Review

# Overview of Tensor-Based Cooperative MIMO Communication Systems—Part 2: Semi-Blind Receivers

Gérard Favier ¹,*,† ID and Danilo Sousa Rocha ²,† ID

¹ I3S Laboratory, Côte d'Azur University, 06903 Sophia Antipolis, France

² Federal Institute of Education, Science, and Technology of Ceará, Fortaleza 60040-531, Brazil; danilo.rocha@ifce.edu.br

* Correspondence: favier@i3s.unice.fr

† These authors contributed equally to this work.

**Abstract:** Cooperative MIMO communication systems play an important role in the development of future sixth-generation (6G) wireless systems incorporating new technologies such as massive MIMO relay systems, dual-polarized antenna arrays, millimeter-wave communications, and, more recently, communications assisted using intelligent reflecting surfaces (IRSs), and unmanned aerial vehicles (UAVs). In a companion paper, we provided an overview of cooperative communication systems from a tensor modeling perspective. The objective of the present paper is to provide a comprehensive tutorial on semi-blind receivers for MIMO one-way two-hop relay systems, allowing the joint estimation of transmitted symbols and individual communication channels with only a few pilot symbols. After a reminder of some tensor prerequisites, we present an overview of tensor models, with a detailed, unified, and original description of two classes of tensor decomposition frequently used in the design of relay systems, namely nested CPD/PARAFAC and nested Tucker decomposition (TD). Some new variants of nested models are introduced. Uniqueness and identifiability conditions, depending on the algorithm used to estimate the parameters of these models, are established. Two families of algorithms are presented: iterative algorithms based on alternating least squares (ALS) and closed-form solutions using Khatri-Rao and Kronecker factorization methods, which consist of SVD-based rank-one matrix or tensor approximations. In a second part of the paper, the overview of cooperative communication systems is completed before presenting several two-hop relay systems using different codings and configurations in terms of relaying protocol (AF/DF) and channel modeling. The aim of this presentation is firstly to show how these choices lead to different nested tensor models for the signals received at destination. Then, by capitalizing on these models and their correspondence with the generic models studied in the first part, we derive semi-blind receivers to jointly estimate the transmitted symbols and the individual communication channels for each relay system considered. In a third part, extensive Monte Carlo simulation results are presented to compare the performance of relay systems and associated semi-blind receivers in terms of the symbol error rate (SER) and channel estimate normalized mean-square error (NMSE). Their computation time is also compared. Finally, some perspectives are drawn for future research work.

**Keywords:** cooperative communication systems; MIMO systems; nested tensor models; relaying systems; semi-blind receivers; tensor codings; tensor decompositions; closed-form algorithms

<figure><img src="image_1.png" /></figure>

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**Citation:** Favier, G.; Rocha, D.S.  
Overview of Tensor-Based  
Cooperative MIMO Communication  
Systems—Part 2: Semi-Blind  
Receivers. *Entropy* **2024**, *26*, 937.  
https://doi.org/10.3390/e26110937

Academic Editor: Stefania Sardellitti

Received: 20 August 2024

Revised: 10 October 2024

Accepted: 24 October 2024

Published: 31 October 2024

<figure><img src="image_2.png" /></figure>

**Copyright:** © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).

1. Introduction

During the last decade, new technologies have emerged for future sixth-generation (6G) wireless communications and networks. Among these promising technologies, we can mention massive multiple-input multiple-output (MIMO) antenna arrays, three-dimensional (3D) polarized antennas, high-frequency millimeter-wave (mmWave), i.e., terahertz (THz), communications, large intelligent surfaces (LISs), and holographic beamforming (HBF) antennas.
To improve signal coverage, mobile connectivity, latency, reliability, and energy con-
sumption, as well as the quality of service (QoS) of future wireless networks, cooperative
communication systems are the subject of tremendous research interests, with the aim of
designing network architectures integrating ground, space, air, and underwater networks.
These cooperative systems can be classified into the following three basic categories, namely
relay-, IRS- (intelligent reflecting surface, also known as reconfigurable intelligent surface
(RIS)) and UAV- (unmanned aerial vehicle, i.e., drones) aided systems. For a comparison
of relay- and IRS-assisted wireless systems, the reader can refer to [1–4]. These types
of assistance can be combined as, for instance, in [5,6], where IRS passive reflection is
combined with active DF (decode and forward) relaying to improve the coverage and rate
performance of conventional IRS-assisted systems.

Wireless propagation channel impairments such as multipath fading, delay, and Doppler spreads, which strongly depend on the environment (urban, rural, and indoor), degrade the quality of reception to recover the transmitted information symbols. Enhanced performance of wireless communication systems can be achieved by combining diverse techniques in multiple domains (space, time, frequency, chip, polarization, etc.) in order to exploit, at the receiver, several versions of transmitted symbols. Such symbol repetition can be achieved via the use of multi-antennas at the transmitter and receiver (space diversity), a repetition of the transmission of the same symbols over several time slots (time diversity), and the use of different subcarriers (frequency diversity). Diversity can also be introduced by means of specific codings, such as space-time (ST), space-frequency (SF), and space-time-frequency (STF) codings.

High-order tensors, also known as multiway arrays, are well suited to taking into account multiple types of diversity via the coding of information symbols to be transmitted, representing multidimensional received signals, designing semi-blind receivers, which is to say without a priori knowledge of the channels, and processing blocks of received signals in order to jointly estimate communication channels and transmitted symbols. Note that semi-blind means that only a few pilot symbols are needed to eliminate the scaling ambiguities inherent to the tensor model of the communication system. Besides the obvious potential to represent, compress, analyze, merge, and classify multidimensional, multimodal, and often incomplete data, tensor decompositions have the property of essential uniqueness (up to trivial indeterminacies in terms of permutation and scaling factor ambiguities in the columns of matrix factors) under milder conditions than matrix decompositions, which offers greater flexibility for the choice of design parameters.

Since the pioneering works of [7–12] for psychometrics, phonetics, and chemomet-
rics applications, blind source separation (BSS), and wireless communication systems,
tensors have been extensively used in various fields of applications, like computer vi-
sion [13], ECG, and EEG applications [14–16], hyperspectral image classification and
anomaly detection [17,18], traffic data completion [19–21], recommendation systems [22,23],
nonlinear system modeling and identification [24], data mining and data fusion [25–28],
and tensor networks to represent and classify very large arrays of data with applications in
machine learning [29,30], among many other applications. For a more complete description
of tensor-based signal processing applications, the reader is referred to [31–33].

In the context of wireless systems, many tensor-based approaches have been proposed to design both point-to-point systems and cooperative systems, with associated semi-blind receivers. Most tensor-based systems have been designed using the popular tensor decomposition known as PARAFAC (parallel factors analysis) [8] or CPD (canonical polyadic decomposition) [34]. However, over the last two decades, various new tensor models have emerged when designing communication systems, such as the CONFAC (constrained PARAFAC) [35,36], PARATUCK-(N₁, N) [37], generalized PARATUCK [38], NCPD (nested CPD) [39,40], nested Tucker decomposition (NTD) [41], coupled NTD [42], and doubly coupled NCPD [43] models. These models mainly depend on the choice of coding, whether or not resource allocation is taken into account, assumptions about communication channels, and the presence or absence of relays, as will be shown in Section 5.
It is fundamental to note that, unlike most tensor-based applications, in the context of wireless communication systems, through construction, all the matrix factors of tensor models are physically interpretable in terms of symbol, coding, or channel matrices, with physical parameters such as fading coefficients, angles of departure and arrival (AoD/AoA), or time delays. In addition, certain matrix or tensor factors of these models are possibly structured, such as, for example, the Vandermonde structure of steering matrices and the orthonormal (Fourier) structure of coding matrices or matrix unfoldings of tensor codings.

The objective of this paper is fourfold:

* To provide a self-contained overview of tensor models used to design wireless communication systems. After a reminder of tensor prerequisites, standard tensor decompositions are first recalled, more particularly the CPD/PARAFAC and Tucker decompositions, as well as some variants. Then, two important classes of tensor decompositions, namely the NCPD and NTD, are presented in an unified and original way, using a new representation by means of graphs and highlighting their link with the tensor train decomposition (TTD) [44]. This link is exploited to demonstrate the uniqueness property of NTD models. Some of the models to be used when designing new relay systems are introduced for the first time.

* To present two families of algorithms that estimate the parameters of NCPD and NTD models: iterative algorithms based on alternating least squares (ALS) and closed-form solutions using Khatri-Rao and Kronecker factorization methods, denoted as KRF and KronF, which consist of SVD-based rank-one matrix or tensor approximations. These closed-form algorithms result from the fact that unfoldings of CPD and TD are expressed in terms of Khatri-Rao and Kronecker products of matrix factors, respectively.

* To provide an overview of tensor-based cooperative MIMO communication systems from a semi-blind receiver perspective, with a focus on two-hop relay systems, as complement of our companion paper [45]. The goal of this presentation is first to show how the choices of coding, relay protocol (AF (amplify-and-forward) or DF), and assumptions made about communication channels (flat fading or frequency-selective fading channels) impact the modeling of relay systems. Then, assuming knowledge of coding tensors and matrices at the destination and exploiting the multilinear structure of the system model, we devise semi-blind receivers to jointly estimate the transmitted symbols and the individual channels for each considered relay system. Uniqueness conditions for the tensor model of each system and necessary conditions for parameter identifiability using the associated receivers are established.

* To compare the performance of main two-hop relay systems and associated semi-blind receivers in terms of the symbol error rate (SER), channel estimate normalized mean-square error (NMSE), and computation time, by means of extensive Monte Carlo simulations, with the aim of showing how the best performance-complexity-identifiability condition tradeoff can be achieved.

The rest of the paper is organized in seven sections. In Section 2, some tensor prerequisites are recalled, including an introduction of index convention and the definition of basic tensor operations in order to make the content as self-contained as possible. Section 3 provides an overview of the main tensor models, with a focus on nested ones, namely the NCPD and NTD, highlighting their link with the TTD and proposing a new representation by means of graphs. New variants of nested models to be used for designing two-hop relay systems are introduced. Uniqueness conditions are established for the nested models considered. In Section 4, several parameter estimation algorithms are described for these models, with particular attention given to closed-form algorithms. The identifiability conditions are established for both ALS and closed-form algorithms. Section 5 is devoted to a comprehensive survey of one-way two-hop relay systems.

After an overview of cooperative systems from a semi-blind receiver point of view, relay systems are classified according to the choice of coding scheme and the resulting tensor model for the signals received at destination. Three classes of coding are considered:
* Tensor-based codings, including TSTF, STSTF, TST, and STST codings (TSTF = tensor space-time frequency; STSTF = simplified TSTF; TST = tensor space-time; STST = simplified TST);

* Matrix-based codings, including DKRSTF and SKRST codings (DKRSTF = doubly Khatri-Rao space-time frequency; SKRST = simplified Khatri-Rao space-time);

* STST and SKRST codings combined with MSMKron and MSMKR codings (MSMKron = multiple symbol matrices' Kronecker product; MSMKR = multiple symbol matrices' Khatri-Rao product), respectively.

In Section 6, by capitalizing on the tensor models of the systems, we design semi-blind receivers to jointly estimate transmitted symbols and individual communication channels. In Section 7, extensive Monte Carlo simulation results are presented to compare the performance of the relay systems considered in this overview and their associated semi-blind receivers in terms of SER, channel-estimate NMSE, and computation time. Finally, Section 8 presents some perspectives for future research work.

The paper is divided into two main parts, entitled “Tensor operations, models, and algorithms” and “Overview of two-hop relay systems and semi-blind receivers”, composed of three sections each, as detailed in the flow chart of Figure 1.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Organization of the paper.</figcaption></figure>

**Notation and acronyms:** Table 1 summarizes the notations used throughout the paper, and a comprehensive glossary of acronyms is provided in Appendix A.
<table><caption>**Table 1.** Notation.</caption><thead><tr><th>Symbols</th><th>Definitions</th></tr></thead><tbody><tr><td>K = R or C</td><td>Set of real or complex numbers</td></tr><tr><td>⟨N⟩ ≜ {1, ..., N}</td><td>Set of first N integers</td></tr><tr><td>i<sub>N</sub> ≜ {i<sub>1</sub>, ..., i<sub>N</sub>}</td><td>Set of N indices</td></tr><tr><td>I<sub>N</sub> ≜ I<sub>1</sub> × ... × I<sub>N</sub></td><td>Size of an Nth-order tensor</td></tr><tr><td>a, a, A, A</td><td>Scalar, column vector, matrix, tensor</td></tr><tr><td>a<sub>i<sub>N</sub></sub> = a<sub>i<sub>1</sub>,i<sub>2</sub>,...,i<sub>N</sub></sub> or [A]<sub>i<sub>1</sub>,i<sub>2</sub>,...,i<sub>N</sub></sub></td><td>(i<sub>1</sub>,i<sub>2</sub>,...,i<sub>N</sub>)-th element of A ∈ K<sup>I<sub>N</sub></sup></td></tr><tr><td>A<sup>T</sup>, A<sup>*</sup>, A<sup>H</sup></td><td>Transpose, complex conjugate, Hermitian transpose of A</td></tr><tr><td>A<sup>†</sup></td><td>Moore–Penrose pseudo-inverse of A</td></tr><tr><td>A<sub>i,•</sub> (A<sub>•,j</sub>)</td><td>i-th (j-th) row (column) of A ∈ K<sup>I×J</sup></td></tr><tr><td>I<sub>I<sub>N</sub></sub></td><td>Identity matrix of size I<sub>N</sub> × I<sub>N</sub></td></tr><tr><td>e<sub>n</sub><sup>(N)</sup></td><td>nth canonical basis vector of the Euclidean space R<sup>N</sup></td></tr><tr><td>vec(·)</td><td>Vectorization operator</td></tr><tr><td>diag(·)</td><td>Diagonalization operator that forms a diagonal matrix from its vector argument</td></tr><tr><td>D<sub>i</sub>(A) = diag(A<sub>i,•</sub>)</td><td>Diagonal matrix whose diagonal entries are the elements of the i-th row of A ∈ K<sup>I×J</sup></td></tr><tr><td>bdiag<sub>i</sub>(A<sub>i,•,•</sub>)</td><td>Block-diagonal matrix whose diagonal blocks are horizontal slices * of A ∈ K<sup>I×J×K</sup></td></tr><tr><td>⟨·,·⟩</td><td>Inner product</td></tr><tr><td>||·||<sub>2</sub>, ||·||<sub>F</sub></td><td>Euclidian and Frobenius norms</td></tr><tr><td>○</td><td>Outer product</td></tr><tr><td>⊙</td><td>Hadamard product</td></tr><tr><td>◇</td><td>Khatri–Rao product</td></tr><tr><td>⊗</td><td>Kronecker product</td></tr><tr><td>×<sub>n</sub></td><td>Mode-n product</td></tr><tr><td>×<sub>p</sub><sup>n</sup></td><td>Modes-(p, n) product</td></tr></tbody></table>

## 2. Tensor Prerequisites

An Nth-order tensor, $\mathcal{X} \in K^{I_N}$, of size $I_N \triangleq I_1 \times \dots \times I_N$, will be denoted as $[x_{i_N}] = [x_{i_1, \dots, i_N}]$, where $i_N \triangleq \{i_1, \dots, i_N\}$, with $K = \mathbb{R}$ or $\mathbb{C}$, depending on whether the tensor is real- or complex-valued. Each index, $i_n \in \langle I_n \rangle \triangleq \{1, \dots, I_n\}$, for $n \in \langle N \rangle \triangleq \{1, \dots, N\}$, is associated with the n-th-mode whose dimension is $I_n$. In the context of wireless communications, each mode of the tensor of encoded or received signals is associated with a particular diversity (space, frequency, chip, time slot, or symbol period).

