IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 6, JUNE 2008
2429
A Constrained Factor Decomposition With
Application to MIMO Antenna Systems
André L. F. de Almeida, Gérard Favier, and João Cesar M. Mota
Abstract—In this paper, we formulate a new tensor decomposition herein called constrained factor (CONFAC) decomposition.
It consists in decomposing a third-order tensor into a triple sum
of rank-one tensor factors, where interactions involving the components of different tensor factors are allowed. The interaction
pattern is controlled by three constraint matrices the columns of
which are canonical vectors. Each constraint matrix is associated
with a given dimension, or mode, of the tensor. The explicit use
of these constraint matrices provides degrees of freedom to the
CONFAC decomposition for modeling tensor signals with constrained structures which cannot be handled with the standard
parallel factor (PARAFAC) decomposition. The uniqueness of this
decomposition is discussed and an application to multiple-input
multiple-output (MIMO) antenna systems is presented. A new
transmission structure is proposed, the core of which consists
of a precoder tensor decomposed as a function of the CONFAC
constraint matrices. By adjusting the precoder constraint matrices, we can control the allocation of data streams and spreading
codes to transmit antennas. Based on a CONFAC model of the
received signal, blind symbol/code/channel recovery is possible
using the alternating least squares algorithm. For illustrating this
application, we evaluate the bit-error-rate (BER) performance for
some configurations of the precoder constraint matrices.
Index Terms—Blind detection, constrained tensor decomposition, multiple-input multiple-output (MIMO) antenna systems,
space-time spreading.
I. INTRODUCTION
D
ECOMPOSITIONS of tensors, or multiway arrays, are extensions of matrix decompositions to orders higher than
two. Among them, the most popular ones are the parallel factor
(PARAFAC) decomposition [1] [also known as canonical decomposition (CANDECOMP) [2]] and the Tucker3 decomposition [3], [4]. PARAFAC decomposes a tensor into a sum of
rank-one tensor factors. The popularity of PARAFAC comes
from its uniqueness properties [5]–[10]. The Tucker3 decomposition can be viewed as a generalization of principal component analysis to three-way data [3]. It has been widely used
Manuscript received May 25, 2007; revised October 28, 2007. The associate
editor coordinating the review of this manuscript and approving it for publication was Dr. Subhrakanti Dey. The work of A. L. F. de Almeida was supported
by CAPES/Brazil.
A. L. F. de Almeida and G. Favier are with the I3S Laboratory, University of
Nice Sophia Antipolis (UNSA), Centre National de la Recherche Scientifique
(CNRS), Les Algorithmes/Euclide B, 06903, Sophia Antipolis, Cedex France
(e-mail: [email protected]; [email protected]).
J. C. M. Mota is with the Wireless Telecom Research Group, Department
of Teleinformatics, Federal University of Ceará, 60455-760, Fortaleza, Ceará,
Brazil (e-mail: [email protected]).
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.2008.917026
in chemometrics to resolve mixtures of chemical data. Contrarily to PARAFAC, the Tucker3 decomposition is generally
nonunique due to rotational indeterminacy.
Constrained tensor decompositions, which can be viewed as
hybrid decompositions between PARAFAC and Tucker3, are
studied for some time in the area of chemometrics [11]–[16]. The
constraints are generally imposed on the equivalent Tucker3 core
tensor, which may have a large majority of zero elements [15].
In some cases, these constraints can be modeled, alternatively,
by means of constraint matrices. This results in a PARAFAC decomposition with rank-deficient structure or linear dependency
[16]. With respect to uniqueness, constrained tensor decompositions may be “partially” unique (or nonunique in a restrictive
sense). Partial uniqueness can be studied from the pattern of
zeros of the core tensor as pointed out in [14] and [15].
In this paper, we present a new third-order tensor decomposition herein called constrained factor (CONFAC)
decomposition. The tensor is decomposed into a triple sum of
rank-one tensor factors, where component combinations, or
interactions, involving the different tensor factors are allowed.
The interaction pattern is captured by three constraint matrices
the columns of which are canonical vectors. Each constraint
matrix is associated with a given dimension or mode of the
tensor. Uniqueness of the CONFAC decomposition is discussed
from its constrained interaction structure. We show that the
proposed decomposition has an inherent “uniqueness tradeoff”:
the essential uniqueness in two modes comes at the price of a
restrictive nonuniqueness (or partial uniqueness) in the third
mode. We present some partial uniqueness conditions and
translate them into equivalences between pairs of constraint
matrices with respect to their pattern of zeros.
An application of the CONFAC decomposition to MIMO
wireless communications with multiple transmit and receive
antennas is presented. A new space-time spreading model is
formulated exploring the constrained structure of the CONFAC
decomposition. The core of the proposed MIMO system is
a precoder tensor that controls the joint coupling of multiple
data streams, spreading codes and transmit antennas to generate the transmitted signal. Based on the resulting CONFAC
model for the received signal, we study the implication of the
partial uniqueness properties of this decomposition to blind
symbol/code/channel recovery by means of the alternating least
squares algorithm. For illustration purposes, we evaluate the
bit-error-rate (BER) performance for some configurations of
the precoder constraint matrices.
The use of tensor decompositions in signal processing problems for wireless communications dates back to the seminal
paper [17], which proposed a blind multiuser detection/sepa-
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ration receiver for direct-sequence code division multiple access (DS-CDMA) systems based on a PARAFAC modeling of
the received signal. Different generalizations of this PARAFACbased DS-CDMA model were proposed in subsequent works
[18]–[23] under different assumptions concerning the multipath
propagation structure (e.g., including frequency-selectivity and
specular multipath). In [20] and [23], a constrained approach is
used for modeling the received signal under frequency-selective
multipath propagation, the constraint matrices being fixed and
dependent on the multipath structure. The modeling approaches
of [19] and [21] are based on block factor decompositions. In
this case, the frequency-selective multipath structure is linked
to the rank of the decomposition blocks [24].
