MIMO multi-relay systems with tensor space-time coding based on coupled nested Tucker decomposition

Accepted Manuscript MIMO multi-relay systems with tensor space-time coding based on coupled nested Tucker decomposition

Danilo S. Rocha, C.Alexandre R. Fernandes, Gérard Favier

PII: DOI: Reference:

S1051-2004(18)30443-3 https://doi.org/10.1016/j.dsp.2019.03.006 YDSPR 2501

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Digital Signal Processing

Please cite this article in press as: D.S. Rocha et al., MIMO multi-relay systems with tensor space-time coding based on coupled nested Tucker decomposition, Digit. Signal Process. (2019), https://doi.org/10.1016/j.dsp.2019.03.006

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MIMO multi-relay systems with tensor space-time coding based on coupled nested Tucker decomposition Danilo S. Rocha a, b,*, C. Alexandre R. Fernandes b, Gérard Favier c a Federal

Institute of Education, Science and Technology, Sobral, CE, Brazil b Federal

c

University of Ceará, Fortaleza, CE, Brazil

I3S Laboratory, University of Côte d’Azur, CNRS, Sophia Antipolis, France

Abstract In this paper, we propose a two-hop MIMO multi-relay system with tensor space-time coding (TSTC) at the source and the relays. The multiple relays use orthogonal channels (parallel relaying) to increase the diversity order, assuming that all the relays can communicate directly with the destination. The signals received at destination form a fifth-order tensor that satisfies a new tensor model, called coupled nested Tucker decomposition (CNTD). This model is a generalization of the nested Tucker decomposition (NTD) and extends the coupling concept, introduced for PARAFAC models, to Tucker-based decompositions. The CNTD consists of a coupling of NTD models that share a common factor, resulting in the concatenation of a generalized Tucker model with a Tucker one. This tensor modeling is exploited to derive a semi-blind receiver for jointly estimating the information symbols and the channels. The uniqueness property of the proposed tensor model and the identifiability conditions of the receiver are established. Monte Carlo simulation results are provided to illustrate the effectiveness of the proposed system and the advantage of exploiting cooperative diversity based on tensor modeling. Keywords: Coupled nested Tucker decomposition, MIMO systems, multi-relay systems, semi-blind receivers, tensor decompositions.

1. Introduction High order tensors (i.e., multiway arrays) and tensor decompositions play an increasing role in signal and image processing applications [1]. In particular, during the last decade, tensor models have been extensively used for designing different types of wireless communication systems, with the aim of developing semi-blind receivers for joint channel and symbol estimation. For instance, one can mention the works [2-5], for point-to-point systems, and [6-10], for relaying systems. Tucker [11] and Parallel Factors Analysis (PARAFAC) [12] are the most commonly used tensor decompositions in signal processing for wireless communications. PARAFAC models have the important property of being essentially unique, which is not the case of the Tucker models, except under certain conditions like a priori knowledge of the core tensor. Essential uniqueness means that the decompositions are unique up to arbitrary scaling and permutation of columns of the factor matrices. However, Tucker models are one of the most important and flexible tensor decompositions [13]. _____________________________________________ Email addresses: [email protected] (Danilo S. Rocha)*, [email protected] (C. Alexandre R. Fernandes), [email protected] (Gérard Favier)

2 Recently, a new tensor model called nested Tucker decomposition (NTD) was introduced in [10]. This decomposition is characterized by the nesting of two third-order Tucker models that share a common matrix factor. NTD can be viewed as an extension of the nesting concept, initially proposed for the PARAFAC decomposition [14], to the Tucker one. The concept of coupled tensor decompositions is gaining interest as a tool for solving data fusion problems. This concept allows a global processing of all the received data taking into account the coupling of multiple tensor decompositions that share a common factor. In [15-17], applications of coupled decompositions based on PARAFAC models are presented. A set of tensor decompositions is said to be “coupled” when at least one of the involved factors is shared by all the decompositions. To the knowledge of the authors, coupled tensor decompositions have not yet been applied in the context of telecommunications. In wireless communication systems, recent studies show important properties of tensor decompositions that bring some advantages over methods based on a 2-D matrix. In the literature, one can find several applications of tensor approaches aiming to increase the capacity of wireless networks. In [18, 19], for instance, one can find tensor completion techniques applied to internet traffic data to solve missing data problems. In [20], a tensor approach is used to get a more accurate algorithm for data anomaly detection. In [21, 22, 23], tensor-based techniques are applied to solve the blind source separation (BSS) of a mixture of signals received by an antenna array. Tensor-based approaches also include the possibility of using tensor coding, with simultaneous spatial multiplexing, spreading spectrum, time spreading and multicarrier modulation, yielding diversity and spectral efficiency gains. In [24], a block coding scheme based on the KhatriRao matrix product was proposed. We can find in [8, 14, 25, 26] examples of applications of this coding scheme to different systems based on different tensor decompositions. In [5], a tensor spacetime coding (TSTC) was proposed aiming to increase the diversity gain by introducing one extra time dimension via a third-order coding tensor in a multiple-input multiple-output (MIMO) system. The TSTC was also applied to cooperative MIMO systems for two-hop [10] and three-hop [27] cases. In the context of tensor-based receivers for the next generations of wireless communications, cooperative networks seem to be a promising solution. In these systems, individual channel estimation is a fundamental problem. Channel state information (CSI) of all links between source, relays and destination plays an important role for optimizing MIMO relay systems in terms of power allocation, decoding and adaptive relaying protocols that must decide when a cooperation is feasible and select a suitable relay [28-29]. Thus, the reliability of systems with cooperative diversity strongly depends on the accuracy of CSI associated with each hop. The authors in [30] propose tensor-based receivers for uplink multiuser systems with cooperative diversity, which consider amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols. In this case, it is assumed that the CSI is not available and, then, blind receivers using the Levenberg-Marquardt (LM) algorithm are proposed to jointly estimate the transmitted symbols and the channels. The authors proposed a tensor formulation of the received signal that

3 unifies all the considered relaying protocols in a PARAFAC-based model. A two-hop AF relay system was introduced in [8], with the source using a simplified KhatriRao space-time (KRST) coding to encode the signals to be transmitted. The considered matrix coding induces the signals received at destination to be a third-order tensor, which satisfies a PARAFAC decomposition when considered the direct link (source-destination) and a PARATUCK2 decomposition when considered the relay-assisted link (source-relay-destination). Thus, three receivers that combine these two models are proposed for a joint and semi-blind estimation of transmitted symbols and channels of both hops. In [10], the authors apply a NTD model to a one-way two-hop half-duplex MIMO relay system with the source and the relay using a TSTC to encode the symbol matrix. The tensor modeling for the signals received at destination satisfies a fourth-order NTD, or NTD(4). Expressions for the unfolded matrices of the NTD(4) were developed, allowing the design four receivers (two semi-blind and two supervised) for symbols and channels joint estimation. Recently, a one-way multi-hop AF relay system that assumes a KRST coding at each relay was addressed in [25]. The system with ‫ ܭ‬relays is modeled by means of ‫ ܭ‬൅ ͳ third-order PARAFAC models. A closed-form semi-blind receiver based on a Khatri-Rao factorization was derived to jointly estimate the symbols and the individual channels. Simulation results show performance gains with an increase of the number relays. Despite these and other applications of tensor models in the context of cooperative systems [9, 31-35], tensor coding applied to multi-relay systems has not yet been exploited. In this paper, we propose a two-hop MIMO multi-relay system with TSTC at the source and the relays. The multiple relays use orthogonal channels (parallel relaying) to increase the diversity order, assuming that all the relays can communicate directly with the destination. This system can be viewed as a generalization of recently proposed systems [5, 8, 10, 34], aiming to exploit the cooperative diversity provided by the multiple relays in a MIMO system with TSTC. Indeed, the present paper extends previous works in different ways, either by proposing a more general tensor decomposition, by using a more general relay coding, by extending these works to the multi-relay case and/or by using a different estimation algorithm. Regarding the use of multiple relays operating in parallel cooperation, we can find many works in the literature that use this kind of approach [9, 30, 32, 35, 36]. However, these works use simpler channel models. Indeed, [9, 35, 36] consider single-antenna nodes, [30, 32] use conventional matrix coding and [30, 32, 35] perform the channel estimation with the use of training sequences. We show that the signals received at destination form a fifth-order tensor, where each mode is linked with a different signal dimension (space, source code, relay code, time and number of relays). Indeed, the proposed system exploits both the space and cooperative diversity, as well as timespreading at the source and relays. This tensor satisfies a new model called coupled nested Tucker decomposition (CNTD), which generalizes the NTD to higher order tensors, admitting generalized Tucker tensors as decomposition factors, while the NTD only admits Tucker tensors as decomposition

4 factors. The proposed CNTD can be viewed as a coupling of multiple NTDs that share a common factor. This model is particularly studied for the case of a fifth-order tensor, corresponding to a concatenation of a fourth-order generalized Tucker decomposition with a third-order Tucker one. The CNTD extends the concept of coupling to Tucker-based models and represents the first application of such coupled tensor models in the context of wireless communications. Indeed, up to now, the coupled decompositions

based

on

PARAFAC

models

[15-17] have

not

found

applications

in

telecommunications. We address uniqueness property of the CNTD, showing that, under certain conditions, the decomposition factors are unique up to scaling ambiguities. It is worth mentioning that the uniqueness is not rigorously proved in [10]. As the CNTD can be viewed as a generalization of the NTD, this means that our demonstration is also valid for NTD, which enhances the analysis of [10]. This tensor modeling is used to develop a receiver algorithm for jointly estimating the symbol matrix and the individual channels with a global processing of all datasets received from multiple relays. The proposed algorithm is a closed-form solution that uses a singular-value decomposition (SVD)-based low-rank approximation algorithm. Monte Carlo simulations are provided to illustrate the effectiveness of the cooperative diversity exploitation and to compare the performances of the proposed receiver with other existing ones. Simulation results show the improvement induced by an increase of the number of relays. The main contributions of this paper are summarized as follows: •

A two-hop MIMO multi-relay system with TSTC at the source and the relays is proposed. Such tensor coding applied to a multiple parallel relays system is original.

•

A new tensor decomposition (CNTD) is introduced as a generalization of the NTD to higher order tensors, generalizing the concept of coupled decompositions to Tucker models.

•

The signals received at destination form a fifth-order tensor which satisfies a new CNTD model, leading to the first application of coupled tensor models in the context of wireless communications.

•

The uniqueness of the CNTD under certain conditions is demonstrated. The proposed uniqueness theorem is also valid for NTD, enhancing the analysis of [10].

•

The CNTD is exploited to develop a semi-blind receiver, based on a closed-form solution, for jointly estimating the symbol matrix and the individual channels. Identifiability conditions of the algorithm are established.

•

Monte Carlo simulation results are presented for illustrating the effectiveness of the proposed system and evaluating the receiver performance. This paper is structured as follows. Section 2 presents the notation and tensor prerequisites. In

Section 3, a new two-hop MIMO multi-relay system with TSTC is described, leading to a CNTD model for the signals received at destination. In Section 4, CNTD is formally defined as a coupling of NTDs and uniqueness properties are established. In Section 5, a receiver based on CNTD is derived for jointly estimating the transmitted symbols and the individual channels. Simulation results are presented in Section 6 and concluding remarks are given in Section 7.

5 2.

Tensor prerequisites In this section, we present the notation, basic operations and properties used throughout the

paper. Scalars, column vectors, matrices and tensors of order higher than two are denoted by lowercase, boldface lowercase, boldface uppercase, and calligraphic letters, i.e., ܽ, ‫܉‬, ‫ ۯ‬and ࣛ, respectively. The transpose, conjugate, Hermitian transpose, Moore-Penrose pseudo-inverse and rank of ‫ ۯ‬are denoted respectively by ‫ ்ۯ‬, ‫ כۯ‬, ‫ۯ‬ு , ‫ۯ‬ற and ‫ ۯݎ‬. ‫ۯ‬௜ή and ‫ۯ‬ή௝ denote the i-th row and the j-th column of ‫ א ۯ‬ԧூൈ௃ . For a ܰ-th order tensor ࣲ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூಿ , the entry at position ሺ݅ଵ ǡ ‫ ڮ‬ǡ ݅ே ሻ is denoted by ‫ݔ‬௜భ ǡ‫ڮ‬ǡ௜ಿ . The matrix ‫܆‬௃భ ‫ڮ‬௃ಿషభ ൈ௃ಿ , where ሼ‫ܬ‬ଵ ǡ ‫ ڮ‬ǡ ‫ܬ‬ே ሽ is any permutation of ሼ‫ܫ‬ଵ ǡ ‫ ڮ‬ǡ ‫ܫ‬ே ሽ, represents a tall

unfolding

of

the

tensor

ࣲ

ൣ‫܆‬௃భ ‫ڮ‬௃ಿషభ ൈ௃ಿ ൧ሺ௝

భ ିଵሻ௃మ ‫ڮ‬௃ಿషభ ା‫ڮ‬ାሺ௝ಿషమ ିଵሻ௃ಿషభ ା௝ಿషభ ǡ௝ಿ

whose

the

entries

are

‫ݔ‬௝భ ǡ‫ڮ‬ǡ௝ಿ ൌ

.

The operator ˜‡…ሺήሻ transforms a matrix into a column vector by stacking the columns of its matrix argument. The operator ݀݅ܽ݃ሺήሻ forms a diagonal matrix from its vector argument. Similarly, the operator ܾ݀݅ܽ݃ሺ‫ۯ‬ଵ ǡ ǥ ǡ ‫ۯ‬௄ ሻ ‫ܾ݃ܽ݅݀ ؜‬ሺ‫ۯ‬௞ ሻ forms a block-diagonal matrix composed of the ‫ܭ‬ matrices ‫ۯ‬௞ , with ݇ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܭ‬. The Kronecker product is denoted by ٔ. Given a set ॺ ൌ ሼͳǡ ‫ ڮ‬ǡ ܰሽ and the matrices ‫ۯ‬ሺ௡ሻ ‫ א‬ԧூ೙ ൈ௃೙ , the multiple Kronecker product is defined by ٔ ‫ۯ‬ሺ௡ሻ ൌ ‫ۯ‬ሺଵሻ ٔ ‫ٔ ڮ‬ ௡‫ॺא‬

ሺேሻ

‫ۯ‬

‫א‬ԧ

ூభ ‫ڮ‬ூಿ ൈ௃భ ‫ڮ‬௃ಿ

. The mode-n product of a tensor ࣛ ‫ א‬ԧ

ூభ ൈ‫ڮ‬ൈூಿ

with a matrix ‫ א ۻ‬ԧ௉೙ ൈூ೙ whose

the second mode is equal to the ݊-th mode of ࣛ, is denoted by ࣛ ൈ௡ ‫ א ۻ‬ԧூభ ൈ‫ڮ‬ൈூ೙షభ ൈ௉೙ ൈூ೙శభ ൈ‫ڮ‬ൈூಿ . The outer product of ‫ ܭ‬vectors ‫܉‬ሺ௞ሻ ‫ א‬ԧூೖ is denoted by ‫܉‬ሺଵሻ ‫܉ ל ڮ ל‬ሺ௄ሻ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூ಼ . Given the matrices ‫ א ۯ‬ԧூൈ௉ , ۰ ‫ א‬ԧ௉ൈெ , ۱ ‫ א‬ԧ௃ൈெ and ۲ ‫ א‬ԧெൈே , we recall some properties that will be used. Property 1.

