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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007 1

On the BER Performance of HierarchicalM -QAM Constellations With Diversity and

Imperfect Channel Estimation

1

2

3

Nuno M. B. Souto, Francisco A. B. Cercas, Rui Dinis, and Jo ao C. M. Silva4

Abstract —The analytical bit error rate of hierarchical quadra-5

ture amplitude modulation formats, which include uniform and6

nonuniform constellations, over at Rayleigh fading environments7

is studied in this paper. The analysis takes into account the effect8

of imperfect channel estimation and considers diversity reception9

with both independent identically and nonidentically distributed0

channels, employing maximal ratio combining.1

Index Terms —Channel estimation, diversity, quadrature ampli-2

tude modulation (QAM), Rayleigh fading.3

I. INTRODUCTION4

I N THE design of wireless communication networks, the5

limitation on spectrum resources is an important restriction6

for achieving high bit rate transmissions. The use of M -ary7

quadrature amplitude modulation ( M -QAM) is considered an8

attractive technique to overcome this restriction due to its high9

spectral efciency, and it has been studied and proposed for 0

wireless systems by several authors [1], [2].1

A great deal of attention has been devoted to obtaining an-2

alytical expressions for the bit error rate (BER) performance3

of M -QAM with imperfect channel estimation. A tight up-4

per bound on the symbol error ratio (SER) of 16-QAM with5

pilot-symbol-assisted modulation (PSAM) in Rayleigh fading6

channels was presented in [3]. An approximate expression for 7

the BER of 16-QAM and 64-QAM in at Rayleigh fading with8

imperfect channel estimates was derived in [4], while in [5],9

exact expressions were obtained for 16-QAM diversity recep-0

tion with maximal ratio combining (MRC). The method used1

in these papers can be extended to general M -QAM constel-2

lations, but the manipulations and development required can3

become quite cumbersome. Recently, exact expressions in [6],4

were published for the performance of uniform M -QAM con-5

stellations with PSAM for identical diversity channels, while6

in [7], expressions valid for nonidentically distributed channels7

Paper approved by X. Dong, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received July 13, 2005; revisedMarch 21, 2006, August 25, 2006, and March 27, 2007.

N. M. B. Souto is with the Instituto Superior T ecnico (IST)/Instituto deTelecomunicacoes, 1049 Lisbon, Portugal (e-mail: [email protected]).

F. A. B. Cercas and J. C. M. Silva are with the Instituto de Ciˆ enciasdo Trabalho e da Empresa (ISCTE)/Instituto de Telecomunicac oes, 1649-026Lisbon, Portugal (e-mail: [email protected]; [email protected]).

R. Dinis is with the Instituto Superior T´ ecnico, Instituto de Sistemas eRobotica (ISR), 1049 Lisbon, Portugal (e-mail: [email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TCOMM.2007.906367

were derived, though in this case, they were not linked to any 38

specic channel estimation method. 39

All the studies mentioned before relate to M -QAM uni- 40

form constellations that can be regarded as a subset of the 41

more general case of nonuniform M -QAM constellations (also 42

called hierarchical constellations). These constellations can 43

be used as a very simple method to provide unequal bit er- 44

ror protection and to improve the efciency and exibility 45

of a network in the case of broadcast transmissions. Nonuni- 46

form 16/64-QAM constellations have already been incorpo- 47

rated in the digital video broadcasting-terrestrial (DVB-T) 48

standard [8]. A recursive algorithm for the exact BER com- 49

putation of hierarchical M -QAM constellations in additive 50

white Gaussian noise (AWGN) and fading channels was pre- 51

sented in [9]. Later on, closed-form expressions were also ob- 52

tained for these channels [10]. As far as we know, the ana- 53

lytical BER performance of these constellations in Rayleigh 54

channels with imperfect channel estimation has not yet been 55

investigated. 56

In this paper, we adopt a different method from [4] and [5] 57

to derive general closed-form expressions for the BER perfor- 58

mance in Rayleigh fading channels of hierarchical M -QAM 59

constellations with diversity employing an MRC receiver. We 60

consider diversity reception with both independent identically 61

and nonidentically distributed channels.PSAM philosophywith 62

channel estimation that accomplished through a nite impulse 63

response (FIR) lter is assumed. 64

This paper is organized as follows. Section II describes the 65

model of the communication system, which includes the deni- 66

tion of nonuniform M -QAM constellations, thechannel, andthe 67

modeling of the channel estimation error. Section III presents 68

the derivation of the BER expressions and Section IV presents 69

some numerical and simulation results. The conclusions are 70

summarized in Section V. 71

II. SYSTEM AND CHANNEL MODEL 72

A. QAM Hierarchical Signal Constellations 73

In hierarchical constellations, there are two or more classes 74

of bits with different error protection levels and to which dif- 75

ferent streams of information can be mapped. By using nonuni- 76

formly spaced signal points (where the distances along the I - or 77

Q- axis between adjacent symbols can differ depending on their 78

positions), it is possible to modify the different error protection 79

levels. As an example, a nonuniform 16-QAM constellation is 80

shown in Fig. 1. The basic idea is that the constellation can be 81

0090-6778/$25.00 © 2007 IEEE

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2 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007

Fig. 1. Nonuniform 16-QAM constellation.

viewed as a 16-QAM constellation if the channel conditions are82

good enough or as a quadrature phase-shifting keying (QPSK)83

constellation otherwise. In the latter situation, the received bit84

rate is reduced by half. This constellation can be characterized85

by the parameter k = D1 / D 2(0 < k ≤ 0.5). If k = 0 .5, the re-86

sulting constellation corresponds to a uniform 16-QAM. For 87

the general case of an M -QAM constellation, the symbols are88

dened as89

s =log 2 (√ M )

l=1±

D l

2+

log 2 (√ M )

l=1±

D l

2 j (1)

and the number of possible classes of bits with different error 90

protection is log2(M ) / 2. In the following derivation, we as-91

sume that the parallel information streams are split into two,92

so that half of each stream goes for the in-phase branch and93

the other half goes to the quadrature branch of the modula-94

tor. The resulting bit sequence for each branch is Gray coded,95

mapped to the corresponding pluggable authentication modules96

(√ M -PAM) constellation symbols, and then, grouped together,97

forming complex M -QAM symbols. The Gray encoding for 98

each (√ M -PAM) constellation is performed according to the99

procedure described in [9]. Firstly, the constellation symbols00

are represented in a horizontal axis where they are numbered01

from left to right with integers starting from 0 to √ M − 1.02

These integers are then converted into their binary representa-03

tions, so that each symbol s j can be expressed as a binary se-04

quence with log2(M )/ 2 digits: b1 j , b2

j , . . . , blog 2 M / 2 j . The corre-05

sponding Gray code [g1 j , . . . , g log 2 M / 2

j ] is then computed using06

(⊕

represents modulo-2 addition)07

g1 j = b1

j , gi j = bi

j

⊕bi−1 j , i = 2 , 3, . . . , log2

M

2 . (2)