Tensor decompositions have also been exploited to model
multiple-input multiple-output (MIMO) antenna systems with
space-time coding/spreading and blind detection [25]–[27]. In
this context, the constraint matrices appear in the tensor model
of [27] as a consequence of the space-time spreading structure proposed therein. In [28], a constrained tensor model with
two constraint matrices having a variable interaction structure is
formulated. The constrained structure of this decomposition is
then fully exploited to design generalized space-time spreading
schemes for DS-CDMA systems using blind detection, which
is in contrast to [27], where the structure of the two constraint
matrices is fixed. The CONFAC decomposition proposed in this
paper is a generalization of [28] by considering constraint matrices in all the three modes. As a consequence, more degrees of
freedom are available for decomposing tensors with more complicated interaction structures.
The rest of this paper is organized as follows. In Section II,
we first introduce the CONFAC decomposition in scalar form
and some links with Tucker3 and PARAFAC decompositions
are established. Then, matrix representations are given and the
interpretation of the constraint matrices in terms of interaction
between modes is illustrated by means of two examples. In
Section III, uniqueness is studied. Two sufficient conditions for
partial uniqueness are presented by focusing on particular cases
of the CONFAC decomposition. In Section IV, we present a
MIMO wireless communication system based on the CONFAC
structure and the physical meaning of the constraint matrices
is illustrated. Section V exploits the partial uniqueness conditions discussed earlier for blind symbol/code/channel recovery
based on a CONFAC model of the received signal. Some simulation results are provided in Section VI for illustrating the BER
performance of the proposed MIMO antenna system using an
alternating least squares based receiver. Section VII concludes
the paper and some perspectives are drawn.
Notation: Some notations and properties are now defined.
and
stand for transpose, inverse, and pseudoinforms a diagonal
verse of , respectively. The operator
forms a diagonal
matrix from its vector argument, while
matrix holding the th row of on its main diagonal. The Kronecker and the Khatri-Rao products are denoted by and ,
respectively
..
.
(1)
.
with
We shall make use of the following property of the Khatri-Rao
product:
(2)
.
for arbitrary
,
Scalars are denoted by lower-case letters
vectors are written as boldface lower-case letters
, ma, and tensors as callitrices as boldface capitals
.
graphic letters
II. CONFAC DECOMPOSITION
, three
Let us consider a third-order tensor
factor matrices
, and three constraint matrices
. The CONFAC decomposition of with
factor combinations is defined in scalar form as
(3)
where
are entries of the factor matrices
and
, respectively. Similarly,
are entries of the constraint matrices
and , respectively. The structure of the constraint matrices is
defined by means of the two following assumptions.
(respectively, and ) are canonA1) The columns of
ical vectors1 belonging to the following canonical bases,
respectively
(4)
and are full-rank matrices.
A2)
Based on these assumptions, the constraint matrices satisfy the
following properties:
(5)
(6)
where
involving the
times that the
denotes the number of combinations
th column of
in (3), i.e., the number of
th column of
is reused for composing the
1A canonical vector e
2
is a unitary vector containing an element
equal to 1 in its nth position and 0’s elsewhere.
DE ALMEIDA et al.: CONFAC DECOMPOSITION
2431
tensor . Similarly,
and
represent the number of combinations involving the th column
of
and the th column of
, respectively. In matrix
form, (6) yields
(7)
where
and
,
. We also have
(8)
Fig. 1. Visualization of the CONFAC decomposition of a third-order tensor.
This property can be demonstrated by noting that
coincides with the
For any
, there is one and only one pair
such
, which implies
.
as
and
, we obtain (8).
Reasoning similarly for
The CONFAC decomposition can be stated in a different
manner, which sheds light on a different way of interpreting its
constrained structure. By exchanging summations in (3), we
obtain
(9)
where
. In this case, the CONFAC decomposition
-factor PARAFAC decomposition [1]
(11)
Matrix Representations: The CONFAC decomposition can
be represented in matrix forms. Two different matrix represenare possible, namely the
tations of the tensor
sliced and unfolded representations. Their construction is similar to that of the standard PARAFAC decomposition [1], [17].
as a matrix obtained by slicing
Let us define
along its first dimension. Similarly,
the tensor
and
are matrices obtained
by slicing the tensor along its second and third dimensions, respectively. These matrix-slices can be expressed as a function
and
, by the following set of
of
equations:
(10)
(12)
tensor
that follows
is an element of a
and .
an -factor PARAFAC decomposition in terms of
, or simply , the constrained core tensor
We call
of the CONFAC decomposition.
Relation With the Tucker3 Decomposition: It is worth noting
that (9) takes the form of a constrained Tucker3 decomposition [11], [13], [14] with the particular characteristic of having
a PARAFAC-decomposed core tensor. The main difference between CONFAC and Tucker3 decompositions is in the following
possible
aspect. In the Tucker3 decomposition, all the
factor combinations exist for composing the tensor , where
each entry of the Tucker3 core tensor defines the “strength”
of each factor combination. Differently, in the CONFAC decomposition, only effective combinations take place for composing the tensor . In this case, the -factor PARAFAC decomposition of the constrained core tensor reveals the pattern of combinations involving the columns of the factor ma, and
. Fig. 1 provides an illustration of
trices
the CONFAC decomposition.
