Improved Iterative Detection And

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008 5039

Improved Iterative Detection andAchieved Throughputs of OFDM Systems under

Imperfect Channel Estimation

Seyed Mohammad Sajad Sadough, Member, IEEE, and Pierre Duhamel, Fellow, IEEE

AbstractAssuming imperfect channel estimation, we pro-pose an improved detector for orthogonal frequency-divisionmultiplexing (OFDM) systems over a frequency-selective fadingchannel. By adopting a Bayesian approach involving the statisticsof the channel estimation errors, we formulate an improvedmaximum-likelihood (ML) detection metric taking into accountthe characteristics of the channel estimates. As a first step, wepropose a modified iterative detector based on a maximum aposteriori receiver formulation which reduces the impact of chan-nel uncertainty on the decoder performance, by an appropriateuse of this metric. The results are compared to those obtainedby using a detector based on a mismatched ML metric, whichuses the channel estimate as if it was the perfect channel. Ina second step, we calculate the information rates achieved byboth the improved and mismatched ML detectors, in terms ofachievable outage rates. These outage rates are compared tothose provided by a theoretical (not practical) decoder definedas the best decoder in the presence of channel estimationerrors. Numerical results over both uncorrelated Rayleigh fadingchannels and realistic ultra wideband channels show that theimproved detector outperforms the classically-used mismatchedapproach in terms of bit error rate and achievable outage rates,without any increase in the receiver complexity.

Index TermsOFDM, BICM, channel estimation errors, mis-matched ML detection, improved iterative BICM detection,achievable outage rates.

I. INTRODUCTION

ORTHOGONAL frequency-division multiplexing

(OFDM) is a spectrally efficient technique for achieving

high data-rate wireless transmission over frequency-selective

fading channels. OFDM offers a simple way to divide the

entire channel into many narrowband subchannels withparallel data transmission, thereby increasing the symbol

duration and reducing the intersymbol interference [1], [2].

Several OFDM based wireless systems employ bit-interleaving combined with a convolutional code as chan-

nel coding. This scheme, referred in the literature as bit-

interleaved coded modulation (BICM) [3], can provide high

Manuscript received August 25, 2007; revised December 7, 2007; acceptedJanuary 26, 2008. The associate editor coordinating the review of this paperand approving it for publication was M. Valenti. The material in this paperwas published in part at the 40th Asilomar Conference on Signals, Systems,and Computers.

S. M. Sajad Sadough is with the Faculty of Electrical and Com-puter Engineering, Shahid Beheshti University (SBU), Tehran, Iran (e-mail:[email protected]).

P. Duhamel is with CNRS/LSS, Suplec - 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/T-WC.2008.070944

diversity order for transmission over Rayleigh fading channels

with coherent detection and perfectknowledge of the channel.

It is well known that reliable coherent data detection is

not possible unless an accurate channel state information is

available at the receiver. A typical scenario for wireless com-

munication systems assumes the channel changes so slowly

that it can be considered time invariant during the transmission

of an entire frame. In such situations, pilot symbol assisted

modulation (PSAM) [4] has been shown to be an effectivesolution for obtaining channel state information at the receiver

(CSIR). It consists of sending some known training (alsocalled pilot) symbols, based on which the receiver estimates

the channel before proceeding to the detection of data sym-bols. Obviously, in practical systems, due to the finite number

of pilot symbols and to noise, the receiver can only obtain an

imperfect(and possibly very poor) estimate of the channel. In

this situation, one may resort to blind detectors which estimate

the transmitted symbols without using any pilot. However,

most of these schemes require a differential modulation whichentails restrictions on the type of the constellation and leads to

about 3 dB loss over Gaussian channels compared to coherentdetection [5].

The performance of BICM over fading channels was stud-

ied, for instance, in [3], [6], [7] under prefect channel knowl-

edge at the receiver. In [8], the authors proposed a blind

iterative receiver for differentially encoded BICM OFDM

systems with unknown frequency-selective fading channels.

A rich literature exists on the impact of imperfect channel

estimation. In [9], the authors showed that in low signal-

to-noise ratio (SNR) situations, the poor quality of channel

estimates may prevent iterative decoding of BICM to be used

at the decoder. In [10], Garg et al. considered a training-based

MIMO system and showed that for compensating the perfor-mance degradation due to imperfect channel estimation, the

number of receive antennas should be increased. Obviously,this may not be always possible in mobile applications.

In order to deal with imperfect channel estimation, the clas-

sical technique, referred to as mismatched detection, consists

in replacing the exact channel by its estimate in the receiver

metric. It is shown, for instance, in [11][13] that this scheme

greatly degrades the detection performance in the presence

of channel estimation errors for a multi-carrier system. Fur-

thermore in [14], authors show that under imperfect channel

estimation, the rates achieved by the mismatched detector are

significantly lower than the channel capacity.

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5040 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008

Fig. 1. Transmitter architecture of the BICM OFDM system.

As an alternative to the aforementioned mismatched ap-

proach, Tarokh et al. in [15] and recently Taricco and Biglieriin [16], proposed an improved maximum likelihood (ML)

detection metric that mitigates the impact of imperfect CSIRand applied it to a space-time coded MIMO system. A similar

investigation was carried out in [17] for trellis-coded mod-

ulation in scalar channels. Very recently, in [18], the authors

derived the maximum possible outage capacity induced by the

channel estimation errors, in the context of Rayleigh channels

and a theoretical decoder. This was further developed in [19],

[20] to address the MIMO situation.

A practical concern for the design of a communication

system is to guarantee a prescribed target rate with small error

probability, irrespective of the accuracy achieved during chan-nel estimation. However, adopting a mismatched approach in

the described scenario raises two important questions : i) What

are the maximal achievable information rates of mismatched

decoding and how close does it perform with respect to therates provided by the best possible decoder in the presence of

channel estimation errors ? ii) What type of practical decoder

can improve the overall system performance under imperfect

CSIR ?

The objective of this paper is to address the above questions

for OFDM channels, estimated by a few number of trainingsequences. To this end, we propose a Bayesian framework

based on the a posteriori probability density function (pdf) ofthe perfect channel conditioned on its estimate. This general

framework enables us to formulate any detection problem by

considering the average over the channel estimations errors,

of the detectors cost function, that would be used if thechannel was perfectly known. In particular, we can recover

the metric of [16] by using our general framework in thecase of ML detection. This ML metric is first used for

deriving an improved maximum a posteriori (MAP) iterative

detector, a technique which is known to be very efficientwhen channel coding and bit interleaving are jointly used.

