IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
1049
Exact Evaluation of Bit- and Symbol-Error Rates for
Arbitrary 2-D Modulation and Nonuniform
Signaling in AWGN Channel
Leszek Szczecinski, Member, IEEE, Sonia Aïssa, Senior Member, IEEE, Cristian Gonzalez, and Marcos Bacic
Abstract—Exact evaluation of bit- and symbol-error rates in a
2-D constellation is a fundamental problem of digital communications, which only for particular modulations and/or bits-to-symbol
mapping has closed-form solutions. Here, we propose a general,
numerically efficient algorithmic method, which yields the exact
results for arbitrary modulation symbols’ set and arbitrary bits-tosymbol mapping, and which deals with the case of nonuniform signaling. These three conditions define any digital transmission using
memoryless modulation, so the proposed method is a general tool
solving all problems tackled in the literature under constraints imposed on one or more of the parameters defining the modulation or
signaling type. Such an evaluation tool is of practical importance
during the design of the modulation. Our analysis and numerical
simulations show the advantages offered by the new method when
compared with the bounding techniques.
Index Terms—Bit-error rate (BER), labeling, maximum a posteriori (MAP) detection, signaling, symbol-error rate (SER), uncoded transmission.
I. INTRODUCTION
E
VALUATION of bit-error rates (BERs) and symbol-error
rates (SERs) is one of the fundamental problems in digital
communications. In this paper, we address the problem of uncoded BER/SER evaluation in memoryless modulation, which
is uniquely determined by two parameters: constellation and labeling (or bits-to-symbol mapping). The constellation is a set of
complex symbols (used further to determine the amplitude and
the phase of the waveforms), and the labeling is a rule assigning
binary codeword (labels) to the symbols in the constellation.
The BER/SER depends also on the signaling, which defines
the probabilities of transmitting the particular symbol or bit. Although signaling is not, strictly speaking, related to the modulation, it must be considered during the performance evaluation.
Exact calculation of BER/SER is required when modulation
is to be designed according to the criteria depending on the
Paper approved by A. Zanella, the Editor for Wireless Systems of the
IEEE Communications Society. Manuscript received April 10, 2005; revised
August 9, 2005; October 4, 2005; and November 30, 2005. This work was
supported in part by the Government of Quebec Province under Grant FCAR
2003-NC-81788, and in part by the Government of Canada under Grant NSERC
249704-02. This paper was presented in part at the IEEE Global Communication Conference (GLOBECOM), St. Louis, MO, November 28–December 2,
2005.
L. Szczecinski and S. Aïssa are with the Institut National de la Recherche Scientifique-EMT, Montréal, QC H5A 1K6, Canada (e-mail: [email protected];
[email protected]).
C. Gonzalez and M. Bacic are with the Universidad Técnica Federico Santa
María, Department of Electronics Engineering, Valparaíso, Chile.
Digital Object Identifier 10.1109/TCOMM.2006.876853
system-level considerations. For example, in [1], the constellation set is designed to minimize the SER (note that uniform
signaling is assumed and the labeling issue is ignored). Often
the design is limited to selecting the most appropriate among
available (predefined) constellations. Therefore, a considerable
effort was deployed to evaluate the uncoded BER/SER, for the
most popular constellations, e.g., pulse or quadrature amplitude modulation (PAM/QAM) [2]–[4] or phase-shift keying
(PSK) [3], [5], [6]. The considerations are mostly limited to
the so-called Gray labeling (which minimizes the Hamming
distance between the closest constellation points) and uniform
signaling (equiprobable symbols in the constellation). These
assumptions are quite practical, because the Gray labeling is
often used (it may be proven to minimize the probability of
error in QAM and PSK [7]), and the uniform signaling is a
reasonable assumption if appropriate source coding is used.
On the other hand, the design of the labeling may be required
when iterative (turbo) decoding is targeted [8] or when signaling
is nonuniform (e.g., due to residual redundancy after source
coding) [9]; the latter problem may be also addressed through
joint design of constellation and labeling [10].
