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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: nuno.souto@lx.it.pt).
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: francisco.cercas@lx.it.pt; joao.carlos.silva@lx.it.pt).
R. Dinis is with the Instituto Superior T´ ecnico, Instituto de Sistemas eRobotica (ISR), 1049 Lisbon, Portugal (e-mail: rdinis@ist.utl.pt).
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
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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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4 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007
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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SOUTO et al. : ON THE BER PERFORMANCE OF HIERARCHICAL M -QAM CONSTELLATIONS 5
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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[3] J. K. Cavers, “An analysis of Pilot symbol assisted modulation for 279
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[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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[6] B. Xia and J. Wang, “Effect of channel-estimation error on QAM systems 288
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[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
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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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[13] G. L. Stuber, Principles of Mobile Communication . Norwell, MA: 311
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6 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007
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: nuno.souto@lx.it.pt).
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: francisco.cercas@lx.it.pt; joao.carlos.silva@lx.it.pt).
R. Dinis is with the Instituto Superior T´ ecnico, Instituto de Sistemas eRobotica (ISR), 1049 Lisbon, Portugal (e-mail: rdinis@ist.utl.pt).
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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4 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007
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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SOUTO et al. : ON THE BER PERFORMANCE OF HIERARCHICAL M -QAM CONSTELLATIONS 5
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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6 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 10, OCTOBER 2007
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