By modifying properly the soft-values at the output of the

MAP detector, we reduce the impact of channel uncertainty

on the decoder performance. In a second step, by using some

tools from information theory, we provide the expression of

the achievable information rates associated to the improved

and mismatched ML metric. Then, we characterize the limits

of reliable information rates achieved by the mismatched andimproved detectors in terms of outage rates, which is more

appropriate for quasi-static channels [21].

Our results may serve in the evaluation of the trade-off

between the required quality of service (in terms of BER and

achieved throughputs) and system parameters (e.g., training

power, transmission power, period of training, outage proba-bility) in the presence of channel estimation errors.

The remainder of this paper is organized as follows. Section

II describes the OFDM system under study and our main

assumptions concerning data transmission and channel esti-

mation. Section III introduces a general Bayesian frameworkfor improved detection in the presence of channel estimation

errors and then formulates the improved ML detection metric.In section IV, we propose the improved MAP detector in

the case of imperfect channel estimation. In section V, we

calculate the achievable outage rates of receivers based on the

improved and mismatched ML metric. Section VI illustrates

via simulations, the performance of the proposed receiver over

both realistic UWB and Rayleigh fading channel environ-

ments. Finally, section VII concludes the paper.

Notational conventions are as follows. IN represents an

(N N) identity matrix; CN denotes a complex Gaussian

distribution; Ex[.] refers to expectation with respect to therandom vector x; det{.}, |.|, and . denote matrix deter-minant, absolute value and vector norm, respectively; (.)T,(.) and (.) denote vector transpose, Hermitian transpose andconjugation, respectively.

I I . SYSTEM DESCRIPTION AND CHANNEL ESTIMATION

A. OFDM System Model

We consider a coded OFDM system with M subcarriersthrough a frequency-selective multipath fading channel, de-

scribed in discrete-time baseband equivalent form by the taps{hl}

L1l=0 . As depicted in Fig. 1, the binary data sequence

is encoded by a non-recursive non-systematic convolutional(NRNSC) code, before being interleaved by a pseudo-random

interleaver. The interleaved bits are gathered in subsequences

of B bits d1k, . . . , dBk and mapped to complex Mc-QAM

(Mc = 2B) symbols sk. At the receiver, after removing

the cyclic prefix (CP) and performing fast Fourier transform

(FFT), the the n-th received OFDM symbol (for brevity weomit n) can be written as [2]

y = Hd s + z, (1)

where the vectors y = [y0, . . . , yM1]T and s =

[s0, . . . , sM1]T respectively denote the received and trans-

mitted symbols, the noise vector z = [z0, . . . , zM1]T is

assumed to be zero-mean circularly symmetric complex Gaus-

sian (ZMCSCG) with distribution z CN(0, 2z IM), andHd diag(H) is the (M M) diagonal channel matrix withdiagonal elements given by vector H = [H0, . . . , H M1]

T,

where Hk =L1

l=0 hl ej2kl/M. The channel coefficients

are assumed to be constant during a frame of OFDM symbols

and change to new independent values from one frame to

another.

For convenience, we also define the diagonal matrix Sd

diag(s). In the following, when there is no confusion, we useH instead ofHd.



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SADOUGH and DUHAMEL: IMPROVED ITERAT IVE DETECTION AND ACHIEVED THROUGHPUTS OF OFDM SYSTEMS 5041

B. Pilot Based Channel Estimation

Consider the estimation of the k-th channel frequency coef-ficient Hk with N pilot symbols si gathered in the row vectorsk = [s0, . . . , sN1], (k = 0,...,Mdata 1) where Mdata isthe number of data subcarriers in the OFDM symbol. In a

practical OFDM system, Mdata is generally smaller than Mdue to the presence of some unmodulated (also called virtual)

subcarriers. In the following, without loss of generality, weassume that Mdata is equal to M.

According to the observation model (1), during a givenchannel training interval, we receive

yk = Hk sk + zk for k = 0,...,M 1, (2)

where the entries of the noise vector zk have the samedistribution as those of z in (1). Moreover, the definition of

yk and zk follow that of sk.The average power ET of the k-th training vector sk is

ET 1

Nsk

2. (3)

Here, we assume equi-powered training vectors for all sub-

carriers. The least-squares estimate of Hk is obtained byminimizing ykHk sk2 with respect to Hk which coincideshere with the ML estimate. This yieldsHMLk = yk sk sksk1

= Hk + Ek for k = 0,...,M 1 (4)

where Ek = zksk

sks

k

1is the channel estimation error

term. For convenience, the samples in (4) can be written in

vector form as

HML = H+ E (5)

where E is a zero-mean Gaussian vector with a covariance

matrix given by

E = EEE

= 2E IM , where 2E =

2zN ET

(6)

Thus, from (5), the conditional pdf of HML given H readsp(HML|H) = CNH,E. (7)

Consider an uncorrelated i.i.d. Rayleigh channel with a prior

distribution H CN(0,H), where H = 2hIM. By usingthe latter pdf and the pdf of(H

ML|H) from (7), we can derive

the posterior distribution of the perfect channel, conditionedon its ML estimate as follows (see Appendix I.A):

p(H|HML) = CN(HML, E), (8)where

= H(E +H)1 = IM and =

2h(2h +

2E)

.

(9)

The availability of the estimation error distribution constitutes

an interesting feature of pilot assisted channel estimation

which is exploited in the next section, resulting in an improved

detector under imperfect channel estimation.For the sake of simplicity, we will not specify hereafter the

superscript ML for H.

