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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1

Cooperative Precoding with Limited Feedback forMIMO Interference Channels

Kaibin Huang and Rui Zhang

AbstractMulti-antenna precoding effectively mitigates theinterference in wireless networks. However, the resultant perfor-mance gains can be significantly compromised in practice if theprecoder design fails to account for the inaccuracy in the channelstate information (CSI) feedback. This paper addresses this issueby considering finite-rate CSI feedback from receivers to theirinterfering transmitters in the two-user multiple-input-multiple-output (MIMO) interference channel, called cooperative feedback,and proposing a systematic method for designing transceiverscomprising linear precoders and equalizers. Specifically, eachprecoder/equalizer is decomposed into inner and outer compo-nents for nulling the cross-link interference and achieving arraygain, respectively. The inner precoders/equalizers are further op-timized to suppress the residual interference resulting from finite-

rate cooperative feedback. Furthermore, the residual interferenceis regulated by additional scalar cooperative feedback signalsthat are designed to control transmission power using differentcriteria including fixed interference margin and maximum sumthroughput. Finally, the required number of cooperative precoderfeedback bits is derived for limiting the throughput loss due toprecoder quantization.

Index TermsInterference channels, multi-antenna system,limited feedback, cooperative communication.

I. INTRODUCTION

I

N wireless networks, multi-antennas can be employed to

effectively mitigate interference between coexisting links

by precoding. This paper presents a new precoding designfor the two-user multiple-input-multiple-output (MIMO) inter-

ference channel based on finite-rate channel-state-information

(CSI) exchange between users, called cooperative feedback.

Specifically, precoders are designed to suppress interference

to interfered receivers based on their quantized CSI feedback,

and the residual interference is regulated by additional coop-

erative feedback of power control signals.

A. Prior Work

Recently, progress has been made on analyzing the capac-

ity of the multi-antenna interference channel. In particular,

interference alignment techniques have been proposed for

achieving the channel capacity for high signal-to-noise ratios(SNRs) [1]. Such techniques, however, are impractical due to

Manuscript received January 11, 2011; revised July 26, 2011 and October17, 2011; accepted December 7, 2011. The associate editor coordinating thereview of this paper and approving it for publication was J. Tugnait.

K. Huang is with the School of Electrical and Electronic Engineering,Yonsei University, S. Korea (e-mail: [email protected]).

R. Zhang is with the Department of Electrical and Computer Engineering,National University of Singapore, and the Institute for Infocomm Research,A*STAR, Singapore (e-mail: [email protected]).

This work was supported in part by the National Research Foundation ofKorea under the grant 2011-8-0740, and the National University of Singaporeunder the grant R-263-000-589-133.

Digital Object Identifier 10.1109/TWC.2012.011812.110063

their complexity, requirement of perfect global CSI, and theirsub-optimality for finite SNRs. This prompts the development

of linear precoding algorithms for practical decentralizedwireless networks. For the time-division duplexing (TDD)

multiple-input-single-output (MISO) interference channel, it

is proposed in [2], [3] that the forward-link beamformers

can be adapted distributively based on reverse-link signal-to-

interference-and-noise ratios (SINRs). Targeting the two-user

MIMO interference channel, linear transceivers are designed in[4] under the constraint of one data stream per user and using

different criteria including zero-forcing and minimum-mean-squared-error. In [5], the achievable rate region for the MISO

interference channel is analyzed based on the interference-temperature principle in cognitive radio, yielding a message

passing algorithm for enabling distributive beamforming. As-

suming perfect transmit CSI, above prior work does not

address the issue of finite-rate CSI feedback though it iswidely used in precoder implementation. Neglecting feedback

CSI errors in precoder designs can result in over-optimisticnetwork performance.

For MIMO precoding systems, the substantiality of CSI

feedback overhead has motivated extensive research on ef-

ficient CSI-quantization algorithms, forming a field called

limited feedback[6]. Various limited feedback algorithms have

been proposed based on different principles such as linepacking [7] and Lloyds algorithm [8], which were applied to

design specific MIMO systems including beamforming [7] andprecoded spatial multiplexing [9]. Recent limited-feedback

research has focused on MIMO downlink systems, where mul-

tiuser CSI feedback supports space-division multiple access

[10]. It has been found that the number of feedback bits per

user has to increase with the transmit SNR so as to bound

the throughput loss caused by feedback quantization [11].Furthermore, such loss can be reduced by exploiting multiuser

diversity [12], [13]. Designing limited-feedback algorithmsfor the interference channel is more challenging due to the

decentralized network architecture and the growth of totalfeedback CSI. Cooperative feedback algorithms are proposed

in [14] for a two-user cognitive-radio network, where the

secondary transmitter adjusts its beamformer to suppress in-

terference to the primary receiver that cooperates by feedbackto the secondary transmitter. This design is tailored for MISO

cognitive radio networks and unsuitable for the general MIMOinterference channel, which motivates this work.