The identity tensor of order N and dimensions R, denoted as $\mathcal{I}_{N,R} = [\delta_{r_1, \dots, r_N}]$, with $r_n \in \langle R \rangle$ for $n \in \langle N \rangle$, is a diagonal tensor whose diagonal elements are equal to 1 and other elements to 0. The generalized Kronecker delta is defined as follows:

$$ \delta_{r_1, \dots, r_N} = \begin{cases} 1 & \text{if } r_1 = \dots = r_N \\ 0 & \text{otherwise} \end{cases}. $$

Table 2 summarizes the notation used for sets of indices and dimensions [33].

<table><caption>**Table 2.** Notation for sets of indices and dimensions.</caption><tr><td>i<sub>P</sub> ≜ {i<sub>1</sub>, ..., i<sub>P</sub>} ; j<sub>N</sub> ≜ {j<sub>1</sub>, ..., j<sub>N</sub>}</td></tr><tr><td>I<sub>P</sub> ≜ {I<sub>1</sub>, ..., I<sub>P</sub>} ; J<sub>N</sub> ≜ {J<sub>1</sub>, ..., J<sub>N</sub>}</td></tr><tr><td>I<sub>P</sub> ≜ I<sub>1</sub> × ... × I<sub>P</sub> ; J<sub>N</sub> ≜ J<sub>1</sub> × ... × J<sub>N</sub></td></tr></table>

<footnote>* See Table 4 for the definition of horizontal slices.</footnote>
## 2.1. Index Convention

We now introduce the index convention, which allows for eliminating the summation symbols in formulae involving multi-index variables. For example, $\sum_{i=1}^{I} a_i b_i$ is simply written as $a_i b_i$. Note that there are two differences relative to Einstein's summation convention:

* Each index can be repeated more than twice in an expression;

* Ordered index sets are allowed.

For example, the index convention allows multiple sums to be abbreviated as follows:

$$ \sum_{i_1=1}^{I_1} \cdots \sum_{i_p=1}^{I_p} x_{i_1, \cdots, i_p} y_{i_1, \cdots, i_p} = \sum_{\mathbf{i}_p=\mathbf{1}}^{\mathbf{I}_p} x_{\mathbf{i}_p} y_{\mathbf{i}_p} = x_{\mathbf{i}_p} y_{\mathbf{i}_p} \quad (1) $$

$$ \sum_{i_1=1}^{I_1} \cdots \sum_{i_p=1}^{I_p} x_{i_1, \cdots, i_p} \prod_{p=1}^{p} a_{i_p}^{(p)} = \sum_{\mathbf{i}_p=\mathbf{1}}^{\mathbf{I}_p} x_{\mathbf{i}_p} \prod_{p=1}^{p} a_{i_p}^{(p)} = \prod_{p=1}^{p} a_{i_p}^{(p)} x_{\mathbf{i}_p}, \quad (2) $$

where $\mathbf{1}$ denotes a set of ones whose number is fixed by the index $P$ of the set $\mathbf{I}_p$. The notation $\mathbf{I}_p$ and $\mathbf{I}_p$ allows us to simplify the expression of the multiple sums into a single sum over an index set, which is further simplified using the index convention.

The index convention can be interpreted in terms of two types of summation, one associated with row indices (subscripts), and one associated with column indices (superscripts), with the following rules [33,46]:

* The order of the column indices is independent of the order of the row indices;

* Consecutive row and column indices (or index sets) can be permuted.

In Table 3, we present examples of vector and matrix Kronecker products and a matrix product using index convention, where $\mathbf{e}_{ij} \triangleq \mathbf{e}_i^{(I)} \otimes \mathbf{e}_j^{(J)}$, and $\mathbf{e}_i^j \triangleq \mathbf{e}_i^{(I)} \otimes (\mathbf{e}_j^{(J)})^T$.

<table><caption>**Table 3.** Vector and matrix Kronecker products and matrix product using the index convention.</caption><tr><td>u</td><td>&infin;</td><td>&infin;</td></tr><tr><td>u &otimes; v</td><td>=</td><td>(u<sub>i</sub>e<sub>i</sub><sup>(I)</sup>) &otimes; (v<sub>j</sub>e<sub>j</sub><sup>(J)</sup>) = u<sub>i</sub>v<sub>j</sub>e<sub>ij</sub> &in; &infin;<sup>IJ</sup></td></tr><tr><td>u &otimes; v<sup>T</sup></td><td>=</td><td>(u<sub>i</sub>e<sub>i</sub><sup>(I)</sup>) &otimes; (v<sub>j</sub>(e<sub>j</sub><sup>(I)</sup>)<sup>T</sup>) = u<sub>i</sub>v<sub>j</sub>e<sub>i</sub><sup>j</sup> &in; &infin;<sup>I&times;J</sup></td></tr><tr><td>A &infin;</td><td>&infin;</td><td>&infin;</td></tr><tr><td>A B</td><td>=</td><td>&sum;<sub>i=1</sub><sup>I</sup> &sum;<sub>k=1</sub><sup>K</sup> (a<sub>ij</sub>b<sub>jk</sub>)e<sub>i</sub><sup>k</sup> = a<sub>ij</sub>b<sub>jk</sub>e<sub>i</sub><sup>k</sup> &in; &infin;<sup>I&times;K</sup></td></tr><tr><td>A &otimes; C</td><td>=</td><td>(a<sub>ij</sub>e<sub>i</sub><sup>j</sup>) &otimes; (b<sub>km</sub>)e<sub>k</sub><sup>m</sup> = a<sub>ij</sub>b<sub>km</sub>e<sub>ik</sub><sup>jm</sup> &in; &infin;<sup>IK&times;JM</sup></td></tr></table>

## 2.2. Notion of Slice

If we fix $N-1$ indices of an $N$th-order tensor, $\mathcal{X} \in \mathbb{K}^{I_N}$, we obtain a vector slice, called a fiber, and for $N-2$ fixed indices, we have a matrix slice. In Table 4, we define three types of vector and matrix slices for a third-order tensor, $\mathcal{X} \in \mathbb{K}^{I \times J \times K}$.

<table><caption>**Table 4.** Vector and matrix slices for a third-order tensor, $\mathcal{X} \in \mathbb{K}^{I \times J \times K}$.</caption><thead><tr><th>Slices</th><th>Definitions</th><th>Dimensions</th></tr></thead><tbody><tr><td>Vectors</td><td>Fibers</td><td></td></tr><tr><td>column</td><td><b>x</b><sub>&bull;,j,k</sub></td><td>I</td></tr><tr><td>row</td><td><b>x</b><sub>i,&bull;,k</sub></td><td>J</td></tr><tr><td>tube</td><td><b>x</b><sub>i,j,&bull;</sub></td><td>K</td></tr><tr><td>Matrices</td><td>Matrix slices</td><td></td></tr><tr><td>frontal</td><td><b>X</b><sub>&bull;,&bull;,k</sub></td><td>I &times; J</td></tr><tr><td>lateral</td><td><b>X</b><sub>&bull;,j,&bull;</sub></td><td>K &times; I</td></tr><tr><td>horizontal</td><td><b>X</b><sub>i,&bull;,&bull;</sub></td><td>J &times; K</td></tr></tbody></table>
Similarly, we can define vector, matrix, and third-order tensor slices for a fourth-order tensor, $\mathcal{X} \in \mathbb{K}^{I \times J \times K \times L}$, such as $\mathbf{x}_{\bullet,j,k,l} \in \mathbb{K}^I$, $\mathbf{X}_{\bullet,\bullet,k,l} \in \mathbb{K}^{I \times J}$, $\mathcal{X}_{\bullet,\bullet,\bullet,l} \in \mathbb{K}^{I \times J \times K}$.

## 2.3. Matrix Unfoldings of a Tensor

A general matrix unfolding formula for an Nth-order tensor, $\mathcal{X} \in \mathbb{K}^{\mathcal{I}^N}$, is given by [38]:

$$ \mathbf{X}_{\mathcal{S}_1;\mathcal{S}_2} = \sum_{i_1=1}^{I_1} \cdots \sum_{i_N=1}^{I_N} x_{i_1,\cdots,i_N} \left( \bigotimes_{n \in \mathcal{S}_1} \mathbf{e}_{i_n}^{(I_n)} \right) \left( \bigotimes_{n \in \mathcal{S}_2} \mathbf{e}_{i_n}^{(I_n)} \right)^T \in \mathbb{K}^{I_1 \times I_2}, \quad (3) $$

where $\mathcal{S}_1$ and $\mathcal{S}_2$ are two disjoint ordered subsets of the set of modes $\mathcal{S} = \langle N \rangle$, composed of $p$ and $N-p$ modes, respectively, with $p \in \langle N-1 \rangle$, and $J_{n_1} = \prod_{n \in \mathcal{S}_{n_1}} I_{n_1}$, for $n_1 = 1$ and $2$.

The subsets $\mathcal{S}_1$ and $\mathcal{S}_2$ contain the modes associated with the rows and the columns of the matrix unfolding $\mathbf{X}_{\mathcal{S}_1;\mathcal{S}_2}$ of $\mathcal{X}$, respectively.

Such a matrix unfolding results from two mode combinations associated with the sets $\mathcal{S}_1$ and $\mathcal{S}_2$, using the convention that the order of the dimensions in a product, $\prod_{p=1}^p I_p \triangleq I_1 \cdots I_p$, corresponding to a combination of P modes, follows the order of variation of the indices, with $i_1$ varying more slowly than $i_2$, which, in turn, varies more slowly than $i_3$, etc.

For example, in the flat mode-1 unfolding $\mathbf{X}_{I \times K J}$ of the third-order tensor $\mathcal{X} \in \mathbb{K}^{I \times J \times K}$, index $k$ varies more slowly than $j$, which implies $x_{ijk} = [\mathbf{X}_{I \times K J}]_{j,(i-1)K+k} = [\mathbf{X}_{K \times J I}]_{k,(j-1)I+i}$.

Transposing $\mathbf{X}_{I \times K J}$ achieves the tall mode-1 unfolding $\mathbf{X}_{K J \times I} = [\mathbf{X}_{I \times K J}]^T$. For a third-order tensor, in addition to $\mathbf{X}_{K J \times I}$, there are five other tall matrix unfoldings, denoted as $\mathbf{X}_{J K \times I}$, $\mathbf{X}_{K I \times J}$, $\mathbf{X}_{I K \times J}$, $\mathbf{X}_{I J \times K}$, $\mathbf{X}_{J I \times K}$.

## 2.4. Basic Tensor Operations

In Table 5, we recall the definitions of some basic operations: the outer product of vectors, the mode-$p$ (multiple-mode-$p$) product of a tensor with a matrix ($P$ matrices), and the modes-$(p, n)$ product of two tensors, respectively denoted as $\circ, \times_p, \times$ and $\times_p^n$.

Note that the outer product of $P$ non-zero vectors $\mathbf{u}^{(p)} \in \mathbb{K}^{I_p}, p \in \langle P \rangle$, gives a rank-one $P$th-order tensor of size $I_p$.

<table><caption>**Table 5.** Definitions of some basic operations.</caption><thead><tr><th>Vectors/Matrices/Tensors</th><th>Operations</th><th>Definitions</th></tr></thead><tbody><tr><td>$\mathbf{u}^{(p)} \in \mathbb{K}^{I_p}, p \in \langle P \rangle$</td><td>$\mathcal{X} = \circ_{p=1}^{p} \mathbf{u}^{(p)}$</td><td>$x_{I_p} = \prod_{p=1}^{p} u_{I_p}^{(p)}$</td></tr><tr><td>$\mathcal{X} \in \mathbb{K}^{I_p}, \mathbf{A} \in \mathbb{K}^{I \times I_p}$</td><td>$\mathcal{Y} = \mathcal{X} \times_p \mathbf{A}$</td><td>$y_{I_1, \dots, I_{p-1}, j, I_{p+1}, \dots, I_p} = \sum_{I_p=1}^{I_p} a_{j, I_p} \mathbf{x}_{I_p} = a_{j, I_p} \mathbf{x}_{I_p}$</td></tr><tr><td>$\mathcal{X} \in \mathbb{K}^{I_p}, \mathbf{A}^{(p)} \in \mathbb{K}^{I_p \times I_p}, p \in \langle P \rangle$</td><td>$\mathcal{Y} = \mathcal{X} \times_{p=1}^{p} \mathbf{A}^{(p)} \triangleq \\ \mathcal{X} \times_1 \mathbf{A}^{(1)} \times_2 \cdots \times_p \mathbf{A}^{(p)}$</td><td>$y_{I_p} = \sum_{I_p=1}^{I_1} \cdots \sum_{I_p=1}^{I_p} \prod_{p=1}^{p} a_{I_p, I_p}^{(p)} \mathbf{x}_{I_p} = \prod_{p=1}^{p} a_{I_p, I_p}^{(p)} \mathbf{x}_{I_p}$</td></tr><tr><td>$\mathcal{X} \in \mathbb{K}^{I_p}, \mathcal{Y} \in \mathbb{K}^{I_N}$<br>with $I_p = J_n = K$</td><td>$\mathcal{Z} = \mathcal{X} \times_p^n \mathcal{Y}$</td><td>$z_{I_1, \dots, I_{p-1}, I_{p+1}, \dots, I_p, J_1, \dots, J_{n-1}, J_{n+1}, \dots, J_N} = \\ \sum_{k=1}^{K} a_{I_1, \dots, I_{p-1}, k, I_{p+1}, \dots, I_p, J_1, \dots, J_{n-1}, k, J_{n+1}, \dots, J_N}$</td></tr></tbody></table>

The mode-$p$ product of a tensor $\mathcal{X} \in \mathbb{K}^{I_p}$ with matrices $\mathbf{A} \in \mathbb{K}^{I_p \times I_p}, \mathbf{B} \in \mathbb{K}^{I_p \times J_p}$ and $\mathbf{A}^{(p)} \in \mathbb{K}^{I_p \times I_p}$ for $p \in \langle P \rangle$ satisfies the following properties:

$$ \mathcal{X} \times_p \mathbf{A} \times_p \mathbf{B} = \mathcal{X} \times_p (\mathbf{B} \mathbf{A}) \quad (4) $$

$$ \mathcal{X}_{p=1}^{P} \times \mathbf{A}^{(\pi(p))} = \mathcal{X}_{p=1}^{P} \times \mathbf{A}^{(p)}. \quad (5) $$
The last equality means that, for any permutation, $\pi(\cdot)$, of the indices $p \in \langle P \rangle$, the order of the mode-$p$ products is irrelevant when the indices are all distinct.

The modes-$(p, n)$ product of the tensors $\mathcal{X} \in \mathbb{K}^{I_p}$ and $\mathcal{Y} \in \mathbb{K}^{I_N}$, with $I_p = J_n = K$, corresponds to a contraction along the modes $p$ of $\mathcal{X}$ and $n$ of $\mathcal{Y}$.

