02e7e51acee67995b5000000

WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob. Comput. 2004; 4:693696Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/wcm.247

Guest Editorial

Special Issue: Multiple-Input Multiple-Output (MIMO)Communications

By Robert W. Heath, Erik G. Larsson, Ross Murch, Arye Nehorai and Murat Uysal, Guest Editors

The idea of using multiple transmit and receive

antennas in wireless communication systems is one

of the most important breakthroughs in communica-

tion theory during the last decade. Popularly referred

to as MIMO technology, this concept can greatly

improve data throughput and link performance in

wireless networks. In principle, a MIMO system can

operate in, or anywhere between, one of the two

possible modes. If the transmitter knows the channel,

then one can use spatial beamforming techniques to

steer RF energy in the direction of the receiver. On the

other hand, if the transmitter does not know the

channel, one can use space-time coding which effect-

ively distributes the transmitted power uniformly in

all directions, and in addition augments the data with

structure that can be used to combat from fading dips.

Sometimes, space-time coding methods are grouped

into two categories: those that focus on throughput

improvement (e.g. Bell-Labs layered space-time

architecture, BLAST), and those that solely aim at

improving link performance (including, most notably,

orthogonal block coding and transmit diversity

schemes); however, this classification is simplistic

and many of the currently best known schemes do

not fall under any of these two groups.

Space-time coding, beamforming and their various

combinations are becoming relatively well understood

in the research communityas evidenced by the

literal explosion of research papers and books (e.g.

[1,2]) on the topic. Nevertheless, researchers are

continuing to explore the more intricate aspects of

combined coding over space and time. The goal of this

special issue has been to collect a few edge-cutting,

high-quality papers that not only capture the state-of-

the-art of the field but also highlight open problems

and current research topics. We are delighted to

present eight papers that survived a very competitive

peer review process. Broadly speaking, the papers can

be grouped into four categories: two papers that deal

with coding for MIMO, two papers on MIMO channel

modeling and simulation, two papers on signal pro-

cessing for MIMO and finally two papers on cross-

layer design for MIMO systems.

The first paper, Iterative receivers for coded MIMO

signaling, by Biglieri, Nordio and Taricco, is a tutorial

on iterative (turbo) processing for systems that con-

catenate a GF(2) channel code with a complex-valued

multidimensional (matrix) channel. The authors pre-

sent the topic using a first-principle approach, and they

also describe how iterative coding and demodulation

for MIMO models work and how they can be analyzed

by using EXIT charts, a powerful technique so far

mostly used for classical turbo coding. The next

paper, Improving the performance of coded FDFR

multi-antenna systems with turbo-decoding, by Wang,

Ma and Giannakis, continues on the theme of iterative

receiver structures. In this work, the authors show how

full-rate full-diversity (FDFR) space-time block codes,

developed by the authors in their previous work, can be

efficiently combined with GF(2) channel coding. This

paper provides valuable insight into the trade-off

between performance and complexity involved in the

choice of linear space-time block codes versus using

powerful GF(2) codes and their combination.

The third paper, by Patzold and Hogstad, A space-

time channel simulator for MIMO channel based on

the geometrical one-ring scattering model describes

a narrowband MIMO channel simulator based on a

geometric model, and evaluates its accuracy. This

topic is certainly important, as MIMO channels in

reality often have different characteristics than toy

models used in purely theoretical studies. The fourth

paper, Performance of MIMO spatial multiplexing

algorithms using indoor channel measurements and

models, is also on performance evaluation of MIMO

systems in realistic environments. The authors,

Copyright # 2004 John Wiley & Sons, Ltd.

Spencer and Swindlehurst, present performance re-

sults for MIMO spatial multiplexing with measured

channels. They examine quantitatively the user spa-

cings necessary to achieve practically decorrelated

channels.

The next paper, Space-time alignment for asyn-

chronous interference suppression in MIMO OFDM

cellular communications by Breinholt, Jung and

Zoltowski, is on signal processing for MIMO. The

paper addresses the important problem of co-channel

interference rejection in asynchronous MIMO-OFDM

systems. The authors present a powerful algorithm

that can improve the link performance and allow for a

more aggressive frequency reuse, thereby addressing

the obviously very significant problem of spectrum

shortage in wireless networks. In Symbol timing

estimation for MIMO correlated flat-fading channels,

Wu and Serpedin continue on the topic of signal

processing for MIMO. This paper proposes efficient

methods for symbol timing estimation in MIMO

systems. Synchronization and channel estimation ac-

count for a large part of the design effort of the silicon

area in todays digital communication receiver, and

the same is likely to hold true for future MIMO

systems. In this paper, the authors address this im-

portant topic by deriving both data-aided and non-

data-aided algorithms, and compare their perfor-

mances to the corresponding Cramer-Rao bounds.

The seventh contribution is Dynamic channel

management in MIMO-OFDM cellular systems by

Lu et al. This paper addresses the problem of power

control and channel allocation for MIMO systems.

This is an important problem that is relevant to study

because in a multiuser system, optimization of chan-

nel and power allocation algorithms may be as im-

portant as optimizing link-level performance. The

final article, Optimum space-time transmission for a

high K-factor wireless channel with partial channel

knowledge by Vu and Paulraj, continues to discuss

cross-layer aspects. This paper deals with the use of

partial channel state information (CSI) at the trans-

mitter in a MIMO system. With no CSI at the

transmitter, space-time coding is optimal, and with

perfect CSI at the transmitter, beamforming is the best

thing to do. In this paper, the authors present a

thorough study of the case of two transmit antennas

and one receive antenna. This study delivers insight

into the fundamental aspects of the problem, and

poses interesting questions for future work.

We would like to thank the contributors of this

special issue for letting us publish their work. Also,

we are indebted to the reviewers who helped select

papers for the issue and provided the authors with

feedback. Finally, we thank editor-in-chief, Prof.

Mohsen Guizani, and the staff at Wiley for their

support.

Robert W. Heath, Guest EditorUniversity of Texas, USA

Erik G. Larsson, Guest EditorUniversity of Florida, USA

Ross Murch, Guest EditorThe Hong Kong University of

Science and Technology, Hong Kong

Arye Nehorai, Guest EditorThe University of Illinois at Chicago, USA

Murat Uysal, Guest EditorUniversity of Waterloo, Canada

References

1. Paulraj A, Nabar R, Gore D. Introduction to Space-TimeWireless Communications. Cambridge University Press:Cambridge, UK, 2003.

2. Larsson E, Stoica P. Space-Time Block Coding for WirelessCommunications. Cambridge University Press: Cambridge,UK, 2003.

Authors Biographies

Robert W. Heath, Jr. received B.S.and M.S. degrees from the Univer-sity of Virginia, Charlottesville, VA,in 1996 and 1997 respectively, andthe Ph.D. from Stanford University,Stanford, CA, in 2002, all in elec-trical engineering. From 1998 to1999, he was a senior member ofthe technical staff at Iospan Wire-less, Inc., San Jose, CA, where heplayed a key role in the design and

implementation of the physical and link layers of the firstcommercial MIMO-OFDM communication system. From1999 to 2001, he served as a senior consultant for IospanWireless, Inc. In 2003, he founded MIMO Wireless, Inc., aconsulting company dedicated to the advancement ofMIMO technology. Since January 2002, he has been withthe Department of Electrical and Computer Engineering atThe University of Texas at Austin, where he serves as anassistant professor as part of the Wireless Networking andCommunications Group. His research interests includeinterference management in wireless networks, sequencedesign and all aspects of MIMO communication includingantenna design, practical receiver architectures, limitedfeedback techniques and scheduling algorithms. Dr. Heathserves as an associate editor for the IEEE Transactions onVehicular Technology.

694 EDITORIAL

Copyright # 2004 John Wiley & Sons, Ltd. Wirel. Commun. Mob. Comput. 2004; 4:693696

Erik G. Larsson received his Ph.D.in electrical engineering fromUppsala University, Sweden, in2002. He held research and teach-ing positions with Ericsson RadioSystems AB (Stockholm, Sweden),Uppsala University (Uppsala, Swe-den) and the University of Florida(Gainesville, FL). Since August2003, he is an assistant professorin the Department of Electrical and

Computer Engineering at the George Washington Univer-sity, Washington, DC. His research interests and experienceinclude space-time diversity for wireless communications,signal processing for communications and radar and loca-tion services for E-911. He has some 20 papers in IEEE andother international journals, he holds several U.S. patentsand is a co-author of the textbook Space-Time Block Codingfor Wireless Communications (Cambridge University Press,2003, with P. Stoica). He is an associate editor for the IEEETransactions on Vehicular Technology.