B. Contributions

The precoder design that maximizes the sum throughput of

the MIMO interference channel is a non-convex optimization

problem and remains open [17]. In practice, sub-optimal linear

1536-1276/12$31.00 c 2012 IEEE

his article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination

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2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

procoders are commonly used for their simplicity, which are

designed assuming perfect transmit CSI and based on various

criteria including interference suppression by zero-forcing orminimum transmission power for given received SINRs [17].

However, existing designs fail to exploit the interference-

channel realizations for suppressing residual interference due

to quantized cooperative feedback. In this work, we consider

the two-user MIMO interference channel with limited feed-

back and propose the decomposed precoder design that makesit possible for precoding to simultaneously regulate residual

interference due to precoder-feedback errors and enhancereceived signal power. For the purpose of exposition, we

consider two coexisting MIMO links where each link employs

transmit and receive antennas to support multiple datastreams. Linear precoding is applied at each transmitter and

enabled by quantized cooperative feedback. Channels are

modeled as i.i.d. Rayleigh block fading.

The contributions of this work are summarized as follows:

1) A systematic method is proposed for jointly designing

the linear precoders and equalizers under the zero-

forcing criterion, which decouples the links in theevent of perfect feedback. To be specific, precoders

and equalizers are decomposed into inner and outer

components that are designed to suppress residual in-

terference caused by feedback errors and enhance array

gain, respectively.

2) Additional scalar cooperative feedback, called interfer-

ence power control (IPC) feedback, is proposed for

controlling transmission power so as to regulate residual

interference. The IPC feedback algorithms are designed

using different criteria including fixed interference mar-

gin and maximum sum throughput.

3) Consider cooperative feedback of inner precoders of thesize with . Under a constraint onthe throughput loss caused by precoder quantization,

the required number of feedback bits is shown to scale

linearly with ( ) and logarithmically with thetransmit SNR as it increases.

Despite both addressing cooperative feedback for the two-

user interference channel, this work differs from [14] in thefollowing aspects. The current system comprises two MIMO

links whereas that in [14] consists of a SISO and a MISO

links. Correspondingly, this paper and [14] concern precoding

and transmit beamforming, respectively. Furthermore, this

work does not consider cognitive radio as in [14], leadingto different design principles. In particular, the current system

requires CSI exchange between two links while that in [14]

involves only one-way cooperative feedback from the primaryreceiver to the secondary transmitter.

C. Organization

The remainder of this paper is organized as follows. Thesystem model is discussed in Section II. The transceiver design

and IPC feedback algorithms are presented in Section III and

IV, respectively. The feedback requirements are analyzed in

Section V. Simulation results are presented in Section VI

followed by concluding remarks.

coop

erative

feedba

ck

local feedback

Fig. 1. The MIMO interference channel with data-link (local) and cooperativefeedback.

Notation: Capitalized and small boldface letters denote

matrices and vectors, respectively. The superscript representsthe Hermitian-transpose matrix operation. The operators [X]and [X] give the -th column and the (, )-th elementof a matrix X, respectively. Moreover, [X]: with represents a matrix formed by columns to of the matrixX. The operator ()+ is defined as ()+ = max(, 0) for R.

I I . SYSTEM MODEL

We consider two interfering wireless links as illustrated

in Fig. 1. Each transmitter and receiver employ and antennas, respectively, to suppress interference as well as

supporting spatial multiplexing. These operations require CSI

feedback from receivers to their interfering and intended trans-mitters, called cooperative feedback and data-link feedback,

respectively. We assume perfect CSI estimation and data-linkfeedback, allowing the current design to focus on suppressing

interference caused by cooperative feedback quantization.1 All

channels are assumed to follow independent block fading. Thechannel coefficients are i.i.d. circularly symmetric complex

Gaussian random variables with zero mean and unit variance,

denoted as (0, 1). Let H be a i.i.d. (0, 1)matrix representing fading in the channel from transmitter to receiver . Then the interference channels are modeled as

{H} with = and the data channels as {H}. Thefactor < 1 quantifies the path-loss difference between thedata and interference links.

Each link supports min(, ) spatial data streamsby linear precoding and equalization. To regulate residual

interference caused by precoder feedback errors, the total

transmission power of each transmitter is controlled by coop-

erative IPC feedback. For simplicity, the scalar IPC feedback

is assumed to be perfect since it requires much less overhead

than the precoder feedback. Each transmitter uses identical

transmission power for all spatial streams, represented by for transmitter with = 1, 2, and its maximum is denoted as

1The errors in data-link feedback decrease received SNRs, which can becompensated by increasing transmission power.