Because the design involves costly iterative and/or combinatory searches with multiple (counted in millions [8], [9]) evaluations of various constellations and/or mappings, closed-form or
algorithmic estimations of BER/SER, eliminating the need for
time-consuming simulations, are very useful. However, no efficient method has been proposed up to now to evaluate exactly
BER/SER when the modulation’s parameters and the signaling
are arbitrarily set. Lack of such a method may be only partially
palliated with bounding techniques [1], [8] because they lose
precision for low values of signal-to-noise ratio (SNR) or, equivalently, for a high value of BER/SER. Note that this region of
BER values is very interesting due to the increasing popularity
of strong error-correcting codes (e.g., turbo codes [11]) which
cope very well with poor-quality (i.e., high BER) input data.
The numerically efficient exact calculation of SER in an arbitrary constellation was shown first in [12], where geometric
considerations led to a simple numerical integrations over decision regions containing the constellation symbols, but the issues
of labeling and signaling were ignored therein.
In this paper, we propose a new method to exactly evaluate
the uncoded BER/SER in arbitrary 2-D constellations (complex baseband symbols) with arbitrary labeling and arbitrary
signaling. As in [12], we also use a geometric approach; however, we allow for arbitrary labeling and nonuniform signaling.
The latter requires problem reformulation because, unlike the
assumption in [12], the symbols do not necessarily belong to
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their respective decision regions [13]. The work of [12] was also
generalized in [14] to the arbitrary constellation case.
The main difference with respect to [12] and [14] is that:
1) trigonometric considerations necessary in [12] and [14] are
avoided; 2) the well-known bivariate Gaussian (BVG) cumulative distribution function (CDF) available in the public domain
and optimized for execution speed is used, so there is no need
for custom numerical integrations from [12] and [14]; and 3) algorithmic steps leading to the required solution are formalized,
i.e., the evaluation procedure is easy to implement.
All the problems of performance evaluation tackled in the
literature under constraints imposed on the constellation, the
labeling, or the signaling may be solved by our method. Its
merit is, therefore, due to absolutely general and yet efficient
numerical solution. Of course the methods developed up to now
(e.g., for QAM or PSK modulations with Gray mapping) are
useful, since they offer closed-form formulas simpler than our
approach.
It is worthwhile to mention that recent bounding techniques
[9], [13] also address the issue of arbitrary constellation
and nonuniform signaling. The advantage of the bounding
techniques is that providing the results without geometric considerations, they may be considered closed-form solutions, in
contrast to the algorithmic approach we propose. In this paper,
we will use the so-called Kuai–Alajaji–Takahara (KAT) bound
[13] to compare it with our method from the point of view of the
implementation complexity and the exactitude of the results.
Other bounds shown in [13] provide much better results, but,
like our method, are algorithmic in nature. Therefore, it is
difficult to reasonably compare them with our method from the
point of view of the implementation complexity.
The paper is organized as follows. In Section II, we introduce the notation and present the data model. In Section III, the
problem is defined and the main algorithmic steps leading to the
solution are outlined. In Section IV, to illustrate the usefulness
of the proposed approach, we contrast it with known expressions, and compare it with the bounding techniques from the
implementation complexity point of view, as well as analyzing
the accuracy of performance evaluation. Conclusions are drawn
in Section V.
II. SYSTEM MODEL
Consider the system where bits
gathered in codewords
of length ,
are
via a memoryless and arbitrary
transformed into symbols
, so that
, where and
mapper
denote discrete times defined for bits and symbols, respectively;
is the set of all codewords,
,
is the modulation constellation, i.e.,
and
.
of generating
A priori probabilities
codewords
are assumed known and are, in general, not equal.
,
The transmission outcome is given by
is a zero-mean, complex, white Gaussian noise with
where
variance . Constellations considered here are zero mean (i.e.,
) and power-normalized (i.e.,
);
such normalization is not a must, but it allows us to conveniently
define the signal-to-noise ratio (SNR) per symbol and per bit,
and SNR
SNR
.
respectively, as SNR
, the receiver takes decision in
Given the observation
favor of the codeword , labeling the constellation with the
which
highest a posteriori probability
may be rewritten employing Bayes’ rule
The transmitted symbols are estimated as
Equation (1) may be also written as
if
(1)
.
(2)
where
symbol
is the decision region corresponding to the
. Knowing that
is Gaussian, i.e.,
, the region is defined as [15, Ch. 2.2]
(3)
where
denotes conjugation,
,
is the
real part, and
is the linear form of
with coefficients
and
. Thus, the decision region
is a convex polygon debetween symbols
fined through the decision lines
and the symbol .