III. DETECTOR DESIGN IN THE PRESENCE OF CHANNEL

ESTIMATION ERRORS

We now provide the formulation of a detection rule thattakes into account the available imperfect CSIR and refer to it

as the improved detector. To this end, we consider the model

(1) and denote by J(y, s,H) the quantity (cost function) thatwould let us to decide in favor of a particular s at the receiver

if the channel was perfectly known. Note that dependingon the detection criteria, the quantity J(y, s,H) can bethe posterior pdf p(s|y,H), the logarithm of the likelihoodfunction p(y|H, s), the mean square error (as in [22]), etc.Under a pilot-based channel estimation characterized by the

posterior pdf of the channel (8), we propose a detector based

on the minimization of a new cost function defined as

J(y, s, H) = H

J(y, s,H)p(H|H) dH= E

H|HJ(y, s,H)H (10)where we have averaged the cost function J over all re-

alizations of the unknown channel H conditioned on its

available estimate H by using the posterior distribution (8).Note that this approach is an alternative to the sub-optimal

mismatched detector, which is based on the minimization of

the cost function J(y, s, H). The latter is obtained by usingthe estimated channel H in the same metric that would beused if the channel was perfectly known, i.e., J(y, s,H). Theapproach in (10) differs from the mismatched detection on

the conditional expectation EH|H[.] which provides a robust

design by averaging the cost function J(y, s,H) over all(true) channel realizations which could correspond to the

available estimate.Consider the problem of detecting symbol s from the

observation model (1) in the ML sense, i.e., so as to maximizethe likelihood function p(y|H, s). The connection of theconsidered detection problem with the MAP detection of

BICM OFDM is discussed in the next section.

It is well known that under perfect channel knowledge and

i.i.d. Gaussian noise, detecting s by maximizing the likelihood

p(y|s,H) is equivalent to minimizing the Euclidean distanceDML as

sML(H) = arg m ins0,...,sM1C

DML(s,y,H)

, (11)

with DML(s,y,H) logp(y|H, s) y Hd s2,where means is proportional to and C denotes the setof constellation symbols.

The detection rule (11) requires the knowledge of the

perfect channel vector H. The sub-optimal mismatched ML

detector consists in replacing the exact channel by its estimateH in the receiver metric assMM(H) = arg min

s0,...,sM1C

DMM(s,y, H)

= arg m ins0,...,sM1C

y Hd s2, (12)

whereDMM(s,y,H) DML(s,y,H)

H=H. (13)

Obviously, the sub-optimality of this detection technique is

due to the mismatch introduced by the channel estimation



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Fig. 2. Block diagram of the improved receiver under imperfect channel estimation.

errors; while the decision metric is derived from the likelihood

function p(y|H, s) conditioned on the perfect channelH, thereceiver uses an estimate H different from H in the detectionprocess.

As an alternative to this mismatched detection, an improved

ML detection metric is proposed in [15], [16]. This metric is

based on modified likelihood p(y|H, s) which is conditionedon the imperfect channel H. The pdfp(y|H, s) can be derivedas follows:

p(y|H, s) = HCM1

p(y,H|H, s) dH=

HCM1

p(y|H, s) p(H|H) dH= E

H|Hp(y|H, s) H, (14)where p(H|H) is the channel posterior distribution of equation(8) and C denotes the set of complex numbers. Actually,

equation (14) shows that p(y|H, s) can be simply derivedfrom the general formulation in (10) by replacing the cost

function J(y, s,H) by the likelihood p(y|s,H) that wouldbe used if the channel H was perfectly known.

By applying Theorem 1.2 of Appendix I.B, the averaged

likelihood in (14) is shown to be a complex Gaussian dis-

tributed vector given by

p(y|H, s) = CNmM,M, (15)where m

M= Hd s, and M = z + 2E SdSd.

Finally, the estimate of the symbol s is

sM

(H) = arg m ins0,...,sM1C

DM

(s,y, H), (16)where

DM

(s,y, H) log p(y|s, H)=

M1k=0

log

2z + 2E |sk|

2

+

yk Hk sk22z +

2E |sk|

2(17)

is referred to as the improved ML decision metric under

imperfect CSIR.

Note that when CSIR tends to the exact value, which is

obtained when the number of pilot symbols tends to infinity,

we have 1, 2E 0 and thus

limN

DM

s,y, HDMM

s,y, H = 1. (18)

Consequently, the improved metric of (17) tends to the mis-

matched metric, when the estimation errors tend to zero.

IV. ITERATIVE DETECTION OF BICM OFDM UNDER

IMPERFECT CHANNEL ESTIMATION

At the receiver, we perform MAP symbol detection and

channel decoding in an iterative manner as proposed for

instance in [23]. As shown in Fig. 2, the receiver mainlyconsists in the combination of two sub-blocks that exchange

soft information with each other. The first sub-block, referred

to as soft detector (also called demapper), produces softinformation in the form of extrinsic probabilities from the

input symbols and send it to the second sub-block which is a

soft-input soft-output (SISO) decoder. Each sub-block can take

advantage of the quantities provided by the other sub-block

as an a priori information. Here, SISO decoding is performedusing the well known forward-backward algorithm [24]. In the

following, we make use of the improved ML metric derived

in the previous section to modify the MAP detector part for

the case of imperfect CSIR.

Let dj,mk be the m-th (m = 1,...,B) coded and interleavedbit corresponding to the constellation symbol sk transmittedat the j-th time slot over the k-th subcarrier. We denote byL(dj,mk ) the coded log-likelihood ratio (LLR) of the bit d

j,mk

at the output of the detector. Conditioned on the imperfect

CSIR Hk, L(dj,mk ) is given byL(dj,mk ) = log

Pdem

dj,mk = 1 yk, Hk

Pdem

dj,mk = 0 yk, Hk , (19)

where Pdem(dj,mk

yk,

Hk) is the probability of transmission of

dj,m

k

at the detector output. We partition the set C that containsall possibly-transmitted symbol sk into two sets Cm0 and Cm1 ,for which the m-th bit of sk equals 0 or 1, respectively.



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We have

L(dj,mk ) = log

sk Cm1

eDM(sk, yk,Hk) B

n=1

n=m

P1dec

dj,nk

sk Cm0

eDM(sk, yk, Hk) Bn=1

n=m

P0dec

dj,nk ,(20)

where P1dec(dj,nk ) and P0dec(dj,nk ) are prior probabilities com-ing from the SISO decoder.

Note that using the metric DM(sk, yk, Hk) for the eval-uation of the LLRs in (20) is an alternative to using the

mismatched ML metric DMM(sk, yk, Hk) which replaces ateach iteration, the exact channel Hk by its estimate Hk inDML(sk, yk, Hk). By doing so, the LLRs are adapted tothe imperfect channel knowledge available at the receiver

and consequently the impact of channel uncertainty on the

SISO decoder performance is reduced. We refer to the latter

approach as improved MAP detector.