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HUANG and ZHANG: COOPE RATIVE PRE CODING WIT H LIMITED FE EDBACK FOR MIMO INTERFERE NCE CHANNE LS 3

max. Assume that all additive white noise samples are i.i.d.(0, 1) random variables. Let G and F denote the linearequalizer used by receiver and the linear precoder appliedat transmitter , respectively, which are jointly designed forseparating the spatial data streams of user with = 1, 2.The received SINR at receiver for the th stream is

SINR[] =

[G]H[F]2

1 + [G]HF2

,

= . (1)

The performance metric is the sum throughput defined as

=2

=1

=1

[log2

(1 + SINR[]

)]. (2)

III. TRANSCEIVER DESIGN

In this section, we propose a decomposition approach for

designing the transceivers (linear precoders and equalizers).

Using this approach, the precoder F is decomposed into an

inner precoder Fi and an outer precoder Fo.

2 Specifically,F = F

iF

o where F

i and F

o are

and

matrices, respectively, with being no smaller than thenumber of data streams and . Similarly, wedecompose the equalizer G as G = GiG

o where G

i is

an inner equalizerand Go an outer equalizerwith . For simplicity, inner/outer precoders andequalizers are constrained to have orthonormal columns. The

inner and outer transceivers are designed to suppress cross-link

interference and achieve array gain, respectively. In the fol-

lowing sub-sections, the transceivers, namely the inner/outer

precoders and equalizers, are first designed assuming perfectcooperative feedback and then modified to mitigate residual

interference caused by feedback quantization.

A. Transceiver Design for Perfect Cooperative Feedback

1) Inner Transceiver Design: A pair of inner precoder and

equalizer (Gi, Fi) with = are jointly designed under

the following zero-forcing criterion:

(Gi)

HFi

= 0, = . (3)The constraint aims at decoupling the links and requires that

+ max(, ). Under the constraint in (3), (Gi, Fi)with = are designed by decomposing H using thesingular value decomposition (SVD) as

H =

V

U

(4)

where the unitary matrices V and U consist of the

left and right singular vectors of H as columns, respec-

tively, and is a diagonal matrix with diagonal ele-

ments {

[]} arranged in the descending order, namely

[1] [2] [min(,)] . Note that is a tall matrix

if and a fat one if < . Definite the indexsets {1, 2, , } and {1, 2, , } such that = and = . The constraint in (3) is satisfied if = and the inner precoder and equalizer are chosen as

Gi

= [V] and Fi

= [U]. (5)

2The decomposed precoder designs have been proposed in the literaturefor other systems such as cellular downlink [18].

Consider the case of + . Each receiver has suf-ficiently many antennas for canceling cross-link interference

and thus cooperative feedback is unnecessary. Specifically,given an arbitrary fixed precoder Fi, the equalizer G

i chosen

as in (5) ensures that the zero-forcing criterion in (3) is

satisfied. Next, consider the case of < + . For thiscase, the receivers have insufficient degrees of freedom (DoF)

for canceling cross-link interference and link decoupling relies

on inner precoding that is feasible given that + .Therefore, with < ,3 cooperative feedback is in generalrequired and the specific design of inner transceiver forquantized cooperative feedback is discussed in the sequel.

2) Outer Transceiver Design: Given (Gi, Fi), the outer

pair (Go, Fo) are jointly designed based on the SVD of the

effective channel Ho =

Gi

HF

i after

inner precoding and equalization:

Ho

= Vo

o

(Uo

)

with the singular values[1],[2], ,[min(,)]

following the descending order. Note that the elements ofHo are i.i.d. (0, 1) random variables and their distribu-tions are independent of (Gi, F

i) since H is isotropic.

Transmitting data through the strongest eigenmodes of Hoenhances the received SNR. This can be realized by choosing

Go and F

o as

Go

= [Vo

]1: and Fo

= [Uo

]1:.

With perfect data-link feedback, the above joint design of

precoders and equalizers converts each data link into decoupled spatial channels. As a result, the receive SNR of

the -th data stream transmitted from transmitter to receiver is given by

SNR[] =

[], = 1, 2, , . (6)

Using the maximum transmission power, the sum capacity can be written as

=2

=1

=1

[log2(1 + max

[])

]. (7)

Last, it is worth mentioning that with , and fixed,maximizing and enhances the array gain of both

links and hence is preferred if perfect CSI is available at thetransmitters. However, for the case of quantized cooperative

feedback, small and allow more DoF to be used forsuppressing residual interference due to precoder-quantization

errors as discussed in the next section.

B. Transceiver Design for Quantized Cooperative Feedback

Consider the case of < + and mitigating cross-linkinterference relies on inner precoding with quantized feedback.