Relation With the PARAFAC Decomposition: Let us consider
and
the CONFAC decomposition (3) with
(13)
(14)
The full information contained in the tensor
can
be organized in three unfolded matrices
, and
. These matrices are constructed
from the sets of matrix-slices
..
.
, and
equations:
..
.
..
.
can be expressed by the following set of
(15)
(16)
(17)
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Proof: The proof of (12) and (17) is simple. It is similar to
the one presented in [29] for the PARAFAC and Tucker decompositions. The CONFAC decomposition (3) can be rewritten as
and
as slices of the constrained core tensor along
its first, second and third dimensions, respectively. In order
as a function of
to factorize the unfolded matrices
, we apply the
the constrained core tensor
property (2) yielding the following expressions:
Let us define
(19)
(18)
(20)
which means that
..
.
..
.
(21)
where
(22)
(23)
(24)
and
..
.
are the three unfolded representations of the constrained core
tensor , with
..
.
Using (1), we straightforwardly obtain
The factorization of
and
can be demonstrated in a similar
way.
Relation With the PARALIND Model: By comparing (18)
with the standard PARAFAC decomposition (11), we remark
that the CONFAC decomposition can be viewed as an -factor
PARAFAC decomposition with equivalent (rank-deficient)
matrices
. The rank-deficient structure due to the repetition of some columns of
and
, is controlled by the constraint matrices
and ,
respectively. A rank deficient tensor model using constraint
matrices is proposed in [16]. This tensor model, which is
called PARALIND, makes use of two constraint matrices to
model interaction patterns between columns of different factor
matrices. In the PARALIND model, the number of factor combinations/interactions is equal to the number of columns of the
factor matrix that is not rank-deficient. The PARALIND model
and
can be obtained from the CONFAC one by making
. Therefore, the CONFAC tensor model can be viewed
as a generalization of the PARALIND one.
Interpretation as a Constrained Tucker3 Decomposition:
As aforementioned, the CONFAC decomposition can be
interpreted as a constrained Tucker3 decomposition with
PARAFAC-decomposed core tensor. We can rewrite (15)–(17)
.
as a function of the constrained core tensor
Taking the property (1) into account, we have
..
.
..
.
(25)
where
(26)
In the same way, we get
(27)
(28)
Definition 1 (Interaction Matrices): The interaction matrices
of the CONFAC decomposition (3) characterize the interaction
pattern involving the factors of different pairs of modes. They
are defined by
(29)
(30)
(31)
DE ALMEIDA et al.: CONFAC DECOMPOSITION
2433
where we have used property (5). Similarly, we define
and
. Using (8),
the three interaction matrices satisfy the following relation:
III. UNIQUENESS
and
are the entries of
where
and
, respectively. We can distinguish the two following
situations:
means that there is no interaction between the
•
th column of
and the th column of
, with
or
;
•
means that there are
interactions
between the th column of
and the th column of
.
Example 1: Let us consider the CONFAC decomposition of
a third-order tensor with
characterized by the following constraint matrices:
of
The uniqueness of the factor matrices
the CONFAC decomposition (up to permutation and scaling)
depends on the particular structure of the constraint matrices
. Specifically, the degrees of freedom introduced in
the decomposition by the three constraint matrices can induce
a transformational ambiguity over (at least a subset of) the
columns of the factor matrices.
Theorem 1 (Identifiability): Let us consider the CONFAC
factor
decomposition (3) of a third-order tensor with
and
are full
combinations. Suppose that
is such
column-rank, and that the joint structure of
, and
that
are also full column-rank, then the decomposition is identifiable
in the least squares (LS) sense from (19)–(21).
Proof: Recall the following properties. For
and
, with full column-rank,
we have
(33)
(35)
(36)
(32)
Let us define the following quantities:
From (29)–(31) we have
indicates that
and
interact twice. The same
is valid for
and
. From
, we can see that
interacts with
while
interacts with
. According to
, there is an interaction
between
and
while
interacts with
, and
with
. Summing the nonzero elements
of
and
yields the number
of factor
combinations.
,
Example 2: Now, consider
with constraint matrices having the following structure:
Identifiability of
, in the LS sense, from
be full column-rank
(19)–(21) requires that
, are full
to be left-invertible. Since
,
column-rank, using (35) implies that
are also full column-rank. From (36), we conclude that
since
is assumed to be
is itself full column-rank. The
full column-rank, and, thus,
.
reasoning is similar for
Identifiability of
and
in the LS sense means
that they are unique up to a multiplication by a nonsingular
yielding the
matrix, i.e., any alternative set
by
same tensor is linked to the true set
(34)
yielding the following interaction matrices:
with
satisfying the following equality:
and
(37)
According to
, each column of
interacts with a different column of
. In particular,
interacts twice with
. We also have
interacting once with
and twice
with
as indicated by
. On the other hand,
interacts twice with
and once with
.
Definition 2 (Admissible Transformation Matrices): The
transformation matrices
and
are called admissible if and only if they preserve the constrained structure of
the decomposition, which means that (37) is satisfied.
Definition 3 (Essential Uniqueness) [6]: Essential uniquegiving
ness means that any alternative set
rise to the same tensor is equal to the set
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up to permutation and scaling of their columns, implying admissible transformation matrices of the form
(38)
where
are diagonal matrices satisfying the relation
, and
are permutation matrices. A proof
of (38) is given in [10] for the PARAFAC decomposition, which
also applies here using a similar reasoning.