Prof. Ross Murch is an associateprofessor in the Department ofElectrical and Electronic Engi-neering at the Hong Kong Univer-sity of Science and Technology.His current research interestsinclude multiple antenna systems,compact antenna design, WLANand Ultra-Wide-Band (UWB) sys-tems for wireless communications.He has several US patents related

to wireless communication, over 150 published papers andacts as a consultant for industry and government. In addi-tion, he is an editor for the IEEE Transactions on WirelessCommunications and was the chair of the Advanced Wire-less Communications Systems Symposium at ICC 2002. Heis also the founding Director of the Center for WirelessInformation Technology at Hong Kong University ofScience and Technology which started in August 1997. Heis also the program director for the M.Sc. in telecommuni-cations at Hong Kong University of Science and Technol-ogy. From AugustDecember 1998, he was on sabbaticalleave at Allgon Mobile Communications (manufactured 1million antennas per week), Sweden and AT&T ResearchLabs, NJ, USA. Prof. Ross Murch received his bachelorsdegree in electrical and electronic engineering from theUniversity of Canterbury, New Zealand where he graduatedin 1986 with first class honors and was ranked first in hisclass. During his bachelors degree he was the recipient ofseveral academic prizes including the John Blackett prize forengineering and also the Austral Standard Cables prize. In1990, he completed his Ph.D., also in electrical and electronicengineering at the University of Canterbury. During his Ph.D.,he was awarded a RGC and also a New Zealand Telecomscholarship. From 1990 to 1992, he was a post-doctoratefellow at the Department of Mathematics and ComputerScience at Dundee University, Scotland. From 1992 to 1998,he was an assistant professor in the Department of Electrical

and Electronic Engineering at the Hong Kong University ofScience and Technology and since 1998 he has been anassociate professor there. He is a senior member of IEEE, aChartered Engineer and a member of IEE. In 1996 and 2001,he won engineering teaching excellence appreciation awards.

Arye Nehorai received his B.Sc.and M.Sc. degrees in electricalengineering from Technion, Israel,and the Ph.D. in electrical engineer-ing from Stanford University,California. After graduation, heworked as a Research Engineer forSystems Control Technology, Inc.,in Palo Alto, CA. From 1985 to1989, he was an assistant professorand from 1989 to 1995, associate

professor with the Department of Electrical Engineering atYale University. In 1995 he joined the Department ofElectrical Engineering and Computer Science at The Uni-versity of Illinois at Chicago (UIC), as a full professor. From2000 to 2001, he was chair of the Departments of Electricaland Computer Engineering (ECE) Division, which is now anew department. In 2001, he was named University Scholarof the University of Illinois. He holds a joint professorshipwith the ECE and Bioengineering Departments at UIC. Hisresearch interests are in signal processing, communicationsand biomedicine. Dr Nehorai is Vice President-Publicationsand Chair of the Publications Board of the IEEE SignalProcessing Society. He is also a member of the Board ofGovernors and of the Executive Committee of this Society.He was editor-in-chief of the IEEE Transactions on SignalProcessing from January 2000 to December 2002, and iscurrently a member of the Editorial Board of it SignalProcessing, the IEEE Signal Processing Magazine, andThe Journal of the Franklin Institute. He is the founderand guest editor of the special columns on leadershipreflections in the IEEE Signal Processing Magazine. Hehas previously been an associate editor of the IEEE Trans-actions on Acoustics, Speech and Signal Processing, theIEEE Signal Processing Letters, the IEEE Transactions onAntennas and Propagation, the IEEE Journal of OceanicEngineering and Circuits, System and Signal Processing.He served as chairman of the Connecticut IEEESignal Processing Chapter from 1986 to 1995, and a found-ing member, vice-chair and later chair of the IEEE SignalProcessing Societys Technical Committee on SensorArray and Multichannel (SAM) Processing from 1998 to2002. He was the co-general chair of the First andSecond IEEE SAM Signal Processing Workshops held in2000 and 2002. He was co-recipient, with P. Stoica, of the1989 IEEE Signal Processing Societys Senior Award forBest Paper, and co-author of the 2003 Young Author BestPaper Award of this Society, with A. Dogandzic. He receivedthe Faculty Research Award from the UIC College ofEngineering in 1999 and was adviser of the UIC OutstandingPh.D. Thesis Award in 2001. He was elected DistinguishedLecturer of the IEEE Signal Processing Society for the term2004 to 2005. He has been a fellow of the IEEE since 1994and of the Royal Statistical Society since 1996.

EDITORIAL 695


Murat Uysal was born in Istanbul,Turkey in 1973. He received theB.Sc. and the M.Sc. degree in elec-tronics and communication engi-neering from Istanbul TechnicalUniversity, Istanbul, Turkey, in1995 and 1998 respectively, andthe Ph.D. in electrical engineeringfrom Texas A&M University, Col-lege Station, Texas, in 2001. From1995 to 1998, he worked as a

research and teaching assistant in the Communication The-ory Group at Istanbul Technical University. From 1998 to

2001, he was affiliated with the Wireless CommunicationLaboratory, Texas A&M University. During the Fall of2000, he worked as a research intern at AT&T Labs-Research, Florham Park, New Jersey. In April 2002, hejoined the Department of Electrical and Computer Engineer-ing, University of Waterloo, Canada, as an assistant profes-sor. His research interests lie in communications theory withspecial emphasis on wireless applications. Specific areasinclude space-time coding, diversity techniques, coding forfading channels and performance analysis over fadingchannels. Dr. Uysal currently serves as an editor for theIEEE Transactions on Wireless Communications and as anassociate editor for the IEEE Communications Letters.


696 EDITORIAL


Invited paper

Iterative receivers for coded MIMO signaling

Ezio Biglieri, Alessandro Nordio and Giorgio Taricco*,y

Politecnico di Torino, Dipartimento di Elettronica, Corso Duca degli Abruzzi, 24-10129 Torino, Italia

Summary

In this tutorial paper, we describe iterative receivers that combine a soft decoder for a general space-time code with

a spatial-interference canceler. The performance of these receivers is analyzed by using EXIT charts, a convenient

graphical description that yields quite accurate results. By properly combining the EXIT characteristics of the

canceler and of the decoder, the convergence behavior of the iterative algorithms can be understood, and design

guidelines derived. The idea here is to observe that both the interference canceler and the decoder can be studied by

examining the evolution of the extrinsic information passed along to the connected blocks. Copyright # 2004 JohnWiley & Sons, Ltd.

KEY WORDS: multiple antennas; space-time coding; iterative receivers; EXIT charts

1. Introduction

Recently, multiple-antenna multiple-input multiple-

output (MIMO) techniques have been recognized to

be capable of greatly increasing the spectral efficiency

of wireless systems. For this reason, a considerable

research effort is being spent to design space-time

codes that approach the impressive values of channel

capacity available. Additional work is directed to-

wards the reduction of the complexity of optimum

decoding: in fact, maximum-likelihood receivers ex-

hibit a complexity that grows exponentially with the

modulation size and the number of antennas, and

hence become quickly unpractical as either parameter

is large. Thus, in addition to search for good space-

time codes, it is important to seek receivers that

achieve a close-to-optimum performance while keep-

ing a moderate complexity: this would remove the

practical restriction to small signal constellations or

few antennas.

Suboptimal receivers may include linear filters,

cancelers of spatial interference or sphere decoders.

In addition, iterative receivers have received a special

attention of late in several contexts: see, for example

[37, 9, 10, 1316, 19, 20, 2225]. In one of its

possible settings, an iterative receiver combines a

soft spatial-interference canceler with a soft-input

soft-output (SISO) decoder, as represented schemati-

cally in Figure 1. Before sending its soft decisions to

the hard decoder, the SISO decoder iteratively feeds

*Correspondence to: Giorgio Taricco, Politecnico di Torino, Dipartimento di Elettronica, Corso Duca degli Abruzzi, 24-10129Torino, Italia.yE-mail: [email protected]

Contract/grant sponsors: Cercom; PRIMO Project within FIRB.


extrinsic information (to be properly defined, which

we will do in the following) back to the soft canceler.

In Reference [2], the combination of a soft canceler

with turbo space-time codes was shown to provide a

good tradeoff between complexity and performance.

There, the received signals are first combined through

a linear minimum mean square error (MMSE) filter,

then spatial interference is reduced by feeding back

soft decisions provided by the decoder. If turbo codes

are used, even the SISO decoder is iterative. This

makes the overall receiver doubly iterative, in the sense

that preliminary results obtained from a few iterations

of the turbo-decoding algorithm are used to reduce

spatial interference. After this reduction, further turbo-

decoding iterations are performed in order to improve

on the interference cancellation, and so on.

In this paper, we elaborate in a tutorial fashion on

the concept of iterative receivers that combine a soft

decoder for a general space-time code with a spatial-

interference canceler. The performance of these re-

ceivers is analyzed by using EXIT charts [17]. By

properly combining the EXIT characteristics of the

canceler and of the decoder, convergence of the

iterative algorithms can be studied, and design guide-

lines derived. The idea here is to observe that, for the

interference canceler as well as for the decoder, their

behavior can be studied by examining how they

transform the extrinsic information passed along to

the connected blocks. One parameter describing this

extrinsic information is, as suggested in Reference

[17], mutual information. Thus, by combining in a

single chart the inputoutput characteristics of two

blocks, the convergence of a turbo-like algorithm can

be given a convenient graphical description, which,

although not exact, yields quite accurate results.

