Constrained Group Decoder for Interference Channels

Constrained Group Decoder for InterferenceChannels

Chen Gong∗, Omar Abu-Ella∗, Xiaodong Wang∗, Ali Tajer†,∗Department of Electrical Engineering, Columbia University, New York, NY 10027, USA

Email: {chengong, omar, wangx}@ee.columbia.edu,†Department of Electrical Engineering, Princeton University, B-311 E-Quad, Olden Street, Princeton, NJ 08544, USA,

Email: [email protected].

Abstract— We provide a general overview of the recentdevelopment on the constrained group decoder (CGD) forinterference channels. We first consider the CGD for a sim-ple system of interference channels with no feedback fromthe receivers to the transmitters, where the transmittersemploy fixed-rate channel codes and modulation schemes.We provide simulation results to show that, compared withthe interference suppression scheme that maximizes thesignal-to-interference-plus-noise ratio (SINR) at the receiverand the one that minimizes the interference leakage tothe signal subspace (i.e., interference alignment), the groupdecoder offers an signal-to-noise ratio (SNR) gain of around5dB at the outage probability 3 × 10−2 and bit error rate(BER) 2 × 10−3. We then discuss the application of theCGD to more sophisticated systems that allow feedbackfrom receivers to transmitters. More specifically, we reviewthe recently developed practical coding schemes for theconstrained partial group decoder (CPGD) and the groupdecoder for the multi-relay-assisted interference channels.Compared with the interference alignment approach, thegroup decoding paradigm not only provides substantialperformance gains, but also eliminates the need of feedingback the channel state information of all users to thetransmitters, thus significantly reducing the system signalingoverhead.

Index Terms— interference channel, group decoder, channelcodes, relay.

I. INTRODUCTIONS

The interference channel is a fundamental buildingblock of wireless networks. Due to the ever-shrinkingnetwork sizes and the increasing demands for achievinghigher spectral efficiency, the emerging wireless networkswill operate in an interference-limited regime. Motivatedby this, various recent developments for further under-standing the fundamental limits of interference channelsbased on interference alignment have appeared [1], [2],[3], [4]. The idea of interference alignment is to processthe transmit signals at the transmitters such that aftersome projection each receiver only sees the signal from itsdesignated transmitter, but not the interference. From thetheoretical point of view, it achieves the optimal degree of

This paper is based on “Group Decoding for Interference Channels,”by C. Gong, A. Tajer, and X. Wang, which appeared in the Proceed-ings of the International Conference on Computing, Networking andCommunications, Maui, Hawaii, USA, Feb.2012. c© 2005 IEEE.

Manuscript received February 15, 2011; revised May 15, 2011;accepted June 15, 2011.

freedom in the asymptotic regime of high signal-to-noiseratio (SNR) [1].

On the other hand, from the practical point of view,various signal processing schemes for interference align-ment have been proposed in [5], [6], [7]. In [5], a practicaldistributed interference alignment scheme is proposed,which requires only local channel knowledge. However,the method has a very high computational complexity andit relies on the assumption of channel reciprocity whichmay not hold true in practice. Thus, beamforming-basedapproaches are still considered viable and practical al-ternatives to interference alignment. Recently, two beam-forming methods have been proposed for interferencealignment, based on maximizing the received signal-to-interference-plus-noise ratio (SINR) [6] and minimizingthe interference leakage to the signal subspace at thereceivers [7], respectively.

Note that the above approaches are all based on the ideaof interference suppression, where all interference shouldbe suppressed and treated as noise through transmitter-endsignal processing. On the other hand, it is well-understoodthat while a receiver is not ultimately interested in de-coding the messages of the interferers, decoding them(fully or partially) is often advantageous for recoveringits desired message [8]. Motivated by this premise somerecent works on interference channels propose that eachreceiver should partition the interfering transmitters intotwo groups; one group to be decoded along with thedesignated transmitter and the other to be treated asGaussian noise [9], [10], [11].

In this paper we provide an overview of the recentdevelopment on such group decoding approach to inter-ference channels. We first consider a simple interferencechannel system without feedback from the receivers tothe transmitters, where due to the lack of feedback allinterference management is performed at the receiver side.We also assume that each transmitter employs a fixed-rate channel code and modulation scheme. Based on theoptimal group decoding order obtained from a greedyalgorithm, each receiver employs a constrained groupdecoder (CGD) to successively decode some interferersand the desired user. Simulation results show that thegroup decoder outperforms the SINR maximization [6]and interference leakage minimization [7] schemes byapproximately 5dB at the outage probability 3 × 10−2

382 JOURNAL OF COMMUNICATIONS, VOL. 7, NO. 5, MAY 2012

© 2012 ACADEMY PUBLISHERdoi:10.4304/jcm.7.5.382-390

and bit error rate (BER) 2× 10−3.We further review two extensions of the above group

decoder, namely, the constrained partial group decoder(CPGD) for a K-user interference channel and the CGDfor a multi-relay-assisted interference channel. For theCPGD, we outline a practical coding scheme for a K-userinterference channel, which further divides each trans-mitter into multiple layers, to provide the receivers withthe freedom of deciding on which interferers to decodeas well as what fraction of each interferer need to bedecoded. For the CGD in multi-relay-assisted interferencechannel, we discuss the rate allocation among the usersin a half-duplex multi-relay-assisted K-user interferencechannel.