B. Received Signal Model 108

Let us consider a transmission over an L diversity branch at 109

Rayleigh fading channel where the branches can have different 110

average powers. Assuming perfect carrier and symbol synchro- 111

nization, each received signal sample can be modeled as 112

r k = α k

·s + n k , k = 1 , . . . , L (3)

where α k is the channel coefcient for diversity branch k, s is 113

the transmitted symbol, and nk represents additive white ther- 114

mal noise. Both αk and nk are modeled as zero mean com- 115

plex Gaussian random variables with E [|α k |2] = 2σ2α k

(2σ2α k

116

is the average fading power of the kth diversity branch) and 117

E [|n k |2] = 2σ2n = N 0 (N 0 / 2 is the noise power spectral den- 118

sity). Due to the Gaussian nature of αk and nk , the probability 119

density function (pdf) of the received signal sample rk , condi- 120

tioned on the transmitted symbol s , is also Gaussian with zero 121

mean. The receiver performs MRC of the L received signals. 122

As a result of the mapping employed, the I and Q branches are 123

symmetric (the BER is the same), and so, our derivation can be 124

developed using only the decision variable for the I branch, i.e., 125

zre = Real L

k=1

r k α∗k . (4)

C. Channel Estimation 126

In this analysis, we consider a PSAM philosophy [3] where 127

the transmitted symbols are grouped in N -length frames with 128

one pilot symbol periodically inserted into the data sequence. 129

The channel estimates for the data symbols can be computed by 130

means of an interpolation with an FIR lter of length W , which 131

uses thereceivedpilot symbols of theprevious (W − 1) / 2 and 132

subsequent W / 2 frames. Several FIR lters were proposed in 133

the literature, such as the optimal Wiener lter interpolator [3], 134

the low pass sinc interpolator [11], or the low-order Gaussian in- 135

terpolator [12]. According to this channel estimation procedure, 136

the estimate α k is a zero-mean complex Gaussian variable. 137

Assuming that the channel’s corresponding autocorrelation 138

and cross-correlation functions can be expressed as in [13], the 139

variance of the channel estimate for symbol t (t = 2, . . . , N ; 140

t = 1 corresponds to the pilot symbol position) in frame u can 141

be written as 142

E

|α k ((u

− 1)N + t)

|2

= 2 σ2α k

W / 2 + u −1

j = − (W −1) / 2 + u−1

W / 2 + u −1

i = − (W −1) / 2 + u−1

× h j −u +1t hi−u +1

t J 0(2πf D |i − j |NT s )

+ 1

|S p|2N 0

W / 2 + u−1

j = − (W −1) / 2 + u −1

(h j −u +1t )2 (5)

where J 0(·) is the zeroth-order Bessel function of the rst kind, 143

f D is the Doppler frequency, h j −u +1t are the interpolation coef- 144

cients of the FIR, lter and S p is a pilot symbol. The second- 145

order moment of rk and α k , which will be required further 146

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SOUTO et al. : ON THE BER PERFORMANCE OF HIERARCHICAL M -QAM CONSTELLATIONS 3

Fig. 2. Impact in a 16-QAM constellation (upper right quadrant shown) of cross-quadrature interference originated by imperfect channel estimation. Onlythe case of phase error is shown.

ahead, is given by7

E [r k ((u − 1)N + t)α∗k ((u − 1)N + t)|s]

= 2 σ2α k s

W / 2 + u −1

j = − (W −1) / 2 + u−1

×h j −u +1

t J 0 (2πf D

|(u

− 1

− j )N + t

− 1

|T s ). (6)

III. BER PERFORMANCE ANALYSIS8

To accomplish this analysis, we start by deriving the bit error 9

probability for each type of bit im (m = 1, . . . , log2(M ) / 2) in0

a constellation. This error probability depends on the position1

t in the transmitted frame, which means that it is necessary2

to average the bit error probability over all the positions in the3

frame toobtain theoverallBER. Although theBERperformance4

of an M -QAM constellation in a Rayleigh channel with perfect5

channel estimation can be analyzed by simply reducing it to6

a √ M -PAM constellation, this simplication is not possible7

in the presence of an imperfect channel estimation. In fact,8

since the channel estimates are not perfect, a residual phase9

error will be present in the received symbols even after channel0

compensation at the receiver. This phase error adds interference1

from the quadrature components to the in-phase components2

and vice versa, as shown in Fig. 2.3

An explicit closed-form expression for the bit error probabil-4

ity of generalized nonuniform QAM constellations in AWGN5

and Rayleigh channels was derived in [10]. It is possible to6

adapt this expression for the case of imperfect channel estima-7

tion, which is a more general case where theconstellation cannot8

be simply analyzed as a PAM constellation. In this situation, as-9

suming equiprobable transmitted symbols, i.e., P (s j, f ) = 1 / M,0

the average BER can be computed as 171

P b (im ) = 1

(N −P )2

√ M

N −P

t =1

√ M / 2−1

j =0

1 −gk j + ( −1)gk

j

×

2( k −1)

l=1

(

−1)l+1 2

√ M ×

√ M / 2−1

f =0

× Prob zre < b m (l)L

k=1|α k |2 |s j, f , t . (7)

where 172

b m (l)

= d

s ((2 l−1) 21 / 2·log 2 M −m )+ ds ((2 l−1) 21 / 2·log 2 M −m +1)