The contracted product $\times_p^n$ is associative; i.e., for any tensors $\mathcal{A} \in \mathbb{K}^{I_p}$, $\mathcal{B} \in \mathbb{K}^{I_N}$ and $\mathcal{C} \in \mathbb{K}^{K_Q}$, such that $I_p = J_n$ and $J_m = K_q$, with $m \neq n$, we have the following:

$$ (\mathcal{A} \times_p^n \mathcal{B}) \times_m^q \mathcal{C} = \mathcal{A} \times_p^n (\mathcal{B} \times_m^q \mathcal{C}) = \mathcal{A} \times_p^n \mathcal{B} \times_m^q \mathcal{C}. \qquad (6) $$

This doubly contracted product yields a tensor of order $P + N + Q - 4$.

When the indices $(m, n, p, q)$ represent numbers of modes in the tensors, the property (6) is no longer valid because the result of the double product $\times_p^n$ and $\times_m^q$ depends on the order in which these products are calculated.

For instance, for $\mathcal{A} \in \mathbb{K}^{I_1 \times J_1 \times R_1}$, $\mathcal{B} \in \mathbb{K}^{R_1 \times I_2 \times J_2 \times R_2}$ and $\mathcal{C} \in \mathbb{K}^{R_2 \times I_3 \times J_3}$, the double contracted product can be written in two different ways:

$$ (\mathcal{A} \times_3^1 \mathcal{B}) \times_5^1 \mathcal{C} = \mathcal{A} \times_3^1 (\mathcal{B} \times_4^1 \mathcal{C}) = \mathcal{A} \times_3^1 \mathcal{B} \times_4^1 \mathcal{C} \in \mathbb{K}^{I_1 \times J_1 \times I_2 \times J_2 \times I_3 \times J_3}. \qquad (7) $$

On the left-hand side of this double equality, the product $\times_3^1$ is performed first, and then, it is followed by the product $\times_5^1$. Meanwhile, in the second term of the equality, the product $\times_4^1$ is first calculated, followed by the product $\times_3^1$. The last term in the equality corresponds to two-by-two contractions of adjacent blocks. The first writing will be used to represent the TTD model by means of Equation (12) with contraction operations performed from left to right, while the third writing will be used in Equation (13).

## 2.5. Inner Product and Frobenius Norm

In the case of two complex-valued tensors, $\mathcal{A}, \mathcal{B} \in \mathbb{C}^{I_N}$, of order $N$ and the same size, their Hermitian inner product is given by the following:

$$ \langle \mathcal{A}, \mathcal{B} \rangle = \sum_{i_1=1}^{I_1} \cdots \sum_{i_N=1}^{I_N} a_{i_1, \cdots, i_N} b_{i_1, \cdots, i_N}^* = \sum_{\mathbf{i}_N=\mathbf{1}}^{I_N} a_{\mathbf{i}_N} b_{\mathbf{i}_N}^*. \qquad (8) $$

We can also write it using the Hermitian inner product of vectorized forms of $\mathcal{A}$ and $\mathcal{B}$:

$$ \langle \mathcal{A}, \mathcal{B} \rangle = \mathrm{vec}^H(\mathcal{B}) \mathrm{vec}(\mathcal{A}), \qquad (9) $$

where $\mathrm{vec}(\mathcal{A})$ and $\mathrm{vec}(\mathcal{B})$ are vectorizations associated with the same mode combination of $\mathcal{A}$ and $\mathcal{B}$.

The Frobenius norm of $\mathcal{A} \in \mathbb{K}^{I_N}$ is the square root of the inner product of the tensor with itself; i.e.,

$$ \| \mathcal{A} \|_F = \langle \mathcal{A}, \mathcal{A} \rangle^{1/2} = \sqrt{\sum_{i_1=1}^{I_1} \cdots \sum_{i_N=1}^{I_N} |a_{i_1, \cdots, i_N}|^2}, \qquad (10) $$

where $| \cdot |$ represents the absolute value or the modulus, depending on whether $\mathcal{A}$ is real ($\mathbb{K} = \mathbb{R}$) or complex ($\mathbb{K} = \mathbb{C}$).

Since the Frobenius norm is equal to the square root of the sum of the squares of the absolute value or modulus of all elements of the tensor, it is also given by the following:

$$ \| \mathcal{A} \|_F = \| \mathrm{vec}(\mathcal{A}) \|_2 = \| \mathbf{A}_{S_1;S_2} \|_F, \qquad (11) $$

that means the Euclidean (if $\mathbb{K} = \mathbb{R}$) or Hermitian (if $\mathbb{K} = \mathbb{C}$) norm of one of its vectorized forms, or as the Frobenius norm of one of its matrix unfoldings $\mathbf{A}_{S_1;S_2}$, defined in (3).
### 3. Overview of Tensor Models

In this section, we provide an overview of the tensor models that will be used in Section 5 when designing two-hop relay systems. We first present the standard Tucker decomposition (TD), the canonical polyadic decomposition (CPD), also known as PARAFAC [8] or CANDECOMP [47], with two variants TD-(N₁, N) and GTD-(N₁, N), and the tensor train decomposition (TTD), with a new generalization that we will call generalized TTD (GTTD). Then, two basic nested decompositions, the so-called NCPD-4 and NTD-4 models, are described in detail. Three new nested models, called NTD-6, NGTD-5 and NGTD-7, are introduced for the first time. Figure 2 shows how the considered nested models are constructed by nesting two basic models: CPD-3 for NCPD-4; TD-(2,3) and TD-(2,4) for NTD-4 and NTD-6, respectively; and GTD-(2,4) and GTD-(2,5) for NGTD-5 and NGTD-7. For the NTD and GNTD models, an interpretation in terms of TTD and GTTD is highlighted.

<figure><img src="image_4.png" /><figcaption>Figure 2. Nested tensor decompositions based on TD and CPD.</figcaption></figure>

#### 3.1. TD and CPD Models

In Tables 6 and 7, we first present various representations of the TD and CPD of a tensor of orders N and three: scalar writing, writings with mode-n products and outer products, and matrix unfoldings.

Comparing Table 7 with Table 6, we can conclude that a CPD can be viewed as a particular TD with an identity core tensor $\mathcal{I}_{N,R}$, implying $R_h = R$ for $n \in \langle N \rangle$ in the case of an Nth-order tensor and $P = Q = S = R$ for a third-order tensor. This results in the sum of R rank-one tensors. When R is minimal, it is called the tensor rank or the canonical rank.

<table><caption>Table 6. TD of an Nth-order and third-order tensor.</caption><thead><tr><th></th><th>Nth-Order Tensor</th><th>Third-Order Tensor</th></tr></thead><tbody><tr><td>Tensors</td><td>$\mathcal{X} \in \mathbb{K}^{I_N}$</td><td>$\mathcal{X} \in \mathbb{K}^{I \times J \times K}$</td></tr><tr><td>Core tensors</td><td>$\mathcal{G} \in \mathbb{K}^{K_N}$</td><td>$\mathcal{G} \in \mathbb{K}^{P \times Q \times S}$</td></tr><tr><td>Matrix factors</td><td>$\mathbf{A}^{(n)} \in \mathbb{K}^{I_n \times R_n}$</td><td>$\mathbf{A} \in \mathbb{K}^{I \times P}, \mathbf{B} \in \mathbb{K}^{I \times Q}, \mathbf{C} \in \mathbb{K}^{K \times S}$</td></tr><tr><td>Scalar writing</td><td>$x_{i_N} = \sum_{r_1=1}^{R_1} \cdots \sum_{r_N=1}^{R_N} g_{r_1, \cdots, r_N} \prod_{n=1}^{N} a_{i_n, r_n}^{(n)}$</td><td>$x_{i,j,k} = \sum_{p=1}^{P} \sum_{q=1}^{Q} \sum_{s=1}^{S} g_{pqs} a_{ip} b_{jq} c_{ks}$</td></tr><tr><td>With mode-n products</td><td>$\mathcal{X} = \mathcal{G} \times_{n=1}^{N} \mathbf{A}^{(n)}$</td><td>$\mathcal{X} = \mathcal{G} \times_1 \mathbf{A} \times_2 \mathbf{B} \times_3 \mathbf{C}$</td></tr><tr><td>With outer products</td><td>$\mathcal{X} = \sum_{r_1=1}^{R_1} \cdots \sum_{r_N=1}^{R_N} g_{r_1, \cdots, r_N} \circ_{n=1}^{N} \mathbf{A}_{r_n}^{(n)}$</td><td>$\mathcal{X} = \sum_{p=1}^{P} \sum_{q=1}^{Q} \sum_{s=1}^{S} g_{pqs} \mathbf{A}_{p} \circ \mathbf{B}_{q} \circ \mathbf{C}_{s}$</td></tr><tr><td>Matrix unfoldings</td><td>$\mathbf{X}_{S_1, S_2} = \begin{pmatrix} \otimes_{n \in S_1} \mathbf{A}^{(n)} \\ \otimes_{n \in S_2} \mathbf{A}^{(n)} \end{pmatrix}^{T}$</td><td>$\mathbf{X}_{I J \times K} = (\mathbf{A} \otimes \mathbf{B}) \mathbf{G}_{P Q \times S} \mathbf{C}^{T}$<br>$\mathbf{X}_{J K \times I} = (\mathbf{B} \otimes \mathbf{C}) \mathbf{G}_{Q S \times P} \mathbf{A}^{T}$<br>$\mathbf{X}_{K I \times J} = (\mathbf{C} \otimes \mathbf{A}) \mathbf{G}_{S P \times Q} \mathbf{B}^{T}$</td></tr></tbody></table>
In Table 9, we illustrate the generalized Tucker-(N₁, N) decomposition, denoted as GTD-(N₁, N), corresponding to a Tucker-(N₁, N) one when some of the factors are tensors instead of matrices, first introduced in [38] to model a wireless communication system using tensor space-time-frequency (TSTF) coding. The case of a fourth-order tensor $\mathcal{X} \in \mathbb{K}^{I_4}$ is considered, with $(N_1, N) = (2, 4)$, a fourth-order core tensor $\mathcal{G} \in \mathbb{K}^{R_1 \times R_2 \times I_3 \times I_4}$, a third-order tensor factor $\mathcal{A} \in \mathbb{K}^{I_1 \times R_1 \times I_3}$, and a matrix factor $\mathbf{B} \in \mathbb{K}^{I_2 \times R_2}$.

<table><caption>**Table 9.** Generalized Tucker models.</caption><tr><td></td><td>GTD-(2,4) model</td></tr><tr><td></td><td>$\mathcal{X} \in \mathbb{K}^{I_4}$</td></tr><tr><td></td><td>$\mathcal{G} \in \mathbb{K}^{R_1 \times R_2 \times I_3 \times I_4}$, $\mathcal{A} \in \mathbb{K}^{I_1 \times R_1 \times I_3}$, $\mathbf{B} \in \mathbb{K}^{I_2 \times R_2}$</td></tr><tr><td>$x_{i_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} g_{r_1,r_2,i_3,i_4} a_{i_1,r_1,r_2,r_3} b_{i_2,r_2'}$</td><td>$\mathcal{X} = \mathcal{G} \times_1^2 \mathcal{A} \times_2 \mathbf{B}$</td></tr></table>

## 3.2. TTD Models

The tensor train (TT) format was proposed in [44] as a reduced parametric complexity tensor representation in order to alleviate the curse of dimensionality with standard tensor decompositions due to an exponential growth of their number of parameters when the order of the tensor increases. This format is very useful for representing high-order data tensors.

A TTD model of an Pth-order tensor $\mathcal{X} \in \mathbb{K}^{I_p}$ consists of decomposing it into a train of two matrices, $\mathbf{G}^{(1)}$ and $\mathbf{G}^{(P)}$, and $(P-2)$ third-order tensors $\mathcal{G}^{(p)}$, $p \in \{2, \dots, P-1\}$, with the following dimensions:

$$ \mathbf{G}^{(1)} \in \mathbb{K}^{I_1 \times R_1} ; \mathbf{G}^{(P)} \in \mathbb{K}^{R_{P-1} \times I_p} ; \mathcal{G}^{(p)} \in \mathbb{K}^{R_{p-1} \times I_p \times R_p} $$

as illustrated in Figure 3.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** TTD of a Pth-order tensor, $\mathcal{X} \in \mathbb{K}^{I_p}$.</figcaption></figure>

The TTD can be written as a sequence, from left to right, of $(P-1)$ contractions between the factors $\mathcal{G}^{(p)}$, for $p \in \{2, \dots, P-1\}$, and $\mathbf{G}^{(P)}$ with the tensor resulting from all contractions to the left of each factor, as follows:

$$ \mathcal{X} = \left( \left( \cdots \left( \mathbf{G}^{(1)} \times_2 \mathcal{G}^{(2)} \times_3 \mathcal{G}^{(3)} \times_4 \cdots \right) \times_{P-1} \mathcal{G}^{(P-1)} \right) \times_P \mathbf{G}^{(P)}, \right) \quad (12) $$

with the modes-(1, p) products calculated from $p=2$ to $p=P$.

Considering two-by-two contractions of adjacent blocks, the TTD can be written as follows:

$$ \mathcal{X} = \mathbf{G}^{(1)} \times_2 \mathcal{G}^{(2)} \times_3 \mathcal{G}^{(3)} \times_3 \mathcal{G}^{(4)} \times_3 \cdots \times_3 \mathcal{G}^{(P-1)} \times_3 \mathbf{G}^{(P)}, \quad (13) $$

where the products $\times_3$ correspond to separate contractions of two adjacent tensors, $\mathcal{G}^{(p)}$ and $\mathcal{G}^{(p+1)}$, along their common mode associated with the index $r_p$ for $p \in \{2, \dots, P-1\}$.

The factors $\mathcal{G}^{(p)}$, $p \in \langle P \rangle$ are called TT cores, and the integer numbers $R_p$, $p \in \langle P-1 \rangle$ are the TT-ranks. For $p=1$ and $p=P$, the TT-cores are two matrices, which implies $R_0 = R_p = 1$. The TT-rank $R_p$ is given by the rank of the matrix unfolding $\mathbf{X}_{I_1 \cdots I_p \times I_{p+1} \cdots I_p}$.