˜‡…ሺ‫ۯ‬۰۱ ் ሻ ൌ ሺ۱ ٔ ‫ۯ‬ሻ˜‡…ሺ۰ሻ ‫ א‬ԧ௃ூ Ǥ

(1)

Property 2.

ሺ‫ ٔ ۯ‬۱ሻሺ۰ ٔ ۲ሻ ൌ ‫ۯ‬۰ ٔ ۱۲ ‫ א‬ԧூ௃ൈெே Ǥ

(2)

Property 3. Let ‫ ۯ‬be a full column rank matrix, then ‫ۯݎ‬۰ ൌ ‫ݎ‬۰ . This implies that ‫ۯ‬۰ is full column rank if and only if A and ۰ are full column rank. Property 4. Given ‫ ۻ‬ൌ ‫ ٔ ۯ‬۱, then ‫ ۻݎ‬ൌ ‫ݎ ۯݎ‬۱ . This implies that ‫ ۻ‬is full column rank if and only if A and C are full column rank. 2.1.

Mode-n product of two tensors Let us consider a ܰ-th order tensor ࣡ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூ೙షభ ൈோ೙ ൈூ೙శభ ൈ‫ڮ‬ൈூಿ and a ܰ௔ -th order tensor

ࣛሺ௡ሻ ‫ א‬ԧூ೙ ൈோ೙ ൈூॺ೙ , with ͵ ൑ ܰ௔ ൑ ܰ ൅ ͳ and ͳ ൑ ݊ ൑ ܰ. The set ॺ௡ is an ordered subset of the set ሼͳǡ ‫ ڮ‬ǡ ݊ െ ͳǡ ݊ ൅ ͳǡ ‫ ڮ‬ǡ ܰሽ. ‫ॺܫ‬೙ and ݅ॺ೙ denote, respectively, short forms for the dimension and the set of indices associated to the modes of the subset ॺ௡ . For instance, if ॺ௡ ൌ ሼ͵ǡͶǡͷሽ, then ‫ॺܫ‬೙ ൌ ‫ܫ‬ଷ ൈ ‫ܫ‬ସ ൈ ‫ܫ‬ହ and ݅ॺ೙ ൌ ሼ݅ଷ ǡ ݅ସ ǡ ݅ହ ሽ. The tensor-tensor mode-n product of ࣡ with ࣛ ሺ௡ሻ , denoted by ࣡ ൈ௡ ࣛሺ௡ሻ , gives a tensor ࣲ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூ೙షభ ൈூ೙ ൈூ೙శభ ൈ‫ڮ‬ൈூಿ defined as

6 ோ೙ ሺ௡ሻ

‫ݔ‬௜భ ǡ‫ڮ‬ǡ௜೙షభ ǡ௜೙ ǡ௜೙శభ ǡ‫ڮ‬ǡ௜ಿ ൌ ෍ ݃௜భ ǡ‫ڮ‬ǡ௜೙షభ ǡ௥೙ ǡ௜೙శభ ǡ‫ڮ‬ǡ௜ಿ ܽ௜೙ ǡ௥೙ ǡ௜ॺ ǡ

(3)

೙

௥೙ ୀଵ

where, by convention, we assume that the second mode of ࣛ ሺ௡ሻ is equal to the ݊-th mode of ࣡. Property 5. Consider the tensors ࣡ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூ೙షభ ൈோ೙ ൈூ೙శభ ൈ‫ڮ‬ൈூಿ , ࣛሺ௡ሻ ‫ א‬ԧூ೙ ൈோ೙ ൈூॺೌ and ࣜ ሺ௡ሻ ‫א‬ ԧ௉೙ ൈூ೙ ൈூॺ್ , where ॺ௔ and ॺ௕ correspond to two ordered subsets of ሼͳǡ ‫ ڮ‬ǡ ݊ െ ͳǡ ݊ ൅ ͳǡ ‫ ڮ‬ǡ ܰሽ, with ܰ௔ െ ʹ and ܰ௕ െ ʹ elements, respectively, such that ॺ௕ ‫ॺ ك‬௔ . We have ൫࣡ ൈ௡ ࣛ ሺ௡ሻ ൯ ൈ௡ ࣜ ሺ௡ሻ ൌ ࣡ ൈ௡ ൫ࣛ ሺ௡ሻ ൈଵ ࣜ ሺ௡ሻ ൯ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூ೙షభ ൈ௉೙ ൈூ೙శభ ൈ‫ڮ‬ൈூಿ Ǥ

(4)

Proof: ூ೙

൫࣡ ൈ௡ ࣛ

ሺ௡ሻ

൯ ൈ௡ ࣜ

ሺ௡ሻ

ோ೙ ሺ௡ሻ

ሺ௡ሻ

ൌ ෍ ෍ ݃௜భ ǡ‫ڮ‬ǡ௜೙షభ ǡ௥೙ ǡ௜೙శభ ǡ‫ڮ‬ǡ௜ಿ ܽ௜೙ ǡ௥೙ ǡ௜ॺ ܾ௣೙ ǡ௜೙ ǡ௜ॺ  ೌ

௜೙ ୀଵ ௥೙ ୀଵ

್

ூ೙

ோ೙

ሺ௡ሻ

ሺ௡ሻ

 ൌ ෍ ݃௜భ ǡ‫ڮ‬ǡ௜೙షభ ǡ௥೙ ǡ௜೙శభ ǡ‫ڮ‬ǡ௜ಿ ቌ ෍ ܽ௜೙ ǡ௥೙ ǡ௜ॺ ܾ௣೙ ǡ௜೙ ǡ௜ॺ ቍ ௥೙ ୀଵ

௜೙ ୀଵ

ೌ

್

ൌ ࣡ ൈ௡ ൫ࣛሺ௡ሻ ൈଵ ࣜ ሺ௡ሻ ൯ ǡ

(5)

which is the desired result. 2.2.

Ƒ

Contraction operation between two tensors Let us consider a ܰ-th order tensor ࣲ ‫ א‬ԧூభ ൈ‫ڮ‬ൈூಿ and a ‫ܯ‬-th order tensor ࣳ ‫ א‬ԧ ௃భ ൈ‫ڮ‬ൈ௃ಾ that

share a common mode (݅௣ ൌ ݆௤ ൌ ݇, with ͳ ൑ ‫ ݌‬൑ ܰ and ͳ ൑ ‫ ݍ‬൑ ‫)ܯ‬. The contraction of ࣲ with ࣳ is defined as a sum over the common mode [37] ௄

‫ݖ‬௜భ ǡ‫ڮ‬ǡ௜೛షభ ǡ௝భ ǡ‫ڮ‬ǡ௝೜షభ ǡ௝೜శభ ǡ‫ڮ‬ǡ௝ಾ ǡ௜೛శభ ǡ‫ڮ‬ǡ௜ಿ ൌ ෍ ‫ݔ‬௜భ ǡ‫ڮ‬ǡ௜೛షభ ǡ௞ǡ௜೛శభ ǡ‫ڮ‬ǡ௜ಿ ‫ݕ‬௝భ ǡ‫ڮ‬ǡ௝೜షభ ǡ௞ǡ௝೜శభ ǡ‫ڮ‬ǡ௝ಾ Ǥ

(6)

௞ୀଵ

This operation will be represented as ࣴ ൌ ࣲ ‫ א ࣳ כ‬ԧூభ ൈ‫ڮ‬ൈூ೛షభ ൈ௃భ ൈ‫ڮ‬ൈ௃೜షభ ൈ௃೜శభ ൈ‫ڮ‬ൈ௃ಾ ൈூ೛శభ ൈ‫ڮ‬ൈூಿ ǡ ௞

(7)

where the operator ‫ כ‬corresponds to the contraction along the modes ‫ ݌‬of ࣲ and ‫ ݍ‬of ࣳ. The ௞

contraction operation yields a tensor of order N + M – 2. It can be written as the following tensormatrix mode-k products by combining modes of ࣲ or ࣳ: ࣴூభ ൈ‫ڮ‬ൈூ೛షభ ൈ௃భ ‫ڮ‬௃೜షభ ௃೜శభ ‫ڮ‬௃ಾ ൈூ೛శభ ൈ‫ڮ‬ൈூಿ ൌ ࣲ ൈ௞ ‫܇‬௃భ ‫ڮ‬௃೜షభ ௃೜శభ ‫ڮ‬௃ಾ ൈ௄ ǡ

(8)

ࣴ௃భ ൈ‫ڮ‬ൈ௃೜షభ ൈூభ ‫ڮ‬ூ೛షభ ூ೛శభ ‫ڮ‬ூಿ ൈ௃೜శభ ൈ‫ڮ‬ൈ௃ಾ ൌ ࣳ ൈ௞ ‫ ܆‬ூభ ‫ڮ‬ூ೛షభ ூ೛శభ ‫ڮ‬ூಿ ൈ௄ ǡ

(9)

where ‫܇‬௃భ ‫ڮ‬௃೜షభ ௃೜శభ ‫ڮ‬௃ಾ ൈ௄ and ‫ ܆‬ூభ ‫ڮ‬ூ೛షభ ூ೛శభ ‫ڮ‬ூಿ ൈ௄ are tall unfoldings of ࣳ and ࣲ, respectively, and ࣴூభ ൈ‫ڮ‬ൈூ೛షభ ൈ௃భ ‫ڮ‬௃೜షభ ௃೜శభ ‫ڮ‬௃ಾ ൈூ೛శభ ൈ‫ڮ‬ൈூಿ and ࣴ௃భ ൈ‫ڮ‬ൈ௃೜షభ ൈூభ ‫ڮ‬ூ೛షభ ூ೛శభ ‫ڮ‬ூಿ ൈ௃೜శభ ൈ‫ڮ‬ൈ௃ಾ are contracted forms of the tensor ࣴ.

7 2.3.

Block Kronecker product The block Kronecker product, introduced by Tracy and Singh [38], also called ʌ-product [39]

or Tracy-Singh product [40], is a Kronecker product of partitioned matrices. Intuitively, a partitioned matrix can be interpreted as a matrix that has a concatenation of horizontal and/or vertical sections called blocks or submatrices. In the case of equally partitioned matrices, the block Kronecker product is said to be balanced [39]. In this paper, we use the notation ‫ ڇ‬proposed in [40]. Given the matrices ‫ ۯ‬ൌ ൣ‫ۯ‬௝ ൧ ‫ א‬ԧெൈ௃ே and ۰ ൌ ൣ۰௝ ൧ ‫ א‬ԧ௉ൈ௃ொ , composed of ‫ ܬ‬blocks ‫ۯ‬௝ ‫ א‬ԧெൈே and ۰௝ ‫ א‬ԧ௉ൈொ , respectively, the balanced block Kronecker product is defined as ‫ ڇ ۯ‬۰ ൌ ሾ‫ۯ‬ଵ ٔ ۰ଵ 2.4.

‫ ۯ‬ଶ ٔ ۰ଶ ‫ڮ‬

‫ۯ‬௃ ٔ ۰௃ ሿ ‫ א‬ԧெ௉ൈ௃ேொ Ǥ

(10)

Generalized Tucker decomposition The generalized Tucker model corresponds to a Tucker model where some (or all) matrix

factors are replaced by tensors, resulting in mode-n products between two tensors. In [3], the authors introduced a generalized Tucker–(ܰଵ ,ܰ) decomposition, which has ܰ െ ܰଵ (for ܰ ൐ ܰଵ ) factors equal to identity matrices. For a fourth-order tensor ࣲ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర , an example of a generalized Tucker– (2,4) decomposition with tensor factors ࣛሺଵሻ ‫ א‬ԧூభ ൈோభ ൈூర and ࣛሺଷሻ ‫ א‬ԧூయ ൈோయ ൈூర is given by ࣲ ൌ ࣡ ൈଵ ࣛሺଵሻ ൈଷ ࣛሺଷሻ , where ࣡ ‫ א‬ԧோభ ൈூమ ൈோయ ൈூర is the core tensor, which can be written in scalar form as ோభ

ோయ ሺଵሻ

ሺଷሻ

‫ݔ‬௜భ ǡ௜మ ǡ௜య ǡ௜ర ൌ  ෍ ෍ ݃௥భ ǡ௜మ ǡ௥య ǡ௜ర ܽ௜భ ǡ௥భ ǡ௜ర ܽ௜య ǡ௥య ǡ௜ర Ǥ

(11)

௥భ ୀଵ ௥య ୀଵ

A tall mode-3 unfolding of ࣲ is given by [3] ሺଵሻ

ሺଷሻ

‫ ܆‬ூమ ூర ூభ ൈூయ ൌ ቂ۷ூమ ٔ ܾ݀݅ܽ݃ ቀ‫ ۯ‬ήή௜ర ቁቃ ۵ூమ ூర ோభ ൈூర ோయ ‫ۯ‬ூర ோయ ൈூయ ǡ ሺଷሻ

(12) ሺଵሻ

ሺଵሻ

ሺଵሻ

where ‫ۯ‬ூర ோయ ൈூయ is a tall mode-1 unfolding of ࣛ ሺଷሻ , ܾ݀݅ܽ݃ ቀ‫ ۯ‬ήή௜ర ቁ ‫ ܾ݃ܽ݅݀ ؜‬ቀ‫ ۯ‬ήήଵ ǡ ‫ ڮ‬ǡ ‫ ۯ‬ήήூర ቁ ‫א‬ ் ்

்

ԧூర ூభ ൈூర ோభ and ۵ூమ ூర ோభ ൈூర ோయ ൌ ቂܾ݀݅ܽ݃൫۵ήଵή௜ర ൯ ‫ܾ݃ܽ݅݀ ڮ‬൫۵ήூమ ή௜ర ൯ ቃ . A second useful unfolding of ࣲ is obtained by combining the first and third modes into one dimension and the second and fourth modes into the other dimension, as follows ሺଵሻ

ሺଷሻ

‫ ܆‬ூభ ூయ ൈூర ூమ ൌ ቂ‫ ۯ‬ூభ ൈூర ோభ ‫ۯ ڇ‬ூయ ൈூర ோయ ቃ ۵ூర ோభ ோయ ൈூర ூమ ǡ ሺଵሻ

(13)

ሺଷሻ

where the unfoldings ‫ۯ‬ூభ ൈூర ோభ and ‫ ۯ‬ூయ ൈூర ோయ are block matrices with ‫ܫ‬ସ column blocks and the balanced ሺଵሻ

ሺଷሻ

ሺଵሻ

ሺଷሻ

ሺଵሻ

ሺଷሻ

block Kronecker product, as defined in (10), is ‫ ۯ‬ூభ ൈூర ோభ ‫ ۯ ڇ‬ூయ ൈூర ோయ ൌ ቂ‫ۯ‬ήήଵ ٔ ‫ۯ‬ήήଵ ‫ ۯڮ‬ήήூర ٔ ‫ۯ‬ήήூర ቃ. The demonstration of (13) is given in Appendix A.