The remainder of this paper is organized as follows.In Section II, we describe the K-user fully connectedinterference channel. In Section III, we discuss the CGDwith fixed-rate channel codes and modulation scheme.In Sections IV and V, we overview the practical CPGDcoding scheme and the CGD for the multi-relay-assistedinterference channel, respectively. Finally, Section VIconcludes this paper.

II. FULLY CONNECTED K-USER INTERFERENCECHANNEL

Consider a fully connected K-user interference chan-nel. We denote the wireless channel from the jth transmit-ter to the ith receiver by hi,j for i, j ∈ K 4

= {1, . . . ,K}.We assume quasi-static block fading channels such thatthe channel coefficients are fixed during the transmissionof a symbol block and change to some other independentstates afterwards. By defining xj as the transmitted signalby the jth transmitter, the received signal by the ith

receiver is given by

yi =K∑

j=1

hi,j xj + vi, for i ∈ {1, . . . , K} , (1)

where vi ∼ NC(0, σ2) accounts for the additive whiteGaussian noise (AWGN) at the ith receiver. The termhi,i xi contains the intended signal for the ith receiverand the remaining summands constitute interference andnoise. Denote the equal transmission power of all trans-mitters as P

4= E(|xj |2). Assume that each receiver i is

only interested in decoding the message of transmitter i,but aware of the codebooks used by all the transmitters.

Note that the single-antenna interference channelmodel can be extended into the multiple-input-single-output (MISO), single-input-multiple-output (SIMO), andmultiple-input-multiple-output (MIMO) models. In thefollowing we describe the SIMO interference channel,where each receiver i employs Ni receiving antennas.Let hi,j be the channel from the jth transmitter to the ith

receiver and yi be the received signals of the ith receiver.We have the following

yi =K∑

j=1

hi,j xj + vi, for i ∈ {1, . . . ,K} (2)

where vi ∼ NC(0, σ2I) accounts for the additive whiteGaussian noise (AWGN) at the ith receiver. Denote theequal transmission power of all transmitters as P

4=

E(|xj |2).

III. CONSTRAINED GROUP DECODER

We consider the CGD for the above K-user interferencechannel, where there is no feedback from the receiversto the transmitters and the transmitters employ fixed-ratechannel codes and modulation schemes, and each user isaware of the coding scheme employed by all other users.

A. Successive Group Decoder

We formalize the successive group decoder (SGD)employed by the receivers. Each stage a subset of theusers are jointly decoded, after subtracting the alreadydecoded users from the received signal, and by treatingthe remaining users as AWGN. In order to control thedecoding complexity, we constrain the number of usersbeing jointly decoded at each stage to be at most µ. LetRj be the rate of transmitter j, and RA = [Rj ]j∈A forA ⊆ K.

For each receiver i, we say that a given orderedpartition Gi 4

= {Gi1, . . . ,Gi

pi,Gi

pi+1} of K is valid if allthe following conditions are satisfied.

1) |Gim| ≤ µ for m ∈ {1, . . . , pi};

2) Transmitter i, i.e., {xi}, is included in Gipi

;3) The rate vector RGi

mis decodable at the mth stage

of the successive decoding procedure for m ∈{1, . . . , pi}.

For a given valid partition Gi of K, the ith receiverdecodes the users included in {Gi

1, . . . ,Gipi} successively

in pi stages while those in Gipi+1 are always treated

as AWGN. More specifically, in the mth stage, the ith

receiver jointly decodes the users in Gim by treating

{Gim+1, . . . ,Gi

pi+1} as AWGN and then subtracts thedecoded messages in Gi

m from the received signal.For the K-user SIMO interference channel [c.f. (2)]

under consideration, we let Hi4= [

√Phi,j ]j∈K and

Hi,A4= [

√Phi,j ]j∈A, and xA

4= [xj ]j∈A for A ⊆ K.

Given the ordered partition Gi 4= {Gi

1, . . . ,Gipi

,Gipi+1},

receiver i performs the following pi-stage successivedecoding.