This paper is organized as follows. Section 2 is

devoted to the definition of the main concepts and

quantities used throughout, from SISO processors to

extrinsic probabilities and EXIT charts. Sections 3

and 4 describe how EXIT charts can be computed for

SISO decoders and for other SISO processors respec-

tively. Section 5 shows how SISO decoders and

interference cancelers can be combined in an iterative

receiver, with its performance evaluated through

EXIT charts. Conclusions are drawn in Section 6.

2. Definitions

2.1. Soft and Hard Decisions

Consider transmission of the n-tuple x x1; . . . ; xnof symbols chosen from an alphabet X . At the outputof the transmission channel a vector y is observed.

Following [8], we call a soft decision for xi the

a posteriori probability distribution of xi given y,that is, pxi j y. Since pxi; y pxi j ypy, andpy is irrelevant to the decision process, one may alsocall soft decision the probability distribution pxi; y,with y interpreted as a parameter. A hard decision for

xi is a probability distribution such that pxi j y isequal either to 0 or to 1.

2.2. Receivers and Interfaces

A receiver is a system accepting as its input the

channel observation y, and generating a hard decision

on each transmitted xj based on a suitable decision

rule (typically, the minimization of an error probabil-

ity). An interface accepts y as its input, and outputs a

soft decision on each xj. An interface is generally a

combination of devices, called SISO processors, that

generate soft decisions (for a detailed definition, see

infra, Section 2.5. Eventually a SISO output is passed

to the hard decoder. This accepts soft decisions at its

input and outputs hard decisions: for example if

X f1g, it chooses pxi 1 j y 1 wheneverpxi 1; y pxi 1; y. The goal of the inter-face, and hence of the SISO processors forming it, is

to process the received data so as to obtain soft

decisions as close as possible to correct hard decisions

before final decoding.

2.3. Extrinsic Probabilities

Turbo processing hinges on the exchange of

extrinsic information. To define properly the latter

Fig. 1. Block diagram of an iterative MIMO receiver.

698 E. BIGLIERI, A. NORDIO AND G. TARICCO


quantity, we use the concept of factor graph (see

Reference [12] and references therein). This represents

in graphical form the factorization of a function

f x1; . . . ; xn of several variables. The sum-productalgorithm allows one to compute (exactly and in a finite

number of steps) the marginals of the function with res-

pect to each variable. These are defined as the functions

fixi,Xxi

f x1; . . . ; xn 1

obtained by summing f x1; . . . ; xn over all its argu-ments consistent with the value of xi. The compact

notation xi denotes the vector x1; . . . ; xi1;xi1; . . . ; xn to be summed over. The sumproductalgorithm computes, for every edge of the factor graph

(which corresponds to one variable), two messages,

one per each direction, whose product yields the

marginal of the function with respect to that variable.

It is often convenient to think of these messages as

probability distributions, which is obtained by prop-

erly normalizing them. For example, with binary

variables a message can be thought of as a pair of

real numbers summing to 1.

A key feature of the sumproduct algorithm is that

a message sent along one direction does not depend on

the message sent along the other one. Thus, if one of

the messages derives directly from the a priori

knowledge of the edge variable, or from its measure-

ment at the channel output, the other message depends

only on the remaining variables: for this reason the

information it carries is called extrinsic. Two exam-

ples illustrate this concept.

Example 1: Soft Decoding

Consider a code with j j words x x1; . . . ; xnand assume that the a priori code word probabilities

are equal. These are transmitted over a stationary

memoryless channel such that the observed n-vector

y is such that

py j x Yni1

pyi j xi 2

Soft decoding of consists of computing the prob-

abilities

pxi; y Xxi

px; y

Xxi

pxpy j x

Xxi

j j 1x 2 Ynj1

pyj j xj 3

where the Iverson function x 2 takes on value 1if vector x is a code word, and 0 otherwise. The last

equation shows the factorization of the function

whose marginals yield the probabilities pxi; y. Thecorresponding factor graph is shown in Figure 2(a).

Application of the sumproduct algorithm yields for

each edge the messages shown in Figure 2(b).

The upward messages are the probabilities pyi j xi,corresponding to channel observations (these are to be

interpreted as functions of xi, with yi as parameters).

The downward messages exi are the extrinsic prob-abilities. Since, after proper message normalization,

the product exipyi j xi yields pxi; y, we define theextrinsic probabilities as the ratios between pxi; yand pyi j xi (suitably normalized):

exi pxi; y=pyi j xiP

xxi2X pxxi; y=pyi jxxi

Notice that, from

pxi; yi Xxi

j j 1x 2 pyi j x

Xxi

j j 1x 2 Yj 6i

pyj j xj

we obtain

exi pxi j yi 4

In other words, the extrinsic probability can also be

interpreted as the probability of the ith code word

symbol conditioned on all other channel observations,

namely yi, through the intermediary of the codestructure. For a simple example [7], examine the

single-parity-check binary code with length 3, whose

symbols are x1, x2 and x3 x1 x2. Informationabout x1 can be gathered from the observation of y1,

and also from the separate observation of y2 and y3,

Fig. 2. (a) Factor graph for soft decoding. (b) Messagespassed along each edge by the sumproduct algorithm.

ITERATIVE MIMO RECEIVERS 699


supplemented by the knowledge of the code structure,

i.e. the fact that x3 x1 x2. Thus, the latter observa-tion generates the extrinsic information.

Example 2: Soft Interference Cancellation

Consider the transmission of n independent binary

symbols x x1; . . . ; xn on a common channel, andthe observation of a noisy vector y whose components

are known functions of all symbols (e.g. linear com-

binations with known coefficients). The channel is

described by the function py j x. Soft estimation ofxi, i 1; . . . ; n, consists of computing

pxi; y Xxi

py; x

Xxi

py j xYnj1

pxj 5

which is tantamount to marginalizing the function

py j xpx1 pxn. The corresponding factor graphis shown in Figure 3, along with the messages ex-

changed by the blocks in the application of the sum

product algorithm. Since, after proper message

normalization}, the product exipxi yields pxi; y,the extrinsic probability is defined as

exi py j xiP

xxi2X py jxxiP

xipy j xpxiP

x py j xpxi6

2.4. The Turbo Algorithm

This consists of coupling SISO processors in such a

way that the extrinsic probability output of one

processor is fed to the input of another. Consider, in

particular, soft interference cancellation of an n-tuple

of coded symbols. The factor graphs of Figures 2 and

3 can be joined so as to share the edges labeled

x1; . . . ; xn. The resulting factor graph is shown inFigure 4. If it exhibits cycles, then the sumproduct

algorithm does not generate the a posteriori probabil-

ities, and an iterative (turbo) algorithm must be used

instead to obtain their approximate values [12]. This

algorithm computes repeatedly the two-way messages

associated with the edges of the graph, until a termi-

nation criterion stops the iterative process. A possible

schedule is illustrated in Figure 4(b): first, y is

observed and the extrinsic probabilities ~eexi, i 1; . . . ; n, are computed and passed along to the codeblock. This computes exi, i 1; . . . ; n, by using~eexi as if they were the channel observationspyi j xi in the algorithm of Figure 2. Next, the lowerblock uses exi, i 1; . . . ; n, as if they were the apriori probabilities pxi in the algorithm ofFigure 3(b) and so forth.

An important feature of the turbo algorithm is its

need for independent messages. This can be met by

using a large-enough value of n and introducing a

random interleaver between the two blocks of

Figure 4, i.e. a device which permutes the components

of the vector x x1; . . . ; xn.

2.5. SISO Processors

For proper definition of the turbo algorithm, it is

convenient to describe the SISO processors as two-

input, two-output devices as shown in Figure 5. A

SISO processor accepts two sets of inputs:

(1) Channel observations, i.e., the conditional prob-

ability distribution py j x, depending on theknowledge of the channel statistics and on the

observation of y and

(2) A priori probabilities, i.e., the marginal probabil-

ities pxi.It outputs:

(1) Soft decisions, i.e. a posteriori probabilities

pxi j y, which will eventually be sent to thedecoder generating hard decisions.

(2) Extrinsic probabilities exi.

Fig. 3. (a) Factor graph for soft interference cancellation. (b)Messages passed along each edge by the sumproduct

algorithm.Fig. 4. (a) Factor graph for soft interference cancellation ofcoded symbols. (b) Messages passed along each edge by the

turbo algorithm.