The summations in (20) are taken over the product of the

likelihood p(yk|sk, Hk) = eDM(sk,yk,Hk) given a symbolsk and the estimated channel coefficient Hk, and of the a

priori probability on sk (the term

Pdec), fed back from theSISO decoder at the previous iteration. In this latter term, thea priori probability of the bit dj,mk itself has been excluded,so as to let the exchange of extrinsic informations between

the channel decoder and the soft detector. Also, note that this

term assumes independent coded bits dj,nk , which is a validapproximation for random interleaving of large size. At the

first iteration, no a priori information is available on bits dj,nk ,therefore the probabilities P0dec(d

j,nk ) and P

1dec(d

j,nk ) are set

to 1/2. The decoder accepts the LLRs of all coded bits and

computes the LLRs of information bits, which are used fordecision, at the last iteration.

V. ACHIEVABLE INFORMATION RATES ASSOCIATED TOIMPROVED AND MISMATCHED DETECTION METRICS

A. Instantaneous Achievable Information Rates Of OFDM

In [25], the authors derive the mismatched capacity of achannel, i.e., the highest rate at which reliable communication

is possible over the channel with a given sub-optimal decodingmetric. Obviously, these achievable rates are lower than the

channel capacity defined by Shannon which assumes a perfect

channel knowldege and optimal decoders at the receiver.Here, we consider an OFDM transmission over the frequency

selective channel H of which the receiver knows an imperfect

estimate H obtained by some training symbols. Our aim is tocalculate the achievable rates CM and CMM associated to anOFDM receiver based respectively on the improved and the

mismatched detection rules of (17) and (13). The following

theorem follows from the direct application of Theorem 1 in

[25] to an OFDM transmission where the receiver uses a givendetection rule D. This transmission is characterized by thelikelihood p(y|s,H) =

M1k=0 CN(Hksk,

2z) which defines

a mapping from the input symbols s S with distributionp(s), to the set of probability measures on the output alphabety Y.

Theorem 1: The highest achievable rate of the transmission

characterized by p(y|s,H) with a receiver using the sub-optimal detection metric D(s,y, H) is given by

CD = maxp(s)

minf

Ip(s), f

, (21)

where the maximization is performed over all probability

distributions p(s) on S, and

Ip(s), f = p(s)f(y|s, )log2

f(y|s, )p(s)f(y|s, )ds

dy ds, (22)

denotes the mutual information functional [26]; in (21),the minimization is over all channels f(y|s, ) =M1

k=0 CN(ksk, 2k) on S Y satisfying the following con-

straints

(c1) : Es

f(y|s, )

= Esp(y|s,H)

, (23)

(c2,k) : Esk

E{yk:f}

D(sk, yk,

Hk)

Esk E{yk:p}D(sk, yk, Hk), (24)for every k = 0, . . . , M 1, where = [0,...,M1]

T;

E{yk :f}[.] and E{yk:p}[.] refer to conditional expectation withrespect to yk by using the pdff(yk|sk, k) and p(yk|sk, Hk),respectively.

1) Case of Improved ML Metric: In the following, we aim

at solving the above constrained minimization problem for

our specific OFDM channel and metric D = DM of (17). Tothis end, we assume that the transmitter has no information

about the channel estimate and consequently uniform power

allocation is done across subcarriers. Furthermore, we assume

a Gaussian i.i.d. input distribution p(s) = CN(0, PIM).Under these conditions, the mutual information (22) of an

OFDM transmission, averaged over all subcarriers is given

by [27]

Ip(s), f

=

1

Mlog2 det

IM + Pd

d

1

=1

M

M1k=0

log2

1 +

P|k|2

2k

, (25)

where = diag([20 ,...,2M1]) and d = diag(). Accord-

ing to Theorem 1, we have to find the covariance matrix

and the optimal channel vector opt (these two characterizethe pdf f(y|s, )), so as to minimize the mutual information(25) and to satisfy constrains (23) and (24).

The unknown diagonal covariance matrix is obtained

from the constraint (c1) of (23). It is easily seen that thisconstraint leads to the equality f(y|) = p(y|H). Moreover,from the likelihoods f(y|s, ) and p(y|s,H), it is clearthat the conditional random vectors (y|) and (y|H) aredistributed as CN(0,+Pd

d) and CN(0,z+PHdH

d),

respectively. Thus the diagonal covariance matrix is ob-

tained as

= P(HdHd d

d) + z, (26)

with its k-th (k = 0,...,M 1) diagonal element equal to2k = P(|Hk|

2 |k|2) + 2z . (27)



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Now, in order to find the optimal vector opt, we have tospecify the set of inequality constraints (c2,k) associated to

the metric DM(sk, yk, Hk) of (17). As shown in AppendixII.A, the k-th inequality constraint set in (24) is equivalent tothe set

VM,k =

k : |k a

Hk|

2 |Hk a

Hk|

2

, (28)

where a = ( 2

z

P 2

E )/( 2

z

P 2

E(1 ) ); =exp

2zP 2

E

E1 2zP 2

E

and E1(x)

+x

exp{u}u du is

defined as the exponential integral.

By using (25), (27) and (28) in Theorem 1, the achievable

rates CM(H, H) associated to the improved metric DM, canbe obtained by solving the following minimization, which is

equivalent to the initial formulation (21)

CM(H, H) =

min

1

M

M1k=0

log2

1 +

P|k|2

2k(|k|)

subject to |k a Hk|2 |Hk a Hk|

2

(29)

for k = 0,...,M 1. Note that the k-th term in (29)is an increasing function of |k|2 and the constraints areconvex. Consequently, the minimum in (29) is obtained at the

border of the sets [28]. Thus, the problem simplifies to thesearch of the optimal channel coefficients k = opt,k fromthe minimization of 2 under the constraint sets resultingfrom the equality in (28). This becomes a classical convex

minimization problem that can be solved by using Lagrange

multipliers. After some algebra provided in Appendix II.B, weget opt = [opt,0,...,opt,M1]

T, where

opt,k = M,kHk, k = 0, . . . , M 1, (30)and

M,k

=

a

|Hk a Hk||Hk| , if a 0

a +|Hk a Hk|

|Hk| , if a < 0.(31)

Finally, the achievable rates associated to metric DM are givenas follows.