Although, in a strict sense, (1) is the maximum a posteriori
(MAP) detection of the symbols, it is also the approximate MAP
detection of the bits when the so-called max-log simplification
is used [16].
(i.e., without
In the following, we will use the symbol
argument ) to denote a set of parameters
. For ex, which
ample, we may write
is a halfmeans that the region (in the complex plane)
. For convenience, we also
plane limited by the line
define the notation
.
III. BER/SER EVALUATION
Errors occur during a transmission if the received signal
falls into the region
while sending the codeword
, i.e.,
,
. The number of bits in error due to this
between
and
event equals the Hamming distance
. Therefore, averaging over the possible transmission of codewords gives the following expression for the average BER
[17]:
BER
(4)
Removing
from (4) results in the required expression for SER, because each erroneous decision provokes
SZCZECINSKI et al.: EVALUATION OF BER/SER FOR 2-D MODULATION AND NONUNIFORM SIGNALING
1051
one symbol error. Therefore, SER and BER are easily found if
may be evaluated, which is the focus
of our development in what follows.
For completeness, we note that a much simpler expression for
the SER may be obtained as follows [12]:
SER
(5)
Noting that
, the probability of
and detecting
is bounded by the probability of
sending
crossing the decision line
separating
from
Fig. 1. Wedges
(L ; L ) (shaded) and
(L ; L ) (hashed) in the
complex plane defined by the lines L (r ) = 0 and L (r ) = 0.
transforming them to be of zero mean, unitary variance, and considering their mutual correlation gives the expression for (11)
[19]
(12)
(6)
where
where
.
Therefore, the upper bound (UB) for the BER is given by
(13)
UB
(7)
and for the SER by
UB
(8)
Note that for uniform signaling (i.e.,
), the argument of the function
in (8) is
, i.e., (8) becomes the familiar union bound
[18, Ch. 4.3.2]. Therefore, we will use this name to denote both
(7) and (8).
Once the decision regions are found (cf. Section III-A),
an efficient and simple method to calculate the probability
is required. With this objective in
mind, we propose to represent the decision plane as a union of
the decision region and disjoint infinite wedges
defined through an intersection of two half-planes
(9)
is the CDF of a BVG random variable efficiently implementable
using Gauss–Legendre integration, e.g., [20].
Consider now the decision region
being a polygon
defined by
lines
,
. The ordering ,
is taken so that the
are enumerated counterclockpolygon’s sides related to
wise, cf. Fig. 2(a). Through simple geometrical considerations,
we may write
(14)
where denotes union of sets.
Since all sets on the left-hand side (LHS) of (14) are disjoint
[as they are also for the right-hand side (RHS) of (14)], the probability of falling into the union of sets defined by RHS/LHS
of (14) is the sum of probabilities of falling into each of its
or the polygon ). Then,
constituents sets (i.e., the wedges
falling into the
applying (12) in (14), the probability of
polygon may be found as
as shown in Fig. 1, where denotes the intersection of sets. In
a similar way, we may write
(10)
which is a consequence of the definition of
, cf. Section II.
Such a “complementary” wedge is shown in Fig. 1.
falling into the wedge is given by
The probability of
(15)
change sign in the calculus of
Note that coefficients
in (12).
For a decision region which extends to infinity, denoted as
, cf. Fig. 2(b), we may write
(11)
Because
linear forms
is Gaussian (conditioned on
), its
and
are also Gaussian, and
(16)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 2. Example of decision regions. (a) Polygon (L
;...;L
) shown shaded, the wedges
(L
;L
),
(L
;L
),
(L
;L
), and
(L
;L
) are shown as arcs. (b) “Infinite” polygon
(L
;L
;L
). In this case, only two wedges identical to those shown in (a) need to be
shown. (c) Infinite stripe defined by lines L
(r ) = 0, L
(r ) = 0 .
So, the probability of
falling into such an infinite set,
, may be obtained straightforwardly from (16) and (12).