2.6. EXIT Charts

Since the turbo algorithm operates on extrinsic prob-

abilities, its convergence behavior can be studied by

examining how these evolve in time. A convenient

graphical description of this process is given by EXIT

charts [17], which yield quite accurate, albeit not

exact, results. An EXIT chart is a graph that illustrates

the inputoutput relation of a SISO processor by

showing the transformations induced on a single

parameter associated with input and output extrinsic

probabilities. Let us focus for simplicity on a binary

alphabet X f1g. The rationale behind EXITcharts stems from the observation that the logarithmic

likelihood ratio (LLR)z

x, log ex 1ex 1

is well approximated by a conditionally normal ran-

dom variable (we write j x N; 2) whoseprobability density function (pdf) p j x satisfiesthe consistency condition

j j 2

27

where and 2 denote conditional mean and variancerespectively. Hence, under this condition, a single

parameter (e.g. 2) completely defines p j x.Corresponding probability distributions: These are

in fact estimated from random observations.

EXIT charts describe the evolution of p j x byshowing the evolution of one parameter derived from

it. There are several possible choices for this para-

meter (a thorough discussion and a comparison can be

found in Reference [21]). A common, convenient

choice is the mutual information Ix; between xand , defined as{

Ix; 12

Xx2f1g

11

p j xlog2p j xp d 8

with p 0:5p j x 1 p j x 1If condition (7) is satisfied, then j x N

x2=2; 2, and hence Ix; depends only on 2.We have, explicitly,

Ix; 1 12

Xx2f1g

11

p j x

log2 1 p j xp j x

d

J2

, 1 11

1ffiffiffiffiffiffi2

p

ez2=22=22

log21 ez dz 9

The behavior of J2, which can be examined bynumerical evaluation of Equation (9), is shown in

Figure 6. If p j x is not known, we still assume that

logp j x

p j x x

zHere, we drop the subscript i to simplify notation.This condition has been first derived in Reference [17] andis a straightforward consequence of the fact that the noise isGaussian-distributed.

Fig. 5. Basic block diagram of a soft-input soft-output(SISO) processor.

{The notation here is not the most felicitous one, as it doesnot distinguish between the random variable x and the valuesit takes on. We put up with it, as it is commonly used in theliterature.

Fig. 6. Plot of the function J2 defined in Equation (9).



Thus, by using the weak law of large numbers, we

can approximate the mutual information Ix; as [18]:

Ix; 1 12

Xx21

11

p j xlog21 ex d

1 1n

Xni1

log21 exii 10

where i are independent samples of the randomvariable corresponding to input values xi.

Refer again to Figure 5, which shows the block

diagram of a SISO processor. Following the factor-

graph notation, since X f1g, we can denote inputand output messages as binary random vectors

lxi xi 1; xi 1 representing prob-ability distribution estimates. Since xi 1xi 1 1, each one of these messages isequivalently represented by the logarithm of the ratio

of its components, i.e. by the LLR

i logxi 1xi 1

This allows us to write Ix; l instead of Ix; .Specifically, we have four types of messages:

(1) input a priori messages: laxi pxi;(2) input channel observation messages: loxi

pyi j xi=P

xxi2X pyi jxxi;(3) output soft decision messages: ldxi pxi j y;(4) output extrinsic messages: lexi exi.

It follows that we can define a priori, channel ob-

servation, decision, and extrinsic mutual informations

as Ia , Ix; la, Io , Ix; lo, Id , Ix; ld and Ie , Ix; le respectively.

We are now ready to describe a SISO processor

by giving its extrinsic information transfer (EXIT)

function

Ie TIa; Io 11

Several examples of EXIT functions can be

found in the literature (see, e.g. References

[9,17,19]), all obtained by Monte-Carlo simulation.

The general algorithm used to derive the values of Ie

from those of Ia; Io, and hence the EXIT function T ,can be outlined as follows (in the next two sections

it will be specialized to SISO decoders and other

processors):

(1) Generate a sample input vector x with random

entries in f1g.(2) Generate the SISO-processor input message

lax satisfying the constraint

Ix; lax Ia

(3) Generate the SISO-processor input message

lox satisfying the constraint

Ix; lox Io

(In this step, the output sample vector y is

generated according to the pdf py j x.)(4) Operate the SISO processor to obtain the ex-

trinsic probabilities exi at its output.(5) Estimate Ie by using the approximation (10).

Notice that the EXIT-chart analysis is approximate,

as it is based on the assumption of independent

extrinsic probabilities, which holds for an infinite-

length interleaver. Thus, some inaccuracies must be

expected [11,17,19]. Nevertheless, the practical use-

fulness of EXIT charts for convergence predictions is

unquestioned.

3. EXIT Charts of SISO Decoders

In this section, we specialize to SISO decoders the

algorithm for the derivation of EXIT functions. Under

the assumption of a stationary memoryless channel

with perfect channel state information (CSI) at the

receiver}, the conditional pdf py j x can be fac-tored into the product

Qi pyi j xi, and we have the

relations

Io J2o Ia J2a

deriving from Equation (9). Here, 2o is the variance ofthe additive noise.

The block diagram of the system used to compute

the mutual information transfer function is depicted in

Figure 7.

A random vector b 2 f1gk of uncoded symbols,k n, is generated, and passed to the encoder. Thisoutputs the code word x 2 f1gn. Vector x is thenpassed to a random generator (labeled 2o) whichoutputs an LLR vector whose entries oi satisfy

oi j xi N xi2o2; 2o



for i 1; . . . ; n. Similarly, the source vector b ispassed to a random generator (labeled 2a) whichoutputs an LLR vector whose entries ai satisfy

ai j bi N bi2a2; 2a

i 1; . . . ; k. The variances 2a and 2o are equal toJ1Ia and J1Io respectively. After operating theSISO decoder on inputs oi and

ai , the mutual

information at the output of the SISO decoder is

computed by applying Equation (10) to e.Notice that the SISO decoder can output extrinsic

information on both uncoded and coded bits, which in

terms of log-likelihood ratios can be denoted by e;u

and e;c. Hence, the mutual informations Ie;u Ib; e;u and Ie;c Ix; e;c can be evaluated.k

The choice of dealing with Ie;u or Ie;c depends on

the application considered. Reference [17], analyzing

the transfer of information between the constituent

decoders of parallel concatenated codes, uses the

mutual informations Ie;u, because the constituent de-

coders share the extrinsic probabilities of uncoded

bits. In contrast, References [9,19], investigating turbo

equalization and MIMO iterative receivers, use the

mutual informations Ie;c, since the SISO equalizer and

the SISO decoder share the extrinsic probabilities of

coded bits. As we deal with coded MIMO systems,

here we use the mutual information on coded bits, Ie;c,

hereafter denoted only by Ie. In the following, to

enhance the simulation efficiency, we implement

the SISO decoder by using the log-MAP BCJR

algorithm [1].

Figures 810 show some examples of EXIT charts

relevant to some recursive systematic convolutional

(RSC) codes and turbo-codes.

Figure 8 refers to a rate-1/2 RSC code with octal

generators (5,7). The curves plot the mutual informa-

tion Ie against Io using Ia as parameter.

Figure 9 refers to several rate-1/3, rate-1/2 and rate-

2/3 RSC codes with different generators and number

of states. Rate-2/3 codes are obtained by puncturing

corresponding rate-1/2 codes. The curves plot the

mutual information Ie against Io assuming Ia 0.Figure 10 refers to a rate-1/2 parallel turbo-code

whose constituent RSC encoders have generators

(5,7). The curves plot the mutual information Ie

against Io assuming Ia 0 for different numbers ofiterations of the turbo-decoding algorithm.

A notable common feature of these EXIT charts is

that, for Ia 0, they can be regarded as smootherversions of a unit step function whose level transition

occurs at a value of Io equal to the code rate . Thiscan be interpreted by observing that, when Ia 0, Iois equivalent to the mutual information exchanged

between the transmitted symbol x and the received

Fig. 7. Block diagram for the derivation of the extrinsicinformation transfer (EXIT) chart of a SISO decoder.

kWe omit again the subscript i here, for sake of simplicity.

Fig. 8. EXIT chart of a rate-1/2 RSC code with octalgenerators (5, 7). Curves plot Ie against Io using Ia as

parameter.

Fig. 9. EXIT charts of rate-1/3, rate-1/2 and rate-2/3 RSCcodes specified in the legend (rate-2/3 codes are obtained bypuncturing corresponding rate-1/2 codes). The curves plot Ie

against Io assuming Ia 0.



signal y, and hence equals the capacity. A capacity-

achieving code can attain reliable communication if

and only if Io > , and hence its EXIT curve wouldexhibit a sharp transition of the extrinsic mutual

information from 0 (unreliable communication) to 1

(reliable communication) in correspondence of

Io . Finite-complexity codes exhibit the smootherbehavior exhibited by the EXIT curves.

Notice also how the transition near , which issymmetric for convolutional decoders, becomes

asymmetric for turbo decoders. These also show a

migration from the unreliable communication con-

dition slower than convolutional codes of similar rate,

but, as the number of iteration increases, a faster

acquisition of the reliable communication condition.