1) Initialize m = 1;2) Compute the covariance matrix of the interference-

plus-noise as follows,

Σi,m = σ2I +∑

j∈∪pi+1k=m+1Gi

j

hi,jhHi,j . (3)

Then, decode the users in Gim from

ri,m = Σ− 12

i,myi = Σ− 12

i,mHi,Gim

xGim

+ vi,m, (4)

where vi,m ∼ NC(0, I) is the additive whiteGaussian noise with unit variance.

JOURNAL OF COMMUNICATIONS, VOL. 7, NO. 5, MAY 2012 383

© 2012 ACADEMY PUBLISHER

3) After decoding Gim, receiver i subtracts the decoded

signals xGim

from yi via updating yi ← yi −Hi,Gi

mxGim

.4) If m = pi+1 then stop; otherwise go to step 2.

We employ channel codes with error decoding detectioncapability, e.g., low-density parity-check (LDPC) codes. Ifa decoding error event in step 2 is detected, stop decodingand claim a decoding outage.

In Fig. 1, we show a simple illustrative example ofthe SGD for an interference channel with four transceiverpairs. We show the SGD at receiver 3. As shown in Fig. 1,user 2 is decoded in the first stage; and users 3 and 4are decoded in the second stage. After the two stages,user 3 are decoded and the decoding terminates. We havep3 = 2, Q3

1 = {2}, Q32 = {3, 4}, Q3

3 = {1}.

3

1{2}Q

3

2{3, 4}Q

3

3{1}Q

Figure 1. A simple illustrative example for SGD.

B. Optimal Successive Group Decoder

In the successive decoding, a rate outage is defined asan event that in some stage the rates of decoded usersfall out of the achievable rate region assuming that theusers employ Gaussian codebooks. More specifically, fortwo disjoint subsets A,B ⊆ K, and the spectrum rateR of the users, we define the following rate margin fordecoding A while treating B as noise,

ε(Hi,A,B, R) , minD⊆A,D6=φ

{∆(Hi,D,B, R)} (5)

for A 6= φ and ε(Hi, ∅,B,R) = 0, where

∆(Hi,D,B, R)

, log∣∣∣∣I + HH

i,D(I + Hi,BHH

i,B)−1

Hi,D

∣∣∣∣−∑

j∈DRj .

(6)

Moreover, for a partition Gi 4= {Gi

1, . . . ,Gipi

,Gipi+1},

we define

ε(Hi,Gi, R) , min1≤m≤pi

{ε(Hi,Gi

m,K \ ∪mj=1Gi

j , R)}

,

(7)as the minimum rate margin through the pi-stage succes-sive decoding. The rate outage at receiver i is equivalentto ε(Hi,Gi,R) < 0. To minimize the rate outageprobability, each receiver i needs to find a partition Gi

which maximizes ε(Hi,Gi,R), i.e., solving the followingoptimization problem

maxGi

ε(Hi,Gi, R). (8)

Algorithm 1 performs the optimal group partition in agreedy manner. In each step, assuming the undecoded setto be S , receiver i finds the optimal set of the decodedusers G∗ as follows

G∗ = arg maxG⊆S,|G|≤µi,G6=∅

ε(Hi,G,S \ G,R). (9)

Based on the submodularity of the achievable rate func-tion, it can be proved that such greedy partition leads toan optimal solution to (8). If in some step, the selectedG∗ leads to the rate margin ε(Hi,G∗,S \ G∗, R) < 0,then declare a rate outage event.

The optimal group search problem (9) can be solvedusing simple exhaustive search via enumerating allpossible nonempty set G ⊆ S with |G| ≤ µi andfinding the one which maximizes ε(Hi,G,S \ G, R).The exhaustive method can be applied for small µi,e.g., µi = 1 or 2, which is the case for most practicalscenarios. For large µi, (9) can be efficiently solvedusing Algorithm 2 as follows.

Algorithm 1 Greedy Partitioning for Fixed Rates R1: Initialize S = K, Gi

opt = φ.2: Identify a group

G∗ = arg maxG⊆S,|G|≤µi,G6=∅ ε(Hi,G,S \ G,R).3: If ε(Hi,G∗,S \ G∗, R) < 0, then4: declare a rate outage and stop;5: else6: update S ← S\G∗ and Gi

opt ← {Giopt,G∗};

7: if i ∈ G∗, then8: output Gi

opt, and stop;9: else

10: go to Step 2;11: end if12: end if

Algorithm 2 Selecting an Optimal Group

1: Let S 4= {G ⊆ S : G 6= φ, |G| = µi or G = S}

and set S1 = φ, δ = −∞.2: For each G ∈ S,3: repeat4: update S1 ←− {S1,G};5: determine

a = minW⊆G,W6=φ ∆(Hi,W,S\G, RW),and let W be the minimizing set withthe smallest cardinality;

6: if δ < a, then set A = G and δ = a;7: update G ←− G\W;8: until G = φ or G ∈ S1;9: end for

10: Output G∗ = A, ε(Hi,G∗,S \ G∗, R) = δ and stop.