Definition 4 (Partial Uniqueness): The CONFAC decomposition is said to be partially unique (or restrictively
nonunique), when a subset of the columns belonging to the set
are essentially unique while the remaining
columns are affected by a linear transformation. Partial uniqueness was first observed in [5], and also investigated in [18]
and [30] in the context of the standard PARAFAC decomposition. For the CONFAC decomposition, the partial uniqueness
property is linked to the structure of the interaction matrices.
It can also be studied from equivalence relations between
pairs of constraint matrices. We now present sufficient (but not
necessary) conditions for partial uniqueness of the CONFAC
decomposition implying essential uniqueness in one or two
modes.
Theorem 2 (Partial Uniqueness): Consider the CONFAC dein terms of factor matrices
composition of
and
, and characterized by interaction matrices
and
. Suppose that the three factor matrices
contain no zeros. We have the following.
, then
is
C.1) If
essentially unique.
, then
is
C.2) If
essentially unique.
, then
is
C.3) If
essentially unique.
Note that
only happens when
.
For
instance,
when
, only
is guaranteed to be essenand
are not guaranteed to be
tially unique, while
unique since only condition 1 of Theorem 2 is satisfied. In this
case, however, partial uniqueness (i.e., essential uniqueness
and
is possible. The
of a subset of columns) of
degree of partial uniqueness depends on the joint interaction
structure of the decomposition. The only case where all the
three above conditions are simultaneously satisfied is the one
, in which the CONFAC decomposition
with
and/or
is close to the PARAFAC one. If
,
is not guaranteed by Theorem
the essential uniqueness of
2. It should be noted that the three above conditions are
sufficient but not necessary for the essential uniqueness of the
factor matrices of the decomposition.
Definition 5 (Equivalent Constraint Matrices): When
(respectively,
or
), the matrix set
(respectively,
or
) is said to be equivalent if
(39)
where
are arbitrary permutation matrices.
Note that the equivalence of two constraint matrices means
that they are identical up to permutation of their rows. For instance, the equivalence between and implies that
and
have the same column repetition pattern
). Consequently, the
(up to permutation of the columns of
defined in (14) can be rewritten
matrix-slice factorization
as
(40)
is a diagonal matrix. Note
where
that (40) is similar to a PARAFAC factorization of the matrixup to permutation and scaling. Similarly, when
slice
or
, we have, respectively
(41)
(42)
in (40), or
Therefore, the essential uniqueness of
or
in (41)–(42) follows from that of
the PARAFAC decomposition.
Partial Uniqueness Corollaries: Using Theorem 2 and the
concept of equivalence between constraint matrices given by
(39), we can deduce the following partial uniqueness corollaries.
and
is an equivalent set,
C.1) When
we have
essentially unique;
•
and
is an equivalent set,
C.2) When
we have
essentially unique;
•
C.3) When
and
is an equivalent set,
we have
essentially unique.
•
From these corollaries, the essential uniqueness in two modes
comes at the expense of a restrictive nonuniqueness in the remaining mode. Such a “uniqueness tradeoff” is inherent to the
CONFAC decomposition. For an illustrative purpose, we can
and
apply corollary C.1 to Example 1 for concluding that
are essentially unique while
is nonunique.
IV. SPACE-TIME SPREADING MIMO SYSTEM
Inspired on the CONFAC decomposition, we present a new
MIMO transmission system. We consider a point-to-point
transmit antennas and
receive anMIMO system with
tennas. At the transmitter, input data streams are transmitted
transmit antennas. The prousing spreading codes and
posed transmission model consists in: i) generating
output
input data streams2
signals to be transmitted by spreading
with the aid of spreading codes and then ii) associating these
output signals with the
transmit antennas. The simultaneous transmission of the data streams across multiple transmit
antennas may use different codes, or fully reuse the same code,
or partially reuse one, or a set of, spreading code(s). Such a
2The R input data streams can be assumed to be associated with R users in
a multiuser model.
DE ALMEIDA et al.: CONFAC DECOMPOSITION
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code reuse pattern will be explicitly modeled by exploiting the
CONFAC structure, as will be clarified in Section IV-A.
denote the spreading factor of the system, i.e., each
Let
secsymbol is composed of chips of duration
onds, where is the symbol period. We assume that is known
or has been estimated (e.g., using cyclostationarity tests). Each
input data stream is a packet of symbols. Let
denote the th symbol of the th data stream,
the
th chip of the th symbol periodic spreading code and
the spatial channel gain between the th transmit antenna and
the th receive antenna. Each transmit-receive antenna pair is
assumed to be characterized by an independent Rayleigh flat
fading. Let us define the following matrices
as the symbol, code, channel matrices,
where
B. Transmitted Signal Model
The precoded signal tensor is represented by the third-order
with entry
. The discrete-time
tensor
baseband version of the precoded signal associated with the
th transmit antenna, th symbol and th chip, is defined
as
. We propose the following
constrained factorization for modeling the effective transmitted
signal:
(44)
By comparing (44) with (9), we deduce the following correspondences:
are the respective entries of these matrices. We assume that the
propagation channel between each pair of transmit and receive
antennas is characterized by a finite number of resolvable multipaths. The multipath channel is assumed to be constant during
symbols. We assume small angle-spread around the receiver,
which arises when the multipath reflectors are in the far field
of the receive antenna array [31]. Intersymbol interference (ISI)
is handled by assuming that the codes include trailing zeros, or
“guard chips” (further details are given in [17]). In this case,
only interchip interference (ICI) exists, and the known codes
are transformed into unknown “effective signature codes,”
given by the convolution of the transmitted spreading codes with
the impulse response of the multipath channel, with denoting
the number of ISI-free chips per symbol. Under the assumption
of independent multipath propagation, the effective signature
codes (henceforth referred to as “spreading codes”) are pseudorandom and mutually independent codes.