CM(H

, H) = 1MM1

k=0 log21 +P 2

M,k|Hk|

2

2z + P(|Hk|2 2M,k |Hk|2) .(32)

2) Case of Mismatched ML Metric: Under the same as-

sumptions as above, we can compute the achievable rates

CMM associated to the mismatched metric DMM of (13) in thesame way. The equality constraint (23) leads to the same result

as equation (27). By following similar steps as in Appendix

II, we can easily derive the k-th inequality constraint setassociated to the metric DMM as

VMM,k

=

k : Re

k

Hk

Re

Hk

Hk

(33)

for k = 0,...,M 1, where Re(.) takes the real part of itsargument. Now, by considering the minimization problem of

equation (29) under the constraints of (33), and using the

Lagrange method, we obtain

CMM(H, H) =1

M

M1k=0

log2

1 +

P 2MM,k

|Hk|22z + P(|Hk|

2 2MM,k

|

Hk|2)

, (34)

where MM,k is given by

MM,k

=Re(Hk

Hk)|Hk|2 . (35)

In order to get insight about the expressions (32) and (34),

assume that perfect CSIR is available at the receiver, i.e.,

Hk = Hk for k = 0,...,M 1. In this situation, 2E = 0, = 1, and we get

MM,k=

M,k= 1 for all k. Consequently,

both detectors provide the instantaneous rate

C(H) =1

M

M1

k=0 log21 +P |Hk|2

2z . (36)

In this case, assuming that there is no delay constraint and the

transmission time is longer than the channel coherence time,

both of the detectors achieve the ergodic capacity [29] which

is defined as

Cerg = EH

C(H)

. (37)

B. Evaluation Of Outage Rates Under Imperfect Channel

Estimation

In several applications in wireless communications, the

channel is chosen randomly at the beginning of the transmis-sion and is held fixed during the whole transmission session.In fact, in this case the transmission time is not longer than

the channel coherence time and consequently the mean of the

maximum mutual information in (37) is in general not equal

to the channel capacity. On the other hand, the capacity in

the Shannon sense does not exist since there is a non-zero

probability that the realized H and its estimate H are notcapable of supporting even a very small rate. In this case, the

capacity is a random entity, as it depends on the instantaneous

channel H. Thus, the concept of capacity-verus-outage [21]

has to be invoked. That is, with any given rate R and channelrealization H, we associate a set of channels (R,H). This setis the largest possible set for which C(H), i.e., the achievedrate for a realization ofH (R,H) satisfies C(H) < R.Formally, we can write (R,H) {H : C(H) < R}.The outage (or failure) probability Pout is then determinedby Pout = Prob

H (R, H)

. This is the definition

of the classical notion of outage capacity, which implicitly

assumes the true channel to be available at the decoder. In

contrast, when this knowledge is not available at the receiver,

but only the channel estimate, we have to invoke the notion

of estimation-induced outage capacity, which has different

properties, as explained in [18].

Following this approach, we make use of the posteriorchannel distribution of (8) that characterizes our channel

estimation process, in order to define the outage probability



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PoutM

(associated to the metric DM

), for an outage rate R 0

and an estimated channel H as [18]PoutM

(R, H) = {H

M(R,H)}p(H|H) dH , (38)

where M

(R,

H)

H : CM(H,

H) < R

. Using this, the

outage rate for an outage probability equal to is given by

CoutM (, H) = supRR 0 : PoutM (R, H) . (39)

The outage rates (39) depend on the random channel estimateH. Thus, we consider as performance measure, the averagedoutage rates over all channel estimates as follows.

CoutM () = EHCoutM (, H). (40)

By using (34) and similar steps as above, we can derive the

averaged outage rates CoutMM() associated to the mismatched

receiver. The outage rates associated to the best possible (but

not practical) decoder under imperfect channel estimation is

derived in [18]. The next section evaluates the outage rates

of the improved and mismatched receivers compared to thoseprovided by this decoder, to which we refer as theoreticaldecoder. In our case, the averaged outage rate of the theoretical

decoder is given by

Cout

T() = EHCoutT (, H), (41)

where the outage rates CoutT

(, H) are computed by using theachievable rates

CT

(H, H) = 1M

M1k=0

log2

1 +

P |Hk|2

2z

(42)

in expressions similar to (38) and (39).

VI . SIMULATION RESULTS

In this section, we provide numerical results to evaluate the

performance provided by the proposed detector in the presence

of channel estimation errors, in comparison with more classi-

cal approaches. We focus on the impact of imperfect channel

estimation on receiver bit error rate (BER) and the achievable

outage rates associated to improved and mismatched detectors.

We consider the multiband OFDM (MB-OFDM) scenario

proposed in the IEEE 802.15.3a WPAN proposal [30] for

UWB applications operating in the range of 3.1 to 10.6 GHz.

In this scheme, the available spectrum is divided into severalsubbands with a conventional OFDM modulation within eachsubband. The only difference is that a frequency hopping

pattern selects the subband over which each OFDM symbol

is sent. Throughout the simulations, one OFDM symbol is

composed of M = 100 data subcarriers and the subbandchannel bandwidth is 528 MHz according to the parametersdefined in [30]. An MB-OFDM system employing only the

first three subbands starting at 3.1 GHz has been considered.For channel coding in our BICM scheme, we consider the

rate 1/2 NRNSC code of constraint length 3 defined in octalform by (5, 7)8. The interleaver is pseudo-random and oper-ates over the entire frame. Data symbols belong to 16-QAMconstellation with the Gray or the set-partition (SP) labeling

specified in [31], and the impact of the labeling is discussed

0 2 4 6 8 10 12 1410

6

105

104

103

102

101

100

Eb

/ N0

(dB)

BER

Mismatched N = 1

Improved N = 1

Mismatched N = 2

Improved N = 2

Mismatched N = 8

Improved N = 8

Perfect CSIR

Fig. 3. BER performance of improved and mismatched MAP detectorover the CM1 channel, training sequence lengths N {1, 2, 8}, 16-QAMmodulation with Gray labeling.

below, since it may enlarge or reduce the improvement brought

by the improved metric. However, note that the considered

labelings are not necessarily the optimal choices that may exist

for a 16-QAM constellation (see [32] and references therein

for the optimization of square QAM labeling).

The performance evaluation is performed over two chan-

nels: i) the realistic UWB channel model CM1 specified in

[33] and ii) the uncorrelated Rayleigh fading channel. For

each frame that contains 100 OFDM symbols, a differentrealization of the channel has been drawn and remains constant

during the whole frame. We use training sequences for channel

estimation and the average pilot-symbol power is equal tothe average data-symbol power. Moreover, the number of

decoding iterations is set to 4.

