IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, …IEEE TRANSACTIONS ON INFORMATION THEORY, VOL....

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 2, FEBRUARY 2012 765

MIMO ARQ With Multibit Feedback:Outage Analysis

Khoa D. Nguyen, Member, IEEE, Lars K. Rasmussen, Senior Member, IEEE,Albert Guillén i Fàbregas, Senior Member, IEEE, and Nick Letzepis, Member, IEEE

Abstract—This paper studies the asymptotic outage perfor-mance of incremental redundancy automatic-repeat-request(INR-ARQ) transmission over multiple-input multiple-output(MIMO) block-fading channels with discrete input constellations.We first show that transmission with random codes using a dis-crete signal constellation across all transmit antennas achievesthe optimal outage diversity given by the Singleton bound. Theoptimal SNR-exponent and outage diversity of INR-ARQ trans-mission over the MIMO block-fading channel are then analysed.We show that a significant gain in outage diversity is obtainedby providing more than one bit feedback at each ARQ round.Thus, the outage performance of INR-ARQ transmission canbe remarkably improved with minimal additional overhead.A practical feedback-and-power-adaptation rule is proposedfor MIMO INR-ARQ, demonstrating the benefits provided bymultibit feedback. Although the rule is sub-optimal in terms ofoutage performance, it achieves the optimal outage diversity.

Index Terms—Automatic-repeat-request, multibit feedback,outage diversity, outage probability, power allocation, SNR-exponent.

I. INTRODUCTION

I N this paper, we take an information-theoretic approachto analyzing and designing multiple-input multiple-output

(MIMO) transmission strategies for incremental redun-dancy (INR) automatic-repeat-request (ARQ) schemes overa block-fading channel. In particular, we propose the use ofmultibit feedback for power adaptation and study the outage

Manuscript received June 01, 2009; revised March 30, 2011; accepted June22, 2011. Date of publication October 06, 2011; date of current version Feb-ruary 08, 2012. This work was presented in part at the 2009 Australian Com-munication Theory Workshop, Sydney, Australia, and the 2009 IEEE Interna-tional Symposium on Information Theory. This work was supported in part bythe Australian Research Council under ARC Grants RN0459498, DP0558861,and DP0881160, in part by the Swedish Research Council under VR Grant621-2009-4666, and in part by the European Research Council under the Eu-ropean Community’s Seventh Framework Programme (FP7/2007-2013)/ERCGrant agreement no. 228044.

K. D. Nguyen and N. Letzepis are with the Institute for Telecom-munications Research, University of South Australia, Australia (e-mail:[email protected]; [email protected]).

L. K. Rasmussen was with the Institute for Telecommunications Research,University of South Australia, Australia. He is now with the CommunicationTheory Laboratory, KTH Royal Institute of Technology, and the ACCESS Lin-naeus Center, Stockholm, Sweden (e-mail: [email protected]).

A. Guillén i Fàbregas is with the with the Institució Catalana de Recerca iEstudis Avançats (ICREA), the Department of Information and CommunicationTechnologies, Universitat Pompeu Fabra, Barcelona, Spain, and the Departmentof Engineering, University of Cambridge, Cambridge, United Kingdom (e-mail:[email protected]).

Communicated by G. Taricco, Associate Editor for Communications.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2011.2170919

diversity of the resulting protocol over the MIMO block-fadingchannel, which characterizes the slope of the outage probabilitycurve at high signal-to-noise ratio (SNR) in log-log scale.

A. Prior Art

The block-fading channel [1], [2] is a useful mathematicalmodel for many practical wireless communication scenarios.The channel consists of a finite number of consecutive orparallel transmission blocks, where each block is affected byan independent fading coefficient. The model approximateswell the characteristics of delay-limited transmission overslowly varying channels, such as Orthogonal Frequency Di-vision Multiplexing (OFDM) transmission over slowly-fadingfrequency-selective multipath channel, as well as narrowbandtransmission with frequency-hopping as encountered in theGlobal System for Mobile Communications (GSM) and theEnhanced Data rate for GSM evolution (EDGE) standards.

Due to the finite number of fading blocks, the information ratesupported by the channel depends on the instantaneous channelrealization and, therefore, is a random variable. When the in-stantaneous mutual information is less than the transmissionrate, transmission is in outage [2]. In this case, it follows fromthe strong converse theorem (see, e.g., [3]–[5]) that messagesare decoded in error with probability one [6], [7]. Furthermore,it is shown in [4], [8] that the use of sufficiently long randomcodes achieves an average error rate equal to the outage proba-bility. Therefore, the outage probability is a fundamental limiton the performance of block-fading channels.

MIMO transmission has revolutionized modern wirelesscommunications, and is now a key technology used in mostcurrent standards, e.g., WiFi (IEEE 802.11) and WiMax (IEEE802.16) [9], [10]. Moreover, due to the randomness of thecommunication rate supported by the channel, it is essential touse adaptive techniques to enable high-rate reliable communi-cation, where the transmission rate and/or power is adjusted tothe channel realization. The use of adaptive techniques dependsstrongly on the availability of channel state information (CSI)at the transmitter and the receiver. In most communicationsystems, CSI can be estimated at the receiver, while CSI isusually not directly available at the transmitter. The use of ARQtransmission techniques is, therefore, a powerful approach forproviding transmitter CSI, which in turn can be used to signif-icantly improve the performance over block-fading channels[11].

The optimal diversity-multiplexing tradeoff for a MIMOchannel with optimal (Gaussian) input distribution has beencharacterized in [12]. For systems with discrete input constel-lations, the rank criterion for the optimal outage diversity was

0018-9448/$26.00 © 2011 IEEE

766 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 2, FEBRUARY 2012

TABLE ISUMMARY OF NOTATIONS

derived in [13] from a worst-case analysis of the pair-wise errorprobability (PEP). References [14], [15] establish the Singletonbound on the optimal SNR-exponent of quasi-static MIMOchannels with discrete input constellations. The Singletonbound is achievable by a wide range of input constellations viaa unified code construction method proposed in [15].