For completeness, we consider also a “degenerate” case when
the decision region is an infinite stripe, as shown in Fig. 2(c),
and
. Such a case may
defined by
be found, e.g., in 1-D constellations. As both lines are parallel,
, so the
the bivariate CDF cannot be calculated because
expression (13) is not defined. Then, the probability of falling
into such region is given by
(17)
Using the above developments, we are ready to evaluate the
BER/SER; however, first the decision regions must be found.
let us reformulate the
linear inequalities
defining the decision region
, cf. (3), as
(18)
where
,
the vector transpose, and
denotes the imaginary part,
is
(19)
to be closed, i.e.,
In what follows, we require the region
are not allowed. To satisfy this require“infinite” polygons
ment, artificial constraints forming a square with arbitrarily
may be added
large dimensions
A. Finding Decision Regions
The problem of finding the polygon’s sides may be efficiently
solved using methods of computational geometry. To this end,
(20)
SZCZECINSKI et al.: EVALUATION OF BER/SER FOR 2-D MODULATION AND NONUNIFORM SIGNALING
1053
where
is a
matrix of ones. These artificial constraints
are kept track of, and may be removed from the final solution.
So, finally, the problem
(21)
and
should be
with
solved, where “;” denotes vertical concatenation of matrices (in
MATLAB-style notation).
Removing the redundant inequalities in (18), which is equivalent to finding the polygon’s sides, is now formulated as a
“classical” problem of computational geometry. To solve it efficiently, first we translate the coordinates
(22)
satisfies all the inequalities in
so that the new origin
(22), i.e., we find such that
is strictly positive (element-wise). Although various methods, cf. [21], may
be used to find a feasible solution of such inequalities [21],
here we opt for simplicity, and define the following constrained
optimization problem:
(23)
which may be solved efficiently using linear programming tools
(available in the public domain). If the solution of (23) does not
exist, it means that the inequalities (18) are contradictory and
is empty.
the set
Next, normalizing (22)
Fig. 3. Symbols and the corresponding decision regions for the QAM used in
the example in Section IV-A.
A. Comparison With Known Expressions
Since we claim our method to be applicable in any constellation and mapping, but we use BVG CDF combined with simple
but not trivial geometric considerations, it is illustrative to show
that the results produced by our method will reduce to the wellknown closed-form expressions (provided the latter exist).
The simple case to consider is 4-QAM with uniform signaling
,
and Gray mapping, shown in Fig. 3 with
,
,
,
,
.
Then, BER is given exactly by [24, Ch. 5.2.9]
BER
(25)
Now, consider using our method to evaluate the BER. Due to
symmetry, we do not need to consider the outer sum in (4), and
we can write
(24)
, we can exploit the duality between the line
where
as points
and points description [22, Ch. 8.2], i.e., we treat
in 2-D space. Then, removing redundant inequalities from (24)
, i.e.,
is equivalent to finding the convex hull for the points
finding the minimum convex region enclosing all the points [22,
Ch. 1.1], [23, Sec. 2.10]. This can be done very efficiently using
the algorithms available, e.g., in MATLAB or Mathematica.
Finally, if one wants to graphically represent the decision regions (as we did in the next section for illustration purposes
only), the vertices of the polygons have to be found, as well. This
vertex enumeration problem is also treated in computational geometry, e.g., [23, Sec. 2.12], and here we obtain it as a byproduct
of finding the convex hull.
IV. EXAMPLES
This section shows particular features of the proposed
method. We contrast it with the well-known closed-form formulas, compare numerical results obtained with those yielded
by means of alternative techniques, and we provide an insight
into the complexity of the implementation.
BER
(26)
and because
,
, and
, cf. Fig. 3
BER
(27)
We can easily verify that
,
, and
, where we used
, cf. (13).1 Finally,
the relationship
and putting in (27), the results for
using
we just obtained yields the exact expression (25).
B. Numerical Results
We compare the developed expression with the results obtained through numerical simulations of the digital transmission
of
symbols
. As the alternative evaluation tool, we use
the union bounds (7) and (8), as well as the so-called KAT bound
1In
fact, these results may be obtained by inspection of Fig. 3.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 5. Decision regions (vertices marked as hollow squares) obtained for
= 0:2. Constellation
SNR = 4 dB in nonuniform signaling with P
symbols are shown with filled circles, labeling is the same as in Fig. 4.
0
Fig. 4. Uniform signaling case. (a) Constellation symbols are shown with filled
circles, labeling is given in square brackets, and the decision regions’ vertices are
marked as hollow squares. (b) BER/SER results obtained for uniform signaling
through the proposed method (exact), simulations, union bounds (7) and (8),
and the KAT bound.
[13], which is a tight lower bound for the SER. The KAT bound
becomes a very loose upper bound for the BER (looser than the
union bound), so we do not show it in the figures (it was not
shown either in [13]).