8

ൌ

Figure 1: Block-diagram of the nested Tucker decomposition of a fourth-order tensor.

2.5.

Nested Tucker decomposition (NTD) Based on tensor train (TT) decompositions [41-42], the NTD introduced in [10] for a fourth-

order tensor ࣲ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర is given by ோభ

ோమ

ோయ

ோర ሺଵሻ

ሺଵሻ

ሺଶሻ

ሺଶሻ

ሺଷሻ

‫ݔ‬௜భ ǡ௜మ ǡ௜య ǡ௜ర ൌ  ෍ ෍ ෍ ෍ ܽ௜భ ǡ௥భ ݃௥భ ǡ௜మ ǡ௥మ ܽ௥మ ǡ௥య ݃௥య ǡ௜య ǡ௥ర ܽ௜ర ǡ௥ర ǡ

(14)

௥భ ୀଵ ௥మ ୀଵ ௥య ୀଵ ௥ర ୀଵ

where ࣡ ሺଵሻ ‫ א‬ԧோభ ൈூమ ൈோమ and ࣡ ሺଶሻ ‫ א‬ԧோయ ൈூయ ൈோర are the core tensors of two Tucker–(2,3) models, and ‫ۯ‬ሺଵሻ ‫ א‬ԧூభ ൈோభ , ‫ۯ‬ሺଶሻ ‫ א‬ԧோమ ൈோయ and ‫ۯ‬ሺଷሻ ‫ א‬ԧூర ൈோర are matrix factors, with ‫ۯ‬ሺଶሻ being the common factor. A block-diagram of this NTD model is illustrated in Figure 1 showing that the NTD model can be viewed as the nesting of two Tucker–(2,3) models ൫࣡ ሺଵሻ ǡ ‫ۯ‬ሺଵሻ ǡ ‫ۯ‬ሺଶሻ ൯ and ൫࣡ ሺଶሻ ǡ ‫ۯ‬ሺଶሻ ǡ ‫ۯ‬ሺଷሻ ൯. It can also be viewed as the contraction of the following ࣮ ሺଵሻ and ࣮ ሺଶሻ tensors ோభ ሺଵሻ ‫ݐ‬௜భ ǡ௜మ ǡ௥య

ோమ ሺଵሻ

ሺଵሻ

ሺଶሻ

ൌ ෍ ෍ ܽ௜భ ǡ௥భ ݃௥భ ǡ௜మ ǡ௥మ ܽ௥మ ǡ௥య ǡ

(15)

௥భ ୀଵ ௥మ ୀଵ ோర

ሺଶሻ ‫ݐ‬௥య ǡ௜య ǡ௜ర

with ࣮

ሺଵሻ

‫א‬ԧ

ூభ ൈூమ ൈோయ

and ࣮

ሺଶሻ

‫א‬ԧ

ሺଶሻ

ሺଷሻ

ൌ ෍ ݃௥య ǡ௜య ǡ௥ర ܽ௜ర ǡ௥ర ǡ

(16)

௥ర ୀଵ

ோయ ൈூయ ൈூర

. Thus, the tensorࣲ is obtained by contracting the tensors

࣮ ሺଵሻ and ࣮ ሺଶሻ along their common mode (‫ݎ‬ଷ ), as defined in (6)-(7), which yields ࣲ ൌ ࣮ ሺଵሻ ‫ ࣮ כ‬ሺଶሻ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర Ǥ ௥య

2.6.

(17)

Coupled tensor decomposition Coupled tensor decompositions have been recently proposed in [15-17] for PARAFAC

models. A set of tensor decompositions is said to be “coupled” when at least one of the involved factors is common to all the decompositions. For instance, given a collection of tensors ࣲሺ௡ሻ ‫א‬ ԧூభ ൈூమ ൈூయ satisfying third-order PARAFAC models, with ݊ ‫ א‬ሾͳǡ ܰሿ for ܰ ൒ ʹ, which yields the fourth-order tensor ࣲ ‫ א‬ԧூభ ൈூమ ൈூయ ൈே , a coupled PARAFAC decomposition is given as ோ ሺଵሻ

ሺଶሻ

ሺଷሻ

ࣲሺ௡ሻ ൌ ෍ ‫܉‬ή௥ሺ௡ሻ ‫܉ ל‬ή௥ሺ௡ሻ ‫܉ ל‬ή௥ ǡ ௥ୀଵ

(18)

9

ൌ

൅

൅

൅

ൌ

൅

൅

൅

Figure 2: Coupled PARAFAC decomposition of a fourth-order tensor.

ሺଵሻ

ሺଷሻ

ሺଶሻ

ሺଵሻ

ሺଵሻ

ሺଵሻ

where ‫܉‬ή௥ሺ௡ሻ ‫ א‬ԧூభ , ‫܉‬ή௥ሺ௡ሻ ‫ א‬ԧூమ and ‫܉‬ή௥ ‫ א‬ԧூయ , with factor matrices ‫ ۯ‬ሺ௡ሻ ൌ ቂ‫܉‬ήଵሺ௡ሻ ǡ ‫ ڮ‬ǡ ‫܉‬ήோሺ௡ሻ ቃ ‫א‬ ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ

ሺଷሻ

ԧூభ ൈோ , ‫ ۯ‬ሺ௡ሻ ൌ ቂ‫܉‬ήଵሺ௡ሻ ǡ ‫ ڮ‬ǡ ‫܉‬ήோሺ௡ሻ ቃ ‫ א‬ԧூమ ൈோ and ‫ۯ‬ሺଷሻ ൌ ቂ‫܉‬ήଵ ǡ ‫ ڮ‬ǡ ‫܉‬ήோ ቃ ‫ א‬ԧூయ ൈோ . Notice that the common matrix factor ‫ۯ‬ሺଷሻ is independent of the index ݊, being common to all the ܰ tensors. Figure 2 illustrates a block-diagram for this coupled decomposition. The models presented in Subsections 2.5 and 2.6 will serve as premises for the new tensor decomposition proposed in Section 4. 3. Two-hop MIMO multi-relay system In this section, we propose a two-hop MIMO multi-relay system with TSTC at the source and the relays. The proposed communication system generalizes the system model of [10] by introducing an additional diversity to the system: the cooperative diversity, obtained through the use of multiple relays. Let us consider the system shown in Figure 3, which is composed of a source (S) sending data to the destination through multiple relays (ଵ ǡ ‫ ڮ‬ǡ  ஽ ), where ‫ ܦ‬is the number of relays. Table 1 gives the definitions and the dimensions of the matrices and the tensors used for modeling this system. The ሺௌோሻ

ሺோ஽ሻ

source-relay (۶ήήௗ ) and relay-destination (۶ήήௗ ) channels, for ݀ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܦ‬, are assumed to be flat fading and quasi-static. All nodes of the system employ multiple antennas and the relays operate in half-duplex mode. In this section, for the sake of simplicity, we consider the noiseless case for describing the system model. The following key assumptions are made: (i) the direct link between the source and destination nodes is not available, corresponding to a link with deep fading; (ii) the code tensors are known at the destination; (iii) the relays use the amplify-and-forward (AF) protocol; (iv) the relays are synchronized at symbol level and they transmit in orthogonal channels in different time slots. The global transmission is composed of ‫ ܦ‬൅ ͳ steps, the first step corresponding to the transmission from the source to the relays and the remaining ‫ ܦ‬steps corresponding to the sequential transmission from the ‫ܦ‬ relays to the destination. While a relay is transmitting to the destination, the other relays remain silent.

10 1

1

R1 1

1

S

D 1

1

RD

Figure 3: Two-hop MIMO multi-relay system. Table 1: Dimensions of matrices and tensors used in the model. Description Transmitted symbols matrix Coding tensor at the source SR channel tensor Coding tensor at the relays

Symbols ‫܁‬

Dimensions ܰൈܴ

ࣝ ሺௌሻ

‫ܯ‬ௌ ൈ ܲ ൈ ܴ

࣢

ሺௌோሻ

‫ܯ‬ோ ൈ ‫ܯ‬ௌ ൈ ‫ܦ‬

ࣝ

ሺோሻ

‫ ்ܯ‬ൈ ‫ ܬ‬ൈ ‫ܯ‬ோ ൈ ‫ܦ‬

RD channel tensor

࣢ ሺோ஽ሻ

‫ܯ‬஽ ൈ ‫ ்ܯ‬ൈ ‫ܦ‬

Received signals tensor

ࣲ ሺௌோ஽ሻ

‫ܯ‬஽ ൈ ‫ ܬ‬ൈ ܲ ൈ ܰ ൈ ‫ܦ‬

In the first hop of transmission, ܴ data streams composed of ܰ symbols each are transmitted by ‫ܯ‬ௌ antennas at the source. TSTC at the source using the coding tensor ࣝ ሺௌሻ ‫ א‬ԧெೄ ൈ௉ൈோ provides temporal spreading with length ܲ, leading to the following tensor of coded signals to be transmitted ࣲ ሺௌሻ ൌ ࣝ ሺௌሻ ൈଷ ‫ א ܁‬ԧெೄ ൈ௉ൈே ǡ

(19)

with entries ோ ሺௌሻ ‫ݔ‬௠ೄ ǡ௣ǡ௡

ሺௌሻ

ൌ ෍ ܿ௠ೄ ǡ௣ǡ௥ ‫ݏ‬௡ǡ௥ ǡ

(20)

௥ୀଵ

with ݉ௌ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܯ‬ௌ , ‫ ݌‬ൌ ͳǡ ‫ ڮ‬ǡ ܲ and ݊ ൌ ͳǡ ‫ ڮ‬ǡ ܰ. The signals received by ‫ܯ‬ோ antennas at the ݀-th relay, during the ݊-th symbol period of the ‫݌‬-th transmission block form a fourth-order tensor given by ெೄ ሺோሻ ‫ݔ‬௠ೃ ǡ௣ǡ௡ǡௗ

ሺௌோሻ

ሺௌሻ

ൌ ෍ ݄௠ೃ ǡ௠ೄ ǡௗ ‫ݔ‬௠ೄ ǡ௣ǡ௡ ǡ

(21)

௠ೄ ୀଵ

ሺோሻ

with ݉ோ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܯ‬ோ . Each relay ݀ reencodes the received signals, using also a TSTC ࣝௗ

‫א‬

ԧெ೅ ൈ௃ൈெೃ with length ‫ܬ‬, before transmitting the coded signals to destination. The signals transmitted by ‫ ்ܯ‬antennas at the ݀-th relay, associated with the ݊-th symbol period of the ‫݌‬-th transmission block at the source and the ݆-th transmission block at the relay are given by ெೃ ሺ்ሻ ‫ݔ‬௠೅ ǡ௝ǡ௣ǡ௡ǡௗ

ሺோሻ

ሺோሻ

ൌ ෍ ܿ௠೅ ǡ௝ǡ௠ೃ ǡௗ ‫ݔ‬௠ೃ ǡ௣ǡ௡ǡௗ Ǥ ௠ೃ ୀଵ

(22)

11 with ݉ ் ൌ ͳǡ ‫ ڮ‬ǡ ‫ ்ܯ‬, ݆ ൌ ͳǡ ‫ ڮ‬ǡ ‫ ܬ‬and ݀ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܦ‬. Finally, after transmission from the ݀-th relay, the noiseless tensor of signals received by ‫ܯ‬஽ antennas at the destination is given by ெ೅ ሺௌோ஽ሻ ‫ݔ‬௠ವ ǡ௝ǡ௣ǡ௡ǡௗ

ሺோ஽ሻ

ሺ்ሻ

ൌ ෍ ݄௠ವ ǡ௠೅ ǡௗ ‫ݔ‬௠೅ ǡ௝ǡ௣ǡ௡ǡௗ ǡ

(23)

௠೅ ୀଵ

ሺ்ሻ

with ݉஽ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܯ‬஽ . Replacing ‫ݔ‬௠೅ ǡ௝ǡ௣ǡ௡ǡௗ by its expression deduced from (20)-(22) gives ெ೅ ሺௌோ஽ሻ ‫ݔ‬௠ವ ǡ௝ǡ௣ǡ௡ǡௗ

ெೄ

ெೃ

ோ ሺோ஽ሻ

ሺௌோሻ

ሺோሻ

ሺௌሻ

ൌ ෍ ෍ ෍ ෍ ݄௠ವ ǡ௠೅ ǡௗ ܿ௠೅ ǡ௝ǡ௠ೃ ǡௗ ݄௠ೃ ǡ௠ೄ ǡௗ ܿ௠ೄ ǡ௣ǡ௥ ‫ݏ‬௡ǡ௥ Ǥ

(24)

௠೅ ୀଵ ௠ೃ ୀଵ ௠ೄ ୀଵ ௥ୀଵ

Thus, the signals received at destination form a fifth-order tensor ࣲ ሺௌோ஽ሻ ‫ א‬ԧெವ ൈ௃ൈ௉ൈࡺൈ஽ , whose modes are associated with space, coding (at source and relays), time and cooperative diversity. Let us define the following generalized Tucker–(2,4) and Tucker–(1,3) models ெ೅ ሺଵሻ ‫ݐ‬௠ವ ǡ௝ǡ௠ೄ ǡௗ