In an INR-ARQ scheme, transmission starts with a high-ratecodeword, and additional redundancy is requested via a feed-back link when the codeword is not successfully decoded.Transmission is in outage if the codeword is not decodablewithin the maximum delay constraint allowed by the system.Traditional INR-ARQ systems implement one-bit feedbackfrom the receiver, indicating whether additional redundancyis required. However, due to the accumulative nature ofINR-ARQ schemes, performance improvements are possiblewhen additional information regarding the status of the currenttransmission is provided through the feedback link. Severalmultibit feedback INR-ARQ schemes have been proposedin the literature. In particular, [16] shows that the throughputperformance of ARQ systems can be improved by multibit feed-back prior to each transmission round. The proposed systemis equivalent to a conventional ARQ system with quantizedCSI at the transmitter (CSIT). For systems with no CSIT, [17],[18] propose transmission using convolutional codes, while[19] proposes a multilayer broadcasting strategy for multibitfeedback ARQ. Both approaches show that multibit feedbackcan significantly improve the throughput performance of ARQtransmission. There is, however, no unified approach for de-signing multibit feedback INR-ARQ transmission schemes.

An important performance measure for INR-ARQ transmis-sion in the MIMO block-fading channel is the rate-diversity-delay tradeoff. This tradeoff has only been studied for INR-ARQsystems with one-bit ACK/NACK feedback in [20]–[22]. In par-ticular, [20] characterizes the rate-diversity-delay tradeoffs ofGaussian input MIMO INR-ARQ systems with both short-termand long-term average power constraints. For systems with dis-crete input constellations, the optimal rate-diversity tradeoff forsystems with short-term power constraints was characterized in[21], [23]. For ARQ systems with discrete input constellationand long-term power constraints, an optimal power allocationrule has been derived in [23], providing significant improvementon outage performance. However, the rate-diversity tradeoff ofthe corresponding system was not studied.

B. Contributions

As a first contribution we consider fixed-rate transmissionover the MIMO block-fading channel. We show that the outagediversity is given by the Singleton bound, and that it is achiev-able with random codes constructed over arbitrary discreteinput constellations. This rigorously proves that the Singletonbound is the optimal SNR-exponent of MIMO transmissionwith discrete input constellations. The result will also prove in-strumental in designing and analyzing INR-ARQ transmissionover the MIMO block-fading channel.

As our main contribution we study the rate-diversitytradeoff of the MIMO ARQ system with multibit feedbackunder long-term power constraints. The analysis shows thatmultibit feedback and optimal power adaptation provide sig-nificant outage diversity gains for ARQ transmission over theblock-fading channel. It is shown that a finite number of feed-back bits is sufficient to achieve the maximal outage diversity.The optimal rate-diversity tradeoff for the one-bit feedbackcase is also presented, which characterizes the asymptoticgains provided by the optimal power allocation rule proposedin [23]. As a further contribution a practically feasible feed-back-and-power-adaptive rule is proposed. Although the rule issub-optimal in terms of outage performance, it can achieve theoptimal outage diversity, thus clearly illustrating the benefitsoffered by multibit feedback.

C. Notation and Organization

The following notations are used in the paper. Boldface up-percase and lowercase variables correspondingly de-note matrices and vectors; while scalar variables are denoted bylightface ( or ). Sets are denoted by calligraphic letters; whilethe sets of natural, real and complex numbers are correspond-ingly denoted with and . The mathematical expectationof a random variable is denoted by . Nonconjugate trans-pose of matrices are denoted by . The operationreturns the maximum (minimum) integer smaller (larger) thana real number. For convenience, the physical meanings of com-monly used parameters are summarized in Table I.

The remainder of the paper is organized as follows. Section IIdescribes the MIMO block-fading channel model. Section IIIproposes the multibit feedback INR-ARQ system based on mu-tual information and information outage. Sections IV and V

NGUYEN et al.: MIMO ARQ WITH MULTIBIT FEEDBACK: OUTAGE ANALYSIS 767

discuss system design and performance analysis. Finally, con-cluding remarks are given in Section VI and proofs are collectedin the Appendices.

II. SYSTEM MODEL

Consider INR-ARQ transmission over a MIMO block-fadingchannel with transmit and receive antennas. Each ARQround is transmitted over additive white Gaussian noise(AWGN) blocks of channel uses each, where block atARQ round is affected by a flat fading channel gain matrix

. The baseband equivalent of the channel in theth ARQ round is given by

(1)

where is the transmit power in roundare correspond-

ingly the transmitted signal, the received signal, and theadditive noise; while is a block diagonalchannel gain matrix at round with

In the INR-ARQ scheme, the receiver attempts to decodeat round based on the received signals collected in rounds

. The entire channel after ARQ rounds is

(2)

where

We consider transmission where the entries of are equiprob-ably drawn from an input constellation of size ,and assume that the constellation has unit average energy,i.e., entries of satisfy . We further as-sume that the entries of and are independently drawnfrom a zero-mean unit-variance complex Gaussian distribution

, and that is available at the receiver. The averageSNR at each receive antenna is then .

Let be the average number of transmission round per code-word, or equivalently the expected interrenewal time [6]. Theaverage transmit power is

(3)

where is adapted to through receiver feedback. For asystem with long-term power constraint , we study feedback-and-power-adaptation rules satisfying

(4)

III. PRELIMINARIES

A. Accumulated Mutual Information

Assuming that the realized channel matrix at round is ,the input-output mutual information of the MIMO channel inround is

(5)

where is the input-output mutual information[5], measured in bits per channel use (bpcu), of an AWGNMIMO channel with input constellation and channel matrix

. More specifically

(6)

where is uniformly drawn from and the entries ofare i.i.d. . The average input-output mutual infor-

mation after ARQ rounds is given by bpcu. Let

(7)

be the accumulated mutual information after ARQ rounds.We now propose the multibit feedback INR-ARQ transmissionscheme based on the accumulated mutual information .

