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Blind Recognition of Linear Space–Time Block Codes:
A Likelihood-Based Approach
Vincent V. Choqueuse, Mélanie Marazin, Ludovic Collin, Ko Clément Yao,
Gilles Burel
To cite this version:
Vincent V. Choqueuse, Mélanie Marazin, Ludovic Collin, Ko Clément Yao, Gilles Burel. Blind
Recognition of Linear Space–Time Block Codes: A Likelihood-Based Approach. IEEE Transactions
on Signal Processing, Institute of Electrical and Electronics Engineers, 2010, 58 (3), pp.1290 - 1299.
�10.1109/TSP.2009.2036062�. �hal-00460048�
1290 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010
Blind Recognition of Linear Space–Time Block
Codes: A Likelihood-Based Approach
Vincent Choqueuse, Member, IEEE, Mélanie Marazin, Ludovic Collin, Koffi Clément Yao, and
Gilles Burel, Senior Member, IEEE
Abstract—Blind recognition of communication parameters
is a research topic of high importance for both military and
civilian communication systems. Numerous studies about carrier
frequency estimation, modulation recognition as well as channel
identification are available in literature. This paper deals with the
blind recognition of the space–time block coding (STBC) scheme
used in multiple-input–multiple-output (MIMO) communication
systems. Assuming there is perfect synchronization at the receiver
side, this paper proposes three maximum-likelihood (ML)-based
approaches for STBC classification: the optimal classifier, the
second-order statistic (SOS) classifier, and the code parameter
(CP) classifier. While the optimal and the SOS approaches require
ideal conditions, the CP classifier is well suited for the blind
context where the communication parameters are unknown at
the receiver side. Our simulations show that this blind classifier
is more easily implemented and yields better performance than
those available in literature.
Index Terms—Maximum-likelihood (ML) detection, mul-
tiple-input–multiple-output (MIMO) systems, space–time block
code (STBC).
I. INTRODUCTION
B
LIND recognition of communication parameters is an
intermediate step between signal detection and signal de-
coding/demodulation. In civilian applications, blind recognition
algorithms are used in software-defined radio (SDR) to cope
with a large panel of communication systems. In electronic
warfare, these algorithms are required for signal interception
and processing, two tasks of key importance in tactical oper-
ations. Usually, the largest part of the algorithms is devoted
to the blind recognition of single-input–single-output (SISO)
communication parameters. Other investigations have dealt
with the development of new technologies aimed at enhancing
the reliability of data transmission in wireless communication
systems. Among them, one of the most promising technologies
is based on the use of multiple-input–multiple-output (MIMO)
Manuscript received February 16, 2009; accepted September 30, 2009. First
published November 06, 2009; current version published February 10, 2010.
The associate editor coordinating the review of this manuscript and approving
it for publication was Prof. Roberto Lopez-Valcarce.
V. Choqueuse was with Lab-STICC, Université de Brest, 29238 Brest Cedex
3, France. He is now with LBMS, UMR CNRS 3192, ISSTB, 29238 Brest Cedex
3, France (e-mail: vincent.choqueuse@gmail.com).
M. Marazin, L. Collin, K. Yao, and G. Burel are with the Laboratory for Sci-
ences and Technologies of Information, Communication and Knowledge (Lab-
STICC—UMR CNRS 3192), Université de Brest, 29238 Brest Cedex 3, France
(e-mail: melanie.marazin@univ-brest.fr; Ludovic.Collin@univ-brest.fr; Koffi-
Clement.Yao@univ-brest.fr; Gilles.Burel@univ-brest.fr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2009.2036062
systems [1] in conjunction with space–time block coding
(STBC) [2]–[4]. The principle of STBC is to take advantage
of channel diversity through a proper encoding of the data
streams into structured blocks. MIMO-STBC communication
systems have been recently standardized in IEEE 802.16e and
IEEE 802.11n and appear as an ideal technology for the next
generation of wireless products.