ெೃ ሺோ஽ሻ

ሺோሻ

ሺௌோሻ

ൌ ෍ ෍ ܿ௠೅ ǡ௝ǡ௠ೃ ǡௗ ݄௠ವ ǡ௠೅ ǡௗ ݄௠ೃ ǡ௠ೄ ǡௗ ǡ

(25)

௠೅ ୀଵ ௠ೃ ୀଵ ோ

ሺଶሻ ‫ݐ‬௠ೄ ǡ௣ǡ௡

ሺௌሻ

ሺௌሻ

ൌ ෍ ܿ௠ೄ ǡ௣ǡ௥ ‫ݏ‬௡ǡ௥ ൌ ‫ݔ‬௠ೄ ǡ௣ǡ௡ ǡ

(26)

௥ୀଵ

which can be expressed in tensor form as ᇱ

࣮ ሺଵሻ ൌ ࣝ ሺோሻ ൈଵ ࣢ ሺோ஽ሻ ൈଷ ࣢ ሺௌோሻ ‫ א‬ԧெವ ൈ௃ൈெೄ ൈ஽ ǡ

(27)

࣮ ሺଶሻ ൌ ࣲ ሺௌሻ ൌ ࣝ ሺௌሻ ൈଷ ‫ א ܁‬ԧெೄ ൈ௉ൈே ǡ

(28)

ᇱ

where the tensor ࣢ ሺௌோሻ ‫ א‬ԧெೄ ൈெೃ ൈ஽ is formed by permuting the first two modes of ࣢ ሺௌோሻ ‫א‬ ሺௌோሻ ᇱ

ԧெೃ ൈெೄ ൈ஽ , i.e. ۶ήήௗ

ሺௌோሻ ்

ൌ ۶ήήௗ

‫ א‬ԧெೄ ൈெೃ . Thus, we can write the received signal model (24) as a

contraction between the tensors ࣮ ሺଵሻ and ࣮ ሺଶሻ ࣲ ሺௌோ஽ሻ ൌ ࣮ ሺଵሻ ‫ ࣮ כ‬ሺଶሻ ‫ א‬ԧெವ ൈ௃ൈ௉ൈேൈ஽ Ǥ ௠ೄ

(29)

Fixing the index ݀ in (27), the generalized Tucker–(2,4) model ࣮ ሺଵሻ ‫ א‬ԧெವ ൈ௃ൈெೄ ൈ஽ can be ሺଵሻ

written as ‫ ܦ‬Tucker–(2,3) models ࣮ሺௗሻ ‫ א‬ԧெವ ൈ௃ൈெೄ , leading to the following coupled NTD model ሺௌோ஽ሻ

ࣲሺௗሻ ሺௌோ஽ሻ

where ࣲሺௗሻ

ሺଵሻ

ൌ ࣮ሺௗሻ ‫ ࣮ כ‬ሺଶሻ ‫ א‬ԧெವ ൈ௃ൈ௉ൈࡺ ǡ ௠ೄ

(30)

is the tensor of signals received at destination from the ݀-th relay, which satisfies a

fourth-order NTD, as introduced in [10] and recalled in Subsection 2.5. Figure 4 shows a block-diagram of the tensor of received signals ࣲ ሺௌோ஽ሻ decomposed into ‫ܦ‬ branches, illustrating that the tensor of coded signals at the source ࣮ ሺଶሻ ൌ ࣲ ሺௌሻ is transmitted to the destination via ‫ ܦ‬relays. Each branch corresponds to a NTD that shares the same tensorࣲ ሺௌሻ with the other branches, characterizing a coupling of ‫ ܦ‬NTD models. Hence, the aim is to jointly estimate the

12

ൌ 

ൌ

Figure 4: Tensor model of noiseless received signals.

transmitted information symbols and the channels from this dataset using a semi-blind receiver. In the next section, we formally define this new coupled tensor model for the case of a fifthorder tensor and, then, we study its uniqueness properties. 4. Coupled nested Tucker decomposition In this section, we introduce a new tensor decomposition that can be viewed as a coupling of multiple NTDs that share common factors. This decomposition is called a coupled nested Tucker decomposition (CNTD) and, in this paper, it is defined for fifth-order tensors, although it may be generalized to higher order tensors. This new tensor model generalizes the NTD by assuming tensor factors and it extends the coupling concept, initially defined for PARAFAC models. 4.1.

CNTD model In the sequel, we consider a generic formulation of the CNTD model for a fifth-order tensor

ࣲ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర ൈூఱ . Defining the tensors ࣛሺଵሻ ‫ א‬ԧூభ ൈோభ ൈூఱ , ࣡ ሺଵሻ ‫ א‬ԧோభ ൈூమ ൈோమ ൈூఱ ,ࣛ ሺଶሻ ‫ א‬ԧோమ ൈோయ ൈூఱ and ࣡ ሺଶሻ ‫ א‬ԧோయ ൈூయ ൈோర , and the matrix ‫ۯ‬ሺଷሻ ‫ א‬ԧூర ൈோర , the CNTD of ࣲ is given by ோభ

ோమ

ோయ

ோర ሺଵሻ

ሺଵሻ

ሺଶሻ

ሺଶሻ

ሺଷሻ

‫ݔ‬௜భ ǡ௜మ ǡ௜య ǡ௜ర ǡ௜ఱ ൌ  ෍ ෍ ෍ ෍ ܽ௜భ ǡ௥భ ǡ௜ఱ ݃௥భ ǡ௜మ ǡ௥మ ǡ௜ఱ ܽ௥మ ǡ௥య ǡ௜ఱ ݃௥య ǡ௜య ǡ௥ర ܽ௜ర ǡ௥ర ǡ

(31)

௥భ ୀଵ ௥మ ୀଵ ௥య ୀଵ ௥ర ୀଵ

which has the following correspondences with the received signals model (24) ൫ࣲ ሺௌோ஽ሻ ǡ ࣢ ሺோ஽ሻ ǡ ࣝ ሺோሻ ǡ ࣢ ሺௌோሻ ǡ ࣝ ሺௌሻ ǡ ‫܁‬൯ ฻ ൫ࣲǡ ࣛ ሺଵሻ ǡ ࣡ ሺଵሻ ǡ ࣛ ሺଶሻ ǡ ࣡ ሺଶሻ ǡ ‫ۯ‬ሺଷሻ ൯

(32)

ሺ‫ܯ‬஽ ǡ ‫ܬ‬ǡ ܲǡ ܰǡ ‫ܦ‬ǡ ‫ ்ܯ‬ǡ ‫ܯ‬ோ ǡ ‫ܯ‬ௌ ǡ ܴሻ ฻ ሺ‫ܫ‬ଵ ǡ ‫ܫ‬ଶ ǡ ‫ܫ‬ଷ ǡ ‫ܫ‬ସ ǡ ‫ܫ‬ହ ǡ ܴଵ ǡ ܴଶ ǡ ܴଷ ǡ ܴସ ሻǤ

(33)

Thus, (27)-(26) become ᇱ

 ࣮ ሺଵሻ ൌ ࣡ ሺଵሻ ൈଵ ࣛ ሺଵሻ ൈଷ ࣛ ሺଶሻ ‫ א‬ԧூభ ൈூమ ൈோయ ൈூఱ ǡ

(34)

࣮ ሺଶሻ ൌ ࣡ ሺଶሻ ൈଷ ‫ۯ‬ሺଷሻ ‫ א‬ԧோయ ൈூయ ൈூర ǡ

(35)

ோభ



ሺଵሻ ‫ݐ‬௜భ ǡ௜మ ǡ௥య ǡ௜ఱ

ோమ ሺଵሻ

ሺଵሻ

ሺଶሻ

ൌ  ෍ ෍ ݃௥భ ǡ௜మ ǡ௥మ ǡ௜ఱ ܽ௜భ ǡ௥భ ǡ௜ఱ ܽ௥మ ǡ௥య ǡ௜ఱ ǡ ௥భ ୀଵ ௥మ ୀଵ

(36)

13

ൌ

ൌ

Figure 5: Coupled nested Tucker decomposition. ோర ሺଶሻ ‫ݐ‬௥య ǡ௜య ǡ௜ర

with the tensor ࣛ

ሺଶሻ ᇱ

ሺଶሻ

ሺଷሻ

ൌ  ෍ ݃௥య ǡ௜య ǡ௥ర ܽ௜ర ǡ௥ర ǡ

(37)

௥ర ୀଵ

‫ א‬ԧோయ ൈோమ ൈூఱ obtained by transposing the mode-3 slices of ࣛሺଶሻ , as defined in

Section 3. The tensors ࣮ ሺଵሻ and ࣮ ሺଶሻ are called components of the CNTD, while ࣛ ሺଵሻ , ࣛሺଶሻ and ‫ۯ‬ሺଷሻ are the decomposition factors, and ࣡ ሺଵሻ and ࣡ ሺଶሻ are the core tensors. The tensor ࣲ in (31) is then obtained by contracting the tensors ࣮ ሺଵሻ and ࣮ ሺଶሻ along their common mode (‫ݎ‬ଷ ) ࣲ ൌ ࣮ ሺଵሻ ‫ ࣮ כ‬ሺଶሻ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర ൈூఱ Ǥ ௥య

(38)

By fixing the index ݅ହ , we get the NTD model ሺଵሻ

ࣲሺ௜ఱ ሻ ൌ ࣮ሺ௜ఱ ሻ ‫ ࣮ כ‬ሺଶሻ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర ǡ ௥య

ሺଵሻ

ሺଵሻ

ሺଵሻ

(39)

ሺଶሻ ்

with ࣮ሺ௜ఱ ሻ ൌ ࣡ሺ௜ఱ ሻ ൈଵ ‫ ۯ‬ήή௜ఱ ൈଷ ‫ۯ‬ήή௜ఱ ‫ א‬ԧூభ ൈூమ ൈோయ . Then, the CNTD (38) can be seen as a coupling of ‫ܫ‬ହ fourth-order NTDs in the sense that the component ࣮ ሺଶሻ is common to all ࣲሺ௜ఱ ሻ , for ݅ହ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܫ‬ହ . A block-diagram for this tensor decomposition is shown in Figure 5. The first level of the figure illustrates the multiple NTDs given in (39) and the second one represents the CNTD defined in (38). From the figure, it is easy to note that the coupling of ‫ܫ‬ହ branches leads to a compact tensor structure, which replaces the matrix factors by tensor factors. Thus, in the case of a fifth-order tensor, CNTD generalizes NTD by coupling ‫ܫ‬ହ NTDs or, equivalently, by contracting a generalized Tucker– (2,4) model with a Tucker–(1,3) one. 4.2.

Uniqueness In this subsection, we first address the uniqueness of the generalized Tucker model and, then,

we deduce the uniqueness properties of the proposed CNTD model.

14 4.2.1. Uniqueness of the generalized Tucker model Let us consider the unfolding (13) for the tensor ࣮ ሺଵሻ defined in (34) ሺଵሻ

ሺଵሻ

ሺଶሻ

ሺଵሻ

‫܂‬ூభ ோయ ൈூఱ ூమ ൌ ቀ‫ ۯ‬ூభ ൈூఱ ோభ ‫ۯ ڇ‬ோయ ൈூఱ ோమ ቁ ۵ூఱ ோభ ோమ ൈூఱ ூమ Ǥ

(40) ሺଵሻ

By fixing the index ݅ହ , ࣮ ሺଵሻ becomes a Tucker–(2,3) model, denoted by ࣮ሺ௜ఱ ሻ , and the unfolding (40) simplifies as ሺଵሻ

ሺଵሻ

ሺଶሻ ்

ሺଵሻ

ቂ‫܂‬ூభ ோయ ൈூమ ቃ ൌ ቀ‫ۯ‬ήή௜ఱ ٔ ‫ ۯ‬ήή௜ఱ ቁ ቂ۵ோభ ோమ ൈூమ ቃ ǡ ௜ఱ

(41)

௜ఱ

where ሾήሿ௜ఱ represents a matrix with the index ݅ହ fixed. The generalized Tucker–(2,4) model is not essentially unique when the core tensor is unknown, since its factors are unique up to nonsingular ்

ഥሺଵሻ ൌ ‫ۯ‬ሺଵሻ ۲ሺଵሻ , ‫ۯ‬ ഥሺଶሻ ൌ transformations. Let us consider an alternative solution for (41) given by ‫ۯ‬ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ሺଶሻ ்

ሺଶሻ

ሺ௡ሻ

‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ , for ݅ହ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܫ‬ହ , and ࣡ҧ ሺଵሻ ൌ ࣡ ሺଵሻ ൈଵ ࣠ ሺଵሻ ൈଷ ࣠ ሺଶሻ , where the nonsingular matrices ۲ήή௜ఱ ‫א‬ ԧோ೙ ൈோ೙ are slices of the ambiguity tensors ࣞ ሺ௡ሻ ‫ א‬ԧோ೙ ൈோ೙ ൈூఱ , and ࣠ ሺ୬ሻ ‫ א‬ԧோ೙ ൈோ೙ ൈூఱ has mode-3 ሺ௡ሻ ିଵ

slices defined as ۲ήή௜ఱ

ഥூሺଵሻ ቃ , for ݊ ൌ ͳǡ ʹ. The alternative unfolding ቂ‫܂‬ భ ோయ ൈூమ

௜ఱ

obtained with the triplet

்

ഥሺଶሻ ǡ ࣡ҧ ሺଵሻ ቁ lead to the same unfolding ቂ‫܂‬ூሺଵሻ ഥሺଵሻ ǡ ‫ۯ‬ ቀ‫ۯ‬ ቃ given in (41). However, if we consider the ήή௜ఱ ήή௜ఱ భ ோయ ൈூమ ௜ఱ

core tensor ࣡ ሺଵሻ known, the following uniqueness theorem is valid for generalized Tucker models. The proof of the Theorem 1 is given in Appendix B. Theorem 1. Consider the fourth-order tensor ࣮ ሺଵሻ ‫ א‬ԧூభ ൈூమ ൈோయ ൈூఱ defined in (34). When the core tensor ࣡ ሺଵሻ ‫ א‬ԧோభ ൈூమ ൈோమ ൈூఱ is known, the factors ࣛ ሺଵሻ ‫ א‬ԧூభ ൈோభ ൈூఱ and ࣛ ሺଶሻ ‫ א‬ԧோమ ൈோయ ൈூఱ are unique ்