B. Multilevel Feedback

We consider an INR-ARQ system with a delay constraint ofARQ rounds, where a feedback index

is delivered after each transmission round through a zero-delayerror-free feedback channel. Power and rate adaptation are per-formed based on receiver feedbacks. The overall system modelis illustrated in Fig. 1.

1) Transmitter: Consider a code book of ratebits per coded symbol, that maps a message

to a codeword . At transmis-sion round of the coded symbols are formatted into

and transmitted via the channel in (1) withpower , where is the vector offeedback indices collected from rounds . The real-ized code rate of a single ARQ round is bpcu, and the realizedcode rate after ARQ rounds is bpcu. If feedback(denoting positive acknowledgment (ACK)) is received aftertransmission rounds, the transmission is successful and trans-mission of the next message starts. Otherwise, the transmittercontinues with new transmission rounds until feedback index

is received or until transmission rounds have elapsed.2) Receiver: Upon receiving round , the receiver attempts to

decode the transmitted message from the received signals col-lected from rounds 1 to . The receiver employs a decoder witherror detection capabilities as described in [6]. The decoder out-puts if there exists a unique messagesuch that and are jointly typical conditioned on


Fig. 1. INR-ARQ system with multibit feedback.

Fig. 2. Example of feedback thresholds.

[5]; then an ACK is delivered to the transmitter via feed-back index . Otherwise, a quantization of the accu-mulated mutual information is delivered via feedback index

satisfying , withpredefined quantization thresholds ,and for . An exampleof the feedback thresholds for the first two rounds of an ARQsystem with is illustrated in Fig. 2. Feedback index3 is used to denote successful transmission. At the first ARQround, the leftmost set of feedback thresholds is used; while atthe second ARQ round, one of the three sets of feedback thresh-olds on the right is employed, depending on which feedbackindex was delivered in the first round. Noting that ,the feedback thresholds in round should be designed suchthat . Thus, the setof quantization thresholds is completely defined by forpractical purposes.

3) Power Constraint: The probability of having feedbackvector at round , denoted as , is recursively expressedas

(8)

(9)

where is given by (5) with . Noting thatdenotes a successful decoding at round ,

the power constraint in (4) can be written as

(10)

C. Information Outage

After ARQ rounds, the input-output mutual information isand the realized code rate is (bpcu). Hence,

transmission is in outage at round if . The probabilityof having an outage at round is then given by

(11)

With an optimal coding scheme, and in the limit of the numberof channel uses , the codeword is correctly decodedwhenever ; otherwise, an error is detected [6]. There-fore, the outage probability is an achievable lower boundon the word error probability at round . For INR-ARQ trans-mission with delay constraint , the overall outage probabilityis .

IV. ASYMPTOTIC ANALYSIS

Consider a power adaptation rule satisfyingthe power constraint in (10). We prove that for large , the op-timal outage probability at round behaves like

(12)

where is the outage diversity at round and the exponen-tial equality indicates [12]

(13)

Subsequently, we determine the optimal rate-diversity-delaytradeoff of ARQ systems with levels feedback andprove that the optimal outage diversity is achievable.

A. MIMO Block-Fading Without ARQ

In order to characterize the outage diversity or achievableSNR-exponent for the MIMO INR-ARQ channel, we first studythe corresponding limits for fixed-rate transmission over theMIMO block-fading channel. These results constitute the keyingredients in proving our main results for multibit ARQ.

Theorem 1: Consider fixed-rate transmission withrate and power over the MIMO block-fading channel in (1)using constellation of size and the transmission schemedescribed in Section III-B. Let be the real-ized input-output mutual information as defined in (5). For large

, we have that(14)

(15)


where is bounded by , and

(16)

(17)

Furthermore, is the SNR-exponent achieved by usingrandom codes with rate , where the code symbols are drawnuniformly from .

Proof: See Appendix A.1

To the best of our knowledge, this is the first rigorous prooffor the outage diversity of a MIMO block-fading channel witha general discrete input constellation. The results of [13], [15]establish as an upper bound for the outage diversity forthe quasi-static fading channel. Code design techniques in [15]show that can be achieved by specifically constructedinput constellations. As a generalization, Theorem 1 shows that

is the outage diversity for MIMO block-fading channelswith any input constellation of size (except when is aninteger). Furthermore, Theorem 1 shows that is achiev-able by using random codes when is noninteger, which isessential for analyzing the performance of INR-ARQ systems.

When , random codes do not achieve the upperbound on outage diversity. Therefore, we can only obtain abound for . This leads to the corresponding bounds onoutage diversity of the INR-ARQ system in the subsequentsections. One might see this as a potential limitation of i.i.d.random codes since there exist specific code constructions thatcan achieve the outage diversity [21], [25]. We conjecturethat similar constructions would achieve the upper bounds onoutage diversity of INR-ARQ systems subsequently presented,even for .

B. Multibit MIMO ARQ

We now consider ARQ transmission over the block-fadingchannel in (1) using input constellation as described inSection III-B1. Using Theorem1, the optimal rate-diversity-delay tradeoff of the MIMOINR-ARQ scheme with multibit feedback is characterize asfollows.

Theorem 2: Consider INR-ARQ transmission over theMIMO block-fading channel in (1) using constellation ofsize and the transmission scheme described in Section III-B,where a codeword is considered successfully delivered at round

if . Assume that the number of feedback levels is. Subject to the power constraint in (10), the

optimal rate-diversity-delay tradeoff is given by

(18)

when is not an integer, where is given in Theorem 1.Proof: See Appendix B for a proof.

1A more general result of the theorem, which deals with power allocation forblock-fading channels with mismatched channel state information, was derivedin [24] after the submission of this paper. The proof given here is simpler andforms a basis for the result in [24].