The development of blind receivers coping with MIMO com-
munication systems is a new challenge. The blind decoding of
the received symbols requires the estimation of several param-
eters like the number of transmitter antennas, the space–time
coding, the modulation, and the channel. Some investigations
have been focused on the blind estimation of either the number
of transmitters [5], [6] or the modulation [7]. Furthermore, other
methods devoted to the channel estimation in space–time coding
systems are also available in [8]–[14]. However, these algo-
rithms require the knowledge of the space–time coding at the
receiver side. The blind STBC recognition is an emerging topic,
which has been recently addressed in [15]–[18]. These feature-
based methods exploit the space–time redundancy of the re-
ceived samples to discriminate between several STBCs. The re-
dundancy is measured through space–time correlations and the
automatic classification is performed through hypothesis testing
[15]–[17] or by minimizing a cost function [18]. In the study re-
ported here, the maximum-likelihood (ML) approach is applied
for STBC recognition. The ML approach is a powerful classi-
fication tool, which has been previously employed for modula-
tion–recognition problems [19]–[21].
The paper is organized as follows. Section II introduces the
signal models and the assumptions made in the study. Section III
describes the optimal classifier in the Bayesian sense. Section IV
presents a low-complexity second-order statistic classifier (SOS)
based on the relaxation of the finite alphabet property of sources.
Finally, Section V describes a simple code parameter (CP) clas-
sifier, which can be employed to discriminate between several
STBCs with different code rate or code length. Finally, the
performances of the three methods are reported in Section VI.
II. S
IGNAL MODELS AND
ASSUMPTIONS
A. Signal Model of Linear STBC
Let us consider a linear STBC
that transmits sym-
bols during
time slots and uses antennas at the transmitter
side. The space–time block encoder generates an
block
matrix, denoted by
, from a block of symbols denoted
. The general expression of is
(1)
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CHOQUEUSE et al.: BLIND RECOGNITION OF LINEAR SPACE–TIME BLOCK CODES: A LIKELIHOOD-BASED APPROACH 1291
where the vector corresponds to
the vertical concatenation of the real and imaginary parts of
.
The
matrices correspond to the coding
matrices and depend only on the STBC employed at the trans-
mitter side. The class of STBC is large and contains, for ex-
ample, the spatial multiplexing, the linear dispersion codes [22],
the quasi-orthogonal codes (QOSTBC) [23], and the orthogonal
codes (OSTBC) [24], [25].
1) Example—Spatial Multiplexing: The
called spatial multiplexing
1
encodes a block of
symbols into
an
matrix as
.
.
.
(2)
The coding matrix
is given by
(3)
where
is the identity matrix of size .
2) Example—Alamouti Code: The STBC
called
Alamouti code [24] encodes a block of two symbols into a
matrix as
(4)
The two coding matrices
and are written as
(5)
(6)
B. Signal Model of the Communication
In this paper, the receiver is assumed to be perfectly synchro-
nized with the transmitter, i.e., one sample per symbol and an
optimum sampling time. In a noncooperative context, the syn-
chronization can be achieved through a symbol-timing estimator
based on the squaring algorithm [26] or by exploiting the cyclic
correlations of the received samples [27].
Let us assume a quasi-static frequency-flat MIMO channel
modeled by an
matrix where corresponds to the
number of receiver antennas. By assuming a perfect synchro-
nization, the
th received block is expressed as
(7)
where the
matrix refers to
the additive noise. By stacking the real and imaginary parts of
the data, one obtains
(8)
where the
matrix is given by
(9)
1
In the strict sense, the spatial multiplexing is not a space–time code since it
does not achieve space–time diversity.
In this paper, the received blocks are represented as column vec-
tors. Let us denote by
and the column vectors of size ,
which are defined as
(10)
where
represents vectorization. Under these notations,
(8) can be expressed as (see [28])
(11)
where
corresponds to the Kronecker product. Finally, the use
of (11) and (1) leads to
(12)
where the
matrix depends only on the STBC
and is defined as
.
.
.
(13)
C. Main Assumptions
In this study, the following conditions are assumed to hold.
AS1) The
equivalent channel is of
full-column rank.
AS2) The noise vector
is a complex, stationary, and er-
godic Gaussian vector process, independent of the sig-
nals, with zero mean and covariance matrix given by
, i.e., where de-
notes a complex multivariate Gaussian distribution with
mean
and covariance .