்

ഥሺଵሻ ൌ ‫ۯ‬ሺଵሻ ۲ሺଵሻ and ‫ۯ‬ ഥሺଶሻ ൌ ‫ۯ‬ሺଶሻ ۲ሺଶሻ , such that ۲ሺ௡ሻ ൌ Ƚሺ௡ሻ ۷ோ , up to the following ambiguities: ‫ۯ‬ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ௜ఱ ೙ ሺଵሻ ሺଶሻ

for ݊ ൌ ͳǡ ʹ, with Ƚ௜ఱ Ƚ௜ఱ ൌ ͳ, ‫݅׊‬ହ ‫ א‬ሾͳǡ ‫ܫ‬ହ ሿ. 4.2.2. Uniqueness of the coupled nested Tucker decomposition The contraction in (38) can be written as a mode-1 product deduced from (9) ሺଵሻ

ࣲூమ ூఱ ூభ ൈூయ ൈூర ൌ ࣮ ሺଶሻ ൈଵ ‫܂‬ூమ ூఱ ூభ ൈோయ ǡ

(42)

ሺଵሻ

where ‫܂‬ூమ ூఱ ூభ ൈோయ is a tall mode-3 unfolding of ࣮ ሺଵሻ and ࣲூమ ூఱ ூభ ൈூయ ൈூర is a third-order contracted form of ࣲ obtained by combining the first, second and fifth modes. Assuming ࣮ത ሺଶሻ ൌ ࣮ ሺଶሻ ൈଵ ઢିଵ and ሺଵሻ ഥூሺଵሻ ൌ ‫܂‬ூమ ூఱ ூభ ൈோయ ઢ as alternative solutions for (42), where ઢ ‫ א‬ԧோయ ൈோయ is a nonsingular ambiguity ‫܂‬ మ ூఱ ூభ ൈோయ

matrix, the tensor ࣲூమ ூఱ ூభ ൈூయ ൈூర becomes ሺଵሻ ഥூ ூ ூ ൈூ ൈூ ൌ ࣮ത ሺଶሻ ൈଵ ‫܂‬ ഥூሺଵሻ ࣲ ൌ ൫࣮ ሺଶሻ ൈଵ ઢି૚ ൯ ൈଵ ቀ‫܂‬ூమ ூఱ ூభ ൈோయ ઢቁ మ ఱ భ య ర మ ூఱ ூభ ൈோయ

(43)

15   

ሺଵሻ

ൌ ࣮ ሺଶሻ ൈଵ ቀ‫܂‬ூమ ூఱ ூభ ൈோయ ઢઢିଵ ቁ ሺଵሻ

ൌ ࣮ ሺଶሻ ൈଵ ‫܂‬ூమ ூఱ ூభ ൈோయ ൌ ࣲூమ ூఱ ூభ ൈூయ ൈூర Ǥ

This identity implies that the contracted form ࣲூమ ூఱ ூభ ൈூయ ൈூర is not unique, since ࣮ത ሺଵሻ ‫࣮ כ‬ത ሺଶሻ ൌ ௥య

࣮

ሺଵሻ

‫࣮כ‬

ሺଶሻ

௥య

. In other words, it is not possible to uniquely decompose ࣲ into the components ࣮ ሺଵሻ and

࣮ ሺଶሻ . Consequently, the tensor factors of the CNTD of ࣲ are not unique. However, in the following theorem, we demonstrate that, under certain conditions, the CNTD (31) is unique up to scaling ambiguities. The proof of Theorem 2 is given in Appendix C. Theorem 2. Consider the fifth-order tensor ࣲ ‫ א‬ԧூభ ൈூమ ൈூయ ൈூర ൈூఱ expressed by means of a CNTD, as in (38), by contracting the tensors ࣮ ሺଵሻ ‫ א‬ԧூభ ൈூమ ൈோయ ൈூఱ and ࣮ ሺଶሻ ‫ א‬ԧோయ ൈூయ ൈூర over their common mode (‫ݎ‬ଷ ). When the core tensors ࣡ ሺଵሻ ‫ א‬ԧோభ ൈூమ ൈோమ ൈூఱ and ࣡ ሺଶሻ ‫ א‬ԧோయ ൈூయ ൈோర are known, the factors ࣛ ሺଵሻ ‫א‬ ഥሺଵሻ ൌ ԧூభ ൈோభ ൈூఱ , ࣛ ሺଶሻ ‫ א‬ԧோమ ൈோయ ൈூఱ and ‫ۯ‬ሺଷሻ ‫ א‬ԧூర ൈோర are unique up to the following ambiguities: ‫ۯ‬ ήή௜ఱ ்

்

ሺଵሻ ሺଵሻ ഥሺଶሻ ሺଶሻ ሺଶሻ ഥ ሺଷሻ ൌ ‫ۯ‬ሺଷሻ ۲ሺଷሻ , such that ۲ሺଵሻ ൌ Ƚሺଵሻ ۷ோ , ۲ሺଶሻ ൌ Ƚሺଶሻ ۷ோ and ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ , ‫ۯ‬ ήή௜ఱ ൌ ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ and ‫ۯ‬ ήή௜ఱ ௜ఱ ήή௜ఱ ௜ఱ భ మ ሺଵሻ ሺଶሻ

۲ሺଷሻ ൌ Ƚሺଷሻ ۷ோర , with Ƚ௜ఱ Ƚ௜ఱ Ƚሺଷሻ ൌ ͳ, ‫݅׊‬ହ ‫ א‬ሾͳǡ ‫ܫ‬ହ ሿ. Remark 1. The uniqueness of NTD under the knowledge of the core tensors was not proved in [10]. However, one can note that (42)-(43), as well as the proof in Appendix C, remain valid if the index ݅ହ is fixed, which corresponds to a NTD. In other words, the uniqueness properties discussed here are also valid for the NTD model. 5. Semi-blind receiver By exploiting the CNTD modeling of the proposed MIMO relaying system, we propose a semi-blind receiver for jointly estimating the symbol matrix and the channel tensors. The receiver is a closed-form solution based on the least squares (LS) estimation of Kronecker products, denoted LSKP. We assume that the coding tensors ࣝ ሺோሻ and ࣝ ሺௌሻ , corresponding to the core tensors of the models (27) and (28), are known by the receiver. Thus, we have the channel tensors ࣢ ሺோ஽ሻ and ࣢ ሺௌோሻ and the symbol matrix ‫ ܁‬to be estimated. The knowledge of the coding tensors ensures that these factors are unique up to scaling ambiguities (see Theorem 2). By using the correspondences (32) and (33), the ambiguity relations given in Theorem 2 ሺோ஽ሻ

become Ƚௗ

ሺௌோሻ

Ƚௗ

ሺோ஽ሻ

Ƚሺ‫܁‬ሻ ൌ ͳ, where Ƚௗ

ሺௌோሻ

, Ƚௗ

and Ƚሺ‫܁‬ሻ are the scalar ambiguity factors of ࣢ ሺோ஽ሻ ,

࣢ ሺௌோሻ and‫܁‬, respectively. Note that the ambiguities on ࣢ ሺோ஽ሻ and ࣢ ሺௌோሻ depend on the relay. The LSKP receiver is a closed-form solution that consists of LS estimations of Kronecker products based on a low-rank approximation algorithm. Applying (8) to the tensor ࣲ ሺௌோ஽ሻ given in (29), we get the following mode-3 product

16 ሺௌோ஽ሻ

ሺଶሻ

ࣲெವ ൈ௃ൈ௉ேൈ஽ ൌ ࣮ ሺଵሻ ൈଷ ‫܂‬௉ேൈெೄ ǡ

(44)

ሺଶሻ

where ‫܂‬௉ேൈெೄ is mode-1 unfolding of (28). By replacing (27) into (44) and using the Property 5, the ሺௌோ஽ሻ

tensor ࣲெವ ൈ௃ൈ௉ேൈ஽ becomes ᇱ

ሺௌோ஽ሻ

ሺଶሻ

ࣲெವ ൈ௃ൈ௉ேൈ஽ ൌ ࣝ ሺோሻ ൈଵ ࣢ ሺோ஽ሻ ൈଷ ቀ࣢ ሺௌோሻ ൈଵ ‫܂‬௉ேൈெೄ ቁ Ǥ ᇱ

(45)

ሺଶሻ

Defining ࣰ ‫ ࣢ ؜‬ሺௌோሻ ൈଵ ‫܂‬௉ேൈெೄ ‫ א‬ԧ௉ேൈெೃ ൈ஽ , the model (45) is a generalized Tucker–(2,4) decomposition, which allows us to write, from (13), the following unfolding ሺௌோ஽ሻ

ሺோ஽ሻ

ሺோሻ

‫ ܆‬ெವ ௉ேൈ஽௃ ൌ ቀ۶ெವ ൈ஽ெ೅ ‫܄ ڇ‬௉ேൈ஽ெೃ ቁ ۱஽ெ೅ ெೃ ൈ஽௃ ǡ

(46)

where ሺଶሻ

ሺௌோሻ

ሺௌሻ

ሺௌோሻ

‫܄‬௉ேൈ஽ெೃ ൌ ‫܂‬௉ேൈெೄ ۶ெೄ ൈ஽ெೃ ൌ ሺ۷௉ ٔ ‫܁‬ሻ۱௉ோൈெೄ ۶ெೄ ൈ஽ெೃ Ǥ

(47)

On the other hand, applying (9) to (29) and replacing ࣮ ሺଶሻ by (28), we can deduce the following mode-1 product ሺௌோ஽ሻ

ሺଵሻ

ࣲ௃஽ெವ ൈ௉ൈே ൌ ࣮ ሺଶሻ ൈଵ ‫܂‬௃஽ெವ ൈெೄ ሺଵሻ

ൌ ࣝ ሺௌሻ ൈଵ ‫܂‬௃஽ெವ ൈெೄ ൈଷ ‫܁‬Ǥ

(48)

The model (48) is a Tucker–(2,3) decomposition, from which we can write the following tall unfolding ሺௌோ஽ሻ

ሺଵሻ

ሺௌሻ

‫ ܆‬ே௃஽ெವ ൈ௉ ൌ ቀ‫܂ ٔ ܁‬௃஽ெವ ൈெೄ ቁ ۱ோெೄ ൈ௉ ǡ

(49)

ሺଵሻ

where the matrix ‫܂‬௃஽ெವ ൈெೄ is the following tall mode-3 unfolding of ࣮ ሺଵሻ deduced from (12) as ሺோ஽ሻ

ሺଵሻ

ሺோሻ

ሺௌோሻ

‫܂‬௃஽ெವ ൈெೄ ൌ ቂ۷௃ ٔ ܾ݀݅ܽ݃ ቀ۶ήήௗ ቁቃ ۱௃஽ெ೅ ൈ஽ெೃ ۶஽ெೃ ൈெೄ Ǥ Defining

the

matrices

ሺோ஽ሻ

૖ ‫ ؜‬۶ெವ ൈ஽ெ೅ ‫܄ ڇ‬௉ேൈ஽ெೃ ‫ א‬ԧெವ ௉ேൈ஽ெ೅ ெೃ

(50) and

ષ‫ٔ܁؜‬

ሺଵሻ

‫܂‬௃஽ெವ ൈெೄ ‫ א‬ԧே௃஽ெವ ൈோெೄ , we deduce the following LS estimates of the Kronecker products from (46) and (49) ற

ሺோሻ ෡ ൌ ‫ ܆‬ሺௌோ஽ሻ ૖ ெವ ௉ேൈ஽௃ ቀ۱஽ெ೅ ெೃ ൈ஽௃ ቁ ǡ

(51)

ற

ሺௌோ஽ሻ ሺௌሻ ෡ ൌ ‫ ܆‬ே௃஽ெ ቀ۱ோெೄ ൈ௉ ቁ Ǥ ષ ವ ൈ௉

(52) ሺோ஽ሻ

The main idea of this receiver is to estimate the matrices ۶ெವ ൈ஽ெ೅ , ‫܄‬௉ேൈ஽ெೃ , ‫ ܁‬and ሺଵሻ

‫܂‬௃஽ெವ ൈெೄ as factors of a balanced block Kronecker product and of a Kronecker product, by applying a ෡ and ષ ෡ in (51) and (52), SVD-based low-rank approximation algorithm [43], using the LS estimates ૖ respectively. In [10], the authors applied the low-rank approximation algorithm to two Kronecker products in the context of MIMO relaying systems. We have adapted the algorithm to a block

17 ሺଵሻ ሺௌோሻ ෡௉ேൈ஽ெ and ‫܂‬ ෡௃஽ெ Kronecker product. Once ‫܄‬ are estimated, the unfolding ۶஽ெೃ ൈெೄ can be ೃ ವ ൈெೄ

estimated from (47) or (50). This algorithm has the interesting advantage of benefitting from the choice of a coding designed from a unitary (or orthogonal in the real case) matrix. This comes from the fact that the LSKP technique uses only one matrix unfolding of each coding tensor. The “unitary coding” consists of a tensor chosen in such a way that one of its unfoldings is a truncated unitary matrix, leading the unfolded coding to be row-orthonormal. Since a row-orthonormal matrix ۱ has its Hermitian transpose equal to its inverse (i.e. ۱۱ு ൌ ۷), the computation of the pseudo-inverses in (51) and (52) are simplified. In addition, unitary transformations are isometric (preserve the norm) and therefore avoid ሺௌሻ

noise enhancement by conserving the energy of the received signals. In the simulation, we use ۱ோெೄ ൈ௉ ሺோሻ

and ۱஽ெ೅ ெೃ ൈ஽௃ equal to a truncated discrete Fourier transform (DFT) matrix. Note that for computing the pseudo-inverses in (51) and (52), as well as for computing ሺௌோሻ ෡ ஽ெ ۶ , some conditions are required to ensure the uniqueness of LS solutions. The unfoldings ೃ ൈெೄ ሺோሻ