Theorem 2 only gives the optimal outage diversity whenis noninteger. When is an integer, the bounds for inTheorem 1 does not coincide; thus, a definite value of isnot known. It can be shown that the optimal outage diversity isbounded by

(19)

An intuitive explanation for the outage diversity gains offeredby multibit feedback is given as follows. At round , the feed-back vector provides the transmitter with the past channelrealizations. This allows raising the transmit power in round

by a factor of without violating the long-term powerconstraint. In the limit of large power constraint, the optimaltransmit power in round satisfies

(20)

where is given by Theorem 1. Since only the exponent issignificant in diversity analysis, the maximum outage diversitycan be achieved if there are sufficient thresholds to feedback

for ( implies successful transmission). ForMIMO block-fading channels with discrete input constellations,the rate-diversity tradeoff is a stair-case function; therefore, a fi-nite number of feedback levels is sufficient to achieve the max-imum outage diversity. Meanwhile, for systems with ,including systems with Gaussian input distribution, is aconstant for . Therefore, no gains in outage diversitycan be obtained by multibit feedback. Conversely, in the mul-tiplexing-diversity tradeoff analysis [12], the outage diversity isa continuous, decreasing function of the multiplexing gain, andthus, an infinite number of feedback levels is required to achievethe optimal diversity-multiplexing tradeoff in our ARQ setting.

Remark 1: The proof of Theorem 2 also gives the followingguidelines to designing the feedback and power allocation rules.

• The optimal outage diversity of INR-ARQ systemsis achievable with feedback levels, wherethe feedback thresholds of each round are fixed at

. Therefore, for systemswith , the optimal outage diversityis achievable if for

.• Furthermore, the outage probability in round is dom-

inated by the events with , where. Therefore, after placing thresholds

at , the remaining feedback thresholds (for systems with) should give higher priority to quantizing

the aforementioned region to improve outage performance.• With the optimal feedback rule, the optimal outage di-

versity can be achieved with power allocation satisfyingfor all , where

is a constant chosen to satisfy the power constraint (10).We now prove that the rate-diversity-delay tradeoff isachievable by using random codes, as given by the followingtheorem.

Theorem 3: Consider INR-ARQ transmission over theMIMO block-fading channel in (1) using constellation of


size and the transmission scheme described in Section III-Bwith power constraint given in (10). Assume that the numberof feedback levels is . With random-codingschemes and , for large , the word error probability

at round satisfies where

(21)

is the achievable SNR-exponent and is given in Theorem 1.Proof: With a random coding scheme and , the

codeword is correctly decoded with probability one at roundif [6], [26], in which case, the receiver feeds back anACK (in contrast to the outage case, where an ACK is fed backif ). The proof then follows similar arguments as theproof of Theorem 2, noting from Theorem 1 that

.

Theorem 3 shows that the rate-diversity-delay tradeoffstated in Theorem 2 is achievable with random codes using thetransmission scheme described in Section III-B when is notan integer; and then, the optimal rate-diversity-delay tradeoffis given by (21). Furthermore, the optimal outage diversity andSNR-exponent of INR-ARQ transmission with delay constraint

is similarly characterized by and given in (18)and (21), respectively.

C. One-Bit MIMO ARQ

In an INR-ARQ system with one-bit ACK/NACK feedback(classical INR-ARQ), the optimal rate-diversity-delay tradeoffis given by the following.

Theorem 4: Consider INR-ARQ transmission over theMIMO block-fading channel in (1) using constellation ofsize and the transmission scheme described in Section III-B,where a codeword is considered successfully delivered at round

if . Assume that the number of feedback levels is. Subject to the power constraint in (10), the optimal

rate-diversity-delay tradeoff is given by

(22)

(23)

for all such that is continuous. Furthermore, the rate-diversity-delay tradeoff is achievable when is not aninteger.

Proof: The proof follows the same arguments as that ofTheorems 2 and 3, with only two feedback levels at 0 and ,respectively.

Theorem 4 characterizes the optimal outage diversity forINR-ARQ systems with when is noninteger. When

is integer, the outage diversity at round is upper boundedby given in (23). A lower bound on the outage diversityis given by the recursive formula in (23) with .

D. Numerical Results

We numerically compare the optimal rate-diversity-delaytradeoff of INR-ARQ systems with , and with

as well as the optimal tradeoff of an INR-ARQ systemwith constant transmit power. The optimal rate-diversity-delaytradeoff and for INR-ARQ transmission with

over the MIMO block-fading channel withare illustrated in Fig. 3(a).

For an INR-ARQ system with delay constraint and constanttransmit power (short-term power constraint), the outage prob-ability is the same as that obtained by transmission withrate over a block-fading channel with fading blocks [21].From Theorem 1, the optimal outage diversity is givenby2

(24)

and is achievable by random codes for all rates such thatis continuous. The rate-diversity-delay tradeoff of the

INR-ARQ system with constant transmit power is plotted inFig. 3(b). Fig. 3 shows an order-of-magnitude improvement inoutage diversity of INR-ARQ when a long-term power con-straint is allowed. Furthermore, significant gains in outage di-versity are provided by multibit feedback, especially at trans-mission rates close to . Since high is particularly rel-evant in ARQ systems, the result suggests that multibit feedbackwill give significant gains in practical implementations.

V. POWER ADAPTATION AND FEEDBACK DESIGN

The design of optimal feedback and transmission rules for anARQ system with multibit feedback includes joint optimizationof the overall set of quantization thresholds

and the corresponding power adaptive rule. The optimal feedback and power adaptation rule is

obtained by minimizing

(25)

subject to the power constraint in (10). To the best of our knowl-edge, the optimization problem is not analytically tractable. We,therefore, propose to partition the design problem into two steps.Step 1: At round , determine a set of feedback thresh-

olds for every feedback vector.

Step 2: Given the set of feedback thresholds in Step 1, deter-mine the corresponding transmit power rule, mini-mizing the outage probability.

The above procedure sub-optimally partitions the joint opti-mization problem into two sequential problems. Moreover,in the following, each individual problem is also sub-opti-mally solved. Nevertheless, this design procedure leads to apractically implementable algorithm that achieves the optimaldiversity derived in the previous section.

2The rate-diversity-delay tradeoff of [21] is larger than that given in (24) sinceit is obtained with rotations, which increase the constellation size, complexityand peak-to-average power ratio.