The TTD can also be concisely written in a scalar way as follows:

$$ x_{i_p} = \sum_{r_1=1}^{R_1} \cdots \sum_{r_{p-1}=1}^{R_{p-1}} g_{i_1,r_1}^{(1)} g_{r_1,i_2,r_2}^{(2)} g_{r_2,i_3,r_3}^{(3)} \cdots g_{r_{p-2},i_{p-1},r_{p-1}}^{(P-1)} g_{r_{p-1},i_p}^{(P)}, \quad (14) $$
or using row vector-matrix, matrix-matrix, and matrix-column vector products as follows:

$$x_{i_p} = g_{i_1, \bullet}^{(1)} G_{i_2, \bullet}^{(2)} G_{i_3, \bullet}^{(3)} \cdots G_{i_{p-1}, \bullet}^{(P-1)} g_{i_p, \bullet}^{(P)} \quad (15)$$

where $g_{i_1, \bullet}^{(1)} \in \mathbb{K}^{1 \times R_1}$ and $g_{i_p, \bullet}^{(P)} \in \mathbb{K}^{R_{p-1}}$ are the $i_1$th row vector of $G^{(1)}$, and the $i_p$th column-vector of $G^{(P)}$, respectively, whereas $G_{i_p, i_p, \bullet}^{(p)} \in \mathbb{K}^{R_{p-1} \times R_p}$ is the $i_p$th lateral slice of $G^{(p)}$, for $p \in \{2, \dots, P-1\}$, i.e.,

$$\mathbf{g}_{i_1, \bullet}^{(1)} = [g_{i_1, 1}^{(1)} \cdots g_{i_1, R_1}^{(1)}]^{\top}, \quad \mathbf{g}_{\bullet, i_p}^{(P)} = [g_{1, i_p}^{(P)} \cdots g_{R_{p-1}, i_p}^{(P)}]^{\top} \\ \mathbf{G}_{\bullet, i_p, \bullet}^{(p)} = \begin{bmatrix} g_{1, i_p, 1}^{(p)} & \cdots & g_{1, i_p, R_p}^{(p)} \\ \vdots & & \vdots \\ g_{R_{p-1}, i_p, 1}^{(p)} & \cdots & g_{R_{p-1}, i_p, R_p}^{(p)} \end{bmatrix}$$

From Equation (15), we can conclude that TTD is not unique since it is invariant for all non-singular transformation matrices, $T^{(p)} \in \mathbb{K}^{R_p \times R_p}$, which transform the lateral slice $G_{\bullet, i_p, \bullet}^{(p)}$ of the core $G^{(p)}$, for $p \in \{2, \dots, P-1\}$, in such a way that the following applies:

$$\mathbf{F}_{\bullet, i_p, \bullet}^{(p)} = [\mathbf{T}^{(p-1)}]^{-1} \mathbf{G}_{\bullet, i_p, \bullet}^{(p)} \mathbf{T}^{(p)}, \quad (16)$$

and, for $p=1$ and $p=P$, $\mathbf{F}^{(1)} = \mathbf{G}^{(1)}\mathbf{T}^{(1)}$ and $\mathbf{F}^{(P)} = [\mathbf{T}^{(P-1)}]^{-1}\mathbf{G}^{(P)}$. Indeed, it is easy to verify that:

$$\mathbf{f}_{i_1, \bullet}^{(1)} \mathbf{F}_{\bullet, i_2, \bullet}^{(2)} \mathbf{F}_{\bullet, i_3, \bullet}^{(3)} \cdots \mathbf{F}_{\bullet, i_{p-1}, \bullet}^{(P-1)} \mathbf{f}_{i_p, \bullet}^{(P)} = \mathbf{g}_{i_1, \bullet}^{(1)} \mathbf{G}_{\bullet, i_2, \bullet}^{(2)} \mathbf{G}_{\bullet, i_3, \bullet}^{(3)} \cdots \mathbf{G}_{\bullet, i_{p-1}, \bullet}^{(P-1)} \mathbf{g}_{i_p, \bullet}^{(P)} = x_{i_p}. \quad (17)$$

The TTD can also be written using outer products of column vectors as:

$$\mathcal{X} = \sum_{r_1=1}^{R_1} \cdots \sum_{r_{p-1}=1}^{R_{p-1}} \mathbf{g}_{\bullet, r_1}^{(1)} \circ \mathbf{g}_{r_1, \bullet, r_2}^{(2)} \cdots \circ \mathbf{g}_{r_{p-2}, \bullet, r_{p-1}}^{(P-1)} \circ [\mathbf{g}_{r_{p-1}, \bullet}^{(P)}]^{\top}, \quad (18)$$

with $\mathbf{g}_{\bullet, r_1}^{(1)} \in \mathbb{K}^{I_1}$, $\mathbf{g}_{r_{p-1}, \bullet}^{(P)} \in \mathbb{K}^{1 \times I_p}$, and $\mathbf{g}_{r_{p-1}, \bullet, r_p}^{(p)} \in \mathbb{K}^{I_p}$, for $p \in \{2, \dots, P-1\}$. In Table 10, we summarize the different writings of a TTD, with $r_0 = r_p = 1$.

<table><caption>**Table 10.** TTD for a $P$-th-order tensor, $\mathcal{X} \in \mathbb{K}^{I_p}$.</caption><thead><tr><th></th><th>χ ∈ K<sup>I<sub>p</sub></sup></th></tr></thead><tbody><tr><td>G<sup>(1)</sup> ∈ K<sup>I<sub>1</sub>×R<sub>1</sub></sup> ; G<sup>(P)</sup> ∈ K<sup>R<sub>p-1</sub>×I<sub>p</sub></sup> ; G<sup>(p)</sup> ∈ K<sup>R<sub>p-1</sub>×I<sub>p</sub>×R<sub>p</sub></sup>, p ∈ {2, ..., P-1}</td><td></td></tr><tr><td>x<sub>i<sub>p</sub></sub> = &sum;<sub>r<sub>p-1</sub>=1</sub><sup>R<sub>p-1</sub></sup> &prod;<sub>p=1</sub><sup>P</sup> g<sub>r<sub>p-1</sub>,i<sub>p</sub>,r<sub>p</sub></sub><sup>(p)</sup> with g<sub>r<sub>0</sub>,i<sub>1</sub>,r<sub>1</sub></sub><sup>(1)</sup> = g<sub>i<sub>1</sub>,r<sub>1</sub></sub><sup>(1)</sup> and g<sub>r<sub>p-1</sub>,i<sub>p</sub>,r<sub>p</sub></sub><sup>(P)</sup> = g<sub>r<sub>p-1</sub>,i<sub>p</sub></sub><sup>(P)</sup></td><td></td></tr><tr><td>x<sub>i<sub>p</sub></sub> = g<sub>i<sub>1</sub>,<sup>(1)</sup></sub> G<sub>i<sub>2</sub>,<sup>(2)</sup></sub> G<sub>i<sub>3</sub>,<sup>(3)</sup></sub> &cdots G<sub>i<sub>p-1</sub>,<sup>(P-1)</sup></sub> g<sub>i<sub>p</sub>,<sup>(P)</sup></sub></td><td></td></tr><tr><td>χ = ((((G<sup>(1)</sup> ×<sup>1</sup><sub>2</sub> G<sup>(2)</sup>) ×<sup>1</sup><sub>3</sub> G<sup>(3)</sup>) ×<sup>1</sup><sub>4</sub>) ×<sup>1</sup><sub>p-1</sub> G<sup>(p-1)</sup>) ×<sup>1</sup><sub>p</sub> G<sup>(P)</sup></td><td></td></tr><tr><td>χ = &sum;<sub>r<sub>1</sub>=1</sub><sup>R<sub>1</sub></sup> &cdots &sum;<sub>r<sub>p-1</sub>=1</sub><sup>R<sub>p-1</sub></sup> g<sub>•,r<sub>1</sub></sub><sup>(1)</sup> &circ; g<sub>r<sub>1</sub>,•,r<sub>2</sub></sub><sup>(2)</sup> &cdots &circ; g<sub>r<sub>p-2</sub>,•,r<sub>p-1</sub></sub><sup>(P-1)</sup> &circ; [g<sub>r<sub>p-1</sub>,•</sub><sup>(P)</sup>]<sup>T</sup></td><td></td></tr></tbody></table>

Note that, for the TTD model, each carriage is characterized by only one inner index, $i_p$. The outer indices, $r_p$, are associated with modes common to two consecutive wagons. They represent the modes on which the tensor contractions operate.

The TTD model can be represented using a graph, as shown in Figure 4, for a fourth-order tensor, $\mathcal{X} \in \mathbb{K}^{I_4}$. The nodes, denoted as $(\bullet)$, represent matrices or third-order tensors
in the train, and the edges are labeled by an index $i_p$ of $\mathcal{X}$, or an index $r_p$ relative to the TT rank $R_p$. The number of edges associated with a node is equal to two or three, depending on whether the node is a matrix or a third-order tensor, respectively. The number of inner indices $i_p$ is equal to the order of $\mathcal{X}$.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Graph of the TTD-4 model for a fourth-order tensor $\mathcal{X} \in \mathbb{K}^{I_4}$.</figcaption></figure>

TTD combines the advantages of CPD and TD in terms of the following: (i) parametric complexity, which is proportional to the tensor order $P$ as for CPD, while it increases exponentially with the order $P$ for TD, due to the dimensionality of the core tensor; and (ii) the existence of a stable parametric estimation algorithm, based on SVD calculations, as for TD.

In Table 11, we summarize the notations and the element $x_{\mathbf{I}_N}$ for the CPD, TD, and TTD models, and their parametric complexity is compared in terms of the size of matrix and tensor factors, assuming $I_n = I$ and $R_n = R$ for all $n \in \langle N \rangle$.

<table><caption>**Table 11.** Parametric complexity of CPD, TD, and TTD.</caption><thead><tr><th>Models</th><th>Notations</th><th>Element x<sub>I<sub>N</sub></sub></th></tr></thead><tbody><tr><td>CPD</td><td>[[A<sup>(1)</sup>, ..., A<sup>(N)</sup>; R]]</td><td>Σ<sub>r=1</sub><sup>R</sup> Π<sub>n=1</sub><sup>N</sup> a<sub>I<sub>n</sub>,r</sub><sup>(n)</sup></td></tr><tr><td>TD</td><td>[[G; A<sup>(1)</sup>, ..., A<sup>(N)</sup>; R<sub>1</sub>, ..., R<sub>N</sub>]]</td><td>Σ<sub>r<sub>1</sub>=1</sub><sup>R<sub>1</sub></sup> ... Σ<sub>r<sub>N</sub>=1</sub><sup>R<sub>N</sub></sup> g<sub>r<sub>1</sub>,...,r<sub>N</sub></sub> Π<sub>n=1</sub><sup>N</sup> a<sub>I<sub>n</sub>,r<sub>n</sub></sub><sup>(n)</sup></td></tr><tr><td>TTD</td><td>[[G<sup>(1)</sup>, G<sup>(2)</sup>, ..., G<sup>(N-1)</sup>, G<sup>(N)</sup>; R<sub>1</sub>, ..., R<sub>N-1</sub>]]</td><td>Σ<sub>r<sub>1</sub>=1</sub><sup>R<sub>1</sub></sup> ... Σ<sub>r<sub>N-1</sub>=1</sub><sup>R<sub>N-1</sub></sup> Π<sub>n=1</sub><sup>N</sup> g<sub>r<sub>n-1</sub>,i<sub>n</sub>,r<sub>n'</sub></sub><sup>(n)</sup> r<sub>0</sub> = r<sub>N</sub></td></tr><tr><th>Models</th><th>Parameters</th><th>Complexity</th></tr><tr><td>CPD</td><td>A<sup>(n)</sup> ∈ K<sup>I<sub>n</sub>×R</sup>, ∀n ∈ ⟨N⟩</td><td>O(NIR)</td></tr><tr><td>TD</td><td>G ∈ K<sup>R<sub>N</sub></sup>; A<sup>(n)</sup> ∈ K<sup>I<sub>n</sub>×R<sub>n</sub></sup>, ∀n ∈ ⟨N⟩</td><td>O(NIR + R<sup>N</sup>)</td></tr><tr><td>TTD</td><td>G<sup>(1)</sup> ∈ K<sup>I<sub>1</sub>×R<sub>1</sub></sup>, G<sup>(N)</sup> ∈ K<sup>R<sub>N-1</sub>×I<sub>N</sub></sup><br>G<sup>(n)</sup> ∈ K<sup>R<sub>n-1</sub>×I<sub>n</sub>×R<sub>n</sub></sup>, ∀n ∈ {2, 3, ..., N-1}</td><td>O(2IR + (N-2)IR<sup>2</sup>)</td></tr></tbody></table>

A generalization of TTD is now introduced in defining a train of tensors of any order. For instance, a tensor $\mathcal{X} \in \mathbb{K}^{I_6}$ can be decomposed into a train of two matrices and two fourth-order tensors, as illustrated with the graph in Figure 5. Such a generalized TTD will be denoted as GTTD-(2,4,4,2), where each number is associated with the order of the corresponding tensor in the train. This type of generalization will be encountered for modeling relay systems based on tensor codings. See Section 5.3.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Graph of the GTTD-(2,4,4,2) model for a sixth-order tensor $\mathcal{X} \in \mathbb{K}^{I_6}$.</figcaption></figure>
### 3.3. Nested Tensor Models

We now present two families of nested tensor models, namely NCPD and NTD. In Sections 3.3.1 and 3.3.2, the standard NCPD-4 and NTD-4 models are described for a fourth-order tensor $\mathcal{X} \in \mathbb{K}^{I_4}$. Then, in Section 3.3.3, two generalizations corresponding to NTD-6 and NGTD-7 models are introduced for sixth- and seventh-order tensors, respectively.

The NCPD-4 and NTD-4 models, summarized in Table 12, will be abbreviated as follows:

$$ \llbracket \mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)}; R_1, R_2 \rrbracket \text{ and } \llbracket \mathbf{A}^{(1)}, \mathcal{G}^{(1)}, \mathbf{U}, \mathcal{G}^{(2)}, \mathbf{A}^{(2)}; R_1, R_2, R_3, R_4 \rrbracket, $$

where, by analogy with the TT ranks, the integers $(R_1, R_2)$ and $(R_1, R_2, R_3, R_4)$ will be called NCPD and NTD ranks, respectively. The TTD-4 model is also presented in Table 12.

<table><caption>**Table 12.** TTD, NCPD, and NTD models for a fourth-order tensor.</caption><tr><td colspan="2">$\mathcal{X} \in \mathbb{K}^{I_4}$</td></tr><tr><td colspan="2">TTD-4 model</td></tr><tr><td>$\mathbf{G}^{(1)} \in \mathbb{K}^{I_1 \times R_1}$ ; $\mathbf{G}^{(4)} \in \mathbb{K}^{R_3 \times I_4}$ ; $\mathcal{G}^{(p)} \in \mathbb{K}^{R_{p-1} \times I_p \times R_p}$, $p \in \{2,3\}$</td></tr><tr><td>$x_{\mathbf{i}_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} \sum_{r_3=1}^{R_3} \delta_{i_1,r_1}^{(1)} \delta_{r_1,i_2}^{(2)} \delta_{r_2,i_3}^{(3)} \delta_{r_3,i_4}^{(4)}$</td></tr><tr><td colspan="2">NCPD-4 model</td></tr><tr><td>$\mathbf{A}^{(1)} \in \mathbb{K}^{I_1 \times R_1}$ ; $\mathbf{B}^{(1)} \in \mathbb{K}^{I_2 \times R_1}$ ; $\mathbf{G} \in \mathbb{K}^{R_1 \times R_2}$ ; $\mathbf{A}^{(2)} \in \mathbb{K}^{I_3 \times R_2}$ ; $\mathbf{B}^{(2)} \in \mathbb{K}^{I_4 \times R_2}$</td></tr><tr><td>$x_{\mathbf{i}_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} a_{i_1,r_1}^{(1)} b_{i_2,r_1}^{(1)} \delta_{r_1,r_2} \delta_{i_3,r_2}^{(2)} b_{i_4,r_2}^{(2)}$</td></tr><tr><td colspan="2">NTD-4 model</td></tr><tr><td>$\mathbf{A}^{(1)} \in \mathbb{K}^{I_1 \times R_1}$ ; $\mathcal{G}^{(1)} \in \mathbb{K}^{R_1 \times I_2 \times R_2}$ ; $\mathbf{U} \in \mathbb{K}^{R_2 \times R_3}$ ; $\mathcal{G}^{(2)} \in \mathbb{K}^{R_3 \times I_3 \times R_4}$ ; $\mathbf{A}^{(2)} \in \mathbb{K}^{I_4 \times R_4}$</td></tr><tr><td>$x_{\mathbf{i}_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} \sum_{r_3=1}^{R_3} \sum_{r_4=1}^{R_4} a_{i_1,r_1}^{(1)} \delta_{r_1,i_2}^{(1)} \delta_{r_2,r_3} \delta_{r_3,i_3}^{(2)} a_{i_4,r_4}^{(2)}$</td></tr></table>

The NCPD-4 and NTD-4 models were first highlighted in [39,41], respectively, in the context of a point-to-point communication system using a DKRSTF coding and a two-hop relay system using a TST coding.