3.1. Error Probabilities on EXIT Charts

Estimates of the error probability of a coded system

can be superimposed to EXIT charts to yield insight

on the receiver performance. By assuming the random

conditional LLR d j x to be Gaussian distributed withmean 2d=2 and variance

2d, the bit error probability

(BER) can be approximated by

Pbe Qdd

Q d

2

12

where Q is the Gaussian tail function. Sinced o a e, the assumption of independentLLRs leads to [17]:

2d 2e 2a 2o

which in turn yields

Pb QffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ1Ie J1Ia J1Io

p2

!13

Figure 11 shows the BER plotted as a function of Io

and Ie with Ia 0.

4. EXIT Charts of Other SISO Processors

Let us consider a MIMO system equipped with t

transmit and r receive antennas. The received signal

can be modeled by the following equation

Y HS Z 14

where S is a t L matrix (space-time code word) ofsymbols belonging to the constellation S with sizej S j 2m, H is a r t channel matrix, and Z is amatrix of iid complex Gaussian noise samples with

zero mean and variance 2z . Denoting by the timeindex ( 1; . . . ; L) and by y, s and z the lthcolumns of Y, S and Z respectively, we can rewrite

Equation (14) as

y Hs z Xti1

hisi; z 15

where hi is the ith column of H. In order to simplify

notation, we shall drop the time index in thefollowing. Moreover, we define the input binary mt-

vector as x, xT1 ; . . . ; xTt , where xi , xi1; . . . ; ximand xij 2 f1g. The binary vector x is mapped tosymbol vector s. The EXIT chart of the SISO receiver

is evaluated according to the block diagram of

Figure 12. Here, vector x is first generated, then

passed through the modulator to yield vector

s mx mx1; . . . ; mx1

which is passed through the channel to obtain the

received vector y providing, in turn, the message lo

consisting of the conditional pdf

py j x2z r

exp jj yHmx jj 2=2z 16

sampled at all possible values of x 2 f1gmt. The otherinput messages are obtained, as LLRs, according to

aij j xij N xij2a2; 2a

Fig. 10. EXIT charts of a rate-1/2 parallel turbo-code whoseconstituent RSC encoders have generators (5, 7). The curvesplot Ie against Io assuming Ia 0 for different numbers of

iterations of the turbo-decoding algorithm.



with 2a J1Ia. The evaluation of the extrinsicprobability distribution (message le) depends on the

type of SISO processor considered. In the following,

we describe three types of SISO processors.

4.1. MAP Equalizer

When a maximum a posteriori (MAP) equalizer is

employed, and for a fixed given matrix H, following

the considerations of Example 2 we obtain the SISO

extrinsic output as**

exij pxij1

Pxij

py j xpxPxxijpxxij1

Pxxij

py j xxpxx

P

xijpy j xpxijP

xx py j xxpxxij17

where py j x is as in Equation (16). Notice that, inthis case, py j x 6

Qi pyi j xi, so that direct evalua-

tion of Io is difficult. Thus, we express the mutual

information Ie as TIa; 2z instead of TIa; Io, andapply again the approximation (10) to the samples eijderived from Equation (17).

The computational complexity of evaluating

Equation (17) (exponential in the product mt) has

led researchers to devise suboptimal, low-complexity

SISO processors based on soft interference cancella-

tion. These processors are based on the combination

of a linear filter and an interference canceler (IC).

4.2. Interference Cancelers with Linear Filtering

Interference cancellation is based on the generation of

soft estimates ss of the transmitted symbol vector s that

are used to eliminate, in an iterative fashion, the spatial

interference. For each transmit antenna, i 1; . . . ; t,the soft estimates are computed as follows:

ssi Xsi2S

si psi 18

where, assuming that the bits contributing to the

transmission of s are independent, psi pxi Qmj1 pxij if si mxi.

Example: Binary PAM

As a special case of interest, let us consider binary

PAM with S f1g and the identity map s x.Since logpx 1=px 1, we have

ss 1 11 e 1

e

1 e tanh

2

Assuming N2=2; 2, the following pdf of ss isobtained:

pss 21 ss2

1ffiffiffiffiffiffi2

p

exp 2arctanhss 2=22

22

!

Figure 13 plots this distribution for some values of 2.Then, the IC block outputs, for each antenna i, the

following soft values

yyi yHss hissi hisi

Xj6i

hj sj ssi

z 19

which are subsequently processed by the antenna-

specific linear filters as described in the following.

(1) MMSE filter: The MMSE filter operates so as to

minimize the mean square error (MSE)

E j fyi yyi xi j 2. As a result, the filter vector f i isobtained as

f i 2z Ir HR2iHy

1

hi 20

Fig. 11. Bit error rate (BER) chart of an iterative receiverplotted as a function of Io and Ie and considering Ia 0.

Fig. 12. Block diagram for the derivation of the EXIT chartof a SISO canceler.

**xij denotes the vector x without the entry xij.



where R2i diag21; . . . ; 2i1; 1; 2i1; . . . ; 2t and the variances 2i are given by

2i E j si ssi j 2Xsi2S

j si j 2psi j ssi j 2 21

Recalling Equation (19), the output of the ith filter

is given by

~yyi , fyi yyi ici i 22

where i fyihi and where i is a complex Gaus-sian random variable with zero mean and variance

2i i 2i

Extrinsic probabilities are finally computed as

follows:

exij Xxij

p~yyi j xiYj0 6j

pxij0 23

The computational complexity involved is linear

in t and exponential in m, the number of bits per

symbol.

(2) Maximum ratio combining filter: The maximum

ratio combining (MRC) filter is based on the filter

vector f i hi. Again, the filter output can bewritten as in Equation (22) where i hyihi and

2i Xj6i

j hyihj j 22j 2zhyihi

Figure 14 shows the EXIT chart of the MAP,

ICMMSE and ICMRC SISO processors con-sidered here. In this case, we assume r t 4(four transmit and receive antennas), a complex

channel matrix H as in Reference [9], a QPSKsignal set, and 1=2z Es=N0 1 dB (solid lines)or 2 dB (dashed lines).yy The curves show that theMAP equalizer outperforms all other processors as

it achieves a better value of Ie at any given Ia.

5. Applications: Iterative MIMO Receivers

5.1. Deterministic Channel

In a turbo device two SISO processors are connected

together so that the extrinsic output of each one is

connected to the others a priori input. Usually,

interleavers are inserted in order to reduce the mes-

sage correlation.

The SISO processors may be both MAP decoders,

which results into a turbo decoder. In this case, the

convergence of the turbo decoder has been extensively

studied by using EXIT charts [17].

Nevertheless, EXIT charts apply with any pair of

SISO processors and portray the behavior of the turbo

device by showing the transfer functions of the mutual

informations involved. Let us focus on the interface

illustrated in Figure 15. The abscissa of the EXIT

chart is a priori input mutual information Iacan of the

Fig. 13. Probability density function (PDF) of the softestimate ss tanh=2 with N2=2; 2. Fig. 14. Example of mutual information transfer function for

different SISO cancelers, static channel, QPSK modulationand Es=N0 1 dB, 2 dB.

yyHere and in the following, Eb and Es denote the averageenergy per transmit antenna per symbol and per informationbit respectively.



SISO canceler coinciding with the extrinsic output

mutual information Iedec of the SISO decoder

(Iacan Iedec). The ordinate is the extrinsic outputmutual information Iecan of the SISO canceler coincid-

ing with the channel observation input mutual infor-

mation Iodec of the SISO decoder (Iecan Iodec). Thus, an

EXIT chart contains the transfer functions

Iecan TcanIacan; Iocan plotted with Iacan on the abscissaand Iecan on the ordinate, and I

edec TdecIadec; Iodec

Iedec TdecIadec 0; Iodec plotted with Iodec on the or-dinate and Iedec on the abscissa. The iterations success-

fully converge if the equilibrium point Iacan Iecan 1is reached.

The SISO processor inputs are the observations

from the r receive antennas and the extrinsic prob-

abilities output by the decoder, while its outputs are

the extrinsic probabilities. These are sent to the input

of the SISO decoder corresponding to the channel

observations. No a priori information is available to

the SISO decoder input, so that Iadec 0.As an example, we consider the combination of a

SISO decoder (based on a rate-1/2 convolutional code

with generators (5,7)) and a SISO canceler (based on

MMSE interference cancellation) on a MIMO system

with four transmit and four receive antennas. Addi-

tionally, QPSK modulation is assumed, and the chan-

nel matrix H is chosen as in Reference [9], with

Es=N0 2 dB. Figure 16 illustrates the first fewiterations of the turbo device operation. The figure

shows the EXIT functions of the SISO decoder

(dashed line) and of the SISO canceler (solid line).

They are taken from Figures 9 (after abscissa-ordinate

inversion) and 14 respectively. The dotted lines plot

the constant-BER curves computed by using Equation

(13). The arrows indicate the first few iterations of the

turbo algorithm: vertical arrows correspond to inter-

ference cancellation, while horizontal arrows corre-

spond to decoding. The points labeled k 0; 1; 2correspond to the extrinsic mutual information at the

output of the SISO decoder after k iterations. Finally,

BER values are reported in the figure (bottom left)

obtained by Monte-Carlo simulation for comparisons

with the values computed by using Equation (19)

(dotted curves). Figure 17 shows the BER for the

same system obtained by simulation (solid lines) and

by EXIT chart analysis (points), for k 0; 1; 2; 8iterations.