2and 173

d s

( j ) =

1/ 2·log 2 M

i=1 2bi

j − 1 D 1/ 2·log 2 M −i+1 .Therefore, to calculate the analytical BER, it is necessary to ob- 174

tain an expression for Prob {zre < b m (l) Lk =1 |α k |2|s j, f , t}, 175

i.e., the probability that the decision variable is smaller 176

than the decision border b m (l) Lk=1 |α k |2 , conditioned on 177

the fact that the transmitted symbol was s j, f and its po- 178

sition in the frame is t. Note that, due to the existing 179

symmetries in the constellations, we only need to compute 180

Prob{zre < b m (l) Lk =1 |α k |2|s j, f , t} over the constellation 181

symbols in one of the quadrants. In the following derivations, 182

we will drop indexes t, i (in-phase branch label of the constella- 183

tion symbol), and f (quadrature branch label of theconstellation 184

symbol), and we replace b m (l) by w, for simplicity of notation. 185

To avoid the explicit dependency of the decision borders on the 186

channel estimate, the probability expression can be rewritten as 187188

Prob zre < wL

k=1|α k |2|s

= Prob Real L

k =1

(r k −wα k )α∗k < 0|s

= Prob(zre < 0|s) (8)

where the modied variables rk = rk

−w

· α k and z

re = 189

Lk =1 zre k =

Lk=1 Real{r k α∗k} are dened. Since r k and α k 190

are complex random Gaussian variables and w is a constant, rk 191

also has a Gaussian distribution. The second moment of rk , the 192

cross moment of r k and α k , and the respective cross-correlation 193

coefcient are given by 194

E [|r k |2

|s] = 2 |s|2 σ2α k + N 0 − 2w Re {E [r k α∗k |s]}

+ w2E [|α k |2] (9)

E [r k α∗k |s] = E [r k α∗k |s] −wE [|α k |2] (10)

µk = E [r k α∗k |s]

E [|r k |2

|s]E [|α k |2

|s]

= |µk |e−ε k · j . (11)

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We will rst compute the pdf of zre conditioned on s for each95

individual reception branch. We start by writing the pdf of zk96

(zk = rk α∗k ) conditioned on s [14]97

p(zk |s) = 1

2πσ 2α k

σ2r k

(1 − |µk |2)

× exp |µk | zre k cos εk + zim k sin εk

σr kσα k (1 − |µk |

2)

×K 0 z 2re k + z 2

im k

σr kσα k (1 − |µk |2)

(12)

where K 0 (·) denotes the modied Hankel function of order 98

zero and99

σ2r k =

E [|r k |2|s]

2 .

Integrating (12) over zim k (zim k

= Imag {zk}) yields the00

marginal pdf of the decision variable zre k01

p(zre k |s) = + ∞

−∞ p(zk |s) dzim k

= 1F k

exp[Gk zre k]exp[−H k ||zre k ||] (13)

where02

F k = 2σr k σα k

1 − |µk |2

(sin εk )2,

Gk = |µk | cos εk

σr kσα k (1 − |µk |2)

, H k = 1 − |µk |2 (sin εk )2

σr kσα k (1 − |µk |2)

.

The decision variable zre is the sum of L independent ran-03

dom variables with pdfs similar to (13). According to [15],04

the resulting pdf can be computed through the inverse Fourier 05

transform of the product of the individual characteristic func-06

tions. The characteristic function of the pdf dened in (13)07

is obtained by applying the Fourier transform, which results08

in09

Ψk (υj ) = − 2H k

F k (υj −Gk −H k ) (υj −Gk + H k ). (14)

If we divide the L diversitybranches into L different sets,where10

each set k contains ρk received branches with equal powers sat-11

isfying L = Lk =1 ρk , then the resulting characteristic function12

is given by the product of the individual characteristic functions,13

and can be written as14

Ψ (υj ) =L

k =1

−2H kF k

ρk

× 1

(υj −Gk −H k )ρk (υj −Gk + H k )ρk . (15)

The pdf of zre is obtained by decomposing (15) as a sum of 215

simple fractions (according to the method proposed in [16]) and 216

applying the inverse Fourier transform. The desired probability 217

can now be computed by integrating this pdf from −∞ to 0 as 218

follows 219

Prob (zre < 0) = 0

−∞ p (zre ) dzre =

L

k =1 −2H kF k

ρk

×L

k=1

ρk

i=1

(−1) i A1k ,i (Gk + H k )−i (16)

where 220

A1k, i =

1(ρk −i)!

∂ ρk −i

∂s ρk −i

L

j =1 j = k

1(s −G j −H j )ρ j

×L

j =1

1(s −G j + H j )ρ j

s = G k + H k. (17)

If all the diversity branches have equal powers, i.e., ρk = L, 221

then L = 1 and F k , Gk , and H k are all equal (index k can be 222

dropped) leading to223

Prob (zre < 0) = 1

(2F H )L

L

k =1

2L −k − 1L −k

2H G + H

k

(18)

which is the same result achieved in [6], though written in 224

a different format. However, if all the diversity branches are 225

different, i.e., ρk = 1 for any k and L = L, then226

A1l, 1=

L

j =1 j = l

1(Gl + H l −G j −H j )θ j ×

L

j =1

1(G l + H l−G j + H j )θ j

(19)

and 227

Prob (zre < 0) =L

k =1

−2H kF k ×

L

k=1

−A1k, 1

(Gk + H k ). (20)

228

IV. NUMERICAL RESULTS 229

To verify the validity of the derived expressions, some sim- 230

ulations were run using the Monte Carlo method. The results 231

obtained are plotted as a function of E S / N 0 (E S —symbol en- 232

ergy) in Figs. 3 and 4. Fig. 3 presents a nonuniform 16-QAM 233

(k = 0 .3) transmission with three equal branch diversity recep- 234

tion, while Fig. 4 presents a nonuniform 64-QAM ( k1 = 0 .4 235

and k2 = 0 .4) transmission with four diversity branches and 236

different relative powers ([0 dB −3 dB −6 dB −9 dB]). In 237

both cases, the frame size is N = 16 , the pilot symbols are 238

S p = 1 + j (transmitted with the same power level of the data 239

symbols),a sinc interpolator [11] is employed with W = 15 ,and 240

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Fig. 3. Analytical and simulated BER performances of nonuniform 16-QAM(k = 0 .3 ) with three equal diversity branches; f d T s = 1 .5

× 10 −2 ; sinc inter-

polation with W = 15 and N = 16 .