#### 3.3.1. NCPD-4 Model

The NCPD-4 model for $\mathcal{X} \in \mathbb{K}^{I_4}$ can be interpreted as the nesting of two CPD-3 models of the following third-order tensors $\mathcal{X}^{(1)} \in \mathbb{K}^{I_1 \times I_2 \times R_2}$ and $\mathcal{X}^{(2)} \in \mathbb{K}^{R_1 \times I_3 \times I_4}$:

$$ \mathcal{X}^{(1)} = \mathcal{I}_{R_1} \times_1 \mathbf{A}^{(1)} \times_2 \mathbf{B}^{(1)} \times_3 \mathbf{G}^T \quad (19) $$

$$ \mathcal{X}^{(2)} = \mathcal{I}_{R_2} \times_1 \mathbf{G} \times_2 \mathbf{A}^{(2)} \times_3 \mathbf{B}^{(2)}, \quad (20) $$

that share the common matrix factor $\mathbf{G}$, as shown in the following equation:

$$ x_{\mathbf{i}_4} = x_{i_1, \dots, i_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} a_{i_1,r_1}^{(1)} b_{i_2,r_1}^{(1)} \delta_{r_1,r_2} a_{i_3,r_2}^{(2)} b_{i_4,r_2}^{(2)} \quad (21) $$

where the red color is associated with the sum over index $r_1$ in the CPD model of the tensor $\mathcal{X}^{(1)}$, while the blue color is used for the sum over index $r_2$ in the CPD model of the tensor $\mathcal{X}^{(2)}$, and the common factor of these two CPDs is in green.

This nesting of the CPD-3 models (19) and (20), represented in Figure 6a, is characterized by five matrix factors $(\mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)})$.
<figure><img src="image_8.png" /><figcaption>Figure 6. NCPD-4 model as (a) a nesting of two CPD-3 models and (b) a cascade of two CPD-3 models.</figcaption></figure>

According to the definition of the CPD-3 models (19) and (20), we have the following:

$$x_{i_1 i_2 r_2}^{(1)} = \sum_{r_1=1}^{R_1} a_{i_1 r_1}^{(1)} b_{i_2 r_1}^{(1)} g_{r_1 r_2} \quad (22)$$

$$x_{r_1 i_3 i_4}^{(2)} = \sum_{r_2=1}^{R_2} g_{r_1 r_2} a_{i_3 r_2}^{(2)} b_{i_4 r_2}^{(2)}. \quad (23)$$

Equation (21) can then be rewritten as follows:

$$x_{i_4} = \sum_{r_1=1}^{R_1} a_{i_1 r_1}^{(1)} b_{i_2 r_1}^{(1)} x_{r_1 i_3 i_4}^{(2)} = \sum_{r_2=1}^{R_2} x_{i_1 i_2 r_2}^{(1)} a_{i_3 r_2}^{(2)} b_{i_4 r_2}^{(2)}. \quad (24)$$

Let us define the matrix unfoldings $X_{I_1 I_2 \times R_2}^{(1)}$ and $X_{I_3 I_4 \times R_1}^{(2)}$, which result from a combination of the first two modes of $\mathcal{X}^{(1)}$ and the last two modes of $\mathcal{X}^{(2)}$, respectively. The writing (24) of $x_{i_4}$ can be interpreted as two CPD-3 models associated with the following third-order contracted forms $\mathcal{X}_{c_1} \in \mathbb{K}^{I_1 \times I_2 \times I_3 I_4}$ and $\mathcal{X}_{c_2} \in \mathbb{K}^{I_1 I_2 \times I_3 \times I_4}$ of the fourth-order tensor $\mathcal{X}$:

$$\mathcal{X}_{c_1} = \mathcal{I}_{R_1} \times_1 \mathbf{A}^{(1)} \times_2 \mathbf{B}^{(1)} \times_3 \mathbf{X}_{I_3 I_4 \times R_1}^{(2)} \quad (25)$$

$$\mathcal{X}_{c_2} = \mathcal{I}_{R_2} \times_1 \mathbf{X}_{I_1 I_2 \times R_2}^{(1)} \times_2 \mathbf{A}^{(2)} \times_3 \mathbf{B}^{(2)}. \quad (26)$$

Figure 6b illustrates another interpretation of the NCPD-4 model, deduced from the contracted form (25), as a cascade of two CPD-3 models. A similar interpretation in terms of cascade can be deduced from the contracted form (26).

Matrix unfoldings associated with the CPD models of the third-order tensors $\mathcal{X}^{(1)}$, $\mathcal{X}^{(2)}$, $\mathcal{X}_{c_1}$ and $\mathcal{X}_{c_2}$, defined in Equations (19), (20), (25) and (26), are deduced from the general formulae recalled in Table 7, as summarized in Table 13. Combining these matrix unfoldings leads to matrix unfoldings for the NCPD-4 model, as given in Table 14.

Applying the property $\mathrm{vec}(\mathbf{ABC}^T) = (\mathbf{C} \otimes \mathbf{A})\mathrm{vec}(\mathbf{B})$ to the last equation in Table 14 gives a vectorized form of the tensor $\mathcal{X}$:

$$\mathbf{x}_{I_3 I_4 I_1 I_2} = \mathrm{vec}(\mathbf{X}_{I_1 I_2 \times I_3 I_4}) = ((\mathbf{A}^{(2)} \diamond \mathbf{B}^{(2)}) \otimes (\mathbf{A}^{(1)} \diamond \mathbf{B}^{(1)})) \mathrm{vec}(\mathbf{G}). \quad (27)$$

Note that the first four matrix unfoldings in Table 14 and the vectorized form (27) contain one isolated factor (to the right of the equations) from $(\mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)})$. These unfoldings will be used separately and iteratively to estimate the matrix factors of the NCPD-4 model, with a five-step alternating least squares (ALS) algorithm, as presented in Section 4.2.
<table><caption>**Table 13.** Matrix unfoldings of CPD models.</caption><tr><td>CPD model of <i>X</i><sup>(1)</sup> = [[<b>A</b><sup>(1)</sup>, <b>B</b><sup>(1)</sup>, <b>G</b><sup>T</sup>; R<sub>1</sub>]] ∈ K<sup>I<sub>1</sub>×I<sub>2</sub>×R<sub>2</sub></sup></td></tr><tr><td><b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>)<b>G</b><br><b>X</b><sup>(1)</sup><sub>I<sub>1</sub>R<sub>2</sub>×I<sub>2</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>G</b><sup>T</sup>)(<b>B</b><sup>(1)</sup>)<sup>T</sup></td></tr><tr><td>CPD model of <i>X</i><sup>(2)</sup> = [[<b>G</b>, <b>A</b><sup>(2)</sup>, <b>B</b><sup>(2)</sup>; R<sub>2</sub>]] ∈ K<sup>R<sub>1</sub>×I<sub>3</sub>×I<sub>4</sub></sup></td></tr><tr><td><b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub> = (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<b>G</b><sup>T</sup><br><b>X</b><sup>(2)</sup><sub>I<sub>4</sub>R<sub>1</sub>×I<sub>3</sub></sub> = (<b>B</b><sup>(2)</sup> ◦ <b>G</b>)(<b>A</b><sup>(2)</sup>)<sup>T</sup></td></tr><tr><td>CPD model of <i>X</i><sub>c<sub>1</sub></sub> = [[<b>A</b><sup>(1)</sup>, <b>B</b><sup>(1)</sup>, <b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub>; R<sub>1</sub>]] ∈ K<sup>I<sub>1</sub>×I<sub>2</sub>×I<sub>3</sub>I<sub>4</sub></sup></td></tr><tr><td><b>X</b><sub>I<sub>1</sub>I<sub>3</sub>I<sub>4</sub>×I<sub>2</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub>) (<b>B</b><sup>(1)</sup>)<sup>T</sup><br><b>X</b><sub>I<sub>2</sub>I<sub>3</sub>I<sub>4</sub>×I<sub>1</sub></sub> = (<b>B</b><sup>(1)</sup> ◦ <b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub>) (<b>A</b><sup>(1)</sup>)<sup>T</sup><br><b>X</b><sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub>I<sub>4</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>)<b>X</b><sup>(2)</sup><sub>R<sub>1</sub>×I<sub>3</sub>I<sub>4</sub></sub></td></tr><tr><td>CPD model of <i>X</i><sub>c<sub>2</sub></sub> = [[<b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub>, <b>A</b><sup>(2)</sup>, <b>B</b><sup>(2)</sup>; R<sub>2</sub>]] ∈ K<sup>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub>×I<sub>4</sub></sup></td></tr><tr><td><b>X</b><sub>I<sub>3</sub>I<sub>1</sub>I<sub>2</sub>×I<sub>4</sub></sub> = (<b>A</b><sup>(2)</sup> ◦ <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub>) (<b>B</b><sup>(2)</sup>)<sup>T</sup><br><b>X</b><sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub></sub> = (<b>B</b><sup>(2)</sup> ◦ <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub>) (<b>A</b><sup>(2)</sup>)<sup>T</sup><br><b>X</b><sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub>I<sub>4</sub></sub> = <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub> (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<sup>T</sup></td></tr></table>

<table><caption>**Table 14.** Matrix unfoldings of the NCPD-4 model.</caption><tr><td>NCPD-4 model of <i>X</i> = [[<b>A</b><sup>(1)</sup>, <b>B</b><sup>(1)</sup>, <b>G</b>, <b>A</b><sup>(2)</sup>, <b>B</b><sup>(2)</sup>; R<sub>1</sub>, R<sub>2</sub>]] ∈ K<sup>I<sub>4</sub></sup></td></tr><tr><td><b>X</b><sub>I<sub>1</sub>I<sub>3</sub>I<sub>4</sub>×I<sub>2</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub>) (<b>B</b><sup>(1)</sup>)<sup>T</sup> = [<b>A</b><sup>(1)</sup> ◦ (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<b>G</b><sup>T</sup>] (<b>B</b><sup>(1)</sup>)<sup>T</sup></td></tr><tr><td><b>X</b><sub>I<sub>2</sub>I<sub>3</sub>I<sub>4</sub>×I<sub>1</sub></sub> = (<b>B</b><sup>(1)</sup> ◦ <b>X</b><sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>1</sub></sub>) (<b>A</b><sup>(1)</sup>)<sup>T</sup> = [<b>B</b><sup>(1)</sup> ◦ (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<b>G</b><sup>T</sup>] (<b>A</b><sup>(1)</sup>)<sup>T</sup></td></tr><tr><td><b>X</b><sub>I<sub>3</sub>I<sub>1</sub>I<sub>2</sub>×I<sub>4</sub></sub> = (<b>A</b><sup>(2)</sup> ◦ <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub>) (<b>B</b><sup>(2)</sup>)<sup>T</sup> = [<b>A</b><sup>(2)</sup> ◦ (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>)<b>G</b>] (<b>B</b><sup>(2)</sup>)<sup>T</sup></td></tr><tr><td><b>X</b><sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub></sub> = (<b>B</b><sup>(2)</sup> ◦ <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub>) (<b>A</b><sup>(2)</sup>)<sup>T</sup> = [<b>B</b><sup>(2)</sup> ◦ (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>)<b>G</b>] (<b>A</b><sup>(2)</sup>)<sup>T</sup></td></tr><tr><td><b>X</b><sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub>I<sub>4</sub></sub> = (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>) <b>X</b><sup>(2)</sup><sub>R<sub>1</sub>×I<sub>3</sub>I<sub>4</sub></sub> = <b>X</b><sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>2</sub></sub> (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<sup>T</sup> = (<b>A</b><sup>(1)</sup> ◦ <b>B</b><sup>(1)</sup>) <b>G</b> (<b>A</b><sup>(2)</sup> ◦ <b>B</b><sup>(2)</sup>)<sup>T</sup></td></tr></table>

When some factors are known, the ALS algorithm can be simplified. That is the case in the context of relay system using the SKRST coding, where the factors (<b>B</b><sup>(1)</sup>, <b>A</b><sup>(2)</sup>) represent coding matrices assumed to be known at the receiver. Then, the other three factors (<b>A</b><sup>(1)</sup>, <b>G</b>, <b>B</b><sup>(2)</sup>) representing the symbol and channel matrices can be estimated using a three-step ALS algorithm. See Equations (67), (68) and (70).

From the cascade structure shown in Figure 6b, and, therefore, from Equation (25), we conclude that the tensor $\mathcal{X}$ can be represented by means of two coupled CPD-3 models associated with the contracted form $\mathcal{X}_{c_1}$ and the tensor $\mathcal{X}^{(2)}$, i.e., Equations (25) and (20). Similarly, the coupled CPD-3 models of the contracted form $\mathcal{X}_{c_2}$ and the tensor $\mathcal{X}^{(1)}$, i.e., Equations (26) and (19), can be used to represent the tensor $\mathcal{X}$.

When (<b>B</b><sup>(1)</sup>, <b>A</b><sup>(2)</sup>) are assumed to be known, each CPD-3 model of $\mathcal{X}^{(1)}$, $\mathcal{X}^{(2)}$, $\mathcal{X}_{c_1}$ and $\mathcal{X}_{c_2}$ contains two unknown matrix factors that can be estimated using a closed-form algorithm based on the Khatri–Rao factorization (KRF) method, as first proposed in [49].

Exploiting each set of coupled CPD-3 models allows for the estimation of the three unknown factors (<b>A</b><sup>(1)</sup>, <b>G</b>, <b>B</b><sup>(2)</sup>) by means of a two-step procedure based on KRF. The resulting two KRF-based closed-form algorithms, which use the unfoldings in green in Table 13, will be described in Section 4.2. See Table 18.
**Remark 1.** The essential uniqueness of a CPD model, i.e., uniqueness up to column permutation and scaling ambiguities in the factor matrices, has been the subject of numerous articles in the literature. From a modeling point of view, uniqueness means that the model $[\mathbf{A}\mathbf{\Pi}\mathbf{\Lambda}^{(\mathbf{A})}, \mathbf{B}\mathbf{\Pi}\mathbf{\Lambda}^{(\mathbf{B})}, \mathbf{C}\mathbf{\Pi}\mathbf{\Lambda}^{(\mathbf{C})}; \mathbf{R}]$ is equivalent to the model $[\mathbf{A}, \mathbf{B}, \mathbf{C}; \mathbf{R}]$, where $\mathbf{\Pi}$ is a permutation matrix, and $(\mathbf{\Lambda}^{(\mathbf{A})}, \mathbf{\Lambda}^{(\mathbf{B})}, \mathbf{\Lambda}^{(\mathbf{C})})$ are $\mathbf{R} \times \mathbf{R}$ diagonal matrices, such that their product is the identity matrix: $\mathbf{\Lambda}^{(\mathbf{A})}\mathbf{\Lambda}^{(\mathbf{B})}\mathbf{\Lambda}^{(\mathbf{C})} = \mathbf{I}_{\mathbf{R}}$.