A. Precoder Decomposition
The signal to be transmitted is modeled as the sum of prebe the
th element
coded signal components. Let
. This tensor determines
of the precoder tensor
the allocation of the th data stream and the th spreading code
to the th transmit antenna. The -factor decomposition of the
precoder tensor is given, in scalar form, by the following “constrained” PARAFAC decomposition:
Hence, the transmitted signal model is a special case of the
CONFAC decomposition, where the third-mode matrix is equal
to the identity matrix.
We can rewrite (44) in the following form:
(45)
is the matrix of stream/code allocation to the th
means that
transmit antenna and
the th transmit antenna transmits the th data stream using the
th spreading code. The precoded signal slice
associated with the th transmit antenna can be expressed as
(46)
or, equivalently, in terms of the constraint matrices
(47)
(43)
are stream allocation,
code allocation and antenna allocation matrices, respectively.
can be viewed as
Therefore, the precoder tensor
a joint stream-code-antenna multiplexer which is decomposed
in terms of elementary stream, code and antenna allocation mameans that the th data
trices. For instance,
stream of the th precoded signal is spread by the th spreading
code and then transmitted by the th transmit antenna.
The block diagram of the proposed MIMO transmission system
is shown in Fig. 2. In this figure, the precoder tensor is shown in
terms of its matrix-slices. The th precoder slice generates a
at the th transmit antenna by
tensor signal component
combining transmitted symbols and spreading codes, the comand
.
bination pattern being determined by
Example 3: In order to illustrate the physical meaning of
the precoder decomposition, let us consider a multiple-antenna
data streams with
spreading
system transmitting
transmit antennas. Let us assume that
codes across
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From (46), we deduce that
Fig. 2. Block-diagram of the proposed MIMO transmission system.
precoded signals are generated using the following constraint matrices:
In this case, the first and second data streams are code-multiplexed at the first transmit antenna. The first and second data
streams are also, respectively, allocated to the second and third
transmit antennas in order to achieve transmit spatial diversity.
Several transmit schemes can be designed for a fixed param, by varying the pattern of 1’s and 0’s of
eter set
the precoder constraint matrices.
C. Received Signal Model
The discrete-time complex baseband received signal at the
th symbol period, th chip and th receive antenna is defined
being the
th
as
collecting
element of the received signal tensor
the received samples associated with symbols, chips and
receive antennas. Using (44),
can be written, in absence
of noise, as
We have
(48)
(50)
resulting in the following precoded signal slices:
which is a CONFAC decomposition of the received signal,
with the symbol, code and channel matrices as factor matrices,
and with the constrained structure determined by the precoder
. The following correspondences can be
tensor
deduced by comparing (9) with (50):
The first data stream is reused at the first and third transmit
antennas with two different spreading codes, while the second
data stream is transmitted by a single antenna using a single
spreading code.
Example 4: Now, let us consider that we have
and
, with the following precoder constraint
matrices:
(51)
be the slice of the received signal tensor associated
Let
with the th receive antenna,
. Using (51) and
(14), we get
We also have
We have
(49)
The unfolded matrices
and
of the received signal tensor can be factored as
in (15)–(17) by taking the correspondences (51) into account.
DE ALMEIDA et al.: CONFAC DECOMPOSITION
V. BLIND SYMBOL/CODE/CHANNEL RECOVERY
The final goal of our MIMO transmission system is the blind
recovery of the transmitted data without using training sequences
and without needing a priori explicit channel knowledge or
estimation. As discussed in Section III, the partial uniqueness
property of the CONFAC decomposition leads to a sort of
“uniqueness tradeoff,” where the essential uniqueness in one (or
two) mode(s) comes at the expense of a restrictive nonuniqueness
in the other mode(s). The implications of such an uniqueness
tradeoff in terms of blind symbol/code/channel recovery are now
studied. Recall that the uniqueness conditions of Section III establish equivalences between pairs of constraint matrices ensuring
essential uniqueness in two factor matrices of the CONFAC
decomposition. Having the correspondences (51) in mind, these
equivalences admit a physical interpretation in terms of allocation
of data streams and spreading codes to transmit antennas, leading
to different blind symbol/code/channel recovery properties. We
shall distinguish the precoder strategies in two groups: those
with reuse in two dimensions only, and those allowing reuse in
all the dimensions. These two cases are considered here.
1) Case 1: Reuse in Two Dimensions Only: We assume that
either i) data streams and spreading codes are reused more than
once in generating the precoded signals, or ii) transmit antennas and data streams are reused more than once. These two
configurations are detailed here.
(no transmit antenna reuse): Each
1.a)
data stream is associated with a different spreading code.
Each data stream/spreading code can be reused more than
once by different transmit antennas (spatial diversity).
(no spreading code reuse): Each
2.a)
transmit antenna is associated with a different data stream.
Each data stream/transmit antenna can be reused more than
once by different spreading codes (code diversity).
2) Case 2: Reuse in All the Dimensions: We assume that data
streams, spreading codes and transmit antennas are reused more
than once in generating the precoded signals. We consider two
different situations.
: Equal number of data streams and
1.b)
codes.
: Equal number of data streams and
2.b)
transmit antennas.
From the partial uniqueness corollaries C.1 [for 1.a) and 1.b)]
and C.3 [for 2.a) and 2.b)] given in Section III, we can conclude
the following.