C. Simulation ResultsWe consider a SIMO K-user interference channel given

by (2) with K = 3 transceiver pairs and each receiver i



employs Ni = 3 receiving antennas. We assume that thechannel hi,j ∼ NC(0, INi) drawn from i.i.d. complexGaussian distribution with zero mean and unit variance,the complex additive white Gaussian noise (AWGN) vi ∼NC(0, INi). All transmitters employ a rate-1/2 LDPCcode with the block length of 5400 bits, and QPSKmodulation; and the receivers employ soft demodulationand soft LDPC decoding to decode the messages of thetransmitters.

We assume that each receiver employs SGD with thegroup size constraint µi = 2 for all users, and Algo-rithms 1 and 2 to determine the optimal decoding orderof the users. We evaluate the performance of the system,in terms of outage rate and BER. We count the systemoutage rate and the BER is computed based on the actualsystem performance, which includes both the rate outagepredicted by the group partition (Algorithm 1) and thedecoding error in the employed LDPC codes. The systemoutage rate and the BER of the OSGD are plotted againstthe SNR P/σ2 from 0dB to 15dB, as shown in Figs. 2and 3, respectively.

We compare the performance of the OSGD with thatof the two interference suppression schemes, the iterativeleast squares (ILS) algorithm [6] and the minimizinginterference leakage (MIL) scheme [7], which employsreceiver filtering to maximize the receiving SINR andminimize the interference leakage, respectively. It is seenthat, compared with the two interference suppressionschemes, the OSGD achieves around 5dB performancegain at the outage probability 3× 10−2 and bit error rate(BER) 2× 10−3.

D. Comparison with Interference Alignment

We have compared the performance of the OSGD withthat of a practical interference alignment scheme (theMIL scheme). Note that the MIL scheme tailors the keyidea of interference alignment for practical scenarios,where it is difficult or impossible to completely removeall interference. Around 7dB performance gain at theoutage probability 5 × 10−2 and BER 2 × 10−3 overthe MIL scheme is observed. Moreover, in terms ofthe practical system implementation, in contrast to theinterference alignment scheme, the group decoder doesnot require the feedback of all channel state information tothe transmitters, i.e., each transmitter either needs to knowonly its own channel to do some simple beamforming. Forexample, each transmitter could perform channel-matchedbeamforming, in which case only its own channel needsto be fedback; or each transmitter could perform randombeamforming, in which case no feedback is needed at all.In both cases, the interference is treated at the receiver bythe group decoder. Hence compared with the interferencealignment approach, the group decoder not only offerssubstantial performance gain, but also significantly re-duces the signaling and feedback overhead of the system.

0 3 6 9 12 15

10−2

10−1

100

SNR (dB)

Out

age

prob

abili

ty

OSGDMILILS

Figure 2. Outage probability performance for OSGD with rate-1/2channel codes and QPSK.

0 3 6 9 12 1510

−4

10−3

10−2

10−1

SNR (dB)

BE

R

OSGDMILILS

Figure 3. BER performance for OSGD with rate-1/2 channel codesand QPSK.

IV. CONSTRAINED PARTIAL GROUP DECODER

In this section we discuss an enhanced version of thegroup decoder, i.e., the constrained partial constrainedgroup decoder for a K-user interference channel [12].

A. Layered Encoding by Signal Superposition

We consider single-antenna K-user interference chan-nel model as in (1). To maximize the sum rate of allusers, the message of each transmitter is split into layerseach drawn from an independent codebook (rate splitting).Denote the number of codebooks (layers) of transmitter j



by Lj . Denoting the signal of layer k by xj,k, we obtain

yi =K∑

j=1

hi,j

Lj∑

k=1

xj,k + vi, i ∈ {1, . . . , K}, (10)

where vi ∼ NC(0, σ2) accounts for the additive whiteGaussian noise (AWGN). Assume that all transmitters em-ploy an equal power P , and that all layers of transmitterj employ an equal power E(|xj,k|2) = P

Lj.

B. Constrained Partial Group Decoding (CPGD)

Each receiver i employs a pi-stage successive de-coding described by an ordered partition Gi 4

={Gi

1, . . . ,Gipi

,Gipi+1} of K (the set of the indices of all

codebooks). All layers of transmitter i, i.e., {xi,k}Li

k=1,are included in {Qi

1, . . . ,Gipi} such that they can be

decoded in the pi-stage decoding. In the mth stage,1 ≤ m ≤ pi, the ith receiver jointly decodes the layersin Gi

m by treating ∪`>mGi` as AWGN and then subtracts

the decoded messages in Gim from the received signal.