The following sufficient uniqueness condition has been established by Kruskal [50] for a third-order CPD model $[\mathbf{A}, \mathbf{B}, \mathbf{C}; \mathbf{R}]$:

$$k_{\mathbf{A}} + k_{\mathbf{B}} + k_{\mathbf{C}} \geq 2\mathbf{R} + 2, \qquad (28)$$

where $k_{\mathbf{A}}$ denotes the k-rank (also known as the Kruskal's rank) of $\mathbf{A}$, i.e., the largest integer $k_{\mathbf{A}}$ such that every set of $k_{\mathbf{A}}$ columns of $\mathbf{A}$ is linearly independent.

When a factor matrix (e.g., $\mathbf{C}$) is known, and the Kruskal condition (28) is satisfied, essential uniqueness is guaranteed without any permutation ambiguity and with only two diagonal ambiguity matrices $(\mathbf{\Lambda}^{(\mathbf{A})}, \mathbf{\Lambda}^{(\mathbf{B})})$, such that $\mathbf{\Lambda}^{(\mathbf{A})}\mathbf{\Lambda}^{(\mathbf{B})} = \mathbf{I}_{\mathbf{R}}$.

Applying Kruskal's condition to the CPD models (19) and (20) leads to the following uniqueness conditions for the NCPD-4 model:

$$k_{\mathbf{A}^{(1)}} + k_{\mathbf{B}^{(1)}} + k_{\mathbf{G}^{T}} \geq 2\mathbf{R}_{1} + 2 \qquad (29)$$

$$k_{\mathbf{A}^{(2)}} + k_{\mathbf{B}^{(2)}} + k_{\mathbf{G}} \geq 2\mathbf{R}_{2} + 2. \qquad (30)$$

As mentioned previously, in the context of relay systems, the factors $(\mathbf{B}^{(1)}, \mathbf{A}^{(2)})$ will be assumed to be known and full column rank, implying $k_{\mathbf{B}^{(1)}} = \mathbf{R}_1$ and $k_{\mathbf{A}^{(2)}} = \mathbf{R}_2$. The Kruskal's conditions (29) and (30) then become

$$k_{\mathbf{A}^{(1)}} + k_{\mathbf{G}^{T}} \geq \mathbf{R}_{1} + 2 \quad \text{and} \quad k_{\mathbf{B}^{(2)}} + k_{\mathbf{G}} \geq \mathbf{R}_{2} + 2, \qquad (31)$$

and the ambiguity relations for the NCPD-4 model (19) and (20) can be simplified as follows:

$$\mathbf{\Lambda}^{(\mathbf{A}^{(1)})} \mathbf{\Lambda}^{(\mathbf{G}^{T})} = \mathbf{I}_{\mathbf{R}_{1}} \quad \text{and} \quad \mathbf{\Lambda}^{(\mathbf{B}^{(2)})} \mathbf{\Lambda}^{(\mathbf{G})} = \mathbf{I}_{\mathbf{R}_{2}}. \qquad (32)$$

### 3.3.2. NTD-4 Model

In this section, we present the NTD-4 model for a fourth-order tensor, $\mathcal{X} \in \mathbb{K}^{I_4}$. This model, introduced in Table 12, corresponds to the nesting of two Tucker-(2,3) decompositions of the third-order tensors $\mathcal{X}^{(1)} \in \mathbb{K}^{I_1 \times I_2 \times R_3}$ and $\mathcal{X}^{(2)} \in \mathbb{K}^{R_2 \times I_3 \times I_4}$:

$$\mathcal{X}^{(1)} = \mathcal{G}^{(1)} \times_1 \mathbf{A}^{(1)} \times_2 \mathbf{I}_{I_2} \times_3 \mathbf{U}^{T} \qquad (33)$$

$$\mathcal{X}^{(2)} = \mathcal{G}^{(2)} \times_1 \mathbf{U} \times_2 \mathbf{I}_{I_3} \times_3 \mathbf{A}^{(2)}, \qquad (34)$$

that share the matrix factor $\mathbf{U}$, as illustrated in the following equation:

$$x_{\mathbf{I}_4} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} \sum_{r_3=1}^{R_3} \sum_{r_4=1}^{R_4} a_{i_1,r_1}^{(1)} g_{r_1,i_2,r_2}^{(1)} u_{r_2,r_3} g_{r_3,i_3,r_4}^{(2)} a_{i_4,r_4}^{(2)}. \qquad (35)$$

The red color is associated with the sums over indices $r_1$ and $r_2$ in the TD-(2,3) model of the tensor $\mathcal{X}^{(1)}$, while the blue color is used for the sums over indices $r_3$ and $r_4$ in the TD-(2,3) model of the tensor $\mathcal{X}^{(2)}$, and the common factor of these two TDs is in green.

Equation (35) is to be compared with Equation (21) of the NCPD-4 model. From this comparison, we deduce that the nesting of two CPD-3 models in the case of NCPD-4 is replaced with the nesting of two TD-(2,3) models for NTD-4.

The NTD-4 model can also be viewed as a particular TTD model, represented in Figure 7a and the graph in Figure 8. This TTD is composed of three matrix factors and two third-order core tensors: $(\mathbf{A}^{(1)}, \mathcal{G}^{(1)}, \mathbf{U}, \mathcal{G}^{(2)}, \mathbf{A}^{(2)})$.
<figure><img src="image_9.png" /><figcaption>Figure 7. NTD-4 model as (a) a particular TTD and (b) a cascade of two TD-(2,3) models.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8. Graph of the NTD-4 model for a fourth-order tensor $\mathcal{X} \in \mathbb{K}^{I_4}$.</figcaption></figure>

Following the same approach as for NCPD-4, the fourth-order tensor $\mathcal{X}$ can be written using the following two third-order contracted forms $\mathcal{X}_{c_1} \in \mathbb{K}^{I_1 \times I_2 \times I_3 I_4}$ and $\mathcal{X}_{c_2} \in \mathbb{K}^{I_1 I_2 \times I_3 \times I_4}$, which result from combinations of the last two modes of $\mathcal{X}^{(2)}$ and the first two modes of $\mathcal{X}^{(1)}$, respectively:

$$ \mathcal{X}_{c_1} = \mathcal{G}^{(1)} \times_1 \mathbf{A}^{(1)} \times_2 \mathbf{I}_{I_2} \times_3 \mathbf{X}_{I_3 I_4 \times R_2}^{(2)} \quad (36) $$

$$ \mathcal{X}_{c_2} = \mathcal{G}^{(2)} \times_1 \mathbf{X}_{I_1 I_2 \times R_3}^{(1)} \times_2 \mathbf{I}_{I_3} \times_3 \mathbf{A}^{(2)}. \quad (37) $$

These contracted decompositions are to be compared with Equations (25) and (26) for the NCPD-4 model where the core tensors $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ are replaced with the identity tensors $\mathcal{I}_{R_1}$ and $\mathcal{I}_{R_2}$, respectively, and the identity matrices $(\mathbf{I}_{I_2}, \mathbf{I}_{I_3})$ are replaced with the matrix factors $(\mathbf{B}^{(1)}, \mathbf{A}^{(2)})$. From Equation (36), we deduce that the NTD-4 model can be viewed as the cascade of two TD-(2,3) models, as illustrated in Figure 7b. A similar cascade structure can be deduced from Equation (37).

Matrix unfoldings associated with the TD models of the third-order tensors $\mathcal{X}^{(1)}, \mathcal{X}^{(2)}, \mathcal{X}_{c_1}$ and $\mathcal{X}_{c_2}$, defined in Equations (33)–(37), are obtained using the general formulae recalled in Table 6, as summarized in Table 15.

Vectorizing $\mathbf{X}_{I_1 I_2 \times I_3 I_4}$ yields the following vectorized form of $\mathcal{X}$:

$$ \mathbf{x}_{I_3 I_4 I_1 I_2} = \mathrm{vec}(\mathbf{X}_{I_1 I_2 \times I_3 I_4}) = \left[ (\mathbf{I}_{I_3} \otimes \mathbf{A}^{(2)}) \mathbf{G}_{I_3 R_4 \times R_3}^{(2)} \otimes (\mathbf{A}^{(1)} \otimes \mathbf{I}_{I_2}) \mathbf{G}_{R_1 I_2 \times R_2}^{(1)} \right] \mathrm{vec}(\mathbf{U}). \quad (38) $$

By combining the unfoldings in Table 15, it is easy to obtain the unfoldings of the NTD-4 model summarized in Table 16, where the last equation is the vectorized form (38).
<table><caption>**Table 15.** Matrix unfoldings of TD models.</caption><tbody><tr><td>TD model of X<sup>(1)</sup> = &#x2713;<sup>(1)</sup>; A<sup>(1)</sup>, I<sub>12</sub>, U<sup>T</sup>; R<sub>1</sub>, R<sub>2</sub> &isin; &#8477;<sup>I<sub>1</sub> &times; I<sub>2</sub> &times; R<sub>3</sub></sup></td></tr><tr><td>X<sub>I<sub>1</sub>I<sub>2</sub>&times;R<sub>3</sub></sub> = (A<sup>(1)</sup> &otimes; I<sub>12</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> U<br>X<sub>I<sub>1</sub>R<sub>3</sub>&times;I<sub>2</sub></sub> = (A<sup>(1)</sup> &otimes; U<sup>T</sup>)G<sub>R<sub>1</sub>R<sub>2</sub>&times;I<sub>2</sub></sub></td></tr><tr><td>TD model of X<sup>(2)</sup> = &#x2713;<sup>(2)</sup>; U, I<sub>3</sub>, A<sup>(2)</sup>; R<sub>3</sub>, R<sub>4</sub> &isin; &#8477;<sup>R<sub>2</sub> &times; I<sub>3</sub> &times; I<sub>4</sub></sup></td></tr><tr><td>X<sub>I<sub>3</sub>I<sub>4</sub>&times;R<sub>2</sub></sub> = (I<sub>3</sub> &otimes; A<sup>(2)</sup>)G<sub>I<sub>3</sub>R<sub>4</sub>&times;R<sub>3</sub></sub> U<sup>T</sup><br>X<sub>I<sub>4</sub>R<sub>2</sub>&times;I<sub>3</sub></sub> = (A<sup>(2)</sup> &otimes; U)G<sub>R<sub>4</sub>R<sub>3</sub>&times;I<sub>3</sub></sub></td></tr><tr><td>TD model of X<sub>c<sub>1</sub></sub> = &#x2713;<sup>(1)</sup>; A<sup>(1)</sup>, I<sub>12</sub>, X<sub>I<sub>3</sub>I<sub>4</sub>&times;R<sub>2</sub></sub>; R<sub>1</sub>, R<sub>2</sub> &isin; &#8477;<sup>I<sub>1</sub> &times; I<sub>2</sub> &times; I<sub>3</sub>I<sub>4</sub></sup></td></tr><tr><td>X<sub>I<sub>2</sub>I<sub>3</sub>I<sub>4</sub>&times;I<sub>1</sub></sub> = [I<sub>2</sub> &otimes; X<sub>I<sub>3</sub>I<sub>4</sub>&times;R<sub>2</sub></sub>] G<sub>I<sub>2</sub>R<sub>2</sub>&times;R<sub>1</sub></sub> (A<sup>(1)</sup>)<sup>T</sup><br>X<sub>I<sub>1</sub>I<sub>3</sub>I<sub>4</sub>&times;I<sub>2</sub></sub> = (A<sup>(1)</sup> &otimes; X<sub>I<sub>3</sub>I<sub>4</sub>&times;R<sub>2</sub></sub>) G<sub>R<sub>1</sub>R<sub>2</sub>&times;I<sub>2</sub></sub></td></tr><tr><td>X<sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>3</sub>I<sub>4</sub></sub> = (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> X<sub>R<sub>2</sub>&times;I<sub>3</sub>I<sub>4</sub></sub> = (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> UG<sub>R<sub>3</sub>&times;I<sub>3</sub>R<sub>4</sub></sub> (I<sub>3</sub> &otimes; A<sup>(2)</sup>)<sup>T</sup></td></tr><tr><td>TD model of X<sub>c<sub>2</sub></sub> = &#x2713;<sup>(2)</sup>; X<sub>I<sub>1</sub>I<sub>2</sub>&times;R<sub>3</sub></sub>, I<sub>3</sub>, A<sup>(2)</sup>; R<sub>3</sub>, R<sub>4</sub> &isin; &#8477;<sup>I<sub>1</sub> I<sub>2</sub> &times; I<sub>3</sub> &times; I<sub>4</sub></sup></td></tr><tr><td>X<sub>I<sub>3</sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>4</sub></sub> = [I<sub>3</sub> &otimes; X<sub>I<sub>1</sub>I<sub>2</sub>&times;R<sub>3</sub></sub>] G<sub>I<sub>3</sub>R<sub>3</sub>&times;R<sub>4</sub></sub> (A<sup>(2)</sup>)<sup>T</sup><br>X<sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>3</sub></sub> = (A<sup>(2)</sup> &otimes; X<sub>I<sub>1</sub>I<sub>2</sub>&times;R<sub>3</sub></sub>) G<sub>R<sub>4</sub>R<sub>3</sub>&times;I<sub>3</sub></sub></td></tr><tr><td>X<sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>3</sub>I<sub>4</sub></sub> = X<sub>I<sub>1</sub>I<sub>2</sub>&times;R<sub>3</sub></sub> G<sub>R<sub>3</sub>&times;I<sub>3</sub>R<sub>4</sub></sub> (I<sub>3</sub> &otimes; A<sup>(2)</sup>)<sup>T</sup> = (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> UG<sub>R<sub>3</sub>&times;I<sub>3</sub>R<sub>4</sub></sub> (I<sub>3</sub> &otimes; A<sup>(2)</sup>)<sup>T</sup></td></tr></tbody></table>

<table><caption>**Table 16.** Unfoldings of the NTD-4 model.</caption><tbody><tr><td>NTD-4 model of X = &#x2713;<sup>(1)</sup>, G<sup>(1)</sup>, U, G<sup>(2)</sup>, A<sup>(2)</sup>; R<sub>1</sub>, R<sub>2</sub>, R<sub>3</sub>, R<sub>4</sub> &isin; &#8477;<sup>I<sub>4</sub></sup></td></tr><tr><td>A<sup>(1)</sup> &isin; &#8477;<sup>I<sub>1</sub> &times; R<sub>1</sub></sup> ; G<sup>(1)</sup> &isin; &#8477;<sup>R<sub>1</sub> &times; I<sub>2</sub> &times; R<sub>2</sub></sup> ; U &isin; &#8477;<sup>R<sub>2</sub> &times; R<sub>3</sub></sup> ; G<sup>(2)</sup> &isin; &#8477;<sup>R<sub>3</sub> &times; I<sub>3</sub> &times; R<sub>4</sub></sup> ; A<sup>(2)</sup> &isin; &#8477;<sup>I<sub>4</sub> &times; R<sub>4</sub></sup></td></tr><tr><td>X<sub>I<sub>3</sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>4</sub></sub> = [I<sub>3</sub> &otimes; (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> U] G<sub>I<sub>3</sub>R<sub>3</sub>&times;R<sub>4</sub></sub> (A<sup>(2)</sup>)<sup>T</sup><br>X<sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub>&times;I<sub>3</sub></sub> = [A<sup>(2)</sup> &otimes; (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> U] G<sub>R<sub>4</sub>R<sub>3</sub>&times;I<sub>3</sub></sub></td></tr><tr><td>X<sub>I<sub>2</sub>I<sub>3</sub>I<sub>4</sub>&times;I<sub>1</sub></sub> = [I<sub>2</sub> &otimes; (I<sub>3</sub> &otimes; A<sup>(2)</sup>)G<sub>I<sub>3</sub>R<sub>4</sub>&times;R<sub>3</sub></sub> U<sup>T</sup>] G<sub>I<sub>2</sub>R<sub>2</sub>&times;R<sub>1</sub></sub> (A<sup>(1)</sup>)<sup>T</sup><br>X<sub>I<sub>1</sub>I<sub>3</sub>I<sub>4</sub>&times;I<sub>2</sub></sub> = [A<sup>(1)</sup> &otimes; (I<sub>3</sub> &otimes; A<sup>(2)</sup>)G<sub>I<sub>3</sub>R<sub>4</sub>&times;R<sub>3</sub></sub> U<sup>T</sup>] G<sub>R<sub>1</sub>R<sub>2</sub>&times;I<sub>2</sub></sub></td></tr><tr><td>x<sub>I<sub>3</sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub></sub> = [(I<sub>3</sub> &otimes; A<sup>(2)</sup>)G<sub>I<sub>3</sub>R<sub>4</sub>&times;R<sub>3</sub></sub> &otimes; (A<sup>(1)</sup> &otimes; I<sub>2</sub>)G<sub>R<sub>1</sub>I<sub>2</sub>&times;R<sub>2</sub></sub> vec(U)</td></tr></tbody></table>

**Remark 2.** We can make the following remarks:

* The equations in Table 16 can be used to build a five-step ALS-based algorithm to separately and iteratively estimate the factors (A<sup>(1)</sup>, G<sup>(1)</sup>, A<sup>(2)</sup>, G<sup>(2)</sup>, U) of the NTD-4 model.