A. Bit Error Rate Analysis

First, the BER performance of the improved and mis-

matched detectors are compared. Let us first address the

case of BICM iterative decoding with 16-QAM and Gray

labeling. It can be observed from Fig. 3 that for N = 1(the shortest possible training sequence), the improvement in

terms of required Eb/N0 in order to attain a given BER isabout 2 dB, compared to the mismatched situation, whilethe perfect channel knowledge would even be 3 dB better.Obviously, it is also observed that these quantities are reduced

(the performance of both the mismatched decoder and the

improved one get closer to the perfect channel one) when

increasing the length of the training sequence. Note that the

performance of the improved receiver with 1 pilot is veryclose to that of the mismatched receiver with 2 pilots. Forcomparison, results obtained with uncorrelated Rayleigh block

fading channel are illustrated in Fig. 4. It can be observed

that for N = 2, the SNR necessary to obtain a BER of 104

is reduced by about 1 dB if the improved MAP detector isused instead of the mismatched detector. The conclusions are

otherwise quite similar on the Rayleigh channel.



8/12


0 2 4 6 8 10 12 1410

6

105

104

103

102

101

100

Eb

/ N0

BER

Mismatched N = 1Improved N = 1

Mismatched N = 2

Improved N = 2

Mismatched N = 8

Improved N = 8

Perfect CSIR

Fig. 4. BER performance of the improved and mismatched MAP detectorover i.i.d. Rayleigh fading channel, training sequence lengths N {1, 2, 8},16-QAM modulation with Gray labeling.

0 2 4 6 8 10 12 1410

6

105

104

103

102

101

100

Eb

/ N0

(dB)

BER

Mismatched N = 1

Improved N = 1

Mismatched N = 2

Improved N = 2

Mismatched N = 8

Improved N = 8

Perfect CSIR

Fig. 5. BER performance of the improved and mismatched MAP detectorover the CM1 channel, training sequence lengths N {1, 2, 8}, 16-QAMmodulation with set-partition labeling.

Similar plots are shown in Fig. 5 for the case of 16-

QAM and SP labeling on the CM1 channel. These show the

behavior of the detectors with respect to the type of labeling.A first observation is that the distance between the mismatcheddetector and the perfect channel knowledge is even larger

(about 6 dB for N = 1 and a BER of 7 104). Anotherobservation is that, even if the global performance is largely

improved by using the SP labeling when perfect channel

knowledge is available, the difference between SP and Gray

labeling is not very large with the mismatched decoder, i.e., the

sensitivity of the iterative decoder to the channel knowledge

seems to be larger for SP labeling. However, the use of the

improved metric allows to recover most of the improvement,

since, even with a single training sample (N = 1), the BER isimproved with SP compared to Gray labeling by about 0.9 dBfor a BER of5104. In other words, iterative decoding withSP labeling benefits more from the improved metric than the

1 2 4 6 8 10 12 14 1610

5

104

103

102

101

N (number of pilot symbols per frame)

BER

Mismatched

Improved

Perfect CSI

Eb

/ N0

= 12 dB

Fig. 6. Reduction of the number of training sequence at Eb/N0 = 12 dBover the CM1 channel, 16-QAM modulation with Gray labeling .

1 2 4 6 8 10 12 1410

6

105

104

103

102


BER

Mismatched

Improved

Perfect CSIR

Eb

/ N0

= 12 dB

Fig. 7. Reduction of the number of training sequence at Eb/N0 = 12 dBover the CM1 channel, 16-QAM modulation with set-partition labeling.

one with Gray labeling. Otherwise, similar conclusions hold

between the SP-labeling curves.

Figures 6 and 7 show the BER performance versus the

number of pilot symbols N at a fixed Eb/N0 of12 dB for 16-

QAM with Gray and set-partition labeling, respectively. Thisallows to evaluate the length of training sequence necessaryto achieve a certain BER. From Fig. 6 we observe that

the improved detector requires 10 pilot symbols per frameto achieve a BER of 104 at Eb/N0 = 12 dB whilethe mismatched detector attains this BER for 12 trainingsymbols. From Fig. 7 we notice that the performance loss

due to the mismatched receiver with respect to the improved

receiver becomes insignificant for N 12 (about 11 % ofthe overall frame of pilot and data symbols). Actually, our

results illustrate that the improved detector outperforms the

mismatched detector especially when a few number of pilot

symbols is dedicated to channel estimation, which is in perfectagreement with equation (18). This makes the improved metric

particularly useful, since training sequences in multicarrier



9/12


0 5 10 15 20 250

1

2

3

4

5

6

7

8

SNR (dB)

ExpectedOutage

Rate(bits/channeluse)

Ergodic Capacity

Theoretical

Improved

Mismatched

Fig. 8. Expected outage rates of MB-OFDM transmission versus SNR forN = 1 pilot per frame and different detection approaches, outage probability= 0.01, i.i.d. Rayleigh fading channel.

0 5 10 15 20 250

1

2

3

4

5

6

7

SNR (dB)

Exp

ectedOutageRates(bits/channeluse)

Ergodic Capacity

Theoretical

Improved

Mismatched

Fig. 9. Expected outage rates of MB-OFDM transmission versus SNR forN = 2 pilots per frame and different detection approaches, outage probability= 0.01, i.i.d. Rayleigh fading channel.

systems are usually of length 1 or 2. Note also that thisimprovement is obtained almost at no computational cost.

B. Achievable Outage Rates

We now analyze the achievable outage rates provided by

a receiver based on either the improved or the mismatched

detection technique for MB-OFDM under imperfect CSIR.

The channel is an uncorrelated i.i.d. Rayleigh fading and

the data symbols are assumed to be distributed as CN(0, 1).Figure 8 shows the expected outage rates (in bits per channel-

use) versus the SNR (in dB), obtained by adopting mismatched

and improved detection approaches. The outage probability

has been fixed to = 0.01 and the channel is estimated bysending N = 1 pilot per frame. For comparison, we alsodisplay the upper bound on the expected outage rates provided

by the theoretical decoder and also the ergodic capacity, given

1 4 8 12 16 201

1.5

2

2.5

3

3.5

4

4.5

5


ExpectedOutage

Rates(bits/channeluse)

Ergodic Capacity

Theoretical

Improved

Mismatched

SNR = 15 dB

SNR = 10 dB

Fig. 10. Expected outage rates of MB-OFDM transmission versus the numberof training sequence for different detection approaches at SNR of 10 and 15dB, outage probability = 0.01, i.i.d. Rayleigh fading channel.

by equations (41) and (37), respectively. The figure clearly

shows the sub-optimality of mismatched detection in terms of

expected outage rates compared to the rates provided by the

theoretical decoder. It can be observed that the mismatchedoutage rate is about 4.8 dB (at a mean outage rates of 5 bits)of SNR far from the rates achieved by the theoretical decoder.

