A Constrained Factor Decomposition With Application to MIMO ...

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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.

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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)

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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.

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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.

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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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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

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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.

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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

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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. REFERENCES [1] R. A. Harshman, “Foundations of the PARAFAC procedure: Model and conditions for an “explanatory” multi-mode factor analysis,” UCLA Working Papers in Phonetics, vol. 16, pp. 1–84, Dec. 1970. [2] J. D. Carroll and J. Chang, “Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition,” Psychometrika, vol. 35, no. 3, pp. 283–319, 1970. [3] L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika, vol. 31, pp. 279–311, 1966.

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[4] P. M. Kroonenberg and J. De Leeuw, “Principal component analysis of three-mode data by means of alternating least squares,” Psychometrika, vol. 45, pp. 69–97, 1980. [5] R. A. Harshman, “Determination and proof of minimum uniqueness conditions for PARAFAC-1,” UCLA Working Papers in Phonetics, no. 22, pp. 111–117, 1972. [6] J. B. Kruskal, “Three-way arrays: Rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics,” Linear Algebra Applicat., vol. 18, pp. 95–138, 1977. [7] J. M. F. ten Berge and N. D. Sidiropoulos, “On uniqueness in CANDECOMP/PARAFAC,” Psychometrika, vol. 67, pp. 399–409, 2002. [8] T. Jiang and N. D. Sidiropoulos, “Kruskal’s permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2625–2636, Sep. 2004. [9] L. De Lathauwer, “A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization,” SIAM J. Matrix Anal. Appl, vol. 28, no. 3, pp. 642–666, 2006. [10] A. Stegeman and N. D. Sidiropoulos, “On Kruskal’s uniqueness condition for the Candecomp/Parafac decomposition,” Linear Algebra Applicat., vol. 420, pp. 540–552, 2007. [11] R. Bro, “Multi-way analysis in the food industry: models, algorithms and applications,” Ph.D. dissertation, Univ. Amsterdam, Amsterdam, The Netherlands, 1998. [12] H. A. Kiers, J. M. F. ten Berge, and R. Rocci, “Uniqueness of threemode factor models with sparse cores: The 3 3 3 case,” Psychometrika, vol. 62, no. 3, pp. 349–374, 1997. [13] H. A. Kiers and A. K. Smilde, “Constrained three-mode factor analysis as a tool for parameter estimation with second-order instrumental data,” J. Chemomet., vol. 12, no. 2, pp. 125–147, Dec. 1998. [14] J. M. F. ten Berge and A. K. Smilde, “Non-triviality and identification of a constrained Tucker3 analysis,” J. Chemomet., vol. 16, pp. 609–612, 2002. [15] J. M. F. ten Berge, “Simplicity and typical rank of three-way arrays, with applications to Tucker3 analysis with simple cores,” J. Chemomet., vol. 18, pp. 17–21, 2004. [16] R. Bro, R. A. Harshman, and N. D. Sidiropoulos, Modeling multi-way data with linearly dependent loadings Univ. of Denmark, Denmark, KVL Tech. Rep. 176, 2005. [17] N. D. Sidiropoulos, G. B. Giannakis, and R. Bro, “Blind PARAFAC receivers for DS-CDMA systems,” IEEE Trans. Signal Process., vol. 48, no. 3, pp. 810–822, Mar. 2000. [18] N. D. Sidiropoulos and G. Z. Dimic, “Blind multiuser detection in WCDMA systems with large delay spread,” IEEE Signal Process. Lett., vol. 8, no. 3, pp. 87–89, Mar. 2001. [19] A. de Baynast and L. De Lathauwer, “Detection autodidacte pour des systemes a acces multiple basee sur 1’analyse PARAFAC,” presented at the 19th GRETSI Symp. Signal Image Process., Paris, France, Sep. 2003. [20] A. L. F. de Almeida, G. Favier, and J. C. M. Mota, “PARAFAC models for wireless communication systems,” presented at the Int. Conf. Phys. Signal Image Process. (PSIP), Toulouse, France, Jan. 31–Feb. 2 2005. [21] D. Nion and L. De Lathauwer, “A block factor analysis based receiver for blind multi-user access in wireless communications,” presented at the ICASSP, Toulouse, France, May 2006. [22] L. De Lathauwer and J. Castaing, “Tensor-based techniques for the blind separation of DS-CDMA signals,” Signal Process., vol. 87, no. 2, pp. 322–336, 2007. [23] A. L. F. de Almeida, G. Favier, and J. C. M. Mota, “PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization,” Elsevier Signal Process., vol. 87, no. 2, pp. 337–351, Feb. 2007. [24] L. De Lathauwer, “The decomposition of a tensor in a sum of rank(Rl,R2,R3) terms,” presented at the Workshop on Tensor Decomposit. Applicat., Marseille, France, 2005. [25] N. D. Sidiropoulos and R. Budampati, “Khatri-Rao space-time codes,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2377–2388, Oct. 2002. [26] A. de Baynast, L. De Lathauwer, and B. Aazhang, “Blind PARAFAC receivers for multiple access-multiple antenna systems,” presented at the VTC Fall, Orlando, FL, Oct. 2003. [27] A. L. F. de Almeida, G. Favier, and J. C. M. Mota, “Space-time multiplexing codes: A tensor modeling approach,” presented at the IEEE Int. Workshop on Signal Process. Adv. in Wireless Commun. (SPAWC), Cannes, France, Jul. 2006. [28] A. L. F. de Almeida, G. Favier, and J. C. M. Mota, “Constrained tensor modeling approach to blind multiple-antenna CDMA schemes,” IEEE Trans. Signal Process., vol. 56, no. 6, Jun. 2008.