• For configurations 1.a) and 1.b), if and are equivalent,
then both and are essentially unique, i.e., joint symbolcode recovery can be achieved.
• For configurations 2.a) and 2.b), if and are equivalent,
then both and are essentially unique, i.e., joint symbolchannel recovery can be achieved.
These results illustrate the link between the space-time precoder
structure with constraints used at the transmitter and the resulting blind symbol/code/channel recovery property at the receiver using the proposed CONFAC model. Several degrees of
freedom for space-time precoder design are available for ensuring the blind symbol recovery.
It is worth noting that for configurations with more transmit
antennas than data streams (meaning that there is one or more
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data streams spatially spread using multiple transmit antennas),
the proposed transmission model is similar to space-time
spreading [32]–[34]. On the other hand, if we have more data
streams than transmit antennas (meaning that two or more data
streams are code-multiplexed at the same transmit antenna),
the proposed transmission model is close to space-time multiplexing [35], [36].
Receiver Algorithm: Alternating Least Squares: The proposed blind detection algorithm is based on the well-known
Alternating Least Squares (ALS) algorithm [11], [17], which
is the classical solution for estimating the factor matrices of a
tensor model in an iterative way. In our case, the ALS algorithm
consists in fitting a CONFAC model to the received signal tensor
represented by means of its unfolded matrices as in (15)–(17) to
estimate the symbol, code and channel matrices in presence of
an additive white Gaussian noise. Since the precoder constraint
and are known at the receiver, they are fixed
matrices
during the whole iterative estimation process.
, as the noisy versions
Define
is an additive complex-valued white Gaussian
of , where
noise matrix. The algorithm consists of the following steps.
;
1) Set
and
;
Randomly initialize
2)
;
, find an LS estimate of
3) Using
4) Using
, find an LS estimate of
5) Using
, find an LS estimate of
6) Repeat steps 2-5 until convergence.
The convergence at the th iteration is declared when the error
between the received signal tensor and its reconstructed version
from the estimated factor matrices does not significantly change
. The conditional update of each
between iterations and
matrix may either improve or maintain but cannot worsen the
current fit. It is worth noting, however, that the ALS algorithm is
strongly dependent on the initialization, and convergence to the
global minimum can be slow. Specifically, the convergence of
this algorithm can sometimes fall in regions of “swamps” during
which the convergence speed is very small and the error between
two consecutive iterations does not significantly decrease [11].
A more efficient initialization strategy consists in first oband using
taining an estimation of the column space of
successive singular value decompositions of
and
, respectively. Note that
, and
can be factored as in (19)–(21) by taking the
correspondences (51) into account. The estimated matrices are
linked to the true ones by nonsingular nonadmissible transformation matrices. They can, however, be used as a starting point
of the ALS algorithm.
In practice, it is reasonable to assume known spreading codes
at the receiver. In this case, the matrix is fixed in the ALS algorithm and Step 4 is skipped. The knowledge of one factor ma-
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 6, JUNE 2008
trix generally accelerates the convergence of the ALS algorithm,
even with small blocks of received samples [25]. Therefore, exploiting the knowledge of the spreading code matrix (whenever
it is available) is beneficial from this viewpoint.
VI. SIMULATION RESULTS
We present some simulation results for illustrating the
performance of the proposed MIMO antenna system based
on CONFAC modeling. The ALS algorithm described in the
previous section is used for this purpose. We are interested in
the symbol and channel recovery with the knowledge or not
of the spreading codes at the receiver. The symbol recovery
performance is evaluated in terms of the BER averaged over
100 Monte Carlo runs. At each run, the additive noise power
is generated according to the signal-to-noise ratio (SNR) value
given by
Fig. 3. Performance of different CONFAC schemes with F = 3.
This SNR measure takes into account the effects of multiple
transmit/receive antennas, fading and multipath. The spatial channel gains are drawn from an i.i.d. complex-valued
Gaussian generator while the transmitted symbols are drawn
from a pseudorandom quaternary phase shift keying (QPSK)
sequence. The BER curves represent the performance averaged
transmitted data streams. Unless otherwise stated,
on the
receive antennas and
signal samples are
assumed throughout the simulations.
The ALS algorithm is strongly dependent on the initialization
and are unknown).
in the completely blind case (where
Indeed, ill-convergence to stationary points generally occurs for
bad initializations. At each run, we consider 10 tentative random
initializations and the one that gives the smallest error is chosen
as initialization of the ALS algorithm. The best initialization
corresponds to the one with the smallest error. The scaling ambiguity affecting the estimate of the symbol matrix is solved by
assuming that the first symbol of each data stream is equal to
1. In this case, the scaling factor is eliminated by normalizing
each column of the estimated symbol matrix by its first element. Optionally differential modulation/encoding can be used
to eliminate this ambiguity [17]. In the unknown spreading code
case, the inherent column permutation ambiguity in is re-column matching alsolved using a greedy least squares
gorithm [17].
A. Performance of Different Precoding Schemes
We evaluate the receiver BER performance for some choices
of the precoder constraint matrices. We begin by considering a
flat-fading channel with the knowledge of the spreading codes
precoded signals,
at the receiver. We assume
or 3 spreading codes, and
or 3 transmit antennas. The
spreading factor is set to
. The orthogonal spreading codes
are columns of a Hadamard(4) matrix. The data stream allocation matrix is the one of Example 3 of Section IV-B, which
is recalled here for convenience
Three different precoding schemes for 2 or 3 spreading codes/
transmit antennas are tested by varying the structure of the code
and antenna allocation matrices and
The BER performance of the three schemes are depicted in
Fig. 3. It can be seen that the first and second schemes have
similar performance. The third scheme provides the best performance due to the fact that the two first schemes reuse one
spreading code which is not the case for the third scheme.
precoded signals,
In a second experiment, we assume
, or orthogonal spreading codes. The number of
and
, and the spreading factor at
transmit antennas is fixed at
. The fixed structure of and is as follows (the same
used in Example 4 of Section IV-B):
(52)
According to the structure of and , we can see that each data
stream is simultaneously transmitted by two transmit antennas.