As mentioned earlier, cooperative feedback is unnecessary if

+ .3This condition usually holds for cellular downlink where a base station

has more antennas than a mobile terminal.


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1) Inner Transceiver Design: In this section, the design of

inner precoders and equalizers in (5) is modified to suppress

residual interference caused by precoder feedback errors.

First, given the inner equalizer Gi in (5), the inner precoderFi in (5) is particularized under the criterion of minimizing

residual interference power. Recall that the precoding at trans-mitter is enabled by quantized cooperative feedback of Fifrom receiver with

= . Let Fi denote the quantized

version of Fi that is also an orthonormal matrix. Define theresultant quantization error as [19]

= 1 (Fi)

Fi2F

, = 1, 2 (8)

where 0 1. The error is zero in the case ofperfect cooperative feedback, namely Fi = F

i. A nonzero

error results in violation of the zero-forcing criterion in (3)

G

i

HF

i

= 0, = . (9)

Given that the inner equalizer designed for perfect feedback is

applied, the residual interference at the output of the equalizer

at receiver has the power

= (Go)(Gi)HFiFo2F, = . (10)

It is difficult to directly optimize Fi for minimizing .Alternatively, Fi can be designed for minimizing an upper

bound on obtained as follows. By rearranging eigenvaluesand eigenvectors, the SVD of H in (4) for > can berewritten as

H

=Gi

B () 0 0

0

() 0 C Fi

(11)

where the diagonal matrices () and () have the diagonal

elements {

[] } and {

[] 1 , / },

respectively. It follows from (10) and (11) that

=

(Go)

()C

F

i

Fo

2

F

(Go)

()C

F

i

Uo

2

F

= (Go)()CFi

2

F(12)

=

=1

=1

[Go]

()C

[F

i

]

2

=1

=1

[Go]2

()C

[F

i]

2

(13)

=

()C

F

i

2

F

(14)

CF

i

2

F

max

[] (15)

where (12) holds since the columns of Uo form a basis

of the space C, (13) applies Schwarzs inequality and (14)

follows from that the columns of Go have unit norms. Next,

the precoder quantization error in (8) can be written as

=1

=1

(1 [Fi]Fi2F

)

=1

=1

[Fi]C2F (16)

=1

CFi2F (17)

where (16) holds since [Fi, C] forms a basis of the spaceC. Substituting (17) into (15) gives

max

[]. (18)

Minimizing the right-hand side of (18) gives that the columns

of Gi should be left eigenvectors of H corresponding

to the smallest singular values. Therefore, the innertransceiver design in (5) for perfect CSI is particularized as

Gi = [V] with = { + 1, , }

Fi

= [U] with = (19)

and Fi is obtained by quantizing Fi such that the quantization

error is minimized [19]. Then (18) can be simplified as

[+1] , = . (20)Next, if > , besides DoF required for inner

equalization, a receiver has ( ) extra DoF that can beused to suppress residual interference. This can be realized

at receiver by redesigning the inner equalizer Gi with

the resultant design denoted as Gi

. To this end, the matrixHF

i is decomposed by SVD as

HFi

= VU

where the singular values along the diagonal of are

denoted as {

[] min(, )} and arranged in

the descending order. The inner equalizer Gi that minimizes

the residual interference should be chosen to comprise theleft eigenvectors of HF

i that correspond to the smallest

singular values and hence Gi = [V](+1): . Another

interpretation of this design is that the inner equalizer Gi is

directed towards the null space ofFi. The resultant residualinterference power after inner precoding is upper bounded as

= GiHFiFo2F GiHFi2F

=

=+1

[]. (21)

Last, if = , the design of Gi remains unchanged and

Gi = Gi.

2) Outer Transceiver Design: Let Go and Fo denote the

outer equalizer and precoder for the case of quantized coopera-tive feedback. The outer transceiver (Go and F

o) is designed

similarly as its perfect-feedback counterpart in Section III-A2.


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Decompose the effective channel matrix(

Gi

)HF

i

after inner precoding/equalization as(G

i

)HF

i

= Vo

o

(Uo

) (22)

where the diagonal matrix o

contains singular values

{

}in the descending order. To maximize the received

SNRs, the outer equalizer and precoder are chosen as

Go

= [Vo

]1: and Fo

= [U]1:.

The corresponding sum throughput follows from (2) as

=

2=1

=1

log2

1+

1 + [G]HF2

(23)

where = , G = GiGo and F = FiFo.