ሺௌሻ

۱஽ெ೅ ெೃ ൈ஽௃ and ۱ோெೄ ൈ௉ must be full row rank and then the conditions ‫ ܬ‬൒ ‫ܯ ்ܯ‬ோ and ܲ ൒ ܴ‫ܯ‬ௌ are ሺௌோሻ

necessary. For the LS estimate of ۶஽ெೃ ൈெೄ from (47) or (50), we must have ܴܲ ൒ ‫ܯ‬ௌ and ܰ ൒ ܴ or ‫ ்ܯܬ‬൒ ‫ܯ‬ோ and ‫ܯ‬஽ ൒ ‫ ்ܯ‬. For elimitanting the scaling ambiguities on the estimates, we assume the knowledge of one pilot symbol (‫ݏ‬ଵǡଵ ). In order to plot the simulation results, we assumed the a priori knowledge of one ሺோ஽ሻ

coefficient of ࣢ ሺோ஽ሻ for each relay (݄ଵǡଵǡௗ ) such that we get the following ambiguity relations ሺோ஽ሻ ሺோ஽ሻ ෡ ሺோ஽ሻ ෡ ήήௗ ۶ ՚ Ƚௗ ۶ ήήௗ ǡ

‫܁‬෠ ՚ Ƚሺ‫܁‬ሻ ‫܁‬෠ǡ ሺோ஽ሻ

with Ƚௗ

෡ήήௗ ՚ ቀȽሺோ஽ሻ ቁ ‫܄‬ ௗ

ିଵ

෡ήήௗ ǡ ‫܄‬

ିଵ ሺଵሻ ሺଵሻ ෡௃஽ெ ෡௃஽ெ ൈெ ǡ ‫܂‬ ՚ ൫Ƚሺ‫܁‬ሻ ൯ ‫܂‬ ವ ൈெೄ ವ ೄ

(53)

ሺோ஽ሻ ሺோ஽ሻ ൌ ݄ଵǡଵǡௗ ൗ݄෠ଵǡଵǡௗ and Ƚሺ‫܁‬ሻ ൌ ‫ݏ‬ଵǡଵ Τ‫ݏ‬Ƹଵǡଵ . In practice, such a priori information can be obtained

by a simple supervised procedure using a pilot-symbol sent from the relays to the destination. This procedure has already been adopted in other works [9, 10, 32, 34] in the context of relaying systems. The LSKP receiver is described in Table 2. The computational complexity of the proposed receiver is ࣩ൫ܰ‫ܯܬܲܦ‬஽ ሺ‫ܯܦ‬ோ ‫ ்ܯ‬൅ ܴ‫ܯ‬ௌ ሻ൯. Assuming ܴ ൌ ‫ܯ‬஽ ൌ ‫ܯ‬ோ ൌ ‫ ்ܯ‬ൌ ‫ܯ‬ௌ ൌ ‫ ܯ‬and ܲ ൌ ‫ܬ‬, it becomes ࣩሺܰ‫ܦ‬ଶ ܲଶ ‫ܯ‬ଷ ሻ. 6.

Simulation results In this section, we provide Monte Carlo simulation results to illustrate the effectiveness of the

proposed two-hop MIMO multi-relay system. In order to evaluate the impact of the choice of the design parameters on the system behavior (regardless of the influence of algorithm), we consider a zero-forcing (ZF) receiver, which is obtained by assuming a perfect knowledge of the channel tensors at destination. The ZF receiver is defined from an unfolding of (48) as

18

1.

Table 2: LSKP receiver. ሺோ஽ሻ Calculate the LS estimate of the block Kronecker product ૖ ൌ ۶ெವ ൈ஽ெ೅ ‫܄ ڇ‬௉ேൈ஽ெೃ : ற

ሺோሻ ෡ ൌ‫܆‬ ෩ ሺௌோ஽ሻ ૖ ெವ ௉ேൈ஽௃ ൫۱஽ெ೅ ெೃ ൈ஽௃ ൯ Ǥ

2. Apply the SVD-based low-rank approximation algorithm to estimate the matrices ሺோ஽ሻ ෡ ෡ெ ෡ ۶ ൈ஽ெ and ‫܄‬௉ேൈ஽ெ from ૖ calculated in step 1. ವ

೅

ೃ

ሺͳሻ

3. Calculate the LS estimate of the Kronecker product ષ ൌ ‫ ܦܯܦܬ܂ ٔ ܁‬ൈ‫ ܵܯ‬: ற

ሺௌோ஽ሻ ሺௌሻ ෡ ൌ‫܆‬ ෩ ே௃஽ெ ષ ൫۱ோெೄ ൈ௉ ൯ Ǥ ವ ൈ௉

4. Apply the SVD-based low-rank approximation algorithm to estimate the matrices ሺଵሻ ෡ calculated in step 3. ෡௃஽ெ ‫܁‬෠ and ‫܂‬ from ષ ವ ൈெೄ 5. Eliminate the scaling ambiguities using (53). 6. Estimate the channel ࣢ ሺௌோሻ , from (47), using ற ሺௌோሻ ෠ ሺௌሻ ෡ ෡ெ ۶ ൈ஽ெ ൌ ൣ൫۷௉ ٔ ‫܁‬൯۱௉ோൈெ ൧ ‫܄‬௉ேൈ஽ெ , ೃ

ೄ

ೃ

ೄ

or from (50), using ሺௌோሻ ሺோሻ ෡ ሺோ஽ሻ ෡ ஽ெ ۶ ൈெ ൌ ൣ൫۷௃ ٔ ܾ݀݅ܽ݃൫۶ήήௗ ൯൯۱௃஽ெ ೃ

ற

೅ ൈ஽ெೃ

ೄ

ሺଵሻ ෡௃஽ெ ൧ ‫܂‬ . ವ ൈெೄ

7. Project the estimated symbols onto the symbol alphabet.

ሺோ஽ሻ

ሺோሻ

ሺௌோሻ

ሺௌሻ

ற

ሺௌோ஽ሻ

‫ ் ܁‬ൌ ൬ቀ۷௉ ۪ ቂቀ۷௃ ۪ ܾ݀݅ܽ݃ ቀ۶ήήௗ ቁቁ ۱௃஽ெ೅ ൈ஽ெೃ ۶஽ெೃ ൈெೄ ቃቁ ۱௉ெೄ ൈோ ൰ ‫ ܆‬௉௃஽ெವ ൈே Ǥ

(54)

Then, the performances of the proposed LSKP receivers is evaluated. 6.1.

Simulation settings The receiver performances were averaged over ͷ ൈ ͳͲସ Monte Carlo runs for various system

configurations. The symbol-error-rate (SER) and the normalized mean square error (NMSE) of the channels were plotted as function of the transmission power to noise spectral density ratio (்ܲ Τܰ଴ ), where ்ܲ is the total transmission power of each node (after TSTC). The transmitted symbols were randomly generated from a unit energy 4-QAM alphabet. Several system configurations were tested for the proposed system. The design parameter values used in the simulations are indicated above each figure. In all the simulations, we consider the same number of receiving and transmitting antennas at the relays (‫ܯ‬ோ ൌ ‫) ்ܯ‬. The channel tensors ࣢ ሺௌோሻ and ࣢ ሺோ஽ሻ are assumed to be Rayleigh flat fading and quasi-static, composed of i.i.d. complex Gaussian entries with zero-mean and unit variance. The channel powers were adjusted taking into account an exponential path-loss model, which inserts an attenuation ܲ௅ that depends on the distance ‫ ܮ‬between each node, given by ܲ௅ ൌ ‫ିܮ‬ସ . The distance ‫ ܮ‬of each hop was considered the same and equal to ‫ܮ‬଴ Τʹ, where ‫ܮ‬଴ is the source-destination distance, arbitrarily chosen equal to 1. Additive white Gaussian noises (AWGN) were added at each receiving node with the same noise variance ܰ଴ . In the noisy model, we consider the global noise tensor ࣨ ሺௌோ஽ሻ ‫ א‬ԧெವ ൈ௃ൈ௉ൈேൈ஽ given by

19 ெ೅ ሺௌோ஽ሻ ߟ௠ವ ǡ௝ǡ௣ǡ௡ǡௗ ሺ஽ሻ

ൌ

ሺ஽ሻ ඥܰ଴ ߟ௠ವ ǡ௝ǡ௣ǡ௡ǡௗ

ெೃ ሺோ஽ሻ

ሺோሻ

ሺோሻ

൅ ෍ ෍ ݄௠ವ ǡ௠೅ ǡௗ ܿ௠೅ ǡ௝ǡ௠ೃ ǡௗ ቀඥܰ଴ ߟ௠ೃ ǡ௣ǡ௡ǡௗ ቁǡ

(55)

௠೅ ୀଵ ௠ೃ ୀଵ

ሺோሻ

where ߟ௠ವ ǡ௝ǡ௣ǡ௡ǡௗ and ߟ௠ೃ ǡ௣ǡ௡ǡௗ are entries of the tensors ࣨ ሺ஽ሻ ‫ א‬ԧெವ ൈ௃ൈ௉ൈேൈ஽ and ࣨ ሺோሻ ‫א‬ ԧெೃ ൈ௉ൈேൈ஽ that represent the noise at the destination and at the relays, respectively. At each run, ܰ଴ was fixed according to the desired ்ܲ Τܰ଴ value. The coding tensors ࣝ ሺௌሻ and ࣝ ሺோሻ were generated with unit amplitude coefficients and random phase drawn from a uniform distribution between Ͳ and ʹߨ for the simulations with the ZF receiver. For the LSKP receiver, the coding tensors were chosen as truncated DFT matrices as discussed in Section 5. In both cases, each coding tensor was multiplied by a fixed scalar gain so that all the transmission nodes have the same mean power and the total transmission power is kept constant, regardless of the number of relays and antennas. Thus, the coding tensors become ࣝ ሺௌሻ ՚ ඥߚ ሺௌሻ ࣝ ሺௌሻ and ࣝ ሺோሻ ՚ ඥߚ ሺோሻ ࣝ ሺோሻ , with ߚ ሺௌሻ ൌ ்ܲ Τ‫ܯ‬ௌ ܴ and ߚ ሺோሻ ൌ ்ܲ Τ‫ܯ ்ܯ‬ோ ሺ்ܲ ܲ௅ ൅ ܰ଴ ሻ, where ்ܲ ൌ ܲ௧௢௧௔௟ Τሺ‫ ܦ‬൅ ͳሻ, with ܲ௧௢௧௔௟ being the total system power arbitrarily chosen equal to 1. 6.2.

Performance evaluation

6.2.1. ZF receiver with perfect channel knowledge In this subsection, we present some simulation results concerning the use of the ZF receiver in order to study the behavior of the proposed system when some parameters are modified. Although some parameter settings used in this subsection do not satisfy the uniqueness conditions given in Section 5 for the LSKP receiver, the presented results aim to evaluate the impact of the choice of these parameters on the system performance, regardless of the estimation algorithm. Figure 6 shows the SER versus ்ܲ Τܰ଴ for ‫ ܦ‬ൌ ሼͳǡʹǡ͵ǡͶሽ in order to evaluate the impact of an increase of the number of relays on the system performance. The case with a single relay (‫ ܦ‬ൌ ͳ) is equivalent to the two-hop MIMO relay system proposed in [10]. A more reasonable comparison between the two systems will be discussed in more details later. For the ZF receiver, the SER performance is clearly improved when the number of relays is increased. This result shows the gain provided by the cooperative diversity. Unless otherwise defined, the next results are obtained with two relays (‫ ܦ‬ൌ ʹ). In Figure 7, we evaluate the impact of the time-spreading lengths ܲ and ‫ ܬ‬on the system performance. Figure 7 (a) shows the SER versus ்ܲ Τܰ଴ for the combinations ሺܲǡ ‫ܬ‬ሻ ൌ ሺʹǡʹሻ, ሺʹǡͶሻ, ሺʹǡͺሻ, ሺͶǡʹሻ and ሺͺǡʹሻ. We can note that any increase of ܲ or ‫ ܬ‬yields a performance improvement. However, an increase of ܲ provides a greater gain than an increase of ‫ܬ‬. That is evidenced by the difference between the curves with ሺܲǡ ‫ܬ‬ሻ ൌ ሺͶǡʹሻ and ሺͺǡʹሻ, and the ones with ሺܲǡ ‫ܬ‬ሻ ൌ ሺʹǡͶሻ and ሺʹǡͺሻ, and it can be justified by the correlation of the relay noise (55) with the temporal spreading ‫ܬ‬, via relay coding. In order to clarify the dependence of the SER with respect to the time-spreading lengths ‫ ܬ‬and ܲ, Figure 7 (b) shows, for fixed levels of ்ܲ Τܰ଴ , SER curves as functions of ܲ or ‫ܬ‬. In

20

Figure 6: ZF receiver performance for different numbers of relays.