We note that by adopting the improved receiver, the above

SNR gap is reduced by about 1.8 dB.Similarly, Fig. 9 shows the outage rates obtained by the

improved and mismatched receivers when the number of

channel-uses for pilot transmission is N = 2 per frame. As

observed, due to a more accurate channel estimation, boththe detectors achieve a higher outage rate as compared to that

obtained with N = 1. However, we note that at a mean outagerate of5 bits, the rate of the mismatched receiver is still about2.8 dB of SNR far from the rate of the theoretical receiver.The figure shows that the improved detector achieves higher

rates and decreases the required SNR by about 1 dB at a meanoutage rate of 5 bits. An open question is still how the gap

to the optimal estimation induced outage capacities could be

filled.

Finally, Fig. 10 compares the achievable rates of the im-proved and mismatched receivers with respect to the number

of training symbols at fixed SNR values of 10 and 15 dB. Ineach case, we also display the corresponding upper bound

on the achievable outage rates provided by the theoretical

decoder and the ergodic capacity. As can be seen, the improved

detector requires fewer pilot symbols in order to provide a

prescribed mean outage rate. For instance, the gain in the

number of channel-uses for pilot transmission is 2 pilots ata mean outage rates of 3.5 bits. These simulation resultsalso confirm the obvious expectation that under near perfect

channel estimation (here for N 16), the performance ofmismatched and improved receivers is almost the same.

VII. CONCLUSION

The problem of signal detection in a practical coded OFDM

communication system where the receiver has only access to



10/12


a noisy estimate of the channel provided by pilot symbols

was investigated. Based on a characterization of the channel

estimation process, we proposed a general detector designthat takes into account the imperfect channel available at the

receiver. In the case of ML detection, this approach led to an

improved ML metric that we used in the derivation of a mod-

ified iterative MAP detector. We also derived the expressions

of the achievable outage rates associated to the improved and

mismatched ML metrics. Our numerical results indicated thatthe mismatched detector is sub-optimal in terms of BER and

achievable outage rates, especially for short training sequence

lengths. They also confirmed the adequacy of the improved

detector under imperfect channel estimation. The impact of

the improved metric was shown to be dependent on the typeof labeling used in the construction of the symbols, and it

was shown that this new metric could save the improvementsbrought by set-partitioning labeling in the case of poor channel

estimates (obtained by very short training sequences). This

performance improvement was obtained without requiring

additional complexity in the receiver. Although the improved

detector was able to reduce the SNR gap between the mis-matched and the theoretical receiver at fixed outage rates, thederivation of other practical detectors providing closer rates to

the theoretical capacity limit is still an open problem.

APPENDIX I

A. Derivation of the A Posteriori Probability (8)

The following theorem is derived in [34].

Theorem 1.1: Let x1 and x2 be circularly symmetric com-

plex Gaussian random vectors with zero means and full-rankcovariance matrices ij = E{xix

j}. Then the conditional

random vector x1

|x2

CN(,) is a circularly symmetriccomplex Gaussian random vector with mean = 12

122 x2

and covariance matrix = 12122 21.

We set x1 = H and x2 = H. From equation (7) and theassumption H CN(0,H), we have 11 = 12 = Hand 22 = H + E in Theorem 1.1. According to thistheorem, the conditional random vector H|H has a circularlysymmetric complex Gaussian distribution with

mean = H where H(H+E)1, and (43)covariance matrix = HH(H+E)

1H = E .

(44)

The equivalence in (44) can be seen by left multiplying both

sides of (E +H) H = E by H(E +H)

1.

As a result, we obtain the a posteriori pdf (8).

B. Evaluation of the Likelihood Function (14)

In order to evaluate the expectation in (14), we use to the

following theorem derived in [35].

Theorem 1.2: For a circularly symmetric complex random

vector u CN(m,) with mean m = E[u] and covariancematrix = E[uu] mm, and a Hermitian matrix A suchthat I+A > 0, we have

Eu exp uAu = expmA(I +A)1mdet{I +A} .(45)

Let us define u = y Hds. Using the a posteriori distri-bution of (8) and after some algebra, we can derive the condi-

tional pdf ofu given s and Hd as u|(s, Hd) CN(mu,u),where mu = yHds and u = ESdSd. We furtherlet A = 1z . By applying Theorem 1.2, we get (14) as

p(y|

H, s) = E

Hd|Hd

exp (y Hds)

z

1(y Hds)

det

z

= 1det

z

I +ESdS

d

1z

exp

y Hds1z

I +ESdSd

1z

1y Hds (46)

Since z, and E are diagonal matrices, the latter

equation is rewritten as

p(y|H, s) = 1det

z + 2ESdS

d

exp y Hdsz + 2ESdSd1y Hds

(47)

= CN

Hds , z + 2ESdSd.

APPENDIX II

A. Details On The Derivation Of The Inequality Constraint

(28)

Using the expression of the modified metric DM from(17), the left-hand side of the k-th constraint in (24) can beexpanded as follows (the index k is omitted for notational

brevity).Es

E{y:f}

DM(s,y, H) = Es log (2z + 2E |s|2)

K

+

Es E{y:f}

|y|2 2Re

y Hs+ 2|H|2|s|2

2z + 2E |s|

2

= K+ Es

E{y:f}[|y|

2] 2ReE{y:f}[y

] Hs+ 2|H|2|s|22z +

2E |s|

2

= K+ Es

2 + ||2|s|2 2Re

H|s|2 + 2|H|2|s|2

2z + 2E |s|2

= K+ Es

2 + |

H|2|s|2

2z + 2E |s|

2 . (48)

Similarly, the right-hand side of (24) can be obtained as

Es

E{y:p}

DM(s,y, H) = K+Es2z + |H H|2|s|2

2z + 2E |s|

2

.