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[29] G. Favier, Calcul matriciel et tensoriel avec applications a 1’automatique et au traitement du signal 2008, under preparation. [30] J. M. F. ten Berge, “Partial uniqueness in CANDECOMP/PARAFAC,” J. Chemomet., vol. 18, no. 1, pp. 12–16, 2004. [31] A.-J. van der Veen, “Algebraic methods for deterministic blind beamforming,” Proc. IEEE, vol. 86, no. 10, pp. 1987–2008, Oct. 1998. [32] B. K. Ng and E. Sousa, “Space-time spreading multilayered CDMA system,” in Proc. IEEE GLOBECOM, Dec. 2000, vol. 3, no. 27, pp. 1854–1858. [33] B. Hochwald, T. L. Marzetta, and C. B. Papadias, “A transmitter diversity scheme for wideband CDMA systems based on space-time spreading,” IEEE J. Sel. Areas Commun., vol. 19, no. 1, pp. 48–60, 2001. [34] R. Doostnejad, T. J. Lim, and E. Sousa, “Space-time spreading codes for a multiuser MIMO system,” in Proc. 36th Asilomar Conf. Signals, Syst. Camp., Pacific Grove, CA, Nov. 2002, pp. 1374–1378. [35] S. Mudulodu and A. J. Paulraj, “A simple multiplexing scheme for MIMO systems using multiple spreading codes,” in Proc. 34th Asilomar Conf. on Signals, Syst. Comp., Pacific Grove, CA, Oct. 2000, pp. 769–774. [36] R. Doostnejad, T. J. Lim, and E. Sousa, “Space-time multiplexing for MIMO multiuser downlink channels,” IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1726–1734, 2006. [37] R. W. Heath and D. J. Love, “Multimode antenna selection for spatial multiplexing systems with linear receivers,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 3042–3056, Aug. 2005. [38] D. J. Love and R. W. Heath, “Multimode preceding for MIMO wireless systems,” IEEE Trans. Signal Process., vol. 53, no. 10, pp. 3674–3687, 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.

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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.