Figure 7: ZF receiver performance for different time-spreading lengths.

this figure, only one of the time-spreading lengths varies at each time, the other one is kept constant. When ܲ varies, we set ‫ ܬ‬ൌ Ͷ, and when ‫ ܬ‬varies, we have ܲ ൌ Ͷ. From the slope of the curves, we can clearly see that an increase of ܲ leads to greater SER gains, as the spread signal at the source is subject to a new spreading at relay node. One can also note a SER floor when ‫ ܬ‬is increased, in scenarios with low ்ܲ Τܰ଴ values, establishing then a trade-off between the performance gain and transmission rate degradation due to an increasing in the time-spreading at the relays. In Figure 8, we evaluate the impact of the number of antennas at the source, relays and destination on the system performance. Figure 8 (a) shows the SER versus ்ܲ Τܰ଴ for the combinations ሺ‫ܯ‬ௌ ǡ ‫ܯ‬ோ ǡ ‫ܯ‬஽ ሻ ൌ ሺʹǡʹǡʹሻ, ሺͶǡʹǡʹሻ, ሺʹǡͶǡʹሻ, ሺʹǡʹǡͶሻ and ሺͶǡͶǡͶሻ. The performance obtained with ሺͶǡͶǡͶሻ, when compared with the one obtained with ሺʹǡʹǡʹሻ, shows the expected improvement by increasing the number of antennas in the MIMO system. From the curves with ሺͶǡʹǡʹሻ, ሺʹǡͶǡʹሻ, ሺʹǡʹǡͶሻ, we also observe that increasing ‫ܯ‬ௌ or ‫ܯ‬ோ leads to a more pronounced improvement than increasing ‫ܯ‬஽ . That can be explained by the dependence of the tensor codings ࣝ ሺௌሻ and ࣝ ሺோሻ with ‫ܯ‬ௌ and ‫ܯ‬ோ . Due to the multiple TSTC, a better performance is expected with an increase of ‫ܯ‬ௌ . However, the similar performance between the curves ሺͶǡʹǡʹሻ and ሺʹǡͶǡʹሻ can be explained from the fact that an increase in the value of ‫ܯ‬ோ implies a greater addition of antennas,

21

Figure 8: ZF receiver performance for different numbers of antennas.

since, globally, the number of relaying antennas depends on the number of relays, i.e. ‫ܯܦ‬ோ . Figure 8 (b) shows the SER versus ்ܲ Τܰ଴ for configurations that keep ‫ܯܦ‬ோ constant. From this figure, we can conclude that exploiting the cooperative diversity (increasing ‫ )ܦ‬is more effective than exploiting the spatial diversity at the relays (increasing ‫ܯ‬ோ ). This comes from the fact the multiple relays use orthogonal channels, i.e. the relays do not interfere with each other, while multiple antennas at the relays interfere each other. 6.2.2. LSKP receiver performance In the next experiments, we evaluate the performance of the LSKP receiver for the proposed MIMO multi-relay system. Firstly, we study the behavior of the proposed semi-blind receiver by changing the number of relays and in the time-spreading lengths, for fixed ்ܲ Τܰ଴ values. Figure 9 (a) shows the SER versus ‫( ܦ‬number of relays) for the LSKP receiver in comparison with the ZF method. This figure shows that the number of relays has a great impact on the SER, illustrating the benefits of the cooperative diversity in the network. Moreover, it is possible to see that the proposed receiver provides a SER performance very close to the one of the ZF method, especially for a low SNR. Figure 9 (b) shows the SER versus ܲ for two values of ‫ ܬ‬and ்ܲ Τܰ଴ . This figure shows that the proposed receiver is able to efficiently exploit the time-spreading at the source and relay in order to improve the SER. Once again, it can be viewed that the LSKP and the ZF receivers provide close SER curves. Figure 10 shows the SER obtained by the LSKP versus ்ܲ Τܰ଴ , for several values of ‫ ܦ‬and ‫ܬ‬, considering the cases (a) ‫ ܦܬ‬ൌ ͺ and (b) ‫ ܦܬ‬ൌ ͳʹ. When ‫ ܦ‬ൌ ͳ, the proposed system is equivalent to the NTD-based MIMO single-relay system introduced in [10]. For a fair comparison between the systems, we plot curves with configurations that preserve the same transmission rate. The transmission rate of the proposed multi-relay system is proportional to ܴȀܲሺ‫ ܦܬ‬൅ ͳሻ. For the single-relay system of [10] (case ‫ ܦ‬ൌ ͳ), the transmission rate is proportional to ܴȀܲሺ‫ ܬ‬൅ ͳሻ. In other words, to keep the same transmission rate for both systems, we use the same values of ܴ and ܲ, and the same value of the product ‫ܦܬ‬. In Figures 10 (a) and 10 (b), one can note significant gains in the SER performance with an

22

Figure 9: SER performance for the LSKP receiver for fixed ்ܲ Τܰ଴ values.

Figure 10: SER performance for the LSKP receiver with (a) ‫ ܦܬ‬ൌ ͺ and (b) ‫ ܦܬ‬ൌ ͳʹ.

increase in the number of relays, for a fixed value of ‫ܦܬ‬. The cooperative diversity exploited by the CNTD allows to improve the symbol estimation in comparison with the NTD-based system. The single-relay system does not exploit cooperative diversity, inducing an increase of ‫ ܬ‬to obtain the same transmission rate. These results show that the cooperative diversity gain is more advantageous than the gain provided by time-spreading. Once again, the LSKP receiver gives performances close to the one obtained with the ZF receiver. In Figures 9 and 10, we can note that the LSKP overcomes the ZF performance in some cases. This degradation can be explained by the fact that the low-rank approximation algorithm used by the LSKP removes the noise subspace, attenuating the effects of the noise. Moreover, the rowሺோሻ

ሺௌሻ

orthonormality of the matrices ۱஽ெ೅ ெೃ ൈ஽௃ and ۱ோெೄ ൈ௉ ǡdiscussed in Section 5, cannot be exploited by the ZF receiver. The above conclusion is reached by analyzing the condition number of the pseudoinverses in (52) and (54), used by the LSKP and ZF receivers for estimating the symbols, as the condition number measures how sensitive a solution is with respect to perturbations in the observed data. It is well known that unitary (isometric) transformations have condition number equal to 1. We have also calculated, by means of simulation, the condition number of the pseudo-inverse in (54). It has an average value of ͳǤ͸ʹ, with a standard deviation of ͲǤͶ͵, for the considered scenario.

23

Figure 11: SER performance comparison for several receivers.

Aiming to illustrate the advantages of the CNTD-based system over other tensor-based approaches, in Figure 11, we provide a SER comparison between the proposed LSKP receiver and two existing systems that consider space-time coding structures. The first one is a receiver based on a Khatri-Rao factorization (KRF) algorithm proposed in [25] for a nested PARAFAC based multi-hop relaying system that uses a simplified KRST coding. The second one is a non-cooperative system using a TSTC at the source proposed in [27], which is equivalent to the one proposed in this paper when relays are absent. In [27], an ALS-based receiver was proposed. For the sake of comparison, we propose a LSKP-based solution for this system, which based on the following relationship ሺௌ஽ሻ

ሺௌሻ

ற

‫۪܁‬۶ ሺௌ஽ሻ ൌ ‫ ܆‬ேெವ ൈ௉ ቀ۱ோெೄ ൈ௉ ቁ ǡ

(56) ሺௌ஽ሻ

where ۶ ሺௌ஽ሻ ‫ א‬ԧெವ ൈெೄ is the channel matrix between the source and destination and ‫ ܆‬ேெವ ൈ௉ is the tall mode-3 unfolding of the received signal tensor ࣲ ሺௌ஽ሻ ‫ א‬ԧெವ ൈ௉ൈே that satisfies a third-order Tucker decomposition. In order to have a fair comparison, we simulated the proposed LSKP with ‫ ܦ‬ൌ ͳ. From Figure 11, we can conclude that the use of TSTC leads to a better performance than the one of [25], due to a more efficient exploitation of the spatial transmit diversity at the source and relay nodes by this coding. When compared to the non-cooperative system, as expected, one can conclude that the proposed relay system provides a remarkable gain in the symbol estimation, which can be explained by the smaller path-loss experienced by each hop in a cooperative system, leading to a less severe channel attenuation. The new TSTC applied by the relay, inserting more diversity, also justify this outperforming. The simulations results presented in this subsection show a significant SER gain of the proposed technique with respect to the techniques of [10, 25, 27]. 6.2.3. Channel estimation performance Concerning to the channel estimation, Figures 12 to 14 provide the NMSE of the estimated

24

Figure 12: NMSE of ࣢ ሺௌோሻ for the LSKP receiver with (a) ‫ ܦܬ‬ൌ ͺ and (b) ‫ ܦܬ‬ൌ ͳʹ.

Figure 13: NMSE of ࣢ ሺோ஽ሻ for the LSKP receiver with (a) ‫ ܦܬ‬ൌ ͺ and (b) ‫ ܦܬ‬ൌ ͳʹ.

Figure 14: Channel NMSE comparison for several receivers.

channels computed as follows ܰ‫ ܧܵܯ‬ൌ

ͳ ෡௠௖ ฮଶ ൗ ԡ࣢௠௖ ԡଶி ቁ ǡ ෍ ቀฮ࣢௠௖ െ ࣢ ி ‫ܥܯ‬

(57)

௠௖

with ݉ܿ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܥܯ‬, being ‫ ܥܯ‬the number of Monte Carlo runs. Figures 12 (a) and 13 (a) show respectively the NMSE of ࣢ ሺௌோሻ and ࣢ ሺோ஽ሻ versus ்ܲ Τܰ଴ for ‫ ܦܬ‬ൌ ͺ and Figures 12 (b) and 13 (b)

25 show respectively the NMSE of ࣢ ሺௌோሻ and ࣢ ሺோ஽ሻ versus ்ܲ Τܰ଴ for ‫ ܦܬ‬ൌ ͳʹ. It can be noted in Figures 12 (a) and 12 (b) that the NMSEs of ࣢ ሺௌோሻ are similar for all the tested configurations, including the two different values of the product ‫ܦܬ‬. However, for the NMSE of ࣢ ሺோ஽ሻ , one can note in Figures 13 (a) and 13 (b) a degradation in the estimate of the channels when the number of relays is increased. That comes from the fact that, with more relays, there are more channel coefficients to be estimated by the receiver with the same number of received signals. Note that, when we fix ‫ܦܬ‬, the system has the same quantity of data arriving at destination. Figure 14 compares the NMSE of ࣢ ሺௌோሻ and ࣢ ሺோ஽ሻ provided by the proposed receiver with the ones provided by the technique of [25] and by the non-cooperative approach of [27], based on (56). Once again, in order to have a fair comparison, we simulated the proposed LSKP with ‫ ܦ‬ൌ ͳ. From this figure, we can conclude that the non-cooperative approach provides NMSE much worst the other methods. Moreover, the proposed receiver and the method of [25] provided roughly similar NMSE curves. 7.

Conclusion A two-hop MIMO multi-relay system using TSTC at the source and the relay nodes has been

proposed. The signals received at destination define a fifth-order tensor, which satisfies a new tensor model, called coupled nested Tucker decomposition (CNTD). This model which generalizes the nested Tucker decomposition (NTD), can be viewed as the coupling of several NTDs that share common factors. The CNTD is characterized by a contraction between a generalized Tucker decomposition and a Tucker one. Assuming the core tensors known, essential uniqueness of a fifth-order CNTD has been established. Based on this tensor model, a semi-blind receiver has been derived for the proposed system, which exploits the cooperative diversity induced by ‫ ܦ‬relays operating in a sequential way. The performance of the proposed receiver has been evaluated by means of extensive Monte Carlo simulations. The simulation results show the effectiveness of exploiting the cooperative diversity by increasing the number of relays. In comparison with the NTD-based single-relay system [10], the proposed CNTD-based system allows to improve the SER performance, keeping the same transmission rate. It is also shown that the CNTD-based system with TSTC at the source and the relays outperform the cooperative system of [25] and the non-cooperative system of [27], which are based on other tensor approaches. A perspective of this work concerns an extension of the proposed tensor decomposition to higher order tensors, combining multi-relay with OFDM modulation, i.e. multi-relay OFDM systems. Other perspectives include the development of other optimization algorithms, including supervised techniques, as well as theoretical analyses of the performance and numerical stability of the algorithm. Appendix A Proof of unfolding (13): Using the element-wise form (11), we can define the unfolding ‫ ܆‬ூభ ூయ ൈூర ூమ as

26 ூర

ூయ

ூభ

ூమ

ሺூ ሻ

ሺூ ሻ

ሺூ ሻ ்

ሺூ ሻ

‫ ܆‬ூభ ூయ ൈூర ூమ ൌ ෍ ෍ ෍ ෍ ‫ݔ‬௜భ ǡ௜మ ǡ௜య ǡ௜ర ቀ‫܍‬௜భభ ٔ ‫܍‬௜యయ ቁ ቀ‫܍‬௜రర ٔ ‫܍‬௜మమ ቁ ௜ర ୀଵ ௜య ୀଵ ௜మ ୀଵ ௜భ ୀଵ ூయ ூర ூమ ூభ ோభ

ோయ ሺଵሻ

ሺூ ሻ

ሺூ ሻ

ሺଷሻ

ሺூ ሻ

ሺூ ሻ ்

ൌ ෍ ෍ ෍ ෍ ෍ ෍ ݃௥భ ǡ௜మ ǡ௥య ǡ௜ర ܽ௜భ ǡ௥భ ǡ௜ర ܽ௜య ǡ௥య ǡ௜ర ቀ‫܍‬௜భభ ٔ ‫܍‬௜యయ ቁ ቀ‫܍‬௜రర ٔ ‫܍‬௜మమ ቁ  ௜ర ୀଵ ௜య ୀଵ ௜మ ୀଵ ௜భ ୀଵ ௥భ ୀଵ ௥య ୀଵ

(58)

with ݃௥భ ǡ௜మ ǡ௥య ǡ௜ర given by ሺூ ሻ

ሺோ ሻ ்

ሺோ ሻ

ሺூ ሻ

ሺூ ሻ

(59)

݃௥భ ǡ௜మ ǡ௥య ǡ௜ర ൌ ቀ‫܍‬௜రర ٔ ‫܍‬௥భ భ ٔ ‫܍‬௥య య ቁ ۵ூర ோభ ோయ ൈூర ூమ ቀ‫܍‬௜రర ٔ ‫܍‬௜మమ ቁǡ ሺூ ሻ

where the vector ‫܍‬௜೙೙ represents the ݅௡ -th vector of the canonical base of the Euclidean space Թூ೙ . Replacing (11) and (59) into (58) gives ூయ

ூర

ூమ

ூభ

ோయ

ோభ

ሺଵሻ

ሺூ ሻ

ሺூ ሻ

ሺଷሻ

‫ ܆‬ூభ ூయ ൈூర ூమ ൌ ෍ ෍ ෍ ෍ ෍ ෍ ܽ௜భ ǡ௥భ ǡ௜ర ܽ௜య ǡ௥య ǡ௜ర ቀ‫܍‬௜భభ ٔ ‫܍‬௜యయ ቁ ௜ర ୀଵ ௜య ୀଵ ௜మ ୀଵ ௜భ ୀଵ ௥భ ୀଵ ௥య ୀଵ ሺூ ሻ

ሺோ ሻ ்

ሺோ ሻ

ሺூ ሻ

ሺூ ሻ

ሺூ ሻ ்

ሺூ ሻ

ή ቀ‫܍‬௜రర ٔ ‫܍‬௥భ భ ٔ ‫܍‬௥య య ቁ ۵ூర ோభ ோయ ൈூర ூమ ቀ‫܍‬௜రర ٔ ‫܍‬௜మమ ቁ ቀ‫܍‬௜రర ٔ ‫܍‬௜మమ ቁ Ǥ

(60)