(49)

Using (48) and (49) in (24), the inequality constraint is written

as

Es

2 + | H|2|s|2

2z + 2E |s|

2

Es

2z + |H H|2|s|2

2z + 2E |s|

2

.

(50)

Now, in order to evaluate the expectation in (50), we introduce

the following lemma.

Lemma 1: Assume s C N(0, ) and x = |s|2 be a centeredChi-squared random variable with two degrees of freedom



11/12


with pdf p(x) = 1 ex/ , and let a, b, c and d be real and

positive scalars. We have

Ex

a + b x

c + d x

=

b

d+

a

d

bc

d2

exp

c

d

E1

c

d

,

(51)

where E1()

exp{u}u du is the exponential integral.

Proof: The proof is easy and thus omitted for brevity.

By using Lemma 1 and replacing a,b,c and d by theirrespective values from (50), we get

| H|22E

+

2

(2E)P

| H|22z(1 2)P

|H H|22E

+

2z

2E P

|H H|22z(1 2)P

(52)

where exp 2z2EP

E1 2z2EP

.

After some algebraic manipulations, (28) can be derived from

(52).

B. Derivation Of The Optimal Solutions in (30)

We have to find the optimal vector opt =[opt,0, . . . , opt,M1]

T that minimizes 2 =M1k=0 |k|

2 subject to M constraints given by

|k a Hk|2 |Hk a Hk|2 = 0, for k = 0, . . . , M 1.Obviously, this problem can be split into M independentconstraint minimization problems for finding each opt,kfrom the minimization of |k|2. According to the Lagrangemultipliers method, we define the Lagrangian Lk as:

Lk = |k|2 + lk

|k a

Hk|

2 |Hk a

Hk|

2

= 0 (53)

for k = 0, . . . , M 1, where lk is an unknown constant. Theoptimal value opt,k is obtained by solving

Lkk

= k + lk (k a Hk) = 0. (54)We get

k =lk

lk + 1a Hk (55)

where lk is fixed by introducing k from (55) into the initialconstraint as follows

|

lk

lk + 1 a Hk a Hk|2 = |Hk a Hk|2. (56)It can be easily verified that (56) leads to two solutions lk,1and lk,2 for lk such that

lk,1

lk,1 + 1= 1

|Hk a Hk||a| |Hk| and lk,2lk,2 + 1 = 1 + |Hk a

Hk||a| |Hk| .

Clearly, the acceptable solution is lk = lk,1 since it minimizes|k|2 in (55).As a result, from (55) we obtain

opt,k = aa |Hk a

Hk|

|a| |Hk| Hk for k = 0, . . . , M 1,(57)which is the expression (30).

ACKNOWLEDGMENT

The authors would like to acknowledge many useful discus-

sions with P. Piantanida in which he provided insightful and

accurate comments.

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Seyed Mohammad Sajad Sadough (S04M08)was born in Paris in 1979. He received the B.Sc.degree in Electrical Engineering (electronic) fromShahid Beheshti University, Tehran, Iran, in 2002,the M.Sc. degree and the PhD degree in ElectricalEngineering (telecommunication), both from ParisSud 11 University, Orsay, France, in 2004 andJanuary 2008, respectively. During his PhD (200407), he has been with ENSTA (National Engineer-ing school in advanced techniques), Paris, Franceand Suplec-CNRS/LSS (Laboratory of Signals and

Systems), Gif-sur-Yvette, France. He has been a lecturer in the Electronics andComputer Engineering Department of ENSTA where his research activitieswere focused on improved reception schemes for Ultra Wideband commu-nication systems. From December 2007 to September 2008, he has been aPostdoctoral researcher at Suplec-CNRS/LSS, Gif-sur-Yvette, France, wherehe was involved in research projects with Alcatel-Lucent on satellite mobilecommunication systems.

Dr Sadough joined the Faculty of Electrical and Computer Engineering ofShahid Beheshti University, Tehran, Iran, in October 2008, as an AssistantProfessor in the Telecommunication Department. His current research interestsinclude Signal processing for wireless communications with particular empha-sis on multicarrier and MIMO systems, joint channel estimation and decoding,iterative reception schemes and interference cancellation under partial channelstate information.

Pierre Duhamel (F98) was born in France in1953. He received the Eng. Degree in ElectricalEngineering from the National Institute for AppliedSciences (INSA) Rennes, France in 1975, the Dr.Eng. Degree in 1978, and the Doctorat s sciencesdegree in 1986, both from Orsay University, Orsay,France.

From 1975 to 1980, he was with Thomson-CSF,Paris, France, where his research interests werein circuit theory and signal processing, including

digitalfi

ltering and analog fault diagnosis. In 1980,he joined the National Research Center in Telecommunications (CNET), Issyles Moulineaux, France, where his research activities were first concernedwith the design of recursive CCD filters. Later, he worked on fast algorithmsfor computing Fourier transforms and convolutions, and applied similartechniques to adaptive filtering, spectral analysis and wavelet transforms.From 1993 to Sept. 2000, he has been professor at ENST, Paris (NationalSchool of Engineering in Telecommunications) with research activities fo-cused on Signal processing for Communications. He was head of the Signaland Image processing Department from 1997 to 2000. He is now withCNRS/LSS (Laboratoire de Signaux et Systmes, Gif sur Yvette, France),where he is developing studies in Signal processing for communications(including equalization, iterative decoding, multicarrier systems, cooperation)and signal/image processing for multimedia applications, including sourcecoding, joint source/channel coding, watermarking, and audio processing. Heis currently investigating the application of recent information theory resultsto communication theory.

Dr. Duhamel was chairman of the DSP committee from 1996 to 1998, anda member of the SP for Com committee until 2001. He was an associateEditor of the IEEE T RANSACTIONS ON SIGNAL PROCESSING from 1989 to1991, an associate Editor for the IEEE S IGNAL PROCESSING LETTERS, anda guest editor for the special issue of the IEEE T RANSACTIONS ON SIGNALPROCESSING on wavelets.

He was Distiguished Lecturer, IEEE, for 1999, and was co-general chair ofthe 2001 International Workshop on Multimedia Signal Processing, Cannes,France. He was also co-technical chair of ICASSP 06, Toulouse, France.The paper on subspace-based methods for blind equalization, which he co-authored, received the Best paper award from the IEEE transactions on SPin 1998. He was awarded the Grand Prix France Telecom by the FrenchScience Academy in 2000.

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