We consider three code reuse patterns. The three choices for the
code allocation matrix are
(53)
The first scheme reuses twice both spreading codes. The second
one reuses only the first spreading code while the third one uses
DE ALMEIDA et al.: CONFAC DECOMPOSITION
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Fig. 4. Performance of different CONFAC schemes with F = 4.
Fig. 5. Performance of two transmit schemes with multipath/delay propagation
and unknown spreading codes, for R = 2 and 3.
different spreading codes. The results are shown in Fig. 4. As expected, the performance improves at the expense of using more
orthogonal codes. From the slope of the BER curves, we remark
that an increased spatial diversity gain is obtained with the third
).
precoding scheme (
B. Performance With Unknown Spreading Codes
We now consider the case where the spreading codes are unknown at the receiver resulting from the presence of ICI due to
multipath/delay propagation. We assume that the channel has
chip-spaced multipath components. The multipaths undergo independent Rayleigh fading. At each run, the multipath
gains are drawn from an i.i.d. complex-valued Gaussian genertrailing zeros (guard chips)
ator. In order to avoid ISI,
are included in each spreading code, as discussed in Section IV.
. The first one
We consider two transmit schemes with
with
and
ISI-free chips (Hadamard(2) codes
trailing zeros) and the second one with
increased by
and
ISI-free chips. Both schemes have the
same antenna reuse pattern and the chosen is the one given
in (52). The two other precoding constraint matrices are given
below for the first and second schemes, respectively
(54)
(55)
Note that both schemes trade off spatial multiplexing and
transmit diversity. In the first one, each data stream is transmitted by two transmit antennas. In the second one, spatial
multiplexing takes place within the first and second antennas.
Both schemes have the same spectral efficiency (the ratio
is constant). According to Fig. 5, the first scheme outperforms
the second one. This is due to the improved signal separation/resolution that is obtained at the receiver when fewer data
streams are transmitted.
Fig. 6. CONFAC schemes versus PARAFAC scheme for M = 4.
C. Comparison With a Parafac Scheme
Now, we consider three transmit schemes with
(i.e.,
) and
. In the first scheme,
given by (54). In the second one, we have
and
are
The third scheme coincides with the PARAFAC scheme of [17],
where
and
(no stream/code/antenna reuse takes place). The results are shown in Fig. 6. The first
and second schemes offer improved performance over the third
(PARAFAC) one. Note that the CONFAC schemes transmit fewer
data streams than transmit antennas, in order to achieve spatial
spreading. Consequently, more degrees of freedom are available
at the receiver for separating the data streams as compared to the
PARAFAC scheme, which is a full spatial multiplexing scheme.
It is worth noting, however, that the PARAFAC scheme has twice
the spectral efficiency as the CONFAC schemes by simultaneously transmitting four data streams instead of two.
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Fig. 7. Blind CONFAC-ALS versus nonblind CONFAC-ZF receivers.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 6, JUNE 2008
Fig. 8. RMSE performance for the blind channel estimation.
D. Comparison With the Nonblind ZF Receiver
In order to provide a performance reference of the proposed
blind receiver (CONFAC-ALS), we have plotted the performance of the nonblind Zero Forcing (ZF) receiver. Contrarily
to the proposed receiver, the nonblind ZF one assumes perfect
knowledge of the channel parameters (fading gains and multipath/delay response) as well as the knowledge of the spreading
codes. We consider a frequency-selective channel with
multipaths. The spreading codes are unknown at the receiver
and
. The results are depicted in Fig. 7. The chosen
and are given by (54) and
. We can observe a gap of,
approximately, 7 dB in terms of SNR between blind ALS and
.
nonblind ZF receivers for
E. Blind Channel Recovery
As aforementioned, joint blind symbol and channel recovery
is possible for some precoder structures with antenna reuse. We
evaluate the accuracy of the blind channel estimation from the
root mean square error (RMSE) measure averaged over 100
Monte Carlo runs and defined as follows:
where
is the channel matrix estimated at the th run. We
and
. Orthogonal
consider two schemes with
. The strucand known spreading codes are assumed with
ture of these matrices for the first and second schemes are as
follows:
Fig. 8 displays the results. The dashed lines are for
and the solid lines for
. The results are shown
Fig. 9. Convergence histogram for CONFAC and PARAFAC for 100 runs.
for
and receive antennas. In all the simulated configurations, a linear decrease in the channel estimation error as a
function of the SNR is observed. The RMSE increases as the
number of data streams/transmit antennas is increased. On the
other hand, the estimation accuracy is improved as the number
of receive antennas is increased.
F. Evaluation of the Convergence
Fig. 9 depicts an ALS convergence histogram (for 100
Monte Carlo runs) for two transmit schemes: i) CONFAC
; ii) PARAFAC scheme
scheme with
. For the CONFAC scheme,
with
and are given by (52). We remark that the convergence of the CONFAC scheme is achieved within 500 iterations
for approximately 90% of runs. In average, the PARAFAC
scheme has a slower convergence, with less than 40% of runs
converging within 500 iterations. Such a difference certainly
comes from the exploitation of the known interaction structure
incorporated into the received signal model by means of the
DE ALMEIDA et al.: CONFAC DECOMPOSITION
constraint matrices. Consequently, the number of parameters to
be estimated is smaller with the CONFAC scheme as compared
to the PARAFAC one.