5.2. Quasi-Static Channel

In quasi-static conditions, the channel matrix H is

random, and changes independently from codeword to

codeword. This implies that the SISO-canceler EXIT

function changes with H, and should be evaluated for

Fig. 15. Block diagram of a turbo interference canceler.

Fig. 16. Decoding path for the combination of a SISOdecoder (based on a rate-1/2 convolutional code with gen-erators (5,7)) and a SISO canceler (based MMSE interfer-ence cancellation on a MIMO system with four transmit andfour receive antennas), with QPSK modulation and

Eb=N0 2 dB.

Fig. 17. Comparison of BER obtained by simulation and byEXIT chart analysis, for a system with t r 4, QPSK,

and a deterministic channel.



a large number of samples in order to estimate the

error performance of the system. On the contrary, the

SISO-decoder EXIT function remains constant at a

fixed value of 2z (or Es=N0). The computationalburden necessary for convergence analysis might be

heavy, but can be substantially alleviated by observing

(through computational experience) that the SISO-

canceler EXIT function exhibits in most cases an

almost linear behavior and, as a consequence, only

two points are needed to plot it as a straight line.

Numerical results showed that the MAP canceler

EXIT function is more linear than the MMSE and

MRC ones. Also, the level of Eb=N0 considered seemsto have little influence on the linearity of the canceler

EXIT function.

The approximation is illustrated by Figure 18,

which considers the same system as that of Figure 16

and plots in addition the straight-line approximation

of the EXIT function of the SISO canceler. The

convergence points (obtained by the intersection of

the SISO canceler and decoder EXIT functions) are

denoted by C and C0 for the proper and approximateSISO-canceler EXIT functions respectively. Ob-

viously, these points lie on the decoder EXIT function

and represent the asymptotic performance attainable

with an infinite number of iterations. It must be noted

that the straight-line approximation leads to non-

conservative convergence estimates, due to the up-

ward convexity of the exact EXIT function of the

SISO canceler. Nevertheless, numerical results show

that the approximation is fairly accurate.

A sample set of convergence points is plotted in

Figure 19 to show their distribution for the same

system parameters. The points have been obtained

by using different, randomly generated matrices H

with iid circularly symmetric complex Gaussian ran-

dom entries with zero mean and unit variance (in-

dependent MIMO Rayleigh fading channel). It is seen

that the distribution of the points is quite concentrated,

thus validating the assumption that their variance is

close enough to zero. Finally, Figure 20 compares the

BER obtained by simulation (solid lines) and by the

linearized EXIT chart analysis (dots) and for k 0; 1; 2; 8 iterations. The figure shows that the EXITchart analysis provides slightly non-conservative re-

sults and, in the case considered, an error of up to

about 0.5 dB. It can also be noticed from the figure

Fig. 18. Approximate and exact decoding trajectories for thecombination of an MMSE interference canceler withr t 4, rate Rc 1=2 CC(5,7) convolutional code,

QPSK modulation, Eb=N0 2 dB.

Fig. 19. Distribution of the convergence points for a systemwith r t 4, MMSE filter, rate Rc 1=2 CC(5,7) con-

volutional code, QPSK modulation, Eb=N0 2 dB.

Fig. 20. BER obtained by simulation and by linearizedEXIT chart analysis for a t r 4 MIMO channel withMMSE interference cancellation, QPSK, and quasi-static

independent Rayleigh fading.



that the error increases slightly with increasing Eb=N0and decreases by increasing the number of iterations

from 1 to 8.

6. Conclusions

After a general introduction on iterative MIMO re-

ceivers, we have analyzed the performance of combi-

nations of a SISO decoder and a spatial-interference

canceler over the MIMO channel. We showed that

EXIT charts can be used in conjunction with linear-

ization of the interference-cancellation characteris-

tics so as to extend their applicability from the case of

constant channel to the case of quasi-static fading

channel. The resulting approximation yields results

that are very close to simulation but are obtained in a

considerably shorter time.

Acknowledgments

The authors are grateful to Tor Aulin, Joseph Boutros

and Joachim Hagenauer for useful discussions con-

cerning some of the topics covered in this paper. They

also thank the guest editor Murat Uysal and Erik

Larsson, Arthur Hashizume for their comments and

support in the preparation of the final manuscript.

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Authors Biographies

Ezio Biglieri (M73SM82F89)was born in Aosta (Italy) in 1944. Hestudied electrical engineering at Poli-tecnico di Torino (Italy), where hereceived his Dr. Engr. degree in 1967.From 1968 to 1975, he was with theInstitute of Electronics and Telecom-munications, Politecnico di Torino,first as a research engineer, then as an

associate professor (jointly with the Institute of Mathe-matics). In 1975, he was made a professor of Electrical



Engineering at the University of Napoli (Italy). In 1977, hereturned to Politecnico di Torino as a professor in theDepartment of Electrical Engineering. From 1987 to 1989,he was a professor of Electrical Engineering at the Uni-versity of California, Los Angeles. Since 1990, he has beenagain a professor with Politecnico di Torino. He has heldvisiting positions with the Department of System Science,UCLA, the Mathematical Research Center, Bell Labora-tories, Murray Hill, NJ, the Bell Laboratories, Holmdel, NJ,the Department of Electrical Engineering, UCLA, the Tele-communication Department of The Ecole Nationale Super-ieure des Telecommunications, Paris, France, the Universityof Sydney, Australia, the Yokohama National University,Japan, the Electrical Engineering Department of PrincetonUniversity, the University of South Australia, Adelaide, theUniversity of Melbourne and the Institute for Communica-tions Engineering, Munich Institute of Technology,Germany. He was elected three times to the Board ofGovernors of the IEEE Information Theory Society, andhe served as its President in 1999. He was an editor of theIEEE Transactions on Communications, the IEEE Transac-tions on Information Theory, the IEEE CommunicationsLetters, the Journal on Communications and Networks andthe Editor in Chief of the European Transactions on Tele-communications. Since 2004, he has been the editor-in-chiefof the IEEE Communications Letters. He has edited threebooks and co-authored five. Among other honors, in 2000 hereceived the IEEE Third-Millennium Medal and the IEEEDonald G. Fink Prize Paper Award, and in 2001 the IEEECommunications Society Edwin Howard ArmstrongAchievement Award.

Alessandro Nordio (S 2000) receivedhis M.Sc. degree in Telecommunica-tions Engineering from Politecnico diTorino, Italy, in July 1998, and thePh.D. from Ecole Politechnique Fed-erale de Lausanne, in April 2002.From August 1998 to December1998, he worked as a consultant forOmnitel. From 1999 to 2002, he waswith the Mobile Communications

Department of Institut Eurecom, Sophia-Antipolis, France,as a Ph.D. student. In April 2002, he joined the Departmentof Electrical Engineering of Politecnico di Torino where heis working as post-doc student. His research interests are inthe field of signal processing, multi-user detection andspace-time coding.

Giorgio Taricco (M91SM03) wasborn in Torino, Italy. He received thedegree of Ingegnere Elettronico (cumlaude) from Politecnico di Torino(Italy) in 1985. In 1985, he joined theTelecom Italian Labs (CSELT) wherehe was involved in the design of thechannel coding subsystem of GSM.Since 1991, he has been with the Dipar-timento di Elettronica of Politecnico di

Torino, currently as an associate professor. In 1996, he was aresearch fellow at ESTEC. He took part in the committees ofseveral IEEE conferences and he is currently an associateeditor of the Journal on Communications and Networks andof the IEEE Communications Letters. Among his researchinterests are the following: error-control coding, multiuserdetection, space-time coding and MIMO communications.He is the author or co-author of about 50 journal papers and100 conference contributions, and holds two internationalpatents in applied error-control coding.




Improving the performance of coded FDFR multi-antennasystems with turbo-decodingz

Renqiu Wang1, Xiaoli Ma2 and Georgios B. Giannakis1*,y

1Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE,

Minneapolis, MN 55455, U.S.A.2Department of Electrical and Computer Engineering, Auburn University, Auburn AL 36849, U.S.A.

Summary

A full-diversity full-rate (FDFR) multi-antenna system was developed recently, enabling uncoded layered space-

time (LST) transmissions to achieve full-diversity (NtNr) and full-rate (Nt symbols per channel use) simulta-

neously, for any number of transmit antennas Nt and receive antennas Nr. In this paper, we investigate the

performance of a coded FDFR design obtained by concatenating an error control coding (ECC) module and FDFR

module with a random interleaver in between. Turbo decoding is performed at the receiver. With Rc denoting the

ECC rate, dmin the minimum Hamming distance of the ECC, and M the constellation size, an overall transfer rate of

RcNtlog2M bits per channel use and a full diversity order dminNtNr are achieved. Different ECC choices are

considered. Approximate analysis reveals that multi-stream ECC and single-stream ECC make no difference when

convolutional codes with long frame length and near-optimal MIMO decoding schemes are adopted. Without

sacrificing rate, the coded FDFR system improves error performance compared with coded V-BLAST, when

relatively weak codes are used. As Nr increases, even strong codes such as rate 1/2 turbo codes can benefit from

FDFR. Specifically, 1.52 dB gain over coded V-BLAST is obtained in a 2 2 antenna setup when convolutionalcodes or rate 3/4 turbo codes are used; 0.5 dB gain is offered in a 2 5 setup when rate 1/2 turbo codes are used.Coded FDFR also outperforms a 16-QAM Alamouti coded scheme by 1 dB when convolutional codes are used.