C. Discussion

In this section, we discuss the robustness of the proposed

quantized feedback precoder by comparison with a conven-

tional design without cooperative feedback. To be specific,

the baseline design is the well-known single-user transceiver

design that provides no cooperative feedback, where interfer-

ence is treated as noise and all spatial DoF are applied to

maximize array gain [20]. Let the data-channel matrix Hbe decomposed by SVD as H = VU

with

{1, 2}. The precoder F and receiver G for thebaseline case as given below transmit data through the strongest eigenmodes of H [20]

F = [V]1: and G = [U]1:. (24)

By using the maximum transmission power, the corresponding

sum throughput is given as

= 2

=1

log2

1 +

max [11], 21 + max[G1]1H12F22

2=1

log2

1 +

[11], 2[G1]H12F22

. (25)

Similarly as (20), it can be proved that the interference power

for the -th data stream, [] , received at receiver is upper

funded as

[] [+1] , = . (26)From (23) and (26) and also using the maximum transmissionpower, the sum throughput for the proposed design can belower bounded as

2=1

log2

1+

max[]11

1 + max[+1]12 2

.

(27)

By comparing (25) and (27), it can be observed that isbounded as max increases but can grow with increasingmax if the quantization errors {} are regulated by adjustingthe number of feedback bits based on max (see Section V).

IV. INTERFERENCE POWER CONTROL FEEDBACK

In this section, we consider the case of = wherereceivers have no extra DoF for suppressing residual in-terference. An alternative solution is to adjust transmission

power for increasing the sum throughput. Two IPC feedbackalgorithms for implementing power control are discussed in

the following sub-sections.

A. Fixed Interference Margin

Receiver sends the IPC signal, denoted as , to trans-mitter/interferer for controlling its transmission power as

= min(, max), = 1, 2. (28)

The scalar is designed to prevent the per-stream interfer-ence power at receiver from exceeding a fixed margin with > 0, namely

[] for all 0 . A sufficient

condition for satisfying this constraint is to bound the right

hand side of (26) by . It follows that

=

[+1]

, = . (29)

Given , a lower bound IM on the sum throughput , calledachievable throughput, is obtained from (23) as

IM =2

=1

=1

log2

1 +

min(, max)[]

1 +

. (30)

It is infeasible to derive the optimal value of for maximiz-ing IM in (30). However, for max being either large or small,simple insight into choosing can be derived as follows.The residual interference power decreases continuously with

reducing max. Intuitively, should be kept small for smallmax. For large max, the choice of is less intuitive sincelarge lifts the constraints on the transmission power butcauses stronger interference and vice versa. We show below

that large is preferred for large max. This requires the resultin [21, Theorem 1] paraphrased as follows.

Lemma 1 ([21]). Let H denote a 1 2 matrix of i.i.d.(0, 1) elements with 1 2. The cumulative distribution

function of the -th eigenvalue of the Wishart matrixH

H

can be expanded as

Pr( < ) = + (), = 1, , 2

where = (1+1)(2+1) and = 1A()B()

with =2

=1(1 )!(2 )!. The matrix A() isdefined for = 1 as

[A()] = (1 2 + + + 2(2 ))!with , = 1, , ( 1) and A(1) = I, B() is defined

for = 2 as

[B()] =2

[(1 2 + + )2 1](1 2 + + )with , = 1, , (2 ) and B(2) = I.To simplify notation, we re-denote (, ) for 1 = and2 = as (, ) and those for 1 = 2 = as (, ).


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Using the above result, we obtain the following lemma that is

proved in the appendix.

Lemma 2. Given finite-rate cooperative feedback and for

large max, the achievable throughput is

IM = 2=1

log2

1+

(1 + )+11

+ (1) .

It can be observed from the above result that the first orderterm of IM attains its maximum as . However, thisterm is finite even for asymptotically large max and , whichis the inherent effect of residual interference.

B. Maximum Achievable Throughput

In this section, an iterative IPC algorithm is designed forincreasing the sum throughput in (2). Since is a non-concave function of transmission power, directly maximizing

does not yield a simple IPC algorithm. Thus, we resort tomaximizing a lower bound ST (achievable throughput) on instead, obtained from (20) and (23) as ST = [] with

=

=1

log2

1 +

1[]11

1 + 2[+1]12 2

+

log2

1 +

2[]22

1 + 1[+1]21 1

.

(31)

The corresponding optimal transmission power pair is

(1 , 2 ) = arg max

1,2[0,max](1, 2). (32)

The objective function remains non-concave and its max-imum has no known closed-form for > 1. Note that for

= 1, it has been shown that the optimal transmission-powerpair belongs to the set {(0, max), (max, 0), (max, max)}[22]. For the current case of > 1, inspired by the messagepassing algorithm in [5], a sub-optimal search for (1 ,

2 )

can be derived using the fact that

(1 , 2 )

= 0, = 1, 2.