Fig. 4. Analytical and simulated BER performances of nonuniform 64-QAM(k1 = 0 .4 and k 2 = 0 .4 ) with four diversity branches and different relativepowers ([0 dB −3 dB −6 dB −9 dB]); f d T s = 1 .5 × 10 −2 ; sinc interpolationwith W = 15 and N = 16 .

Rayleigh fading is considered with f d T s = 1 .5 × 10−2 . The an-1

alytical results computed using expressions (7), (18), and (20),2

as well as curves corresponding to perfect channel estimation3

are drawn in both gures. In both cases, it is clear that the an-4

alytical results accurately match the simulated ones. We can5

also notice in both gures a considerable difference between6

the performance with perfect channel estimation and with im-7

perfect channel estimation, the existence of irreducible BER8

oors (more noticeable in the least protected bit since they ap-9

pear at higher BER values) being visible. This is a consequence0

of the effect of the time-varying fading channel that results1

in a reduced quality of the channel estimates. Moreover, these2

gures also show that the nonuniformity of the constellations3

clearly results in differentiated performances for thedifferentbit4

classes.5

V. CONCLUSION 256

In this paper, we have derived general analytical expressions 257

for the evaluation of the exact BER performance for the individ- 258

ualbit classes of any nonuniform square M -QAMconstellation, 259

in the presence of imperfect channel estimation. These expres- 260

sions can be applied to Rayleigh fading environments, with 261

either equal or unequal receiving diversity branches, and MRC 262

receivers. 263

ACKNOWLEDGMENT 264

The authors would like to thank the Editor and the anony- 265

mous reviewers for their helpful constructive comments which 266

improved the paper signicantly. Special thanks are due to the 267

Editor, Prof. X. Dong, for pointing out relevant reference [7] 268

of which we were not aware when we submitted the rst ver- 269

sion of the paper. This work was elaborated as a result of the 270

participation in the C-MOBILE project (IST-2005-27423). 271

REFERENCES 272

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fading channels,” IEEE Trans. Commun. , vol. 45, no. 10, pp. 1218–1230, 277

Oct. 1997. 278

[3] J. K. Cavers, “An analysis of Pilot symbol assisted modulation for 279

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pp. 686–693, Nov. 1991. 281

[4] X. Tang, M.-S. Alouini, andA. J. Goldsmith, “Effect of channel estimation 282

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[5] L. Cao and N. C. Beaulieu, “Exact error-rate analysis of diversity 285

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no. 6, pp. 1019–1029, Jun. 2004. 287

[6] B. Xia and J. Wang, “Effect of channel-estimation error on QAM systems 288

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488, Mar. 2005. 290

[7] L. Najazadeh and C. Tellambura, “BER analysis of arbitrary QAM for 291

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Nuno M. B. Souto received the B.Sc. degree in20

aerospace engineering—avionics branch, in 2000,21

andthe Ph.D.degreein electricalengineeringin 2006,22

both from the Instituto Superior T´ ecnico, Lisbon,23

Portugal.24

From November 2000 to January 2002, he was25

a researcher with the Instituto de Engenharia e26

Sistemas de Computadores, Lisbon, where he was27

engaged in research on automatic speech recog-28

nition. He is currently with the Instituto Supe-29

rior Tecnico (IST)/Instituto de Telecomunicacoes,30

Lisboa, Portugal. His current research interests include wideband code-division31

multiple-access (CDMA) systems, orthogonal frequency-division multiplex-32

ing (OFDM), channel coding, channel estimation, and multiple-input multiple-33

output (MIMO) systems.34

35

FranciscoA. B. Cercas receivedthe Dipl.-Ing., M.S.,36

andPh.D. degrees fromthe InstitutoSuperior T ecnico37

(IST), Technical University of Lisbon, Lisbon, Por-38

tugal, in 1983, 1989, and 1996, respectively.39He worked for the Portuguese Industry as a Re-40

search Engineer and as an invited researcher at the41

Satellite Centre of the University of Plymouth, Ply-42

mouth, U.K. He was a lecture at the IST and became43

Associate Professorin 1999 at ISCTE, Lisbon, where44

he is currently the Director of telecommunications45

and informatics. He is the author or coauthor of more46

than 60 papers published in international journals. His current research interests47

include mobile and personal communications, satellite communications, chan-48

nel coding, and ultra wideband communications.49

50

Rui Dinis received the Ph.D. degree from the Insti- 351

tuto Superior T ecnico (IST), Technical University of 352

Lisbon, Lisbon, Portugal, in 2001. 353

From 1992 to 2001, he was a member of the re- 354

search center Centro de An´alise e Processamento de 355

Sinais (CAPS/IST). During 2001, he was a Professor 356

at theIST. Since 2002, he is a member of theresearch 357

centerInstituto de Sistemas e Rob otica(ISR/IST).He 358

has been involved in several research projects in the 359

broadband wireless communications area. His cur- 360

rent research interests include modulation, equaliza- 361

tion, and channel coding. 362

363

Jo ao C. M. Silva received the B.S. degree in 364

aerospace engineering in 2000, and the Ph.D. de- 365

gree in 2006, both from the Instituto Superior 366

Tecnico (IST), Lisbon Technical University, Lisbon, 367

Portugal. 368

From 2000–2002, he worked as a business con- 369

sultant with McKinsey and Company. He was en- 370gaged in research on spread spectrum techniques, 371

multiuser detection schemes, and multiple-input 372

multiple-output systems. He is currently with the 373

Instituto de Ci encias do Trabalho e da Empresa 374

(ISCTE)/Instituto de Telecomunicac oes, Lisbon, Portugal. 375

376

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QUERIES 377

Q1. Author: Please provide the complete educational information (degree, etc.) of R. Dinis. 378

Q2. Author: Please provide the subject of the Ph.D. degree of J. C. M. Silva. 379