Fig. 3. Optimal rate-diversity-delay tradeoff of ARQ transmission with(a) long-term power constraint and (b) and constant power. 16-QAM is usedover a MIMO block-fading channel with .Thick and thin lines in (a) represent the optimal tradeoffs achievedby multibit feedback and achieved by one-bitfeedback , respectively. Crosses and circles correspond to the ratepoints where the SNR-exponent of random codes does not achieve the optimaldiversity.

A. Selecting the Set of Feedback Thresholds

From the observations in Remark 1, we propose the followingchoice of feedback thresholds. Consider the feedback levels atround for a given feedback vector . Let

and . The feedback thresholds in round ,given is then determined as follows.

1) Place a threshold at , and at;

2) Place thresholds at ;3) Place the remaining thresholds sequentially

within

until no more thresholds are left to place, and such that thethresholds uniformly partition each region.

The procedure for choosing the thresholds , given the feed-back vector , is illustrated in Fig. 4. More particularly, thefeedback thresholds for INR-ARQ transmission over the block-fading channel with , and

using 16-QAM constellations are illustrated in Fig. 2,where , and the values of are re-ported in Table II.

B. Power Adaptation

The sub-optimal power adaptation rule is obtained from thefollowing simplifications.

• We consider a power constraint more stringent than theconstraint in (10),

(26)

for , whereby definition.

• When feedback is received, we have that. Then, the feedback probability is approximated

from (9) by replacing with ; and the outageprobability can be upper bounded as

(27)

where is given by (5) with .• To further simplify the problem, we consider minimizing

sequentially.Based on the simplifications, the corresponding power adapta-tion rule is obtained by solving

(28)

The optimization problem is separable and, thus, can be solvedvia a branch-and-bound simplex algorithm using piece-wiselinear approximation [27]. For single-input multiple-output(SIMO) channels, the probabilities and in(28) can be approximated numerically by shifting the outageprobability bounds in [28] according to the gap between thebound and the corresponding simulation curve at high SNR.For MIMO channels, solving (28) requires tabulating the prob-abilities and , which can be obtained fromMonte-Carlo simulations.

It follows from (27) that the objective function in (28) upperbounds the outage probability for a given power allocation rule.The bound is useful to evaluate the system performance, espe-cially at low outage probability where Monte Carlo simulationis too computational demanding.

C. Numerical Results

First consider SISO INR-ARQ transmissionwith at rate over the block-fading channel in(1) with using 16-QAM input constellations. The outage


Fig. 4. Example of feedback threshold design .

Fig. 5. Outage performance of ARQ transmission schemes for a 16-QAM input block-fading channel with . The graylines represent the optimal asymptotic slope.

TABLE IIFEEDBACK THRESHOLDS FOR

performance of systems with is illustrated inFig. 5. The simulation shows that the slope of the outage curvesapproach the optimal outage diversity, which is 3 for constantpower, 4 for and 5 for . Due to the diversity gain,we observe significant gains in outage performance when powerallocation is allowed, especially in systems with multibit feed-back. Particularly, power allocation for the system withprovides approximately 2 dB gain in power at outage probability

; while an additional 2 dB gain is observed when .

The outage performance of MIMO INR-ARQ transmissionover the block-fading channel in (1) with

using 16-QAM input constellations is illustrated inFigs. 6. Similar to the SISO case, Fig. 6 shows that systemswith power allocation significantly outperform that with con-stant transmit power. A 6 dB gain in power is observed at outageprobability when power allocation is employed in an ARQsystems with one bit feedback. Further performance gains areobtained with multibit. For , an additional 1.5 dB gain inpower is observed at outage probability .

In both cases, the simulation results suggest that increasingbeyond 8 does not substantially improve the outage perfor-

mance; and thus, the figures show that even for , the sub-optimal choice of feedback thresholds in Section V-A performswithin 1 dB of systems with large and optimal thresholds.

The simulation ranges in Figs. 5 and 6 do not reveal the outagediversity of the ARQ systems. We employ the bounds obtainedfrom solving (28) to numerically illustrate the outage diversity


Fig. 6. Outage performance of ARQ transmission using the 16-QAM input constellation over the block-fading channel with. Systems with constant transmit power, and systems employing power adaptation with are considered.

Fig. 7. Upper bound on outage performance of ARQ transmission schemes using 16-QAM constellation over the block-fading channel withand . The gray lines represent the asymptotic bound with diversity 5.

achieved by multibit feedback. Figs. 7 and 8 plot the boundsof the outage performance presented in Figs. 5 and 6. The plotsshow that in both the SISO and MIMO systems, outage diversity5 is achieved when , as predicted by Theorem 2.

VI. CONCLUSION

We have studied the outage performance of MIMOblock-fading channels with and without employing theINR-ARQ strategy. An information-theoretic multibit feedbackINR-ARQ scheme is proposed based on the accumulative

mutual information, which potentially improves the perfor-mance of INR-ARQ transmission with minimal extra overheadrequirement compared to classical INR-ARQ. The study onpower adaptation has revealed large gains in outage diversityprovided by multibit feedback in INR-ARQ systems witha long-term power constraint. More generally, the multibitfeedback INR-ARQ based on accumulated mutual informationmay prove useful in obtaining the fundamental limit of multibitfeedback INR-ARQ systems. Furthermore, since the proposedscheme is a generalization to that in [17] and [19], it promises


Fig. 8. Upper bound on outage performance of ARQ transmission using 16-QAM input constellations over the block-fading channel withand . The gray lines represent the asymptotic bound with diversity 5.

further gain from the throughput performance obtained in [17],[19].