To this end, the slopes of are obtained using (31) as

(1, 2)

= + (33)

where

= log2

=1

[]

1 + [+1] +

[]

= log2

=1

[+1]

1 + [+1] +

[]

=log2

2

[+1]

1 + [+1]

.

Note that based on estimated CSI, has to be computed at and (, ) at with = . Therefore, using (33),we propose the following iterative IPC feedback algorithm.

Algorithm 1:

1) Transmitter 1 and 2 arbitrarily select the initial valuesfor 1 and 2, respectively.

2) The transmitters broadcast their choices of transmission

power to the receivers.

3) Given (1, 2), the receiver 1 computes (1, 2, 2) andfeeds back 1 and (2 2) to transmitter 1 and 2,respectively. Likewise, receiver 2 computes (2, 1, 1)and feeds back 2 and (1 1) to transmitter 2 and1, respectively.

4) Transmitter 1 and 2 update 1 and 2, respectively,

using (33) and the following equation

(+1) = min

{() +

(1, 2)

+, max

}

where is the iteration index and a step size.5) Repeat Steps 2) 4) till the maximum number of

iterations is performed or the changes on (1, 2) aresufficiently small.

Note that the IPC-feedback overhead increases linearly with

the number of iterations. By choosing an appropriate step size,

the convergence of the above iteration is guaranteed but theconverged throughput need not be globally maximum.

V. PRECODER FEEDBACK REQUIREMENTS

In this section, consider the case of = as in the pre-ceding section and the number of bits for cooperative precoderfeedback is derived under a constraint on the throughput loss

due to precoder quantization.The expected precoder quantization errors is related to the

number of feedback bits as follows. Consider the codebook of orthonormal matrices that is used by each receiverto quantize the inner precoder for the corresponding interferer.Given , the quantization error in (8) is minimized byselecting the quantized precoder Fi

as

Fi

= arg minW

1 W

Fi2F

, = 1, 2. (34)

The above operation is equivalent to minimizing the Chordal

distance between Fi and Fi: [7]

Fi = arg min

W(W, F

i) (35)

where the chordal distanc is defined as

(W, Fi

) =1

2WW Fi(Fi)F

= WFi

2F

.

The codebook selection in (35) motivates the codebook design

based on minimizing the maximum chordal distance between

every pair of codebook members [9]. For such a design, the

expected quantization error can be upper bounded as [24]

[] 1

1 2

+ (2

)1 (36)

where = ( ), the number of feedback bits =log2 , =

1!

=1

()!()!

, (0, 1) is a given constant,and denotes the gamma function.

We consider a constraint on the minimum throughput loss

due to quantized cooperative precoder feedback, namely

= max

() (37)


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with and given in (7) and (23), respectively, and de-notes a power-control policy. To satisfy the constraint with > 0, it is sufficient to equate the following upper boundon to :

(max, max) (38)where (max, max) corresponds to a sub-optimal power-

control policy that fixes the power of both transmitters atthe maximum. The above upper bound has a similar form asthe throughput loss for multi-antenna downlink with limited

feedback as defined in [19]. Thus, the following result can be

proved following a similar procedure as [19, Theorem 2].

Corollary 1. For large max, choosing the number of bits forcooperative precoder feedback as

= log2(max) log2(

2

2 1)

+ (39)

ensures that

+ (1), max (40)where = log2

( 1 )[+1]

1

.

A few remarks are in order:

1) For large max, log2 max. For small max, thenetwork is noise limited and the number of precoder

feedback bits can be kept small.

2) For the case offixed interference margin, it can be also

proved that log2 max for large max followinga similar procedure as Corollary 1.

3) The upper bound on the capacity loss approaches asmax

increases.

4) The feedback-bit scaling obtained in [19] for the MIMO

downlink system is similar to that in (39) despite the dif-

ference in system configuration. Specifically, it is shown

in [19, Theorem 2] that the number of precoder-feedbackbits per user should scale as ( )log for large so as to constrain the sum-throughput loss,where and are the numbers of antennas at thebase station and each mobile, respectively, and is thetotal transmission power at the base station. The above

similarity rises from the fact that both the proposed

precoding and the block-diagonalization precoding in

[19] are designed using the zero-forcing criterion to null

multiuser interference.

VI . SIMULATION RESULTS

In the simulation, the codebook for quantizing the feedback

precoders is randomly generated as in [28] and the system

performance is averaged over a larger number of codebook

realizations. The simulation parameters are set as follows

unless specified otherwise. The numbers of antennas at eachtransmitter and receiver are = 6 and = 3, respectively.The number of data stream per user is = 2. The size ofinner precoder is fixed as 6

3. The path-loss factor is set

as = 0.5. Power control based on interference margin uses = 2. The number of cooperative-feedback bit is = 8.

