 2004-06,Communications Society, 2008-09). He served on as IEEE Transactions onWireless Communications Steering Committee Member (2007-2009) and itsChair (2009-2010). Dr. Ding received the 2012 IEEE Wireless CommunicationRecognition Award from the IEEE Communications Society and is a coauthorof the text: Modern Digital and Analog Communication Systems, 4th edition,Oxford University Press, 2009.

Xin Liu (S’99-M’03) received the Ph.D. degree inelectrical engineering from Purdue University, WestLafayette, IN, in 2002. She is currently an AssociateProfessor with the Computer Science Department,University of California (UC), Davis. Before joiningUC Davis, she was a Postdoctoral Research As-sociate with the Coordinated Science Laboratory,University of Illinois at UrbanaChampaign. Herresearch is on wireless communication networkswith a focus on resource allocation and dynamicspectrum management. Dr. Liu received the Best

Paper of Year Award from Computer Networks in 2003 for her work onopportunistic scheduling. She received the US National Science FoundationCAREER Award in 2005 for her research on smart radio-technology-enabledopportunistic spectrum utilization. She received the Outstanding EngineeringJunior Faculty Award from the College of Engineering, UC Davis, in 2005.

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