APPENDIX A*PROOF OF THEOREM 1

We first assume a genie-aided receiver that perfectly elimi-nates the interference between the transmit antennas. This re-sults in parallel SIMO block-fading channels, each withreceive antennas. Let be the realized input-output mutualinformation of the genie-aided channel, then . Further-more, from the analysis in [26], [28], [29], we have that

(29)

Therefore

(30)

The proof is thus completed by proving that

(31)

Following the arguments in [26], [28], [29], we have that

(32)

and therefore

(33)

We now prove that . Considering trans-mission over the block-fading channel in (1) with random codes

of rate , where the coded symbols in are drawn uni-formly random from the constellation . Let be the worderror probability achieved by random coding. We have from therandom-coding achievability and the strong converse theorem[3]–[5] that for a channel realization

(34)

when . Therefore, the word error probability of randomcodes satisfies

(35)

We now prove that . Consider encoding and trans-mitting a message as a random codeword . Assuming thatthe channel realization is , the pairwise error probability be-tween and is bounded by [25]

(36)

where, by letting

(37)

Here, is the channel gain from transmit antenna to re-ceive antenna in block , and is the coded symbol trans-mitted by antenna at time instant of block . Let us write


, where . Further define a ma-trix of normalized fading gains where

, then

(38)

By averaging (36) over the random coding ensemble, the pair-wise error probability of random codes is

(39)

(40)

where is the th entry of vector and

(41)

By summing over the possible error events, the unionbound on the word error probability is given by

(42)

For any , denote , and ,where

(43)

Then, for any given , and letting, we can write

(44)

(45)

Since the ’s are uniformly drawn from , we havethat

(46)

with probability 1 if there exists such that .

Noting that , it follows from (41) that

(47)

Thus, the error probability in (42) is asymptoticallyupper-bounded by

(48)

Let be the outageset. By averaging over the fading matrix and letting ,the error probability is bounded by

(49)

where is the joint pdf of the random vector . Followingthe analysis in [29], and letting , the SNR-exponent forthe case of using random codes is lower bounded by

(50)

(51)

Thus, by letting , the outage diversity is achievableusing random codes. Therefore, we have from (35) that

(52)

Thus, (31) is obtained from (33).


APPENDIX BPROOF OF THEOREM 2

A sketch of the proof is given as follows. We first lower-bound the outage diversity by considering a sub-optimal ARQsystem with feedback levels, where the quan-tization thresholds are placed at

. Using Theorem 1, we prove by induction that theoutage diversity of the sub-optimal ARQ system at round is

.Conversely, consider an optimal INR-ARQ system with

feedback levels. The outage performanceof the system can be improved by adding extraquantization thresholds (and corresponding feedback indices)at . Using Theorem 1, we prove by induc-tion that the outage diversity at round of the improved systems(with feedback levels) is also given by .Therefore, is the optimal outage diversity at round foran ARQ system with feedback levels.

A. Lower Bound on the Optimal Outage Diversity

To get a lower bound to the outage diversity, consider an ARQsystem with feedback levels, where the fol-lowing (sub-optimal) set of feedback thresholds is employed,

(53)

with . In this case, feedback index is delivered atround if , regardless of the realized feedbackindices of the previous rounds. At round , the transmit poweris sub-optimally adapted to the feedback index as

, where

(54)

The power adaptation rule in (54) satisfies the power constraintin (10). We now derive the outage diversity achieved by theaforementioned system.

For , we have from Theorem 1 that

(55)

where .For and a given , we now

prove by induction that for

(56)

where is given in (18).

Equation (55) shows that (56) is correct at round 1. Assumenow that (56) is correct at round . From (54) we have that

(57)

Therefore, for

(58)

Given and , we have that

. Therefore, by applying Theorem 1,and noting the transmit power in (57), we have that

(59)

Since (56) is assumed at round , the first summation dominatesin (58). Thus, from (59), we have that

(60)

The asymptotic exponent in (60) is given by

(61)

(62)

(63)

(64)

where (62) follows from assumption in (56),and (63) follows since the minimum in (62) is achieved witheither or . Therefore, from (60)

(65)

where in (18). Thus, (56) is cor-rect for by induction. Consequently, for any

, we have that

(66)


and thus, the diversity in (18) is achieved by the ARQ systemwith feedback levels.

Noting when , the outage prob-ability at round is dominated by the events with and

in (60), which correspond to the events with. The observation is useful for designing the

feedback thresholds for the system, as summarized in Remark 1.

B. Upper Bound on the Optimal Outage Diversity

Conversely, we derive an upper bound to the outage diver-sity achieved by a system with optimal feedback thresholdwith levels per transmission round. We first assume that

for some . Consider im-proving the performance of the system by employing a feed-back threshold set with feedback levelsper ARQ round by adding levels to the optimal feed-back threshold set . The extra levels are located at

.Let

, and further let. Then, given that the feed-

back vector at round is , the receiver deliversfeedback index if ;otherwise, it delivers index , where is chosen such that

.From the power constraint (10), the optimal power allocation

rule is upper-bounded by

(67)

Meanwhile, the power adaptation rule

(68)

satisfies the power constraint in (10). Therefore, theoptimal power allocation rule asymptotically satisfies

given in (67).For , let

.

Since for , belongs to the set ofthresholds

for some . Applying

Theorem 1, for any and , we havethat

(69)

(70)

where given in (18).

For the induction proof, assume that whenand , we have

(71)

where given in (18). The assumption is correctfor . We prove that (71) is also valid at round . In fact,considering , we have

From assumption (71) and power allocation rule (67), when, the transmit power in round is

for all . Therefore,

when . Thus, with similararguments that are used to derive (59), we have that

(72)

as given in (60). Therefore, following the steps used to derive(65), we have that

(73)

for . It follows that

(74)

for all . The results in (73) and (74) prove that as-sumption (71) is valid at round , and thus, by mathematicalinduction, (71) is valid for .

Since

(75)

which proves that the outage diversity of the system with-level feedback is the same as that given in (18).

REFERENCES

[1] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoreticconsiderations for cellular mobile radio,” IEEE Trans. Veh. Technol.,vol. 43, no. 2, pp. 359–378, May 1994.

[2] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Informatic-theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol.44, no. 6, pp. 2619–2692, Oct. 1998.


[3] S. Arimoto, “On the converse to the coding theorem for discrete mem-oryless channels,” IEEE Trans. Inf. Theory, vol. IT-19, pp. 357–359,1973.

[4] E. Malkamäki and H. Leib, “Coded diversity on block-fading chan-nels,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 771–781, Mar.1999.