-5 0 5 10 15 20

Transmit SNR (dB)

4

6

8

10

12

14

16

Su

mT

hroughput(bit/s/Hz)

inner equalizer size = 3x3

inner equalizer size = 3x2

Fig. 2. Effect of the width of inner equalizer matrices, , on the sumthroughput for different SNRs. The transmission power is fixed at max.

-5 0 5 10 15 20

Transmit SNR (dB)

4

6

8

10

12

14

SumT

hroughput(bit/s/Hz)

B = 4, 6, 8, 10 bit

Fig. 3. Comparison of achievable sum throughput of the proposed precoding

design for different numbers of cooperative-feedback bit . The transmissionpower is fixed as max and the inner equalizer has the size of= 33.

A. Performance Evaluation

The size of inner equalizer determines the allocation of DoF

at each receiver for mitigating residual interference and for

enhancing array gain. The sum throughput for two different

inner-equalizer sizes, namely 3 2 and 3 3, is comparedin Fig. 2 with transmission power of each transmitter fixed as

max. The larger inner precoder (3 3) allocates more DoFfor achieving array gain and it can be observed to increase

the throughput for low SNRs where noise dominates residual

interference. However, the smaller inner precoder is preferred

for high SNRs since it allows more DoF for mitigating residual

interference.

Fig. 3 compares the sum throughput of the proposed

transceiver design for the cooperative-feedback bit ={4, 6, 8, 10}. The transmission power of each transmitter isfixed as max and the inner equalizer has the size of =3 3. It can be observed that the increment of every 2cooperative-feedback bits increases the sum throughput by

about 0.7 bit/s/Hz for high SNRs.

Fig. 4 compares the sum throughput of different IPC feed-back algorithms where the inner equalizer size is set as 3 3.Residual interference between links is observed to decrease the


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throughput dramatically with respect to perfect CSI feedback.

For large max, the IPC feedback Algorithm 1 designedfor maximizing the achievable throughput is observed toprovide substantial gain over that based on fixed interference

margin. Moreover, iterative IPC feedback using Algorithm 1are observed to give significant gain only at high SNRs.

B. Comparison with a Conventional Transceiver DesignThe proposed precoding algorithm is compared with the

conventional interference coordination [17] in terms of sum

throughput with quantized cooperative feedback. The interfer-

ence coordination algorithm attempts to align the interferenceby precoding such that at each receiver interference is ob-

served only over the last antennas and the signals receivedover the first () antennas are free of interference. To thisend, the precoder of user , denoted as F, is chosento be orthogonal to the channel sub-matrix [H]

row1:()

with = where [X]row: denotes a sub-matrix comprisingrows to of a matrix X; the quantized version F of F

is fed back from receiver to transmitter for precoding.Furthermore, the receiver G of user decouples the datastreams by zero-forcing:

G =

([H]

row1:()F

)(

[H]row1:()F

) ([H]

row1:()F

)1.

Also considered in the comparison is the case of no CSIT

where the precoder {F} are arbitrarily chosen to be indepen-dent with all channels. The achievable sum throughput for theproposed and conventional algorithms are compared in Fig. 5.

The transmission power of each transmitter is max; for theproposed deign, the inner equalizer size is set as 33. It can beobserved from Fig. 5 that the proposed cooperative-feedback

design yields dramatic throughput gains over the conventional

algorithms. In particular, the gain over interference coordi-

nation is as large as about 4 bit/s/Hz for high SNRs. Theperformance gains of the proposed design result from the

joint tuning of precoders and equalizers for simultaneously

suppressing residual interference and harvesting diversity gain.

VII. CONCLUSION

We have proposed a systematic design of linear precodersand equalizers for the two-user MIMO interference channelwith finite-rate cooperative precoder feedback. This design

suppresses residual interference due to feedback precoder

quantization. Building upon the above design, we have further

proposed scalar cooperative feedback algorithms for control-

ling transmission power based on different criteria including

fixed interference margin and maximum sum throughput.

Finally, we have derived the scaling of the number of co-

operative precoder-feedback bits under a the constraint onthe sum throughput loss. Possible extensions of the current

work include generalizing the proposed algorithms to the

interference channel with an arbitrary number of users and

relaxing the current zero-forcing criterion on the precoder

design.

0 5 10 15 20

Maximum Transmit SNR (dB)

8

10

12

14

16

Su

mT

hroughput(bit/s/Hz)

fixed interf. margin

max throughput(1 iteration)

perfect CSI feedback

max throughput(0 iteration)

Fig. 4. Comparison of achievable sum throughput between different IPCfeedback algorithms. The inner equalizer has the size of = 3 3.For IPC feedback Algorithm 1 that maximizes achievable throughput, the stepsize for updating transmission power is max; the initial transmissionpower is set as max/2.