As an example we consider constellation
with
symbols,
shown in Fig. 4(a). The set is known to be a minimum energy
eight-point constellation maximizing the minimum squared Euclidean distance between constellation points [24, Ch. 5.2.9].
Parameters and are calculated to normalize the constellation power and to remove the mean, cf. Section II. The labeling is shown in binary format, and the symbols/codeword
are indexed according to the labeling, i.e.,
, where
is the th bit of the codeword
.
The decision regions shown in Fig. 4(a) correspond to uni,
), so they
form signaling (i.e.,
, because
in (4).
are independent of the noise level
The comparison between the estimated and simulated
BER/SER is shown in Fig. 4(b). The proposed expressions
perfectly match the results of simulations for all ranges of
considered SNR. The union bound overestimates significantly
the results for low SNR, while, as expected, it becomes tight for
high SNR. For low SNR, the KAT bound offers a much better
fit than the union bound.
Next, we consider nonuniform signaling defined through unand
equal probabilities of sending ones
so, assuming independence between bits
zeros
, the symbol probability is calculated as
;
. The constants
for the sake of example, we chose
and are appropriately calculated so the constellation is not centered on the origin, as shown in Fig. 5, where we may observe
also the form of the decision regions defined for SNR
dB. Note that unlike in uniform signaling, the terms
are
different from zero, so the regions’ shapes depend not only on
the constellation but also on the value of SNR. We may see
that the symmetry of regions forms is lost and their shapes are
irregular.
Observe as well that: 1) a symbol does not need to belong to
belongs
its own decision regions [in fact, only the symbol
to its decision region ]; 2) a region may contain more than
one symbol; and 3) the region may be an empty set (e.g.,
), which means that its corresponding symbol will never be
detected. In fact, 3) is a consequence of 2). The situation 3)
when the decisions are
happens for significant noise level
strongly affected by the a priori information. In the example,
of generating the symbol is small, while
is
probability
becomes the largest decision region “taking over”
large, so
the region .
We do not show the results, but again, the perfect match between the results of simulations and those yielded by the proposed method was observed.
For more numerical results obtained in different constellations, mappings, and signaling scenarios, we refer the reader to
[25], which is also accompanied by the MATLAB programs implementing the proposed method in a fully automated manner,
based solely on the model defined in Section II.
SZCZECINSKI et al.: EVALUATION OF BER/SER FOR 2-D MODULATION AND NONUNIFORM SIGNALING
TABLE I
COMPLEXITY MEASURED BY THE NUMBER OF NUMERICAL INTEGRATIONS
REQUIRED BY THE PROPOSED METHOD, UNION BOUND, AND THE
KAT BOUND; THE COMPLEXITY OF THE LATTER CORRESPONDS
TO COMPUTATION OF THE SER
C. Implementation Complexity
To make a comparison between the implementation complexity of the methods for evaluation of the BER/SER, a
reference setup must be adopted; namely, we propose to:
• ignore the complexity of the geometrical considerations required to determine the decision regions, cf. Section III-A;
• consider only uniform signaling, as then the form of the
decision regions does not depend on the SNR;
• compare only the numbers of integrals which have to be
function and BVG CDF
evaluated; this includes the
(note that both are based on 1-D numerical integration);
• do not take into account the symmetry of the constellation.
Using the above simplifications, in Table I, we compare the
implementation complexity of the proposed method, the union
bound, and the KAT bound for -ary QAM.
function
times,
The union bound calls the
while the KAT bound needs
evaluations of
function and/or BVG CDF.
We can observe that the increase in the number of numerical integrations required by our method is less than quadrupled
integrations
when compared with the union bound, i.e.,
are required. This is because the decision regions are squares,
half-infinite stripes, or quarter-planes. On the other hand, the
integrations, i.e., is much more
KAT bound requires
complex to implement, particularly for large constellations.
Considering -ary PSK modulation with uniform signaling,
we note that the decision regions are wedges, so only one BVG
CDF is needed to calculate the probability of falling into each
of them. As a result, our method has the same complexity as the
union bound; this is quite notable!
And finally, let us comment on the complexity of finding the
decision regions.