[5] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nded. Hoboken, NJ: Wiley, 2006.

[6] G. Caire and D. Tuninetti, “The throughput of hybrid-ARQ protocolsfor the Gaussian collision channel,” IEEE Trans. Inf. Theory, vol. 47,no. 5, pp. 1971–1988, Jul. 2001.

[7] G. Caire, G. Taricco, and E. Biglieri, “Optimal power controlover fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp.1468–1489, Jul. 1999.

[8] G. Caire, D. Tuninetti, and S. Verdú, “Variable-rate coding for slowlyfading Gaussian multiple-access channels,” IEEE Trans. Inf. Theory,vol. 50, no. 10, pp. 2271–2292, Oct. 2004.

[9] S. Nanda, R. Walton, J. Ketchum, M. Wallace, and S. Howard, “A high-performance MIMO OFDM wireless LAN,” IEEE Commun. Mag., vol.43, no. 2, pp. 101–109, Feb. 2005.

[10] A. Ghosh, D. Wolters, J. Andrews, and R. Chen, “Broadband wirelessaccess with WiMax/802.16: Current performance benchmarks and fu-ture potential,” IEEE Commun. Mag., vol. 43, no. 2, pp. 129–136, Feb.2005.

[11] D. J. Costello, J. Hagenauer, H. Imai, and S. B. Wicker, “Applicationsof error-control coding,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp.2531–2560, Oct. 1998.

[12] L. Zheng and D. N. Tse, “Diversity and multiplexing: A fundamentaltradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol.49, no. 5, pp. 1073–1096, May 2003.

[13] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codesfor high data rate wireless communications: Performance criterionand code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp.744–764, Mar. 1998.

[14] H. F. Lu and P. V. Kumar, “Rate-diversity tradeoff of space-timecodes with fixed alphabet and optimal constructions for PSK mod-ulation,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2747–2751,Oct. 2003.

[15] H. F. Lu and P. V. Kumar, “A unified construction of space-time codeswith optimal rate-diversity tradeoff,” IEEE Trans. Inf. Theory, vol. 51,no. 5, pp. 1709–1730, May 2005.

[16] J. Perret and D. Tuninetti, “Repetition protocols for block fadingchannels that combine transmission requests and state information,”in Proc. IEEE Int. Conf. Communications, Beijing, China, May 2008,pp. 1297–1301.

[17] E. Visotsky, Y. Sun, and V. Tripathi, “Reliability-based incrementalredundancy with convolutional codes,” IEEE Trans. Commun., vol. 53,no. 6, pp. 987–997, Jun. 2005.

[18] Z. Yiqing and W. Jiangzhou, “Optimal subpacket transmission for hy-brid ARQ systems,” IEEE Trans. Commun., vol. 54, no. 5, pp. 934–942,May 2006.

[19] A. Steiner and S. Shamai (Shitz), “Multi-layer broadcasting hy-brid-ARQ strategies for block-fading channels,” IEEE Trans. WirelessCommun., vol. 7, no. 7, pp. 2640–2650, Jul. 2008.

[20] H. El Gamal, G. Caire, and M. O. Damen, “The MIMO ARQ channel:Diversity-multiplexing-delay tradeoff,” IEEE Trans. Inf. Theory, vol.52, no. 8, pp. 3601–3621, Aug. 2006.

[21] A. Chuang, A. Guillén i Fàbregas, L. K. Rasmussen, and I. B.Collings, “Optimal throughput-diversity-delay tradeoff in MIMOARQ block-fading channels,” IEEE Trans. Inf. Theory, vol. 54, no. 9,pp. 3968–3986, Sep. 2008.

[22] K. D. Nguyen, L. K. Rasmussen, A. Guillén i Fàbregas, and N. Let-zepis, “Diversity-rate-delay tradeoff for ARQ systems over the MIMOblock-fading channels,” in Proc. Aus. Comm. Theory Workshop,Sydney, Australia, Feb. 2009, pp. 116–121.

[23] H. Liu, L. Razoumov, N. Mandayam, and Spasojevic, “An optimalpower allocation scheme for the STC hybrid-ARQ over energy lim-ited networks,” IEEE Trans. Wireless Commun., vol. 8, no. 12, pp.5718–5722, Dec. 2009.

[24] T. Kim, K. Nguyen, and Guillén i Fàbregas, “Coded modulation withmismatched CSIT over MIMO block-fading channels,” IEEE Trans.Inf. Theory, vol. 56, no. 11, pp. 5631–5640, Dec. 2010.

[25] A. J. Viterbi and J. K. Omura, Principles of Digital Communications.New York: McGraw-Hill, 1979.

[26] R. Knopp and P. A. Humblet, “On coding for block fading channels,”IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 189–205, Jan. 2000.

[27] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Program-ming: Theory and Algorithms, 3rd ed. New York: Wiley, 2006.

[28] K. D. Nguyen, A. Guillén i Fàbregas, and L. K. Rasmussen, “A tightlower bound to the outage probability of block-fading channels,” IEEETrans. Inf. Theory, vol. 53, no. 11, pp. 4314–4322, Nov. 2007.

[29] A. Guillén i Fàbregas and G. Caire, “Coded modulation in the block-fading channel: Coding theorems and code construction,” IEEE Trans.Inf. Theory, vol. 52, no. 1, pp. 91–114, Jan. 2006.

Khoa D. Nguyen (S’06–M’10) was born in Vietnam in 1982. He received theBachelor of Engineering degree (electrical and electronics engineering) fromthe University of Melbourne, Australia, in December 2005 and the Ph.D. de-gree from the Institute for Telecommunications Research, University of SouthAustralia, in March 2010.

Since 2009, Dr. Nguyen has been a Research Fellow at the Institute forTelecommunications Research. He was a summer research scholar at theAustralian National University in 2004 and held a visiting appointment at theUniversity of Cambridge, Cambridge, U.K., in 2007. His research interests arein the areas of information theory and communication theory, coding theory,and adaptation and signal processing for wireless applications.