To control the decoding complexity, we constrain that|Gi

m| ≤ µ for m ∈ {1, . . . , pi}.A key problem for CPGD is how to partition the groups

Gi for 1 ≤ i ≤ K and allocate the rates for all layers. Aniterative joint rate allocation and group partition algorithmis developed in [12], that maximizes the minimum rateincrements of the layers.

C. Practical Coding Scheme

A practical coding scheme based on the CPGD is givenin [12], which consists of the following three steps:• user inactivation and message layering;• transmission mode selection;• rate enhancement.We describe the three-step practical coding scheme

based on an illustrative example shown in Fig. 4. As-sume that we have a discrete spectrum rate table T ={d1, d2, d3, d4}.

In the first step we perform tentative rate allocation tothe users and divide the users into multiple layers, andinactivate the users (layers) with rate smaller than d1 (notimplementable). Assume that after the user inactivationand message layering, based on the threshold rate d1, user2 is inactivated and users 1 and 3 are divided into 2 layers,(1, 1), (1, 2), (3, 1), (3, 2), and (4, 1).

In the second step, in order to find some implementablerates close to those yielded by the first step we quan-tize the rates according to the quantization rate tableT = {d1, ..., dT }. Assume that in the transmission modeselection, the rates of the five layers (1, 1), (1, 2), (3, 1),(3, 2), and (4, 1) are quantized to be d1, d2, d3, d1,and d4, respectively, i.e., the number of information bitsassigned to these layers are Nd1, Nd2, Nd3, Nd1, andNd4, respectively, where N is the nominal number oftransmitted symbols. As the quantized rates must bedecodable, each quantized rate is smaller than its originalcounterpart which incurs some loss in spectral efficiency.

In the final step, we compensate for such loss byincreasing the rates of layers beyond their quantizedvalues and make them as close as possible to the originalrates. The rate enhancement is performed via reducing thenumber of transmitted symbols to η∗N for some η∗ < 1,and thus the rate of the five layers are enhanced to d1/η∗,d2/η∗, d3/η∗, d1/η∗, and d4/η∗, respectively.

The quantization table T = {d1, ..., dT } needs to beoptimized to maximize the sum rate of all layers obtainedfrom the above three-step scheme. The optimization isdone offline based on a training set of channel samples.

D. Raptor Codes

We use the doped Raptor code [13] with a rate-0.95IRA precode for the following two reasons. Firstly, itexhibits near-capacity performance for both single-userchannel and two-user multiple-access channel. Secondly,the online fine tuning can be easily implemented by theemployed doped Raptor codes. All transmitters performincremental transmission, via first transmitting the numberof channel symbols given by the fine tuning, and theneach time transmitting a certain number of channel sym-bols until all receivers successfully decode their desiredmessages.

The profiles of Raptor codes are optimized using the ex-trinsic information transfer (EXIT) functions, to minimizethe gap of the practical coding scheme to the Gaussianmodulation and infinite-length codes. An interesting resultfor the profile optimization is that the gap for µ = 1 issignificantly smaller than that for µ = 2. This is becausefor µ = 2, one user may be decoded individually or jointlywith another user, so that we need to find a good codeprofile for both the single-user decoding and the two-userjoint decoding. Due to the large gap for µ = 2 usingpractical codes, it suffices to have a group size of one inCPGD, which has a low complexity to achieve most ofthe performance gain for practical implementations.

E. Results

Consider a single-antenna interference channel withK = 6 transceivers. For i, j ∈ {1, . . . ,K}, the channelcoefficients hi,j are distributed as NC(0, 1).

Assume that the size of the rate quantization table is|T | = 4. We let the group size µ = 1 and employthe optimized code profile and rate quantization tablesof the layered and un-layered coding. Fig. 5 shows thecorresponding throughput, denoted as “layered, optimum”and “un-layered, optimum”, respectively, for the channelSNR P/σ2 from 0dB to 9dB. At each channel SNR, 1000channel realizations are simulated and the throughputis the total number of information bits divided by thetotal number of transmitted symbols of the incrementaltransmission.

For comparison, we plot the throughput performanceof the rate table T = {0.40, 0.80, 1.20, 1.60} for thelayered and un-layered coding schemes, denoted as “lay-ered uniform” and “un-layered uniform”, respectively. It



(1,2)

(1,1)

(3,2)

(3,1)

(4,1)

User 2

User 4

User 3

User 1

User Inactivation and

Message Layering

d1

Transmission

Mode Selection

T = {d1,d2,...,dT}

(1,2)

(1,1)

(3,2)

(3,1)

(4,1)

Number of

Information Bits

N

Nd1

Nd2

Nd3

Nd4

Nd1

Rate

Enhancement

(1,2)

(1,1)

(3,2)

(3,1)

(4,1)

*N

rate d1/ *

rate d2/ *

rate d3/ *

rate d4/ *

rate d1/ *

*N: number of

transmitted symbolsDesign

T = {d1,d2,...,dT}

Design d1Design

{d2,...,dT}

Figure 4. A simple illustrative example for the practical coding scheme for CPGD.

is seen that the optimized T provides larger throughputs.Furthermore, to show the performance gain of the codeprofile optimization, we plot the throughput of Luby’sprofile for the layered and un-layered coding schemeswith the same optimized T as that for the optimizedprofile. It is seen that the optimized profile significantlyoutperforms Luby’s profile.