-5 0 5 10 15 20

Transmit SNR (dB)

4

6

8

10

12

14

16

SumT

hroughput(bit/s/Hz)

proposed design

interf. coordination

no CSIT

Fig. 5. Comparison of sum throughput between the proposed and two con-ventional precoding algorithms for the MIMO interference channel, namelyinterference coordination and the case of no CSIT. The transmission poweris fixed at max. Moreover, for the proposed deign, the inner equalizer sizeis set as 3 3.

APPENDIX

Lemma 2 is proved as follows. For convenience, the achiev-

able throughput in (30) can be written as

IM =2

=1

=1

[] (41)

where

[] =

log2

1 +

min(, max)[]

1 +

. (42)

Expand [] as

[] =

log2

1 +

[]

1 +

max

Pr( max) +

log2

1 +

max[]

1 +

Pr( > max).(43)


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Using (29), the first term 1 of [] in (43) is rewritten as

1 =

log2

1 +

(1 + )+1

3 (44)

with 3 defined as

3 =

max

0

log2

1 +

(1 + )

+1()]

.

As max , 3 can be simplified as

3 =

max

0

(1) + log2

1

+1()

()=

max

0

(1)[+122 + (

22)]

= (

2+1

max)

(45)

where () is obtained using Lemma 1 and log21

= (1)

for 0. Similarly, we obtain the second term 2 of []in (43) as

2 = (

2+1max

), max . (46)

Substituting (44), (45), and (46) into (43) and then (41) yields

the desired result.

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Kaibin Huang (S05M08) received the B.Eng.(first-class hons.) and the M.Eng. from the NationalUniversity of Singapore in 1998 and 2000, respec-tively, and the Ph.D. degree from The University ofTexas at Austin (UT Austin) in 2008, all in electricalengineering.

Since Mar. 2009, he has been an assistant pro-fessor in the School of Electrical and ElectronicEngineering at Yonsei University, Seoul, Korea.From Jun. 2008 to Feb. 2009, he was a PostdoctoralResearch Fellow in the Department of Electrical

and Computer Engineering at the Hong Kong University of Science andTechnology. From Nov. 1999 to Jul. 2004, he was an Associate Scientistat the Institute for Infocomm Research in Singapore. He frequently serveson the technical program committees of major IEEE conferences in wirelesscommunications. Recently, he is the technical co-chair of IEEE CTW 2013and the track chairs of IEEE Asilomar 2011 and IEEE WCNC 2011. He is aneditor for the IEEE W IRELESS COMMUNICATIONS LETTERS and also the

Journal of Communication and Networks. Dr. Huang received the OutstandingTeaching Award from Yonsei, Motorola Partnerships in Research Grant, theUniversity Continuing Fellowship at UT Austin, and the Best Paper awardat IEEE GLOBECOM 2006. His research interests focus on multi-antennalimited feedback techniques and the analysis and design of wireless networks

using stochastic geometry.


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Rui Zhang (S00M07) received the B.Eng. (First-Class Hons.) and M.Eng. degrees from the NationalUniversity of Singapore in 2000 and 2001, respec-tively, and the Ph.D. degree from the Stanford Uni-versity, Stanford, CA USA, in 2007, all in electricalengineering.

Since 2007, he has worked with the Institute forInfocomm Research, A*STAR, Singapore, where heis now a Senior Research Scientist. Since 2010, hehas also held an Assistant Professorship position

with the Department of Electrical and Computer En-gineering at the National University of Singapore. He has authored/coauthoredover 100 internationally refereed journal and conference papers. His currentresearch interests include wireless communications (e.g., multiuser MIMO,cognitive radio, cooperative communication, energy efficiency and energy har-vesting), wireless power and information transfer, smart grid, and optimization

theory for applications in communication and power networks.Dr. Zhang was the co-recipient of the Best Paper Award from the IEEE

PIMRC (2005). He was Guest Editors of the EURASIP Journal on AppliedSignal Processing Special Issue on Advanced Signal Processing for CognitiveRadio Networks (2010), the Journal of Communications and Networks (JCN)Special Issue on Energy Harvesting in Wireless Networks (2011), and theEURASIP Journal on Wireless Communications and Networking Special Issueon Recent Advances in Optimization Techniques in Wireless CommunicationNetworks (2012). He has also served for various IEEE conferences asTechnical Program Committee (TPC) members and Organizing Committee

members. He was the recipient of the 6th IEEE ComSoc Asia-Pacifi

c BestYoung Researcher Award (2010), and the Young Investigator Award ofNational University of Singapore (2011). He is an elected member for IEEESignal Processing Society SPCOM Technical Committee.


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