AS3) The transmitted symbols
are independent and iden-
tically distributed (i.i.d.) and belong to a constellation
composed of states. The average symbol energy
is normalized
2
to 1 and the real and imaginary
parts of
are assumed to be uncorrelated with variance
.
AS4) The receiver intercepts a whole number
of
space–time blocks
, i.e., the
first and last intercepted samples correspond to the start
and the end of a space–time block, respectively.
In our study, the assumption AS1) is only required when the
channel and noise power are unknown. As
for most of the well-designed STBCs,
3
this assumption holds if
.If , the inequality is
a necessary condition to meet the requirement AS1). The condi-
tion AS2) is usual and AS3) holds for most of the complex con-
stellations including
4 phase-shift keying (PSK) and square
2
Without loss of generality, the average symbol energy can be absorbed into
the channel matrix
H
.
3
If
M
fails to have full-column rank, the rank deficiency represents the
number of wasted transmit diversity degrees. Such a code design is undesir-
able in practice [10].
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1292 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010
quadrature amplitude modulation (QAM). Finally, AS4) is as-
sumed to allow simplifications of the following mathematical
expressions. However, extensions of the proposed methods can
be easily obtained when AS4) does not hold.
D. STBC Recognition With Likelihood Ratio Tests
The study deals with the recognition of the STBC in order
to answer the following question: Given a measurement of
received column vectors , how can one find the STBC of
the transmitted signals among a set
of STBC candidates?
In the ML-based approach, the problem is viewed as a mul-
tiple-hypothesis problem. Under the assumption that the a priori
probabilities of the STBC candidates are equal, the recognized
STBC,
is the one that maximizes the log-likelihood function,
i.e.,
(14)
where
is the log-likelihood function of con-
ditioned on the STBC
and on the communication parameters
. Given that the conditions AS3) and AS4) hold, the log-like-
lihood function can be expressed as
(15)
where
is the total number of received blocks under
the assumption
and is the log-likelihood
function of the received block
.
III. T
HE OPTIMAL
CLASSIFIER
In this section, the optimal ML-based approach for modula-
tion–classification [20], [21] is adapted to the STBC-recognition
problem. If the channel matrix
, the modulation , and the
noise power
are perfectly known at the receiver side, then
this method provides an upper bound on the performances of
any classifier.
A. Likelihood Function
Let us model the unknown transmitted symbols
as random
variables with a probability density function (pdf) equal to
.
The likelihood function
is determined by
averaging the conditional likelihood function
with respect to the prior distribution of so that
(16)
The
elements of the vector belong to a discrete alphabet
composed of states [assumption AS3)]. As these elements
are i.i.d., one obtains
for and
elsewhere. The use of these equalities in (16) leads to
(17)
where
is the likelihood function of condi-
tioned on
, , , and . In the case of Gaussian distributed
noise [assumption AS2)], one gets
(18)
where
is the Frobenius norm. Finally, the STBC chosen
is the one that maximizes the function
over the set .
B. Discussion
One should note that, though this classifier maximizes the av-
erage probability of correct classification, it has several draw-
backs. First, the log-likelihood function is computation time
consuming. Let us denote by
the complexity of the el-
ementary operations. It can be shown that the computation of
the likelihood function has complexity
.
Therefore, the optimal classifier is computationally too complex
in the case of high-order modulation
and/or a large number
of symbols per block
[21]. Furthermore, the computation of
requires the knowledge of several pa-
rameters, which are usually unknown in a noncooperative en-
vironment. When these parameters are unknown, this classifier
is impractical since the maximization of the likelihood function
with respect to
, , , and is computationally cost pro-
hibitive.
IV. T
HE SOS-STBC C
LASSIFIER
The i.i.d. assumption AS3) and the signal model in (12) both
show that the received vector
is the sum of i.i.d. random
variables plus the additive noise. According to the central
limit theorem, the distribution of
can be approximated by
a Gaussian distribution for
whatever the modulation
[13]. Although this approximation is only strictly correct in the
asymptotic case, a recent study [29] showed that the Gaussian
approximation provides the optimum second-order solution
when: 1) the SNR is very low or 2) the symbols belong to a
multilevel constellation. These considerations lead us to relax
the finite alphabet constraint of the sources, which are modeled
as i.i.d. Gaussian variables. One should note that this relaxation
has been previously used with success in SOS-based channel
estimation problems [11], [13]. In this study, this relaxation is
employed to propose an SOS-STBC classifier.