Lars K. Rasmussen (S’92–M’93–SM’01) was born on March 8, 1965 inCopenhagen, Denmark. He got his M.Eng. in 1989 from the Technical Uni-versity of Denmark (Lyngby, Denmark) and his Ph.D. degree from GeorgiaInstitute of Technology (Atlanta, GA) in 1993. From 1993 to 1995, he wasa Research Fellow at the Institute for Telecommunications Research (ITR),University of South Australia (Adelaide, Australia). From 1995 to 1998, hewas a Senior Member of Technical Staff with the Centre for Wireless Com-munications at the National University of Singapore (Singapore). From 1999to 2002, he was an Associate Professor at Chalmers University of Technology(Gothenburg, Sweden), where he maintained a part-time appointment until2005. From 2002 to 2008, he held a position as Research Professor at ITR,University of South Australia (Adelaide, Australia), where he was the Con-venor of the Australian Research Council (ARC) Communications ResearchNetwork (ACoRN), and a co-founder of Cohda Wireless Pty Ltd. He has heldvisiting positions at University of Pretoria (Pretoria, South Africa), SouthernPoro Communications (Sydney, Australia), and Aalborg University (Aalborg,Denmark). He now holds a position as Professor in Communications Theory,School of Electrical Engineering, and the ACCESS Linnaeus Center at the KTHRoyal Institute of Technology (Stockholm, Sweden). He is a Senior Memberof the IEEE, a member of the IEEE Information Theory and CommunicationsSocieties and served as Chairman for the Australian Chapter of the IEEEInformation Theory Society 2004–2005, and has been a board member of thejoint IEEE Communications Society and IEEE Vehicular Technology Chapterin Sweden since 2010. He is an associate editor for IEEE TRANSACTIONS ONCOMMUNICATIONS and was a guest editor for IEEE JOURNAL ON SELECTEDAREAS IN COMMUNICATIONS (2007). He was also a member of the organizingcommittees for the IEEE 2004 International Symposium on Spread SpectrumSystems and Applications (Sydney, Australia), and the IEEE 2005 InternationalSymposium on Information Theory (Adelaide, Australia), as well as the TPCco-chair of the Communications Theory Symposium at the IEEE GlobalCommunications Conference (Globecom) 2009. His research interests includetransmission strategies and coding schemes for wireless communications,coding for delay-constrained applications, ad hoc wireless networks, coop-erative communications, communications and control, communications andpositioning, and vehicular communication systems.


Albert Guillén i Fàbregas (S’01–M’05–SM’09) was born in Barcelona,Catalunya, Spain, in 1974. He received the Telecommunications EngineeringDegree and the Electronics Engineering Degree from Universitat Politècnica deCatalunya, Barcelona, Catalunya, Spain, and the Politecnico di Torino, Torino,Italy, respectively, in 1999, and the Ph.D. in communication systems fromEcole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in2004.

From August 1998 to March 1999, he conducted his Final Research Projectat the New Jersey Institute of Technology, Newark, NJ. He was with TelecomItalia Laboratories, Italy, from November 1999 to June 2000 and with the Euro-pean Space Agency (ESA), Noordwijk, The Netherlands, from September 2000to May 2001. During his doctoral studies, from 2001 to 2004, he was a Researchand Teaching Assistant at Institut Eurcom, Sophia-Antipolis, France. From June2003 to July 2004, he was a Visiting Scholar at EPFL. From September 2004to November 2006, he was a Research Fellow at the Institute for Telecommu-nications Research, University of South Australia, Mawson Lakes, Australia.Since 2007, he has been a Lecturer in the Department of Engineering, Univer-sity of Cambridge, Cambridge, U.K., where he is also a Fellow of Trinity Hall.He has held visiting appointments at Centrum Wiskunde & Informatica, Ams-terdam, The Netherlands; Ecole Nationale Supérieure des Télécommunications,Paris, France; Texas A&M University, Doha, Qatar; Universitat Pompeu Fabra,Barcelona, Spain; and the University of South Australia, Australia. His researchinterests are in communication theory, information theory, coding theory, digitalmodulation, and signal processing techniques with wireless applications.

Dr. Guillén i Fàbregas is currently an Editor of the IEEE TRANSACTIONS ONWIRELESS COMMUNICATIONS. He received a pre-doctoral Research Fellowshipof the Spanish Ministry of Education to join ESA. He received the Young Au-thors Award of the 2004 European Signal Processing Conference EUSIPCO2004, Vienna, Austria and the 2004 Nokia Best Doctoral Thesis Award in Mo-bile Internet and 3rd Generation Mobile Solutions from the Spanish Institutionof Telecommunications Engineers. He is also a member of the ARC Communi-cations Research Network (ACoRN) and a Junior Member of the Isaac NewtonInstitute for Mathematical Sciences.

Nick Letzepis (M’03) received the Bachelor’s Degree in electrical and elec-tronic engineering with First Class Honours from Flinder’s University of SouthAustralia in 1998, and the Ph.D. Degree in telecommunications from the In-stitute for Telecommunications Research, University of South Australia, 2006.From 1998 to 2003, he worked as a Research Engineer for Dspace Pty., Ltd.,specializing in the research and development of digital satellite communica-tion systems. From 2003 to 2010 he was with the ITR, first as a Ph.D. candi-date (2003–2006) and then as a Research Fellow in the Coding and Informationtheory group (2006–2010). Since 2010, he has been in the Command ControlCommunications and Intelligence Division of the Defence Science and Tech-nology Organization as a Senior Communications Research Engineer. His re-search experience encompasses a broad range of fields, including: coding andinformation theory, communications theory and signal processing. More specif-ically, his research interests include: digital satellite communications, multi-an-tenna communications, multi-user communications, multi-carrier communica-tions (OFDM), spread spectrum, vehicle-to-vehicle communications, iterativedecoding, block-fading channels, and free-space optical communications.

Date post:	21-Jan-2021
Category:	Documents
Upload:	others
View:	13 times
Download:	0 times

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, …IEEE TRANSACTIONS ON INFORMATION THEORY, VOL....

Documents