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007 1

On the BER Performance of HierarchicalM -QAM Constellations With Diversity and

Imperfect Channel Estimation

1

2

3

Nuno M. B. Souto, Francisco A. B. Cercas, Rui Dinis, and Jo ao C. M. Silva4

Abstract —The analytical bit error rate of hierarchical quadra-5

ture amplitude modulation formats, which include uniform and6

nonuniform constellations, over at Rayleigh fading environments7

is studied in this paper. The analysis takes into account the effect8

of imperfect channel estimation and considers diversity reception9

with both independent identically and nonidentically distributed0

channels, employing maximal ratio combining.1

Index Terms —Channel estimation, diversity, quadrature ampli-2

tude modulation (QAM), Rayleigh fading.3

I. INTRODUCTION4

I N THE design of wireless communication networks, the5

limitation on spectrum resources is an important restriction6

for achieving high bit rate transmissions. The use of M -ary7

quadrature amplitude modulation ( M -QAM) is considered an8

attractive technique to overcome this restriction due to its high9

spectral efciency, and it has been studied and proposed for 0

wireless systems by several authors [1], [2].1

A great deal of attention has been devoted to obtaining an-2

alytical expressions for the bit error rate (BER) performance3

of M -QAM with imperfect channel estimation. A tight up-4

per bound on the symbol error ratio (SER) of 16-QAM with5

pilot-symbol-assisted modulation (PSAM) in Rayleigh fading6

channels was presented in [3]. An approximate expression for 7

the BER of 16-QAM and 64-QAM in at Rayleigh fading with8

imperfect channel estimates was derived in [4], while in [5],9

exact expressions were obtained for 16-QAM diversity recep-0

tion with maximal ratio combining (MRC). The method used1

in these papers can be extended to general M -QAM constel-2

lations, but the manipulations and development required can3

become quite cumbersome. Recently, exact expressions in [6],4

were published for the performance of uniform M -QAM con-5

stellations with PSAM for identical diversity channels, while6

in [7], expressions valid for nonidentically distributed channels7

Paper approved by X. Dong, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received July 13, 2005; revisedMarch 21, 2006, August 25, 2006, and March 27, 2007.

N. M. B. Souto is with the Instituto Superior T ecnico (IST)/Instituto deTelecomunicacoes, 1049 Lisbon, Portugal (e-mail: [email protected]).

F. A. B. Cercas and J. C. M. Silva are with the Instituto de Ciˆ enciasdo Trabalho e da Empresa (ISCTE)/Instituto de Telecomunicac oes, 1649-026Lisbon, Portugal (e-mail: [email protected]; [email protected]).

R. Dinis is with the Instituto Superior T´ ecnico, Instituto de Sistemas eRobotica (ISR), 1049 Lisbon, Portugal (e-mail: [email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TCOMM.2007.906367

were derived, though in this case, they were not linked to any 38

specic channel estimation method. 39

All the studies mentioned before relate to M -QAM uni- 40

form constellations that can be regarded as a subset of the 41

more general case of nonuniform M -QAM constellations (also 42

called hierarchical constellations). These constellations can 43

be used as a very simple method to provide unequal bit er- 44

ror protection and to improve the efciency and exibility 45

of a network in the case of broadcast transmissions. Nonuni- 46

form 16/64-QAM constellations have already been incorpo- 47

rated in the digital video broadcasting-terrestrial (DVB-T) 48

standard [8]. A recursive algorithm for the exact BER com- 49

putation of hierarchical M -QAM constellations in additive 50

white Gaussian noise (AWGN) and fading channels was pre- 51

sented in [9]. Later on, closed-form expressions were also ob- 52

tained for these channels [10]. As far as we know, the ana- 53

lytical BER performance of these constellations in Rayleigh 54

channels with imperfect channel estimation has not yet been 55

investigated. 56

In this paper, we adopt a different method from [4] and [5] 57

to derive general closed-form expressions for the BER perfor- 58

mance in Rayleigh fading channels of hierarchical M -QAM 59

constellations with diversity employing an MRC receiver. We 60

consider diversity reception with both independent identically 61

and nonidentically distributed channels.PSAM philosophywith 62

channel estimation that accomplished through a nite impulse 63

response (FIR) lter is assumed. 64

This paper is organized as follows. Section II describes the 65

model of the communication system, which includes the deni- 66

tion of nonuniform M -QAM constellations, thechannel, andthe 67

modeling of the channel estimation error. Section III presents 68

the derivation of the BER expressions and Section IV presents 69

some numerical and simulation results. The conclusions are 70

summarized in Section V. 71

II. SYSTEM AND CHANNEL MODEL 72

A. QAM Hierarchical Signal Constellations 73

In hierarchical constellations, there are two or more classes 74

of bits with different error protection levels and to which dif- 75

ferent streams of information can be mapped. By using nonuni- 76

formly spaced signal points (where the distances along the I - or 77

Q- axis between adjacent symbols can differ depending on their 78

positions), it is possible to modify the different error protection 79

levels. As an example, a nonuniform 16-QAM constellation is 80

shown in Fig. 1. The basic idea is that the constellation can be 81

0090-6778/$25.00 © 2007 IEEE

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Fig. 1. Nonuniform 16-QAM constellation.

viewed as a 16-QAM constellation if the channel conditions are82

good enough or as a quadrature phase-shifting keying (QPSK)83

constellation otherwise. In the latter situation, the received bit84

rate is reduced by half. This constellation can be characterized85

by the parameter k = D1 / D 2(0 < k ≤ 0.5). If k = 0 .5, the re-86

sulting constellation corresponds to a uniform 16-QAM. For 87

the general case of an M -QAM constellation, the symbols are88

dened as89

s =log 2 (√ M )

l=1±

D l

2+

log 2 (√ M )

l=1±

D l

2 j (1)

and the number of possible classes of bits with different error 90

protection is log2(M ) / 2. In the following derivation, we as-91

sume that the parallel information streams are split into two,92

so that half of each stream goes for the in-phase branch and93

the other half goes to the quadrature branch of the modula-94

tor. The resulting bit sequence for each branch is Gray coded,95

mapped to the corresponding pluggable authentication modules96

(√ M -PAM) constellation symbols, and then, grouped together,97

forming complex M -QAM symbols. The Gray encoding for 98

each (√ M -PAM) constellation is performed according to the99

procedure described in [9]. Firstly, the constellation symbols00

are represented in a horizontal axis where they are numbered01

from left to right with integers starting from 0 to √ M − 1.02

These integers are then converted into their binary representa-03

tions, so that each symbol s j can be expressed as a binary se-04

quence with log2(M )/ 2 digits: b1 j , b2

j , . . . , blog 2 M / 2 j . The corre-05

sponding Gray code [g1 j , . . . , g log 2 M / 2

j ] is then computed using06

(⊕

represents modulo-2 addition)07

g1 j = b1

j , gi j = bi

j

⊕bi−1 j , i = 2 , 3, . . . , log2

M

2 . (2)