0 1 2 3 4 5 6 7 8 9

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

P/σ2 (dB)

syst

em tr

houg

hput

un−layered, optimumlayered, optimumun−layered, uniformlayered, uniformun−layered, robust Solitonlayered, robust Soliton

Figure 5. The simulated throughput with QAM and optimizedrateless codes.

V. CONSTRAINED GROUP DECODER FORMULTI-RELAY ASSISTED INTERFERENCE CHANNEL

We next review the application of the CGD in a multi-relay-assisted interference channel [14].

A. System Model

Consider a communication system with K sources, Kdestinations, and N relays, all employing single antenna.

Let fn,j , hi,j , and gi,n denote the channel gain fromsource j to relay n, from source j to destination i, andfrom relay n to destination i, respectively. We assumequasi-static block fading channels, i.e., the channel gainis fixed during one transmission period and changes toanother independent state afterwards.

We assume half-duplex relay transmission with syn-chronized relays that operate in the same frequency bandas the sources do. In the first phase, all transmitterstransmit to the relays and the destinations; and in thesecond phase, all sources and relays transmit to thedestinations.

Let x1j , x2

j denote the transmitted signal of sourcej in the first and second phases, respectively, and xr

n

denote the transmitted signal of relay n. Let yrn denote

the received signal of relay n, and y1i and y2

i denote thereceived signal of destination i in the first and secondphases, respectively. Based on the transmission protocol,in the first phase the received signals yr

n and y1i are given

respectively by

yrn =

K∑

j=1

fn,jx1j + vr

n, (11)

and y1i =

K∑

j=1

hi,jx1j + v1

i , (12)

where vrn and v1

i ∼ NC(0, σ2) are the additive whiteGaussian noise (AWGN) at relay n and destination i,respectively. The received signal y2

i by destination i inthe second phase is given by

y2i =

K∑

j=1

hi,jx2j +

N∑n=1

gi,nxrn + v2

i , (13)

where v2i ∼ NC(0, σ2) is the AWGN at destination i in

the second phase.

B. Relay Assignment and Relaying Modes

Each intermediate relay assists a group of source-destination pairs, such that each source-destination pair



is assisted by at most one relay. Let Sn be the set ofsources assisted by relay n ∈ N , and S0 be the set ofsources not assisted by any relay. Let c(j) = n if andonly if j ∈ Sn.

In the second phase, relay n re-encodes the informationof the sources in Sn, and employs analog network coding(ANC) to combine the re-encoded signals as follows

xrn =

∑

j∈Sn

xj , (14)

where xj is the re-encoded signal of source j. From (13)and (14), we have

y2i =

K∑

j=1

hi,jx2j +

K∑

j=1

gi,c(j)xj + v2i , (15)

where gi,c(j) for 1 ≤ i, j ≤ K denotes the gain fromsource j to destination i of the equivalent interferencechannel formed by the relay-destination link. We assumethat each source employs a power P , and each relayemploys the same power P for each source when forward-ing the signal of the assigned sources, i.e., E

(|x1

j |2)

=

E(|x2

j |2)

= E(|xj |2

)= P for 1 ≤ j ≤ K.

We consider two types of relaying schemes. First, weconsider hopping relays where there is no direct source-destination link (∀i, j, hi,j = 0) and destination i decodessource i through the received signals from the relays in thesecond phase. Secondly, we consider inband relays wherethe relays and sources share the same frequency band anddestination i decodes source i by receiving both the directand relayed transmissions.

C. User Rate Allocation

We assume that the receivers at both the relays anddestinations perform constrained group decoding, i.e.,they decode a subset of the interferers along with thedesired messages. Denote R1

j and R2j as the rates of

the source messages in the first and second phases,respectively, and define t as the fraction of the durationof the first phase. Then, the overall rate of source j ∈ Kis given by Rj = tR1

j + (1 − t)R2j . Note that according

to (15) the interference channel at the destination is theone with 2K transmitters and K receivers, with the ratesof the transmitters [R1

jR2j ]j∈K. We are interested in the

max-min rate allocation for the relay assisted interferencechannels, which maximizes the minimum rate among allsources, i.e., maxminj∈KRj , all valid relay assignments,all possible group decoding strategies at the relays andthe destinations, and all rate vectors such that [R1

j ]j∈Kdecodable by the relays and [R1

jR2j ]j∈K decodable by the

destinations.In [14], we have developed rate allocation schemes for

the following four relay types.• hopping relay system with fixed relay assignment;• hopping relay system with dynamic relay assign-

ment;• inband relay system with fixed relay assignment;• inband relay system with dynamic relay assignment.