It is known that finding the convex hull with the Quick Hull al[22, Ch. 1.1]; basic
gorithm has the complexity
arithmetic operations are necessary, which can be roughly compared with the complexity of the numerical integration evaluated above. Finding all the decision zones will have the com. Thus, for
of practical interest,
plexity of
, the complexity of finding the decision zones is
e.g.,
comparable to the complexity of numerical integration, which,
in turn, is reasonably close to that of the union bound, as shown
above in the particular case of -QAM.
V. CONCLUSIONS
This paper presents a method for exact calculation of the uncoded BER/SER in a 2-D (complex) constellation, based on the
decomposition of the observation space into decision regions
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with polygonal shapes over which integration must be carried
out. Two separate problems are solved. The first is related to
defining the decision regions, and the second, to finding the numerical integrals. We explain how to efficiently solve both of
them.
A comparison with the competing bounding techniques is
given, showing that the proposed method is not only exact,
but also computationally efficient. Since the new method deals
with any constellation, labeling, and allows for nonuniform
signaling, it is a perfect tool to solve in a uniform manner all the
problems addressed up to now in the literature for evaluation
of the uncoded BER/SER.
ACKNOWLEDGMENT
The authors thank Prof. D. Avis (McGill University, Canada)
for the useful insight into the problems of computational geometry, Mr. R. Bettancourt (UTFSM, Chile) for numerical implementation of the KAT bound and the procedures described in
Section III-A, and the anonymous reviewers for their suggestions which helped the authors improve the paper.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Sonia Aïssa (S’93–M’00–SM’03) received the Ph.D.
degree in electrical and computer engineering from
McGill University, Montreal, QC, Canada, in 1998.
She is currently an Associate Professor with the
Institut National de la Recherche Scientifique-EMT,
University of Quebec, Montreal, QC, Canada, and
Adjunct Professor with Concordia University, Montreal, QC, Canada. From 1996 to 1997, she was a
Visiting Researcher at the Department of Electronics
and Communications, Kyoto University, Kyoto,
Japan, and at the wireless systems laboratories of
NTT, Kanagawa, Japan. From 1998 to 2000, she was a Research Associate
at INRS-Telecommunications, Montreal, QC, Canada. From 2000 to 2002,
she was a Principal Investigator in the major program of personal and mobile
communications of the Canadian Institute for Telecommunications Research,
conducting research in radio resource management in CDMA systems. Her
research interest includes radio resource management, cross-layer design for
wireless networks, and MIMO systems.
Dr. Aïssa is currently serving as Editor for the IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS, Associate Editor for the IEEE Communications
Magazine, and Technical Editor for the IEEE Wireless Communications
Magazine. She served as Guest Editor for the 2006 EURASIP Journal on
Wireless Communications and Networking Special Issue on Radio Resource
Management in 3G+ Systems. She is the Chair of the Montreal Chapter of the
IEEE Women In Engineering Society, and Co-Chair of the IEEE Wireless Communication Symposium of the International Conference on Communications
2006. She also holds the Quebec government FQRNT fellowship “Strategic
Program for Professors-Researchers” at the Institut National de la Recherche
Scientifique-EMT
Cristian Gonzalez obtained the B.Sc. and M.Sc. degrees in electronics engineering from the Universidad
Técnica Federico Santa María, Valparaiso, Chile, in
2005.
Between 2003–2004, he spent 14 months of internship with the Institut National de la Recherche
Scientifique-Énergie, Matériaux et Télécommunications, Montreal, QC, Canada.
Leszek Szczecinski (M’98) received the M.Eng.
degree from the Technical University of Warsaw,
Warsaw, Poland, in 1992, and the Ph.D. degree from
INRS-Telecommunications, Montreal, QC, Canada,
in 1997.
From 1998 to 2000, he was with the Department
of Electrical Engineering, University of Chile, Santiago, Chile. Since 2001, he has been an Assistant
Professor with the Institut National de la Recherche
Scientifique-EMT, Montreal, QC, Canada. His
research interests are in the area of digital signal
processing for wireless communications, with emphasis on iterative processing.
Marcos Bacic obtained the B.Sc. and M.Sc. degrees
in electronics engineering from the Universidad
Técnica Federico Santa María, Valparaiso, Chile, in
2005.
From 2003 to 2004, he participated in a research
project with the Institut National de la Recherche
Scientifique-Énergie, Matériaux et Télécommunications, Montreal, QC, Canada. His research interests
are in the area of wireless and mobile communications.