* When the tensors G<sup>(1)</sup> and G<sup>(2)</sup> are known, the equations in Table 16 can be used to estimate the unknown matrix factors (A<sup>(1)</sup>, A<sup>(2)</sup>, U) of the NTD-4 model using a three-step ALS algorithm. In Section 4.3, equations in Table 15 will be exploited to derive two closed-form algorithms based on the Kronecker factorization (KronF) method. These algorithms will be used to develop semi-blind receivers for relay systems where the tensors G<sup>(1)</sup> and G<sup>(2)</sup> are STST coding tensors assumed to be known at the destination.

* Using the writing (15) of a TTD, the NTD-4 model (35) can be rewritten as follows:

$$x_{i_4} = a_{i_1, \bullet}^{(1)} G_{\bullet, i_2, \bullet}^{(1)} U G_{\bullet, i_3, \bullet}^{(2)} a_{i_4, \bullet}^{(2)}, \quad (39)$$

which allows for the conclusion that the NTD-4 model is unique up to non-singular matrices $\Lambda^{(n)} \in \mathbb{K}^{R_n \times R_n}$, $n \in \langle 4 \rangle$, such that the following applies:
$$ \tilde{\mathbf{A}}^{(1)} = \mathbf{A}^{(1)} \mathbf{\Lambda}^{(1)}, \quad \tilde{\mathbf{U}} = [\mathbf{\Lambda}^{(2)}]^{-1} \mathbf{U} \mathbf{\Lambda}^{(3)}, \quad \tilde{\mathbf{A}}^{(2)} = [\mathbf{\Lambda}^{(4)}]^{-1} \mathbf{A}^{(2)} \quad (40) $$

$$ \tilde{\mathbf{G}}_{\bullet, i_2, \bullet}^{(1)} = [\mathbf{\Lambda}^{(1)}]^{-1} \mathbf{G}_{\bullet, i_2, \bullet}^{(1)} \mathbf{\Lambda}^{(2)}, \quad \tilde{\mathbf{G}}_{\bullet, i_3, \bullet}^{(2)} = [\mathbf{\Lambda}^{(3)}]^{-1} \mathbf{G}_{\bullet, i_3, \bullet}^{(2)} \mathbf{\Lambda}^{(4)}. \quad (41) $$

Indeed, it is easy to verify that $\tilde{\mathbf{a}}_{i_1, \bullet}^{(1)} \tilde{\mathbf{G}}_{\bullet, i_2, \bullet}^{(1)} \tilde{\mathbf{U}} \tilde{\mathbf{G}}_{\bullet, i_3, \bullet}^{(2)} \tilde{\mathbf{a}}_{i_4}^{(2)} = \mathbf{a}_{i_1, \bullet}^{(1)} \mathbf{G}_{\bullet, i_2, \bullet}^{(1)} \mathbf{U} \mathbf{G}_{\bullet, i_3, \bullet}^{(2)} \mathbf{a}_{i_4}^{(2)} = x_{i_4}$.

* When the core tensors $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ are perfectly known, as will be the case with the coding tensors in the context of relay systems, the ambiguity matrices $\mathbf{\Lambda}^{(n)}$ become identity matrices multiplied by a scalar:

$$ \mathbf{\Lambda}^{(n)} = \alpha_n \mathbf{I}_{R_n} \text{ for } n \in \langle 4 \rangle, \text{ with } \prod_{n=1}^{4} \alpha_n = 1; \alpha_2 = \alpha_1, \alpha_3 = \alpha_4 \text{ and } \tilde{\mathbf{U}} = \frac{\alpha_4}{\alpha_1} \mathbf{U}. \quad (42) $$

The scaling factors $\alpha_1$ and $\alpha_4$ can be determined using the knowledge of one element in $\mathbf{A}^{(1)}$ and $\mathbf{A}^{(4)}$, respectively.

### 3.3.3. NTD-6 and NGTD-7 Models

In Table 17, we present two generalizations of the NTD-4 model: the NTD-6 and NGTD-7 models, for tensors of order 6 and 7, respectively. These models will be used to represent relay systems with TST and TSTF codings, respectively, in Section 5.

<table><caption>Table 17. NTD-6 and NGTD-7 models.</caption><tr><td colspan="2">X ∈ K<sup>I<sub>6</sub></sup></td></tr><tr><td colspan="2">NTD-6 model</td></tr><tr><td>A<sup>(1)</sup> ∈ K<sup>I<sub>1</sub>×R<sub>1</sub></sup> ; G<sup>(1)</sup> ∈ K<sup>R<sub>1</sub>×I<sub>2</sub>×I<sub>3</sub>×R<sub>2</sub></sup> ; U ∈ K<sup>R<sub>2</sub>×R<sub>3</sub></sup> ; G<sup>(2)</sup> ∈ K<sup>R<sub>3</sub>×I<sub>4</sub>×I<sub>5</sub>×R<sub>4</sub></sup> ; A<sup>(2)</sup> ∈ K<sup>I<sub>6</sub>×R<sub>4</sub></sup></td><td></td></tr><tr><td>x<sub>i<sub>6</sub></sub> = &sum;<sub>r<sub>1</sub>=1</sub><sup>R<sub>1</sub></sup> &sum;<sub>r<sub>2</sub>=1</sub><sup>R<sub>2</sub></sup> &sum;<sub>r<sub>3</sub>=1</sub><sup>R<sub>3</sub></sup> &sum;<sub>r<sub>4</sub>=1</sub><sup>R<sub>4</sub></sup> a<sub>i<sub>1</sub>,r<sub>1</sub></sub><sup>(1)</sup> g<sub>r<sub>1</sub>,i<sub>2</sub>,i<sub>3</sub>,r<sub>2</sub></sub><sup>(1)</sup> u<sub>r<sub>2</sub>,r<sub>3</sub></sub><sup>(2)</sup> g<sub>r<sub>3</sub>,i<sub>4</sub>,i<sub>5</sub>,r<sub>4</sub></sub><sup>(2)</sup> a<sub>i<sub>6</sub>,r<sub>4</sub></sub><sup>(2)</sup></td><td></td></tr><tr><td colspan="2">X ∈ K<sup>I<sub>7</sub></sup></td></tr><tr><td colspan="2">NGTD-7 model</td></tr><tr><td>A<sup>(1)</sup> ∈ K<sup>I<sub>1</sub>×I<sub>2</sub>×R<sub>1</sub></sup> ; G<sup>(1)</sup> ∈ K<sup>R<sub>1</sub>×I<sub>2</sub>×I<sub>3</sub>×I<sub>4</sub>×R<sub>2</sub></sup> ; U ∈ K<sup>R<sub>2</sub>×I<sub>2</sub>×R<sub>3</sub></sup> ; G<sup>(2)</sup> ∈ K<sup>R<sub>3</sub>×I<sub>2</sub>×I<sub>5</sub>×I<sub>6</sub>×R<sub>4</sub></sup> ; A<sup>(2)</sup> ∈ K<sup>I<sub>7</sub>×R<sub>4</sub></sup></td><td></td></tr><tr><td>x<sub>i<sub>7</sub></sub> = &sum;<sub>r<sub>1</sub>=1</sub><sup>R<sub>1</sub></sup> &sum;<sub>r<sub>2</sub>=1</sub><sup>R<sub>2</sub></sup> &sum;<sub>r<sub>3</sub>=1</sub><sup>R<sub>3</sub></sup> &sum;<sub>r<sub>4</sub>=1</sub><sup>R<sub>4</sub></sup> a<sub>i<sub>1</sub>,i<sub>2</sub>,r<sub>1</sub></sub><sup>(1)</sup> g<sub>r<sub>1</sub>,i<sub>2</sub>,i<sub>3</sub>,r<sub>4</sub>,r<sub>2</sub></sub><sup>(1)</sup> u<sub>r<sub>2</sub>,i<sub>2</sub>,r<sub>3</sub></sub><sup>(2)</sup> g<sub>r<sub>3</sub>,i<sub>2</sub>,i<sub>5</sub>,i<sub>6</sub>,r<sub>4</sub></sub><sup>(2)</sup> a<sub>i<sub>7</sub>,r<sub>4</sub></sub><sup>(2)</sup></td><td></td></tr></table>

The NTD-6 model corresponds to a GTTD-(2,4,2,4,2), which means a train of three matrix factors ($\mathbf{A}^{(1)}, \mathbf{U}, \mathbf{A}^{(2)}$) and two fourth-order core tensors ($\mathcal{G}^{(1)}, \mathcal{G}^{(2)}$), illustrated in the graph in Figure 9, to be compared with Figure 8 for NTD-4. It can be viewed as the nesting of the following two TD-(2,4) models (see Table 8 with $(N_1, N) = (2, 4)$) of the fourth-order tensors $\mathcal{X}^{(1)} \in K^{I_1 \times I_2 \times I_3 \times R_3}$ and $\mathcal{X}^{(2)} \in K^{R_2 \times I_4 \times I_5 \times I_6}$, which share the matrix factor $\mathbf{U}$:

$$ \mathcal{X}^{(1)} = \mathcal{G}^{(1)} \times_1 \mathbf{A}^{(1)} \times_4 \mathbf{U}^T \quad (43) $$

$$ \mathcal{X}^{(2)} = \mathcal{G}^{(2)} \times_1 \mathbf{U} \times_4 \mathbf{A}^{(2)}. \quad (44) $$

<figure><img src="image_11.png" /><figcaption>Figure 9. Graph of the NTD-6 model for a sixth-order tensor $\mathcal{X} \in K^{I_6}$.</figcaption></figure>
The NGTD-7 model is a GTTD-(3,5,3,5,2), represented in Figure 10. It corresponds to the nesting of the following two GTD-(2,5) models:

$$ \mathcal{X}^{(1)} = \mathcal{G}^{(1)} \times_1^3 \mathcal{A}^{(1)} \times_5^1 \mathcal{U} \in \mathbb{K}^{I_1 \times I_2 \times I_3 \times I_4 \times R_3} \quad (45) $$

$$ \mathcal{X}^{(2)} = \mathcal{G}^{(2)} \times_1^3 \mathcal{U} \times_5 \mathbf{A}^{(2)} \in \mathbb{K}^{R_2 \times I_2 \times I_5 \times I_6 \times I_7}, \quad (46) $$

which share the mode $i_2$ and the third-order tensor factor $\mathcal{U}$. Equations (45) and (46) are to be compared with Equations (33) and (34) for the NTD-4 model. From this comparison, we deduce the correspondences ($I_1, I_2, I_3, I_4$) $\leftrightarrow$ ($I_2 I_1, I_2 I_3 I_4, I_2 I_5 I_6, I_7$), and the contracted forms (36) and (37) become the following:

$$ \mathcal{X}_{c_1} = \mathcal{G}^{(1)} \times_1^3 \mathcal{A}^{(1)} \times_5 \mathbf{X}_{I_2 I_5 I_6 I_7 \times R_2}^{(2)} \in \mathbb{K}^{I_1 \times I_2 \times I_3 \times I_4 \times I_5 I_6 I_7} \quad (47) $$

$$ \mathcal{X}_{c_2} = \mathcal{G}^{(2)} \times_1 \mathbf{X}_{I_1 I_2 I_3 I_4 \times R_3}^{(1)} \times_5 \mathbf{A}^{(2)} \in \mathbb{K}^{I_1 I_2 I_3 I_4 \times I_5 \times I_6 \times I_7}. \quad (48) $$

<figure><img src="image_12.png" /><figcaption>Figure 10. Graph of the NGTD-7 model for a seventh-order tensor $\mathcal{X} \in \mathbb{K}^{I_7}$.</figcaption></figure>

Sharing mode $i_2$ among the factors ($\mathcal{A}^{(1)}, \mathcal{G}^{(1)}, \mathcal{U}, \mathcal{G}^{(2)}$) means that the Kronecker products in matrix unfoldings can be calculated separately for each value of $i_2 \in \langle I_2 \rangle$, using block-diagonal matrices composed of $I_2$ blocks, denoted as $\text{bdiag}_{i_2}(\cdot)$. From the GTD-(2,5) models (45) and (46) of $\mathcal{X}^{(1)}$ and $\mathcal{X}^{(2)}$, we deduce the following block-diagonal matrix unfoldings:

$$ \text{bdiag}_{i_2}(\mathbf{X}_{I_1 R_3 \times I_3 I_4}^{(1)}(i_2)) = \text{bdiag}_{i_2}[\mathbf{A}_{I_1 \times R_1}^{(1)}(i_2) \otimes \mathbf{U}_{R_3 \times R_2}(i_2)]\text{bdiag}_{i_2}(\mathbf{G}_{R_1 R_2 \times I_3 I_4}^{(1)}(i_2)) \quad (49) $$

$$ \text{bdiag}_{i_2}(\mathbf{X}_{I_7 R_2 \times I_5 I_6}^{(2)}(i_2)) = [\mathbf{A}^{(2)} \otimes \text{bdiag}_{i_2}(\mathbf{U}_{R_2 \times R_3}(i_2))]\text{bdiag}_{i_2}(\mathbf{G}_{R_4 R_3 \times I_5 I_6}^{(2)}(i_2)), \quad (50) $$

where the first term in the right side of (49) denotes a block-diagonal matrix for which each block on the diagonal is the Kronecker product of unfoldings of $\mathcal{A}^{(1)}$ and $\mathcal{U}$ calculated for a given $i_2$.