B. Received Signal Model 108

Let us consider a transmission over an L diversity branch at 109

Rayleigh fading channel where the branches can have different 110

average powers. Assuming perfect carrier and symbol synchro- 111

nization, each received signal sample can be modeled as 112

r k = α k

·s + n k , k = 1 , . . . , L (3)

where α k is the channel coefcient for diversity branch k, s is 113

the transmitted symbol, and nk represents additive white ther- 114

mal noise. Both αk and nk are modeled as zero mean com- 115

plex Gaussian random variables with E [|α k |2] = 2σ2α k

(2σ2α k

116

is the average fading power of the kth diversity branch) and 117

E [|n k |2] = 2σ2n = N 0 (N 0 / 2 is the noise power spectral den- 118

sity). Due to the Gaussian nature of αk and nk , the probability 119

density function (pdf) of the received signal sample rk , condi- 120

tioned on the transmitted symbol s , is also Gaussian with zero 121

mean. The receiver performs MRC of the L received signals. 122

As a result of the mapping employed, the I and Q branches are 123

symmetric (the BER is the same), and so, our derivation can be 124

developed using only the decision variable for the I branch, i.e., 125

zre = Real L

k=1

r k α∗k . (4)

C. Channel Estimation 126

In this analysis, we consider a PSAM philosophy [3] where 127

the transmitted symbols are grouped in N -length frames with 128

one pilot symbol periodically inserted into the data sequence. 129

The channel estimates for the data symbols can be computed by 130

means of an interpolation with an FIR lter of length W , which 131

uses thereceivedpilot symbols of theprevious (W − 1) / 2 and 132

subsequent W / 2 frames. Several FIR lters were proposed in 133

the literature, such as the optimal Wiener lter interpolator [3], 134

the low pass sinc interpolator [11], or the low-order Gaussian in- 135

terpolator [12]. According to this channel estimation procedure, 136

the estimate α k is a zero-mean complex Gaussian variable. 137

Assuming that the channel’s corresponding autocorrelation 138

and cross-correlation functions can be expressed as in [13], the 139

variance of the channel estimate for symbol t (t = 2, . . . , N ; 140

t = 1 corresponds to the pilot symbol position) in frame u can 141

be written as 142

E

|α k ((u

− 1)N + t)

|2

= 2 σ2α k

W / 2 + u −1

j = − (W −1) / 2 + u−1

W / 2 + u −1

i = − (W −1) / 2 + u−1

× h j −u +1t hi−u +1

t J 0(2πf D |i − j |NT s )

+ 1

|S p|2N 0

W / 2 + u−1

j = − (W −1) / 2 + u −1

(h j −u +1t )2 (5)

where J 0(·) is the zeroth-order Bessel function of the rst kind, 143

f D is the Doppler frequency, h j −u +1t are the interpolation coef- 144

cients of the FIR, lter and S p is a pilot symbol. The second- 145

order moment of rk and α k , which will be required further 146

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Fig. 2. Impact in a 16-QAM constellation (upper right quadrant shown) of cross-quadrature interference originated by imperfect channel estimation. Onlythe case of phase error is shown.

ahead, is given by7

E [r k ((u − 1)N + t)α∗k ((u − 1)N + t)|s]

= 2 σ2α k s

W / 2 + u −1

j = − (W −1) / 2 + u−1

×h j −u +1

t J 0 (2πf D

|(u

− 1

− j )N + t

− 1

|T s ). (6)

III. BER PERFORMANCE ANALYSIS8

To accomplish this analysis, we start by deriving the bit error 9

probability for each type of bit im (m = 1, . . . , log2(M ) / 2) in0

a constellation. This error probability depends on the position1

t in the transmitted frame, which means that it is necessary2

to average the bit error probability over all the positions in the3

frame toobtain theoverallBER. Although theBERperformance4

of an M -QAM constellation in a Rayleigh channel with perfect5

channel estimation can be analyzed by simply reducing it to6

a √ M -PAM constellation, this simplication is not possible7

in the presence of an imperfect channel estimation. In fact,8

since the channel estimates are not perfect, a residual phase9

error will be present in the received symbols even after channel0

compensation at the receiver. This phase error adds interference1

from the quadrature components to the in-phase components2

and vice versa, as shown in Fig. 2.3

An explicit closed-form expression for the bit error probabil-4

ity of generalized nonuniform QAM constellations in AWGN5

and Rayleigh channels was derived in [10]. It is possible to6

adapt this expression for the case of imperfect channel estima-7

tion, which is a more general case where theconstellation cannot8

be simply analyzed as a PAM constellation. In this situation, as-9

suming equiprobable transmitted symbols, i.e., P (s j, f ) = 1 / M,0

the average BER can be computed as 171

P b (im ) = 1

(N −P )2

√ M

N −P

t =1

√ M / 2−1

j =0

1 −gk j + ( −1)gk

j

×

2( k −1)

l=1

(

−1)l+1 2

√ M ×

√ M / 2−1

f =0

× Prob zre < b m (l)L

k=1|α k |2 |s j, f , t . (7)

where 172

b m (l)

= d

s ((2 l−1) 21 / 2·log 2 M −m )+ ds ((2 l−1) 21 / 2·log 2 M −m +1)