A. Likelihood Function
As the additive noise
and the transmitted symbols have
zero mean,
. Furthermore as AS3) holds, one obtains
(19)
As
[assumption AS2)], it can be shown
that
(20)
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CHOQUEUSE et al.: BLIND RECOGNITION OF LINEAR SPACE–TIME BLOCK CODES: A LIKELIHOOD-BASED APPROACH 1293
From (12), (19), and (20), the covariance matrix of the
received samples can be expressed as
(21)
By using the Gaussian approximation,
where corresponds to a multivariate Gaussian distribution.
Therefore, the likelihood function of
conditioned on the co-
variance matrix
is expressed as
(22)
where
corresponds to the matrix determinant. From (22) and
(15), the likelihood function of
is equal to
(23)
Given that
(24)
where
is the trace function, it follows that
(25)
where
is the estimated covariance matrix
(26)
Finally, the chosen STBC
is the one that maximizes the func-
tion
over the set .
B. Discussion
One should note that the SOS-STBC classifier has several
advantages over the optimal one: indeed, the knowledge of the
modulation being not a prerequisite, it is more easily imple-
mented since the likelihood function solely depends on SOS.
Furthermore, the SOS-STBC classifier can be extended to the
blind context as discussed below.
In the blind context, the log-likelihood function in (25) cannot
be assessed directly since the covariance matrix
is un-
known at the receiver side. However, the log-likelihood function
can be approximated by using the procedure proposed hereafter.
First,
is assumed to be true and the unknown parameters
and are estimated. Then, these estimates, denoted and ,
are used to compute
. Finally, the STBC chosen is the
one that maximizes the function
. This pro-
cedure is sometimes called hybrid-likelihood ratio test (HLRT)
[21]. To obtain the estimate of
and , several techniques
can be employed. If
is an OSTBC, the Kullback-matching
approach in [13] leads to the optimal ML joint estimate of the
channel and noise variance. In the general STBC context, the
joint estimation is a challenging problem and a suboptimal al-
ternative relies on the separate estimation of
and . Let us
introduce the following proposition.
Proposition 1: Let us denote the eigenvalues of
by
. The smallest eigenvalues of
are all equal to , i.e.,
(27)
Proof: On condition that AS1), AS3), and AS4) hold, (21)
shows that the rank of
is or, which is equivalent, that
the
smallest eigenvalues of are equal to zero.
It follows, therefore, that the smallest
eigenvalues of
are all equal to .
Using the above proposition, an estimate of the noise variance
can be obtained as [30]
(28)
where
are the eigenvalues of the estimated
covariance matrix
. Concerning the estimation of , several
techniques are available in [8]–[11], [13], and [14], however, the
method employed must not introduce additional ambiguities to
those associated to the blind channel estimation from SOS. This
requirement guarantees a correct estimation of
,evenin
the presence of channel indetermination. As pointed out in [31],
the approaches [8]–[10] do not meet this requirement.
V. T
HE CP C
LASSIFIER
In many STBC classification problems, the blind identifica-
tion of the three code parameters
, , and is sufficient to
distinguish between several STBCs. For example, the Alamouti
code and an OSTBC3 can be distinguished through detection
of the number of transmitter antennas
. Furthermore, the spa-
tial multiplexing and the Alamouti code can be differentiated by
their code length
. Finally, two codes with the same code length
and using the same number of antennas at the transmitter side
can be identified through detection of the number of symbols
per space–time block. The blind detection of the number of
transmitter antennas is a well-known problem, which has been
investigated in numerous papers and reviewed in [6]. This sec-
tion focuses on the blind recognition of both code length
and
on the number of encoded symbols per block
. This CP classi-
fier only exploits a small portion of the redundancy introduced
by the STBC, but it is well suited for the blind scenario.
Let us consider an STBC
that transmits symbols
during
time slots. From Proposition 1, the covariance matrix
can be modeled as
(29)
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