D. Numerical Results

We simulate the rate allocation for the proposed groupdecoder for the relay system with K = 6 sources anddestinations. Assume that for 1 ≤ i ≤ 6, the positionsof source i and destination i are (−0.5, 0.5 × i) and(0.5, 0.5× i), respectively. For fixed relay assignment, weassume that there are 6 relays and the position of relay i is(0, 0.5×i) for 1 ≤ i ≤ 6, where each relay i is assigned toassist source i. For dynamic relay assignment, we assumethat there are N = 3 relays and the position of relay i is(0, 0.75+0.5×i) for 1 ≤ i ≤ 3. The channel gain betweentwo nodes is assumed to be complex Gaussian distributedwith the mean zero and the variance 1/d2 where d is thedistance between the two nodes. We assume that the firstphase occupies half of the entire transmission period, i.e.,t = 0.5, and consider the SNR values, P/σ2, from 0dBto 9dB. For each SNR value, 1000 channel realizationsare simulated.

For the hopping relay system with fixed relay as-signment, we plot the minimum rates and sum rates ofsources in Fig. 6, respectively, for group decoders withthe group size µ denoted as “GD, µ = i” for 1 ≤ i ≤ 3,and for comparison the linear MMSE decoder where allinterference is treated as noise. It is seen that the groupdecoder provides significantly larger minimum and sumrates over the linear MMSE decoder.

We also compare the performance of the proposeddecode and forward (DF) scheme with the relay amplifiedand forward (AF) scheme where the group decoder isemployed only at the receivers, denoted as “AF + GD”.Since the relays do not decode the source messages andthus superimpose their signals in the second phase, foreach channel realization we randomly select a relay toforward the received noisy signals using a scaled power ofKP . It is seen that, the proposed DF scheme significantlyoutperforms the AF scheme for the group sizes µ = 1,2, and 3, because the AF scheme also forwards thenoise to the destinations. Moreover, we tailor the multi-relay AF scheme proposed in [15] for a single source-destination pair to the multiple source-destination pairscenario using a time-division mode, where each sourcetransmits in the 1/K of the total duration (denoted as“Multi, AF, TDMA”). For fair comparison, each sourcetransmits using power KP and each relay forwards usingpower P . It is seen that the proposed group decodersignificantly outperforms the multi-relay AF scheme.

We have also evaluated the performance of the pro-posed rate allocation scheme for the other three relayingtypes. Numerical results show that for fixed relay assign-ment, the minimum rate obtained from the proposed rateallocation scheme significantly outperforms that obtainedfrom the MMSE decoding; and for dynamic relay as-signment, the minimum rate obtained from the proposedscheme performs within 90% of the computationally in-tractable optimal minimum rate obtained from exhaustivesearch.



0 1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

P/σ2 (dB)

Min

rat

e

MMSE decoderGD, µ = 1GD, µ = 2GD, µ = 3AF + GD, µ = 1AF + GD, µ = 2AF + GD, µ = 3Multi, AF, TDMA

Figure 6. Minimum rates of sources for the hopping relay systemwith fixed relay assignment.

VI. CONCLUSIONS

We have provided an overview of the recent develop-ment on the group decoding techniques for interferencechannels. The basic idea is to let each receiver decode,either fully or partially, some interfering signals, in orderto optimize the overall system performance, in terms ofeither total throughput or outage rate. It is seen that forsystems that employ fixed-rate coding and modulation, thegroup decoder can offer significant performance gain overthe traditional interference suppression approaches. Onthe other hand, for systems that employ adaptive codingand modulations, an optimal greedy algorithm can provideboth the decoding schedule for each receiver, and therate allocation for the corresponding transmitter. Practicaldesigns under such group decoding paradigm taking intoaccount practical modulation formats and channel codeshave been developed. Moreover, the group decoders alsofind applications in more sophisticated systems such asrelay-assisted interference channels. Compared with theinterference alignment approach which performs transmit-ter side signal processing and requires the channel stateinformation of all users be fedback to the transmitters,the group decoding is a receiver-centric paradigm that notonly offers superior performance, but also eliminates theoverhead of channel feedback.

REFERENCES

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[2] M. Maddah-Ali, A. Motahari, and A. Khandani, “Commu-nication over MIMO X channels: Interference alignment,decomposition, and performance analysis,” IEEE Trans.Info. Theory, vol. 54, no. 8, pp. 3457–3470, Aug. 2008.