The price paid is increased complexity. Copyright # 2004 John Wiley & Sons, Ltd.

KEY WORDS: space-time; diversity; V-BLAST; FDFR; turbo decoding

1. Introduction

High transmission rate and low error rate are the

ultimate goals of modern wireless communication

modems, which are challenged by multiplicative

channel fading and additive Gaussian noise (AGN)

effects. Traditional error control coding (ECC) over

the Galois field (GF) deals with AGN and fading by

adding redundancy. Allowing for long block or large

interleaver sizes, thus assuming unconstrained encod-

ing and decoding complexity, low-density parity

check (LDPC) codes and turbo codes approach the

*Correspondence to: Georgios B. Giannakis, Department of Electrical and Computer Engineering, University of Minnesota,200 Union Street SE, Minneapolis, MN 55455, U.S.A.yE-mail: [email protected] of the results in this paper was presented at IEEE International Symposium on Signal Processing and InformationTechnology December 1417, 2003, Darmstadt, Germany. Guest Editor: Dr. E.G. Larsson, email: [email protected]

Contract/grant sponsors: ARL/CTA; contract/grant number: DAAD 19-01-2-0011.


bit error rate (BER) limit dictated by channel capacity

[13]. However, when delay or complexity is con-

strained, alternative low-complexity ECC options be-

come more practical, among which convolutional

codes (CC) are often preferable due to their simple

yet flexible structure and mature low-complexity

Viterbi decoding [4]. Although ECC is a well-

documented means of improving error performance,

it reduces spectral efficiency due to the redundancy

inserted. Bandwidth-efficient means of mitigating

channel fading by exploiting diversity flavors in other

dimensions are thus well motivated. Linear complex

field (LCF) coding and space-time (ST) coding are

two such flavors, in the precoded modulation and

spatial dimensions respectively. LCF coding (LCFC)

is the counterpart of GF coding. With each entry of the

generator matrix chosen from the complex field,

LCFC has been shown capable of enabling maximum

diversity with small or no rate loss; see for example

References [59] and references thereof. The princi-

ple is to construct a P P encoder matrix whichproduces codewords with any pairwise Hamming

distance equal to P. Relying on multiple (Nt) transmit

and multiple (Nr) receive antennas, multi-input multi-

output (MIMO) spatial wireless links are created. It

has been shown that the MIMO capacity of indepen-

dent Rayleigh fading ergodic channels increases ap-

proximately linearly with the minimum of (Nt, Nr),

implying that MIMO can potentially boost both diver-

sity and data rate [10]. There have been many ad-

vances in this field, which in general fall into two

classes: the first class aims at improving error perfor-

mance by exploiting spatial diversity, while the second

one targets high data rate. ST orthogonal designs

[11,12] and ST trellis codes [13] are two examples in

the first class. BLAST-type ST codes [14,15] and linear

dispersion (LD) codes [16] belong to the second class.

Although, it is still worthwhile to fully explore the

potential of each ST code design, jointly exploiting

merits from two or more designs often leads to more

desirable tradeoffs in rate-diversity-complexity. By

concatenating an LCF coder with a layered ST

(LST) mapper properly, the recently developed full-

diversity full rate (FDFR) design [17] enables an

uncoded LST system to have full diversity (NtNr)

and full-rate (Nt symbols per channel use) simulta-

neously (see also Ref. [18]). Joint consideration of

ECC and LCFC in ST setups was pursued also in

Reference [19]. Although the triangular ST mapper

developed in Reference [19] enables full diversity

order dminNtNr, where dmin is the minimum Hamming

distance or free distance of the ECC, the overall

transmission rate is only about half of the maximum

possible.

The performance of uncoded FDFR and Reference

[19] motivate us to investigate the performance of a

joint ECC and FDFR system in this paper. We will

particularly consider relatively weakly coded FDFR

architectures, which rely on the concatenation of

ECC, LCFC and ST multiplexing at the transmitter,

along with soft-to-hard sphere decoding (SHD-SD)

[20,21] with iterative detection at the receiver. After

developing the system model in Section 2, we will

analyze the diversity order of coded FDFR under the

assumptions of near-perfect interleaving and near-

optimal decoding. A few special cases, including

CC and turbo coding (TC), will be considered in

choosing a single-stream coding structure over its

multi-stream counterpart. We will use coded V-

BLAST as a reference in our performance compar-

isons. In Section 4, we will illustrate by simulations

that the FDFR design offers notable performance

improvement by enabling full spatial diversity without

sacrificing rate, when CC or high rate TC is used, at

the expense of increased complexity.

Notation: Upper (lower) bold face letters will be usedfor matrices (column vectors). Superscript * will

denote Hermitian transpose and T indicates transpose.

We will use to stand for the Kronecker product;diag(v) will stand for a diagonal matrix with entries of

the vector v on its main diagonal.

2. System Model

As depicted in Figure 1, the coded FDFR system

concatenates an ECC module and an FDFR module

with a random interleaver in between. Soft turbo

decoding between the ECC decoding module and the

FDFR decoding module is performed at the receiver

end. Both MIMO channels and the FDFR code are

decoded at the same time. We will use the term FDFR

block to denote an FDFR processing unit, ECC

stream for an ECC encoder unit, and frame for a

set of information bits that will be processed by the

ECC module, the interleaver, and the FDFR module

serially without dependence on another frame. We can

also think of a frame as the systems processing unit.

2.1. The Transmitter and EquivalentMIMO Channels

A frame of information bits b with length Kc is firstencoded by an ECC module to yield c, and then goes

712 R. WANG, X. MA AND G. B. GIANNAKIS


through a random interleaver P. The ECC modulewith interleaver can implement either a single-stream

coding structure as depicted in Figure 2, where

information bits are processed serially by a single

encoder, or, they can implement a multi-stream cod-

ing structure as depicted in Figure 3(a), where infor-

mation bits are divided into n sub-streams and each

sub-stream is encoded independently. As a special

case of the multi-stream structure, the multi-steam per

layer transmission is depicted in Figure 3(b), where

instead of using one interleaver, the coded bits per

sub-stream are scrambled with a sub-interleaver ma-

trix Pi independently. In this case, the equivalentoverall interleaver matrix Po diagP1 . . .Pn) isno longer a random interleaver although each sub-

interleaver Pi can be random.Interleaved bits ~cc are mapped to a frame of symbols

f with frame length Nc adhering to a certain constella-

tion; f is then fed to the FDFR module. Frame f isdivided first into FDFR blocks fskgKk1 with blocklength N2t symbols, where k indexes the FDFR block,

and K is the number of blocks. Let us temporarily

omit the block index k to explain the FDFR design.

We will come back to it in Section 3. Each FDFR

block s is then divided into Nt sub-blocks with sub-

block length equal to Nt. Let sg denote the gth Nt 1sub-block (g 1; . . . ;Nt), whose entries fsg;kgNtk1 are

drawn from a complex finite alphabet set S. The sub-block sg is first coded to obtain

ug Hgsg; g 1; . . . ;Nt 1

where fHg : g1HgNtg1 is the set of LCF encoders, is a scalar and H is chosen from the class of unitaryVandermonde matrices:

H 1ffiffiffiffiffiNt

p FNt diag1; ; . . . ; Nt1 2

where FNt is the Nt Nt FFT matrix withm 1; n 1st entry ej2mn=Nt, and is a scalar.Three design approaches for and H (or equivalently) have been derived to enable full-diversity and full-rate in Reference [17]. As an example, when Nt 2k,with k being a natural number, design A selects ej=2Nt and Nt ej=4N2t ; design B selects ej=N3t and Nt ; and design C selects ej=2 and Nt or as in the design A, butwith Nt ej=2.

The LCF coded symbols fuggNtg1 then go throughan LST mapper, and are transmitted through Ntantennas as follows:

V

u1;1 uNt;2 . . . u2;Ntu2;1 u1;2 . . . u3;Nt... ..

.. . . ..

.

uNt;1 uNt1;2 . . . u1;Nt

26664

37775 ! time# space 3

Fig. 1. The coded full-diversity full-rate (FDFR) system model.

Fig. 2. The single-stream error control coding (ECC) model.

IMPROVING THE PERFORMANCE OF CODED FDFR MULTI-ANTENNA SYSTEMS 713


where ug;i denotes the ith entry of vector ug.