2and 173

d s

( j ) =

1/ 2·log 2 M

i=1 2bi

j − 1 D 1/ 2·log 2 M −i+1 .Therefore, to calculate the analytical BER, it is necessary to ob- 174

tain an expression for Prob {zre < b m (l) Lk =1 |α k |2|s j, f , t}, 175

i.e., the probability that the decision variable is smaller 176

than the decision border b m (l) Lk=1 |α k |2 , conditioned on 177

the fact that the transmitted symbol was s j, f and its po- 178

sition in the frame is t. Note that, due to the existing 179

symmetries in the constellations, we only need to compute 180

Prob{zre < b m (l) Lk =1 |α k |2|s j, f , t} over the constellation 181

symbols in one of the quadrants. In the following derivations, 182

we will drop indexes t, i (in-phase branch label of the constella- 183

tion symbol), and f (quadrature branch label of theconstellation 184

symbol), and we replace b m (l) by w, for simplicity of notation. 185

To avoid the explicit dependency of the decision borders on the 186

channel estimate, the probability expression can be rewritten as 187188

Prob zre < wL

k=1|α k |2|s

= Prob Real L

k =1

(r k −wα k )α∗k < 0|s

= Prob(zre < 0|s) (8)

where the modied variables rk = rk

−w

· α k and z

re = 189

Lk =1 zre k =

Lk=1 Real{r k α∗k} are dened. Since r k and α k 190

are complex random Gaussian variables and w is a constant, rk 191

also has a Gaussian distribution. The second moment of rk , the 192

cross moment of r k and α k , and the respective cross-correlation 193

coefcient are given by 194

E [|r k |2

|s] = 2 |s|2 σ2α k + N 0 − 2w Re {E [r k α∗k |s]}

+ w2E [|α k |2] (9)

E [r k α∗k |s] = E [r k α∗k |s] −wE [|α k |2] (10)

µk = E [r k α∗k |s]

E [|r k |2

|s]E [|α k |2

|s]

= |µk |e−ε k · j . (11)

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We will rst compute the pdf of zre conditioned on s for each95

individual reception branch. We start by writing the pdf of zk96

(zk = rk α∗k ) conditioned on s [14]97

p(zk |s) = 1

2πσ 2α k

σ2r k

(1 − |µk |2)

× exp |µk | zre k cos εk + zim k sin εk

σr kσα k (1 − |µk |

2)

×K 0 z 2re k + z 2

im k

σr kσα k (1 − |µk |2)

(12)

where K 0 (·) denotes the modied Hankel function of order 98

zero and99

σ2r k =

E [|r k |2|s]

2 .

Integrating (12) over zim k (zim k

= Imag {zk}) yields the00

marginal pdf of the decision variable zre k01

p(zre k |s) = + ∞

−∞ p(zk |s) dzim k

= 1F k

exp[Gk zre k]exp[−H k ||zre k ||] (13)

where02

F k = 2σr k σα k

1 − |µk |2

(sin εk )2,

Gk = |µk | cos εk

σr kσα k (1 − |µk |2)

, H k = 1 − |µk |2 (sin εk )2

σr kσα k (1 − |µk |2)

.

The decision variable zre is the sum of L independent ran-03

dom variables with pdfs similar to (13). According to [15],04

the resulting pdf can be computed through the inverse Fourier 05

transform of the product of the individual characteristic func-06

tions. The characteristic function of the pdf dened in (13)07

is obtained by applying the Fourier transform, which results08

in09

Ψk (υj ) = − 2H k

F k (υj −Gk −H k ) (υj −Gk + H k ). (14)

If we divide the L diversitybranches into L different sets,where10

each set k contains ρk received branches with equal powers sat-11

isfying L = Lk =1 ρk , then the resulting characteristic function12

is given by the product of the individual characteristic functions,13

and can be written as14

Ψ (υj ) =L

k =1

−2H kF k

ρk

× 1

(υj −Gk −H k )ρk (υj −Gk + H k )ρk . (15)

The pdf of zre is obtained by decomposing (15) as a sum of 215

simple fractions (according to the method proposed in [16]) and 216

applying the inverse Fourier transform. The desired probability 217

can now be computed by integrating this pdf from −∞ to 0 as 218

follows 219

Prob (zre < 0) = 0

−∞ p (zre ) dzre =

L

k =1 −2H kF k

ρk

×L

k=1

ρk

i=1

(−1) i A1k ,i (Gk + H k )−i (16)

where 220

A1k, i =

1(ρk −i)!

∂ ρk −i

∂s ρk −i

L

j =1 j = k

1(s −G j −H j )ρ j

×L

j =1

1(s −G j + H j )ρ j

s = G k + H k. (17)

If all the diversity branches have equal powers, i.e., ρk = L, 221

then L = 1 and F k , Gk , and H k are all equal (index k can be 222

dropped) leading to223

Prob (zre < 0) = 1

(2F H )L

L

k =1

2L −k − 1L −k

2H G + H

k

(18)

which is the same result achieved in [6], though written in 224

a different format. However, if all the diversity branches are 225

different, i.e., ρk = 1 for any k and L = L, then226

A1l, 1=

L

j =1 j = l

1(Gl + H l −G j −H j )θ j ×

L

j =1

1(G l + H l−G j + H j )θ j

(19)

and 227

Prob (zre < 0) =L

k =1

−2H kF k ×

L

k=1

−A1k, 1

(Gk + H k ). (20)

228

IV. NUMERICAL RESULTS 229

To verify the validity of the derived expressions, some sim- 230

ulations were run using the Monte Carlo method. The results 231

obtained are plotted as a function of E S / N 0 (E S —symbol en- 232

ergy) in Figs. 3 and 4. Fig. 3 presents a nonuniform 16-QAM 233

(k = 0 .3) transmission with three equal branch diversity recep- 234

tion, while Fig. 4 presents a nonuniform 64-QAM ( k1 = 0 .4 235

and k2 = 0 .4) transmission with four diversity branches and 236

different relative powers ([0 dB −3 dB −6 dB −9 dB]). In 237

both cases, the frame size is N = 16 , the pilot symbols are 238

S p = 1 + j (transmitted with the same power level of the data 239

symbols),a sinc interpolator [11] is employed with W = 15 ,and 240

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Fig. 3. Analytical and simulated BER performances of nonuniform 16-QAM(k = 0 .3 ) with three equal diversity branches; f d T s = 1 .5

× 10 −2 ; sinc inter-

polation with W = 15 and N = 16 .

Fig. 4. Analytical and simulated BER performances of nonuniform 64-QAM(k1 = 0 .4 and k 2 = 0 .4 ) with four diversity branches and different relativepowers ([0 dB −3 dB −6 dB −9 dB]); f d T s = 1 .5 × 10 −2 ; sinc interpolationwith W = 15 and N = 16 .