VII. CONCLUSION AND PERSPECTIVES
In this paper, we have presented a new constrained factor
decomposition of a third-order tensor, the so-called CONFAC
decomposition which consists in a “constrained triple sum” of
rank-one third-order tensors, where interactions involving the
factors of the decomposition are allowed. The interaction pattern is controlled by three constraint matrices, each one being
associated with one dimension, or mode, of the tensor. The
uniqueness tradeoffs of the CONFAC decomposition have been
studied and conditions for partial uniqueness, which ensure essential uniqueness in one or two modes, have been presented.
We have used the CONFAC decomposition to design space-time
spreading schemes for MIMO antenna systems with a meaningful physical interpretation of the constrained structure of this
decomposition. A space-time precoder model fully exploiting
the constrained structure of this new decomposition has been
presented. We have shown that the CONFAC constraint matrices
define the allocation of the data streams and spreading codes
to transmit antennas. Based on the CONFAC modeling of the
received signal, we have discussed blind symbol/code/channel
recovery from the partial uniqueness properties of this decomposition.
Perspectives of this work are multifold. In the area of
chemometrics, we expect that the CONFAC decomposition
can be exploited to model mixtures of chemical processes with
more complicated interaction structures not covered by the
PARALIND model [16]. The study of necessary and sufficient
conditions in the general case is a topic for future work.
Another perspective concerns the development of efficient
algorithms for fitting the CONFAC model, specially in cases
where the number of factor combinations is large, or when
the constraint matrices of the decomposition are unknown.
With respect to the proposed MIMO antenna system, we
must note that the design of the precoder constraint matrices has
only focused on uniqueness aspects and has not considered performance optimization at the receiver. Several multiple-antenna
transmit schemes can be derived by appropriately choosing the
structure of these constraint matrices. For fixed precoder param, there exists a finite-set of feasible constraint
eters
matrices
ensuring blind symbol/channel/code recovery. We conjecture that limited feedback precoding methods
such as those of [37] and [38] can be exploited for selecting
the best set of precoder constraint matrices using the estimated
MIMO channel. The optimization of the proposed precoder
structure from these methods is an interesting research topic to
be addressed in a future work.
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Oct. 2005.
André L. F. de Almeida was born in Teresina,
Brazil, in 1978. He received the B.Sc. and M.Sc.
degrees in electrical engineering from the Federal
University of Ceará, Brazil, in 2001 and 2003,
respectively, and the double Ph.D. degree in science
and teleinformatics engineering from the University
of Nice, Sophia Antipolis, France, and the Federal
University of Ceará, Fortaleza, Brazil, in 2007.
He is currently a Postdoctoral Fellow with the I3S
Laboratory, Sophia Antipolis. He is also affiliated to
the Department of Teleinformatics Engineering of
the Federal University of Ceará as an Associated Researcher with the Wireless
Telecom Research Group. His research interest lies in the area of signal
processing for communications, and include array processing, blind signal
separation and equalization, multiple-antenna techniques, and multicarrier
and multiuser communications. He has focused on the development of tensor
decompositions with applications in MIMO wireless communication systems.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 6, JUNE 2008
Gérard Favier was born in Avignon, France, in
1949. He received the engineering diplomas from
the Ecole Nationale Supérieure de Chronométrie et
de Micromécanique (ENSCM), Besançon, France,
and Ecole Nationale Supérieure de l’Aéronautique
et de l’Espace (ENSAE), Toulouse, France, in 1973
and 1974, respectively. He received the Engineering
Doctorate and State Doctorate degrees from the
University of Nice, Sophia Antipolis, in 1977 and
1981, respectively.
In 1976, he joined the Centre National de la
Recherche Scientifique (CNRS) where he now works as a Research Director
of CNRS, I3S Laboratory, Sophia Antipolis. From 1995 to 1999, he was the
Director of the I3S Laboratory. His present research interests include nonlinear
process modelling and identification, blind equalization, tensor decompositions, and tensor approaches for wireless communication systems.
João Cesar M. Mota was born in Rio de Janeiro,
Brazil, in 1954. He received the B.Sc. degree in
physics from the Universidade Federal do Ceará
(UFC), Brazil, in 1978, the M.Sc. degree from
Pontifícia Universidade Católica (PUC-RJ), Brazil,
in 1984, and the Ph.D. degree from the Universidade
Estadual de Campinas—UNICAMP, Brazil, in 1992,
all in telecommunications engineering.
Since August 1979, he has been with the UFC,
and currently is Professor with the Teleinformatics
Engineering Department. He was with the Institut
National des Télécommunications and Institut de Recherche en Communications et Cybernetique de Nantes, both in France, as an Invited Professor during
1996–1998 and spring 2006, respectively. His research interests include digital
communications, adaptive filter theory, and signal processing.
Dr. Mota was General Chairman of the 19th Brazilian Telecommunications
Symposium—SBrT’2001 and the International Symposium on Telecommunications—ITS’2006. He is responsible for the international mobility program for
engineering students of UFC. He is a member and counselor of the Sociedade
Brasileira de Telecomunicaões, and a member of the IEEE Communications Society and IEEE Signal Processing Society. He is a counselor of the IEEE Student
Branch in UFC.