[3] R. Tresch, M. Guillaud, and E. Riegler, “On the achievabil-ity of interference alignment in the K-user constant MIMOinterference channel,” in Statistical Signal Processing,IEEE 15th Workshop on, 31 2009-sept. 3 2009, pp. 277–280.

[4] M. Yamada and T. Ohtsuki, “Interference alignment in the2× (1 + N) MIMO model,” in GLOBECOM Workshops,2010 IEEE, Dec. 2010, pp. 1963 –1967.

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[6] H. Yu and Y. Sung, “Least squares approach to joint beamdesign for interference alignment in multiuser multi-inputmulti-output interference channels,” IEEE Trans. SignalProcess., vol. 58, no. 9, pp. 4960 –4966, Sept. 2010.

[7] P. Mohapatra, K. E. Nissar, and C. R. Murthy, “Interferencealignment algorithms for the K user constant MIMO in-terference channel,” IEEE Trans. Signal Process., vol. 59,no. 11, pp. 5499 –5508, Nov. 2011.

[8] T. S. Han and K. Kobayashi, “A new achievable rate regionfor the interference channel,” IEEE Trans. Info. Theory,vol. 27, no. 1, pp. 49 – 60, Jan. 1981.

[9] N. Prasad and X. Wang, “Outage minimization and rateallocation for the multiuser Gaussian interference chan-nels with successive group decoding,” IEEE Trans. Info.Theory, vol. 55, no. 12, pp. 5540–5557, Dec. 2009.

[10] A. Tajer, N. Prasad, and X. Wang, “Beamforming and rateallocation in MISO cognitive radio networks,” IEEE Trans.Signal Process., vol. 58, no. 1, pp. 362–377, Jan. 2010.

[11] A. Motahari and A. Khandani, “To decode the interferenceor to consider it as noise,” IEEE Trans. Info. Theory,submitted.

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[14] C. Gong, A. Tajer, and X. Wang, “Interference channelwith constrained partial group decoding,” submitted toIEEE Journal Select. Area Commun.

[15] H. Bolcskei, R. U. Nabar, O. Oyman, and A. J. Paulraj,“Capacity scaling laws in MIMO relay networks,” IEEETrans. Wireless Commun., vol. 5, no. 6, pp. 1433–1444,Jun. 2006.

Chen Gong received the B.S. degree in electrical engineeringand mathematics (minor) from Shanghai Jiaotong University,Shanghai, China in 2005, M.S. degree in electrical engineeringfrom Tsinghua University, Beijing, China in 2008. Now heis pursuing his Ph.D degree in Columbia University, NewYork City, New York, USA. His research interests are in thearea of channel coding and modulation techniques for wirelesscommunications.

Omar Abu-Ella received the B.Sc. degree in electrical en-gineering (with honor) from Academy of Aero Studies andSciences, Misurata, Libya, in 2001, and his MS.c. degree fromthe Higher Institute of Industry (HII), Misurata, Libya, in 2007.He is working toward the Professional degree at the ColumbiaUniversity in the City of New York. Since 2003, he has joinedthe Department of Electrical Engineering at Misurata University.Also, he has worked as adjunct lecturer assistant at the HIIbetween 2007 and 2008. His current research interests includeseveral topics in wireless Communications and signal processing



for Communications.

Xiaodong Wang (S’98-M’98-SM’04-F’08) received the Ph.Ddegree in Electrical Engineering from Princeton University. Heis a Professor of Electrical Engineering at Columbia Universityin New York. Dr. Wang’s research interests fall in the generalareas of computing, signal processing and communications, andhas published extensively in these areas. Among his publicationsis a recent book entitled “Wireless Communication Systems:Advanced Techniques for Signal Reception”, published by Pren-tice Hall in 2003. His current research interests include wire-less communications, statistical signal processing, and genomicsignal processing. Dr. Wang received the 1999 NSF CAREERAward, and the 2001 IEEE Communications Society and Infor-mation Theory Society Joint Paper Award. He has served as anAssociate Editor for the IEEE Transactions on Communications,the IEEE Transactions on Wireless Communications, the IEEETransactions on Signal Processing, and the IEEE Transactionson Information Theory. He is a Fellow of the IEEE.

Ali Tajer received the B.Sc. and M.Sc. degrees in ElectricalEngineering from Sharif University of Technology, Tehran, Iran,in 2002 and 2004, respectively, and the M.Phil. and the Ph.D.degrees in electrical engineering from Columbia University,New York, NY in 2010. He is currently a Postdoctoral ResearchAssociate in the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ, and an Adjunct Assistant Professor inthe Department of Electrical Engineering, Columbia University,New York, NY. He spent the summers 2008 and 2009 as aResearch Assistant with NEC Laboratories America, Princeton,NJ. His research interests lie in the general areas of networkinformation theory and statistical signal processing with appli-cations in wireless communication.



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Constrained Group Decoder for Interference Channels