Using the property (2) and separating the summations, we get ூర

‫ ܆‬ூభ ூయ ൈூర ூమ ൌ ቎ ෍ ௜ర ୀଵ ூయ

ோయ

෍ ෍ ௜య ୀଵ ௥య ୀଵ

ሺଷሻ ܽ௜య ǡ௥య ǡ௜ర

ோభ

ூభ

ሺூ ሻ ் ቌ‫܍‬௜రర

ሺூ ሻ ሺோ ሻ ்

ሺଵሻ

ٔ ෍ ෍ ܽ௜భ ǡ௥భ ǡ௜ర ൬‫܍‬௜భభ ‫܍‬௥భ భ ൰ ٔ ௜భ ୀଵ ௥భ ୀଵ

ሺூ ሻ ሺோ ሻ ் ቀ‫܍‬௜యయ ‫܍‬௥య య ቁቍ቏ ۵ூర ோభ ோయ ൈூర ூమ

ூర

቎෍ ௜ర ୀଵ

ሺூ ሻ ሺூ ሻ ் ‫܍‬௜రర ‫܍‬௜రర

ூమ

ሺூ ሻ ሺூ ሻ ்

ٔ ෍ ‫܍‬௜మమ ‫܍‬௜మమ ቏ǡ ௜మ ୀଵ

(61)

which simplifies as ூర

ሺூ ሻ ்

‫ ܆‬ூభ ூయ ൈூర ூమ ൌ ෍ ൬‫܍‬௜రర ௜ర ୀଵ

ሺଵሻ

ሺଵሻ

ሺଷሻ

ٔ ‫ ۯ‬ήή௜ర ٔ ‫ ۯ‬ήή௜ర ൰ ۵ூర ோభ ோయ ൈூర ூమ ൫۷ூర ٔ ۷ூమ ൯

ሺଷሻ

ሺଵሻ

 ൌ ቀ‫ۯ‬ήήଵ ٔ ‫ۯ‬ήήଵ

ሺଷሻ

‫ ۯ‬ήήଶ ٔ ‫ ۯ‬ήήଶ ‫ڮ‬

ሺଵሻ

ሺଷሻ

‫ ۯ‬ήήூర ٔ ‫ ۯ‬ήήூర ቁ۵ூర ோభ ோయ ൈூర ூమ ǡ

(62)

which corresponds to the unfolding (13). Note that the term between parentheses is equivalent to a block Kronecker product as defined in (10). Appendix B ்

்

ഥሺଶሻ ൌ ‫ۯ‬ሺଶሻ ۲ሺଶሻ and ഥሺଵሻ ൌ ‫ۯ‬ሺଵሻ ۲ሺଵሻ , ‫ۯ‬ Proof of Theorem 1: Considering the alternative solutions ‫ۯ‬ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ήή௜ఱ ሺଵሻ ିଵ

࣡ҧ ሺଵሻ ൌ ࣡ ሺଵሻ ൈଵ ࣠ ሺଵሻ ൈଷ ࣠ ሺଶሻ , with ൣ࣡ҧ ሺଵሻ ൧௜ ൌ ൣ࣡ ሺଵሻ ൧௜ ൈଵ ۲ήή௜ఱ ఱ

ఱ

ሺଶሻ ିଵ

ൈଶ ۲ήή௜ఱ

for any value of ݅ହ , (41)

becomes ்

ഥோሺଵሻோ ൈூ ቃ ഥሺଵሻ ٔ ‫ۯ‬ ഥሺଶሻ ቁ ቂ۵ ഥூሺଵሻ ቃ ൌ ቀ‫ۯ‬ ቂ‫܂‬ ήή௜ఱ ήή௜ఱ భ ோయ ൈூమ భ మ మ ௜ఱ

௜ఱ



(63)

27 ሺଵሻ

ሺଶሻ ்

ሺଵሻ

ሺଵሻ ିଵ

ሺଶሻ

 ൌ ቀ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ ٔ ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ ቁ ቀ۲ήήହ ሺଵሻ

ሺଵሻ

ሺଵሻ ିଵ

ൌ ቀ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ ۲ήή௜ఱ

ሺଶሻ ்

ሺଵሻ

ሺଶሻ ்

ሺଶሻ ିଵ

ٔ ۲ήή௜ఱ

ሺଶሻ ିଵ

ሺଶሻ

ٔ ‫ ۯ‬ήή௜ఱ ۲ήή௜ఱ ۲ήή௜ఱ ሺଵሻ

ሺଵሻ

ቁ ቂ۵ோభ ோమ ൈூమ ቃ

௜ఱ

ሺଵሻ

ቁ ቂ۵ோభ ோమ ൈூమ ቃ

௜ఱ

ሺଵሻ

ൌ ቀ‫ ۯ‬ήή௜ఱ ٔ ‫ ۯ‬ήή௜ఱ ቁ ቂ۵ோభ ோమ ൈூమ ቃ ൌ ቂ‫܂‬ூభ ோయ ൈூమ ቃ ǡ ௜ఱ

௜ఱ

்

ഥሺଶሻ ǡ ࣡ҧ ሺଵሻ ቁ lead to the same unfolding ቂ‫܂‬ூሺଵሻ ഥሺଵሻ ǡ ‫ۯ‬ ቃ . showing that the alternative solutions ቀ‫ۯ‬ ήή௜ఱ ήή௜ఱ భ ோయ ൈூమ ௜ఱ

However, if we consider the core tensor ࣡ ሺଵሻ known, we get ்

ሺଵሻ ሺଵሻ ሺଶሻ ሺଶሻ ሺଵሻ ഥூሺଵሻ ቃ ൌ ቀ‫ۯ‬ήή௜ఱ ۲ήή௜ఱ ٔ ‫ۯ‬ήή௜ఱ ۲ήή௜ఱ ቁ ቂ۵ோభ ோమ ൈூమ ቃ ቂ‫܂‬ భ ோయ ൈூమ ௜ఱ

ሺଶሻ ்

ሺଵሻ

ሺଵሻ

ሺଶሻ



௜ఱ

ሺଵሻ

 ൌ ቀ‫ ۯ‬ήή௜ఱ ٔ ‫ ۯ‬ήή௜ఱ ቁ ቀ۲ήή௜ఱ ٔ ۲ήή௜ఱ ቁ ቂ۵ோభ ோమ ൈூమ ቃ Ǥ ௜ఱ

(64)

By comparing (64) with (41), we can conclude that the tensors ࣮ ሺଵሻ and ࣮ത ሺଵሻ are identical if ሺଵሻ

ሺଶሻ

۲ήή௜ఱ ٔ ۲ήή௜ఱ ൌ ۷ோభ ோమ ǡ ሺ௡ሻ

ሺ௡ሻ

(65)

ሺଵሻ ሺଶሻ

ሺ௡ሻ

which implies ۲ήή௜ఱ ൌ Ƚ௜ఱ ۷ோ೙ , for ݊ ൌ ͳǡ ʹ, with Ƚ௜ఱ Ƚ௜ఱ ൌ ͳ. Each Ƚ௜ఱ is the scalar ambiguity for the ݅ହ -th mode-3 slice of ࣛ ሺ௡ሻ . Appendix C Proof of Theorem 2: as ࣮ ሺଵሻ is a generalized Tucker decomposition, the factors ࣛሺଵሻ and ࣛ ሺଶሻ admit ்

்

ഥሺଵሻ ൌ Ƚሺଵሻ ‫ۯ‬ሺଵሻ and ‫ۯ‬ ഥሺଶሻ ൌ Ƚሺଶሻ ‫ۯ‬ሺଶሻ , for ݅ହ ൌ ͳǡ ‫ ڮ‬ǡ ‫ܫ‬ହ , as shown in Subsection the ambiguities ‫ۯ‬ ήή௜ఱ ௜ఱ ήή௜ఱ ήή௜ఱ ௜ఱ ήή௜ఱ 4.2.1. On the other hand, ࣮ ሺଶሻ is a Tucker decomposition and the factor ‫ۯ‬ሺଷሻ admits the ambiguity ഥ ሺଷሻ ൌ Ƚሺଷሻ ‫ۯ‬ሺଷሻ . In (43), we have assumed ࣮ത ሺଶሻ as an alternative solution for ࣮ ሺଶሻ . We can then write ‫ۯ‬ ഥ ሺଷሻ ൯ ൈଵ ઢିଵ ǡ ࣮ത ሺଶሻ ൌ ൫࣡ ሺଶሻ ൈଷ ‫ۯ‬

(66)

where ઢ ‫ א‬ԧோయ ൈோయ is a nonsingular ambiguity matrix. Note that the ambiguity matrix ઢ acts on the first mode of ࣮ ሺଶሻ . However, ࣮ ሺଶሻ only has a factor in third mode, which means that the decomposition factor of first mode of ࣮ ሺଶሻ is known and equal to identity matrix ۷ோయ . Then, we conclude that ઢ ൌ ۷ோయ , which implies that the contracted form ࣲூమ ூఱ ூభ ൈூయ ൈூర in (42) is unique. ሺଵሻ

ሺଶሻ

Now we prove the relation between the scalar ambiguities Ƚ௜ఱ , Ƚ௜ఱ

and Ƚሺଷሻ of the

decomposition factors. By fixing the index ݅ହ in (43), we have ഥூ ூ ൈூ ൈூ ൧ ൌ ࣮ത ሺଶሻ ൈଵ ቂ‫܂‬ ഥூሺଵሻ ቃ ǡ ൣࣲ మ భ య ర మ ூభ ൈோయ ௜ఱ

௜ఱ

(67)

ሺଵሻ

ഥூ ூ ൈோ ቃ is be deduced from (12) as where ቂ‫܂‬ మ భ య ௜ఱ

ഥሺଵሻ ቁ ቂ۵ூሺଵሻோ ൈோ ቃ ‫ۯ‬ ഥሺଶሻ Ǥ ഥூሺଵሻ ቃ ൌ ቀ۷ூమ ٔ ‫ۯ‬ ቂ‫܂‬ ήή௜ఱ ήή௜ఱ మ ூభ ൈோయ మ భ మ ௜ఱ

௜ఱ

(68)

28 Considering ઢ ൌ ۷ோయ and replacing (66) and (68) into (67) gives ഥூ ூ ൈூ ൈூ ൧ ൌ ൫࣡ ሺଶሻ ൈଷ ‫ۯ‬ ഥ ሺଷሻ ൯ ൈଵ ቀ۷ூ ٔ ‫ۯ‬ ഥሺଵሻ ቁ ቂ۵ூሺଵሻோ ൈோ ቃ ‫ۯ‬ ഥሺଶሻ ǡ ൣࣲ ήή௜ఱ ήή௜ఱ మ భ య ర మ మ భ మ ௜ఱ

௜ఱ

(69)

which yields the following tall unfolding ഥூ ூ ூ ൈூ ൧ ൌ ቈቆቀ۷ூ ٔ ‫ۯ‬ ഥሺଵሻ ቁ ቂ۵ூሺଵሻோ ൈோ ቃ ‫ۯ‬ ഥሺଶሻ ቇ ٔ ‫ۯ‬ ഥ ሺଷሻ ቉ ۵ோሺଶሻோ ൈூ  ൣࣲ ήή௜ఱ ήή௜ఱ ర మ భ య మ మ భ మ ర య య ௜ఱ

௜ఱ

ሺଵሻ ሺଵሻ

ሺଵሻ

ሺଶሻ ሺଶሻ

ሺଶሻ

ൌ ቈቆቀ۷ூమ ٔ Ƚ௜ఱ ‫ۯ‬ήή௜ఱ ቁ ቂ۵ூమ ோభ ൈோమ ቃ Ƚ௜ఱ ‫ۯ‬ήή௜ఱ ቇ ٔ Ƚሺଷሻ ‫ۯ‬ሺଷሻ ቉ ۵ோర ோయ ൈூయ  ௜ఱ

ሺଵሻ ሺଶሻ

ሺଵሻ

ሺଵሻ

ሺଶሻ

ሺଶሻ

ൌ Ƚ௜ఱ Ƚ௜ఱ Ƚሺଷሻ ቈቆቀ۷ூమ ٔ ‫ۯ‬ήή௜ఱ ቁ ቂ۵ூమ ோభ ൈோమ ቃ ‫ ۯ‬ήή௜ఱ ቇ ٔ ‫ۯ‬ሺଷሻ ቉ ۵ோర ோయ ൈூయ Ǥ ௜ఱ

(70)

ሺଵሻ ሺଶሻ

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[ ]Danilo S. Rocha was born in Fortaleza, Brazil, in 1986. He is graduated in physics from the Universidade Federal do Cear´a (UFC), Brazil (2008) and got his Ph.D. degree in teleinformatics engineering from the UFC in 2019. From May 2017 to April 2018, he was at the I3S Laboratory, Sophia Antipolis, France, as an exchange (sandwich) PhD student. In 2012, he joined the Instituto Federal de Educac¸a˜ o, Ciˆencia e Tecnologia do Cear´a (IFCE), where he works as a Full Professor with the Department of Physics, in Sobral. His research interest deals with the area of signal processing for communication systems, including tensor models/decompositions for MIMO communications, cooperative communications and channel estimation and equalization.

[ ] G´erard Favier received the engineer diplomae from Ecole Nationale Sup´erieure de Chronom´etrie et de Microm´ecanique (ENSCM), Besanc¸on, and Ecole Nationale Sup´erieure de l’A´eronautique et de l’Espace (ENSAE), Toulouse, the Dr.-Ing. (Ph.D.) and State Doctorate degrees from the University of Nice-Sophia Antipolis, France, in 1973, 1974, 1977, and 1981, respectively. In 1976, he joined the French National Center for Scientific Research (CNRS). He is presently an Emeritus Research Director of CNRS, at the I3S Laboratory, Sophia Antipolis, France. From 1995 to 1999, he was the Director of the I3S Laboratory. His research interests include nonlinear system modeling and identification, tensor models/decompositions, and tensor approaches for MIMO communication systems. He is coauthor or author of more than 300 papers published in international scientific conferences and journals, and 20 books or book chapters.

]C. Alexandre R. Fernandes was born in Fortaleza, Brazil, in 1981. [ He received the B.Sc. degree in electrical engineering from the Universidade Federal do Cear´a (UFC), Brazil, in 2003, M.Sc. degrees from the UFC and University of Nice Sophia-Antipolis (UNSA), France, in 2005, and the double Ph.D. degree from the UFC 1

and UNSA, in 2009, in telecommunications engineering. In 2008 and 2009, he was a Teaching Assistant with the UNSA/FR and, from July 2009 to February 2010, he was a Postdoctoral Fellow with the Department of Teleinformatics Engineering, UFC. In 2010, he joined the UFC, where he works as a Full Professor with the Department of Computer Engineering, in Sobral. His research interest lies in the area of signal processing for communications, and include tensor decompositions, multiliner algebra, channel estimation and equalization, multicarrier systems, cooperative communications, tensor decompositions and nonlinear systems.

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