Let s : sT1 ; . . . ; sTNt T

denote one FDFR block and

hTi denote the ith row of H. By defining the permuta-tion matrix Pi and the diagonal matrix Drespectively, as:

Pi :0 Ii1

INti1 0

and D : diag1; ; . . . ; Nt1

we obtain the equivalent FDFR encoder for the entire

block s as

U :P1D hT1

..

.

PNtD hTNt

264

375 4

and the FDFR output vector as x Us.We use Hl to denote the Nr Nt MIMO channel

coefficient matrix during the lth time slot that trans-

mitted vector Vl is facing, where vl denotes the lth

column of matrix v given as in Equation (3). Thus, the

channel matrix H for the transmitted vectorv : vT1 . . . vTNt

Tcan be written as:

H :

H1 0 . . . 00 H2 . . . 0

..

. ...

. . . ...

0 0 . . . HNt

26664

37775 5

When MIMO channels are invariant over each FDFR

block, that is Hl H; l 1; . . .Nt, the resultingFDFR-block-fading channel matrix can be written

in a simple form as H INt H with INt denoting theNt Nt identity matrix.

Let yl denote the lth Nr 1 received vector,y : yT1 ; . . . ; yTNr

T, nl denote the kth Nr 1 noise

vector and n : nT1 ; . . . ;nTNr T. The input-output re-

lationship is then [17]:

y HUs n Heqs n 6

where the equivalent channel matrix for the entire

FDFR block is Heq HU.

2.2. The Receiver With Turbo Decoding

At the receiver end, turbo decoding is carried out to

achieve an overall near-ML performance. Two mod-

ules, indexed by subscripts 1 and 2, perform soft

decoding of the FDFR-MIMO and ECC parts respec-

tively (see Figure 1). Extrinsic information about

c, denoted as kE, from one decoding module is(de-)interleavered to yield a priori information about

c, denoted as kA, for the other module. After a certainnumber of iterations or after a certain BER is

achieved, a hard decision bb is obtained based on the

a posteriori information about b, denoted as kD2, fromthe ECC decoding module.

Inside each module, the optimal maximum a pos-

teriori (MAP) decoder, whether it operates over the

GF or over the real/complex field (RCF), requires

complexity that increases exponentially with the pro-

blem size in general (e.g. the memory length for CC or

the block size and the constellation size for RCF

code). Several near-optimal algorithms with polyno-

mial complexity have been developed for decoding

GF and RCF codes respectively. Those for decoding

over GF are well documented when CC or TC is used.

We adopt the so-called log-MAP algorithm to decode

CC and TC in Reference [22]. To decode RCF coded

transmissions over FDFR-MIMO channels, hard

sphere decoding (HD-SD) [2325] and semi-definite

programming (SDP) [26] offer two well-known near-

ML schemes to generate hard decisions. Other sub-

optimal decoding schemes with lower complexity

Fig. 3. (a) The multi-stream ECC model; (b) the multi-stream per layer model.

714 R. WANG, X. MA AND G. B. GIANNAKIS


include zero-forcing (ZF), minimum mean-square

error (MMSE) and nulling-cancelling (NC) alterna-

tives [27]. Compared with hard decoding, the soft

decoding problem for the real/complex block model

has been looked upon only recently. A soft version

SD, known as list SD (LSD) [28] was recently

proposed to perform soft MIMO channel decoding

and was shown to enable MIMO capacity approaching

performance. A soft version SDP has also been

developed to this end [29]. Recent soft-to-hard SD

(SHD-SD) transformation schemes [20,21] achieve

comparable performance as LSD at reduced complex-

ity. In this paper, we will use the near-optimal SHD-

SD scheme 1 of Reference [20] to decode QPSK

modulated FDFR transmissions. Since SHD-SD

schemes only work for binary constellations, we will

not consider other constellations in this paper.

We now briefly explain the FDFR-MIMO decoding

process with SHD-SD schemes. First, by separating

the real and imaginary parts of the matrices and

vectors in Equation (6), we obtain a real equivalent

model as

~yy yryi

Heq;r Heq;i

Heq;i Heq;r

srsi

nr

ni

~HH~ss ~nn

7

Each entry of ~ss, ~ssk (k 1; . . . ; 2N2t ), is equal to either1 or 1. Define the a priori information, the aposteriori information given ~yy, and the extrinsicinformation of ~ssk respectively as:

A~ssk : lnP~ssk 1P~ssk 1

;

D~sskj~yy : lnP~ssk 1j~yyP~ssk 1j~yy

;

E~sskj~yy : D~sskj~yy A~ssk

Let kA : A~ss1; . . . ; A~ss2N2t T

denote the a priori

vector of ~ss. With the AWGN assumption and the max-log approximation [22], we can approximate the

extrinsic information of ~ssk as [21,28]:

E~sskj~yy 1

2max

x2Xk;1 12

jj~yy ~HHxjj2 xTkA

12

maxx2Xk;1

12

jj~yy ~HHxjj2 xTkA

A~ssk

where x is the candidate of ~ss, Xk;1 : fxjxk 1gand Xk;1 : fxjxk 1g.

Relying on the spatially independent channel as-

sumption, ~HH has full column rank almost surely.Therefore, we can find a vector yA satisfying

2~HHTyA 2kA 8

Using Equation (8), we can rewrite the extrinsic

information of ~ssk as:

E~sskj~yy 1

22min

x2Xk;1jj~yy yA ~HHxjj2

122

minx2Xk;1

jj~yy yA ~HHxjj2 A~ssk 9

Let X denote the union of Xk;1 and Xk;1. Ifssmap : arg min

x2Xjj~yy yA ~HHxjj2, and ssk : arg min

x2Xk;ssk;mapjj~yy yA ~HHxjj2 for k 1; . . . ; 2N2t , then Equation(9) can be further simplified as

E~sskj~yy ssk;map

22jj~yy yA ~HHssmapjj2

ssk;map22

jj~yy yA ~HHsskjj2 A~ssk 10

Hard sphere decoding (SD) can be used to find ssmapand fsskg2N

2t

k1. The soft max-MAP decoding problem isthus converted to a set of hard SD problems. Based on

this max-MAP decoder, so termed SHD-SD Scheme 1

in Reference [20], additional approximate schemes

have been developed in Reference [20] to trade-off

error performance with complexity.

3. Performance Analysis

In this section, we will analyze the error performance

of coded FDFR, and show it is capable of enabling a

multiplicative diversity effect; namely that the diver-

sity order of coded FDFR is the product of that

enabled by ECC and by FDFR respectively. We will

also compare the two ECC structures: single-stream

CC versus multi-stream CC. The comparison will

suggest a single-stream structure that we will further

test with simulations presented in the next section.

3.1. Diversity Order

We here resort to a pairwise error probability (PEP)

approach to analyze the performance of coded FDFR.

Let us assume for now that the MIMO channel

remains constant over an entire FDFR block but is

IMPROVING THE PERFORMANCE OF CODED FDFR MULTI-ANTENNA SYSTEMS 715


allowed to vary independently from block to block.

Consider two different information bit frames bh1i andbh2i, each with length Kc. They yield two codewordsch1i and ch2i with length Nc after ECC, with either asingle-stream or a multi-stream structure. These two

codewords differ from each other in d positions and so

do the interleaved codewords. Although, it is possible

that these d positions could be in close proximity for a

certain interleaver and a certain pair of codewords,

considering the fact that the interleaver P is randomwith a different realization per frame, these d posi-

tions will most likely be sufficiently far apart provided

that the interleaver size is sufficiently long. Under this

assumption, we can henceforth consider that after

constellation mapping, the two symbol sequences

fh1i and fh2i still have d different symbols, and inany FDFR block the vectors skh1i and skh2i differin at most one symbol, where k 2 1;K indexes theFDFR block and K is the number of FDFR blocks.

After LCF coding, LST mapping and propagation

through the channel Hk, the equivalent channelmatrix for the kth FDFR block is Heqk. The resultingsymbol vectors are fzkh1i Heqkskh1igKk1 andfzkh2i Heqkskh2igKk1. Out of K blocks, only dof them are different. Without causing confusion, we

will use fzih1igdi1 and fzih2igdi1 to denote them.

When sih1i and sih2i are different in the mthsymbol, the Euclidean distance between zih1i andzih2i is:

jjzih1i zih2ijj2 jjheq;mijj2jsmij2 11

where heq;mi is the mth column of the equivalentchannel matrix Heqi, and jsmij2 is the Euclideandistance between the two different symbols smih1iand smih2i. With 2 standing for the minimum Eu-clidean distance between two symbols, we have that

jsmij2 2 12

Since Heq HU, by the definitions of H in Equa-tion (5) and U in Equation (4), if the mth symbol is inthe gth FDFR sub-block, we then have

jjheq;mijj2 jjHimjj2 XNrl1

XNtj1

jhl; jij2jg; j;mj2

1Nt

XNrl1

XNtj1

jhl; jij2

13

where m is the mth column of U, g; j;m is

02e7e51acee67995b5000000

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