Rayleigh fading is considered with f d T s = 1 .5 × 10−2 . The an-1

alytical results computed using expressions (7), (18), and (20),2

as well as curves corresponding to perfect channel estimation3

are drawn in both gures. In both cases, it is clear that the an-4

alytical results accurately match the simulated ones. We can5

also notice in both gures a considerable difference between6

the performance with perfect channel estimation and with im-7

perfect channel estimation, the existence of irreducible BER8

oors (more noticeable in the least protected bit since they ap-9

pear at higher BER values) being visible. This is a consequence0

of the effect of the time-varying fading channel that results1

in a reduced quality of the channel estimates. Moreover, these2

gures also show that the nonuniformity of the constellations3

clearly results in differentiated performances for thedifferentbit4

classes.5

V. CONCLUSION 256

In this paper, we have derived general analytical expressions 257

for the evaluation of the exact BER performance for the individ- 258

ualbit classes of any nonuniform square M -QAMconstellation, 259

in the presence of imperfect channel estimation. These expres- 260

sions can be applied to Rayleigh fading environments, with 261

either equal or unequal receiving diversity branches, and MRC 262

receivers. 263

ACKNOWLEDGMENT 264

The authors would like to thank the Editor and the anony- 265

mous reviewers for their helpful constructive comments which 266

improved the paper signicantly. Special thanks are due to the 267

Editor, Prof. X. Dong, for pointing out relevant reference [7] 268

of which we were not aware when we submitted the rst ver- 269

sion of the paper. This work was elaborated as a result of the 270

participation in the C-MOBILE project (IST-2005-27423). 271

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fading channels,” IEEE Trans. Commun. , vol. 45, no. 10, pp. 1218–1230, 277

Oct. 1997. 278

[3] J. K. Cavers, “An analysis of Pilot symbol assisted modulation for 279

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[5] L. Cao and N. C. Beaulieu, “Exact error-rate analysis of diversity 285

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[6] B. Xia and J. Wang, “Effect of channel-estimation error on QAM systems 288

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488, Mar. 2005. 290

[7] L. Najazadeh and C. Tellambura, “BER analysis of arbitrary QAM for 291

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[8] Digital Video Broadcasting (DVB) Framing Structure, Channel Coding 295

and Modulation for Digital Terrestrial Television (DVB-T) V1.5.1, ETSI 296

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[9] P. K. Vitthaladevuni and M.-S. Alouini, “A recursive algorithm for the 298

exact BER computation of generalized hierarchical QAM constellations,” 299

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[10] P. K. Vitthaladevuni and M.-S. Alouini, “A closed-form expression for the 301

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[11] Y.-S. Kim, C.-J. Kim, G.-Y. Jeong, Y.-J. Bang, H.-K. Park, and S. S. Choi, 304

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[13] G. L. Stuber, Principles of Mobile Communication . Norwell, MA: 311

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[15] Athanasios Papoulis , Probability, Random Variables, and Stochastic Pro- 316

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Nuno M. B. Souto received the B.Sc. degree in20

aerospace engineering—avionics branch, in 2000,21

andthe Ph.D.degreein electricalengineeringin 2006,22

both from the Instituto Superior T´ ecnico, Lisbon,23

Portugal.24

From November 2000 to January 2002, he was25

a researcher with the Instituto de Engenharia e26

Sistemas de Computadores, Lisbon, where he was27

engaged in research on automatic speech recog-28

nition. He is currently with the Instituto Supe-29

rior Tecnico (IST)/Instituto de Telecomunicacoes,30

Lisboa, Portugal. His current research interests include wideband code-division31

multiple-access (CDMA) systems, orthogonal frequency-division multiplex-32

ing (OFDM), channel coding, channel estimation, and multiple-input multiple-33

output (MIMO) systems.34

35

FranciscoA. B. Cercas receivedthe Dipl.-Ing., M.S.,36

andPh.D. degrees fromthe InstitutoSuperior T ecnico37

(IST), Technical University of Lisbon, Lisbon, Por-38

tugal, in 1983, 1989, and 1996, respectively.39He worked for the Portuguese Industry as a Re-40

search Engineer and as an invited researcher at the41

Satellite Centre of the University of Plymouth, Ply-42

mouth, U.K. He was a lecture at the IST and became43

Associate Professorin 1999 at ISCTE, Lisbon, where44

he is currently the Director of telecommunications45

and informatics. He is the author or coauthor of more46

than 60 papers published in international journals. His current research interests47

include mobile and personal communications, satellite communications, chan-48

nel coding, and ultra wideband communications.49

50

Rui Dinis received the Ph.D. degree from the Insti- 351

tuto Superior T ecnico (IST), Technical University of 352

Lisbon, Lisbon, Portugal, in 2001. 353

From 1992 to 2001, he was a member of the re- 354

search center Centro de An´alise e Processamento de 355

Sinais (CAPS/IST). During 2001, he was a Professor 356

at theIST. Since 2002, he is a member of theresearch 357

centerInstituto de Sistemas e Rob otica(ISR/IST).He 358

has been involved in several research projects in the 359

broadband wireless communications area. His cur- 360

rent research interests include modulation, equaliza- 361

tion, and channel coding. 362

363

Jo ao C. M. Silva received the B.S. degree in 364

aerospace engineering in 2000, and the Ph.D. de- 365

gree in 2006, both from the Instituto Superior 366

Tecnico (IST), Lisbon Technical University, Lisbon, 367

Portugal. 368

From 2000–2002, he worked as a business con- 369

sultant with McKinsey and Company. He was en- 370gaged in research on spread spectrum techniques, 371

multiuser detection schemes, and multiple-input 372

multiple-output systems. He is currently with the 373

Instituto de Ci encias do Trabalho e da Empresa 374

(ISCTE)/Instituto de Telecomunicac oes, Lisbon, Portugal. 375

376

8/13/2019 1780 Tcomm Souto


QUERIES 377

Q1. Author: Please provide the complete educational information (degree, etc.) of R. Dinis. 378

Q2. Author: Please provide the subject of the Ph.D. degree of J. C. M. Silva. 379

1780 Tcomm Souto

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