Similarly, from the models (47) and (48) of $\mathcal{X}_{c_1}$ and $\mathcal{X}_{c_2}$, we have the following block-diagonal matrix unfoldings:

$$ \text{bdiag}_{i_2}(\mathbf{X}_{I_1 I_5 I_6 I_7 \times I_3 I_4}(i_2)) = \text{bdiag}_{i_2}[\mathbf{A}_{I_1 \times R_1}^{(1)}(i_2) \otimes \mathbf{X}_{I_5 I_6 I_7 \times R_2}^{(2)}(i_2)]\text{bdiag}_{i_2}(\mathbf{G}_{R_1 R_2 \times I_3 I_4}^{(1)}(i_2)) \quad (51) $$

$$ \text{bdiag}_{i_2}(\mathbf{X}_{I_7 I_1 I_3 I_4 \times I_5 I_6}(i_2)) = [\mathbf{A}^{(2)} \otimes \text{bdiag}_{i_2}(\mathbf{X}_{I_1 I_3 I_4 \times R_3}^{(1)}(i_2))]\text{bdiag}_{i_2}(\mathbf{G}_{R_4 R_3 \times I_5 I_6}^{(2)}(i_2)). \quad (52) $$

The matrix unfoldings (49)–(52) will be used in Section 4.5 to devise two closed-form parameter estimation algorithms.

In the cases of the NTD-6 and NGTD-7 models, Equation (39) becomes, respectively, the following:

$$ x_{i_6} = \mathbf{a}_{i_1, i_2, i_3, i_4, i_5, i_6}^{(1)} \mathbf{G}_{i_2, i_3, i_4, i_5, i_6}^{(1)} \mathbf{U} \mathbf{G}_{i_2, i_3, i_4, i_5, i_6}^{(2)} \mathbf{a}_{i_6}^{(2)} \quad (53) $$

$$ x_{i_7} = \mathbf{a}_{i_1, i_2, i_3, i_4, i_5, i_6, i_7}^{(1)} \mathbf{G}_{i_2, i_3, i_4, i_5, i_6, i_7}^{(1)} \mathbf{U}_{i_2, i_3, i_4, i_5, i_6, i_7}^{(2)} \mathbf{G}_{i_2, i_3, i_4, i_5, i_6, i_7}^{(2)} \mathbf{a}_{i_7}^{(2)}. \quad (54) $$

From these equations, we can draw the same conclusions as for the NTD-4 model, namely that the NTD-6 and NGTD-7 models are unique up to non-singular matrices $\mathbf{A}^{(n)} \in \mathbb{K}^{R_n \times R_n}$, $n \in \langle 4 \rangle$, and that, when the tensors $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ are known, the ambiguity matrices $\mathbf{A}^{(n)}$ become identity matrices multiplied by a scalar.
**Remark 3.** A simplified version of the NGTD-7 model, called NGTD-5 and represented in the graph in Figure 11, will be used in Section 5 to model the tensor of received signals in a relay system employing a simplified TSTF coding, denoted as STSTF. This model for a fifth-order tensor, $\mathcal{X} \in \mathbb{K}^{I_5}$, corresponds to a generalized tensor train decomposition GTTD-(3,4,3,4,2), which results from the nesting of the following two GTD-(2,4) models:

$$ \mathcal{X}^{(1)} = \mathcal{G}^{(1)} \times_1^3 \mathcal{A}^{(1)} \times_4^1 \mathcal{U} \in \mathbb{K}^{I_1 \times I_2 \times I_3 \times I_5} \quad (55) $$

$$ \mathcal{X}^{(2)} = \mathcal{G}^{(2)} \times_1^3 \mathcal{U} \times_5 \mathbf{A}^{(2)} \in \mathbb{K}^{R_2 \times I_2 \times I_4 \times I_5}, \quad (56) $$

with

$$ \mathcal{A}^{(1)} \in \mathbb{K}^{I_1 \times I_2 \times R_1} ; \quad \mathcal{G}^{(1)} \in \mathbb{K}^{R_1 \times I_2 \times I_3 \times R_2} ; \quad \mathcal{U} \in \mathbb{K}^{R_2 \times I_2 \times R_3} \quad (57) $$

$$ \mathcal{G}^{(2)} \in \mathbb{K}^{R_3 \times I_2 \times I_4 \times R_4} ; \quad \mathbf{A}^{(2)} \in \mathbb{K}^{I_5 \times R_4}. \quad (58) $$

Note that the tensor factors $(\mathcal{A}^{(1)}, \mathcal{G}^{(1)}, \mathcal{U}, \mathcal{G}^{(2)})$ have the dimension $I_2$ in common, and the core tensors $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(2)}$ are of fourth-order instead of fifth-order for NGTD-7.

<figure><img src="image_13.png" /><figcaption>Figure 11. Graph of the NGTD-5 model for a fifth-order tensor $\mathcal{X} \in \mathbb{K}^{I_5}$.</figcaption></figure>

In Figure 12, we summarize two families of tensor models using two trees, based on the basic CPD/PARAFAC and Tucker decompositions. For the definition of acronyms and associated references, see Appendix A.

<figure><img src="image_14.png" /><figcaption>Figure 12. Two families of TD- and CPD-based decompositions.</figcaption></figure>

Note that the NGTD and some variants of nested models are introduced, for the first time, in the present paper as generalizations of the NTD-4 model [41]. Nested tensor
models will be used in Section 5 to represent two-hop relay systems using different tensor- and matrix-based codings. See Table 25.

**Remark 4.** The standard PARATUCK-2 model for a third-order tensor combines the properties of the PARAFAC and Tucker models. It was originally introduced in the context of psychometric applications [51]. In the context of wireless communications, a first application of the PARATUCK-2 model was proposed for blind joint identification and equalization of Wiener–Hammerstein nonlinear communication channels [52]. Then, it was used to model a two-hop relay system using simplified Khatri–Rao space-time (SKRST) coding [53]. An extension, denoted as PARATUCK-(2, 4), was introduced in [37] to model the fourth-order tensor of signals received in a MIMO communication system with a tensor space-time (TST) coding.

## 4. Parameter Estimation Algorithms

In this section, we present two families of algorithms to estimate the parameters of the tensor models introduced in the previous section: alternating least-squares (ALS)-based algorithms and closed-form algorithms. These latter use the Khatri–Rao and Kronecker factorization methods, respectively denoted as KRF and KronF, described in Appendices B and C. Closed-form algorithms can be applied when some factors of the models are known a priori, as will be the case with coding matrices and tensors, in the context of relay systems studied in Section 5. The standard ALS algorithm is first recalled for a third-order CPD/PARAFAC model.

### 4.1. CPD/PARAFAC Model

The ALS algorithm was first proposed to estimate the parameters of a CPD/PARAFAC model in [8,47]. Such a model for a third-order tensor $\mathcal{X}$ is trilinear in its parameters in the sense that it is linear with respect to each of its three matrix factors ($\mathbf{A}, \mathbf{B}, \mathbf{C}$). When the relations (11) are taken into account, the idea behind the ALS algorithm is to replace the global minimization of the quadratic error between the data tensor and its CPD/PARAFAC model

$$ \min_{\mathbf{A},\mathbf{B},\mathbf{C}} \|\mathcal{X} - [\mathbf{A}, \mathbf{B}, \mathbf{C}; \mathbf{R}]\|_F^2 \qquad (59) $$

through an alternating minimization of three quadratic cost functions deduced from matrix unfoldings of $\mathcal{X}$, each one being quadratic with respect to one factor, while the other two are fixed with their previously estimated values.

At iteration $t$, the minimization of each cost function provides an estimated matrix factor:

$$ \left\{ \begin{array}{l} \min_{\mathbf{A}} \|\mathbf{X}_{JK \times I} - (\mathbf{B}_{t-1} \diamond \mathbf{C}_{t-1}) \mathbf{A}^T\|_F^2 \Rightarrow \mathbf{A}_t^T = (\mathbf{B}_{t-1} \diamond \mathbf{C}_{t-1})^\dagger \mathbf{X}_{JK \times I} \\ \min_{\mathbf{B}} \|\mathbf{X}_{KI \times J} - (\mathbf{C}_{t-1} \diamond \mathbf{A}_t) \mathbf{B}^T\|_F^2 \Rightarrow \mathbf{B}_t^T = (\mathbf{C}_{t-1} \diamond \mathbf{A}_t)^\dagger \mathbf{X}_{KI \times J} \\ \min_{\mathbf{C}} \|\mathbf{X}_{IJ \times K} - (\mathbf{A}_t \diamond \mathbf{B}_t) \mathbf{C}^T\|_F^2 \Rightarrow \mathbf{C}_t^T = (\mathbf{A}_t \diamond \mathbf{B}_t)^\dagger \mathbf{X}_{IJ \times K} \end{array} \right. \qquad (60) $$

where $\mathbf{A}_t$ denotes the LS estimate of $\mathbf{A}$ at iteration $t$.

### 4.2. NCPD-4 Model

In the next two subsections, we first present a five-step ALS-based algorithm to estimate the five matrix factors $(\mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)})$ of the NCPD-4 model (21). The case where the factors $(\mathbf{B}^{(1)}, \mathbf{A}^{(2)})$ are known is also considered. Then, under this assumption, two closed-form algorithms using the KRF method are described. Necessary identifiability conditions are given for each identification algorithm.
### 4.2.1. ALS Algorithm

Considering the first four matrix unfoldings in Table 14 with Equation (27), and applying the same approach as for the CPD model, we derive the following five LS cost functions to estimate the parameters of a NCPD-4 model:

$$ \min_{\mathbf{B}^{(1)}} \|\mathbf{X}_{I_1 I_3 I_4 \times I_2} - [\mathbf{A}_{t-1}^{(1)} \diamond (\mathbf{A}_{t-1}^{(2)} \diamond \mathbf{B}_{t-1}^{(2)}) \mathbf{G}_{t-1}^{T}] (\mathbf{B}^{(1)})^{T}\|_{F}^{2} \Rightarrow \mathbf{B}_{t}^{(1)} \quad (61) $$

$$ \min_{\mathbf{A}^{(1)}} \|\mathbf{X}_{I_2 I_3 I_4 \times I_1} - [\mathbf{B}_{t}^{(1)} \diamond (\mathbf{A}_{t-1}^{(2)} \diamond \mathbf{B}_{t-1}^{(2)}) \mathbf{G}_{t-1}^{T}] (\mathbf{A}^{(1)})^{T}\|_{F}^{2} \Rightarrow \mathbf{A}_{t}^{(1)} \quad (62) $$

$$ \min_{\mathbf{B}^{(2)}} \|\mathbf{X}_{I_3 I_1 I_2 \times I_4} - [\mathbf{A}_{t-1}^{(2)} \diamond (\mathbf{A}_{t}^{(1)} \diamond \mathbf{B}_{t}^{(1)}) \mathbf{G}_{t-1}] (\mathbf{B}^{(2)})^{T}\|_{F}^{2} \Rightarrow \mathbf{B}_{t}^{(2)} \quad (63) $$

$$ \min_{\mathbf{A}^{(2)}} \|\mathbf{X}_{I_4 I_1 I_2 \times I_3} - [\mathbf{B}_{t}^{(2)} \diamond (\mathbf{A}_{t}^{(1)} \diamond \mathbf{B}_{t}^{(1)}) \mathbf{G}_{t-1}] (\mathbf{A}^{(2)})^{T}\|_{F}^{2} \Rightarrow \mathbf{A}_{t}^{(2)} \quad (64) $$

$$ \min_{\mathbf{G}} \|\mathbf{x}_{I_3 I_4 I_1 I_2} - ((\mathbf{A}_{t}^{(2)} \diamond \mathbf{B}_{t}^{(2)}) \otimes (\mathbf{A}_{t}^{(1)} \diamond \mathbf{B}_{t}^{(1)})) \mathrm{vec}(\mathbf{G})\|_{F}^{2} \Rightarrow \mathbf{G}_{t}. \quad (65) $$

Minimizing the above LS criteria leads to the following equations for the ALS algorithm:

$$ (\mathbf{B}_{t}^{(1)})^{T} = [\mathbf{A}_{t-1}^{(1)} \diamond (\mathbf{A}_{t-1}^{(2)} \diamond \mathbf{B}_{t-1}^{(2)}) \mathbf{G}_{t-1}^{T}]^{\dagger} \mathbf{X}_{I_1 I_3 I_4 \times I_2} \quad (66) $$

$$ (\mathbf{A}_{t}^{(1)})^{T} = [\mathbf{B}_{t}^{(1)} \diamond (\mathbf{A}_{t-1}^{(2)} \diamond \mathbf{B}_{t-1}^{(2)}) \mathbf{G}_{t-1}^{T}]^{\dagger} \mathbf{X}_{I_2 I_3 I_4 \times I_1} \quad (67) $$

$$ (\mathbf{B}_{t}^{(2)})^{T} = [\mathbf{A}_{t-1}^{(2)} \diamond (\mathbf{A}_{t}^{(1)} \diamond \mathbf{B}_{t}^{(1)}) \mathbf{G}_{t-1}]^{\dagger} \mathbf{X}_{I_3 I_1 I_2 \times I_4} \quad (68) $$

$$ (\mathbf{A}_{t}^{(2)})^{T} = [\mathbf{B}_{t}^{(2)} \diamond (\mathbf{A}_{t}^{(1)} \diamond \mathbf{B}_{t}^{(1)}) \mathbf{G}_{t-1}]^{\dagger} \mathbf{X}_{I_4 I_1 I_2 \times I_3} \quad (69) $$

$$ \mathrm{vec}(\mathbf{G}_t) = ((\mathbf{A}_t^{(2)} \diamond \mathbf{B}_t^{(2)}) \otimes (\mathbf{A}_t^{(1)} \diamond \mathbf{B}_t^{(1)}))^{\dagger} \mathbf{x}_{I_3 I_4 I_1 I_2}. \quad (70) $$

This algorithm requires the pseudo-inversion of five matrices, implying the following necessary identifiability conditions on the dimensions to guarantee the uniqueness of the pseudo-inverses:

$$ I_1 I_3 I_4 \geq R_1; \quad I_2 I_3 I_4 \geq R_1; \quad I_3 I_1 I_2 \geq R_2; \quad I_4 I_1 I_2 \geq R_2; \quad I_3 I_4 I_1 I_2 \geq R_1 R_2. \quad (71) $$

As is well known, the main drawback of iterative algorithms based on ALS is slow convergence, with the possibility of converging to a local minimum, which strongly depends on the choice of initialization.

When the factors $(\mathbf{B}^{(1)}, \mathbf{A}^{(2)})$ are known, as will be the case with coding matrices in the context of relay systems, a three-step ALS algorithm composed of Equations (67), (68) and (70) can be used to estimate the unknown factors $(\mathbf{A}^{(1)}, \mathbf{B}^{(2)}, \mathbf{G})$.

### 4.2.2. KRF-Based Closed-Form Algorithms

As mentioned in Section 3.3.1, when the factors $(\mathbf{B}^{(1)}, \mathbf{A}^{(2)})$ are known, two closed-form algorithms using the KRF method can be devised as shown hereafter. These algorithms, composed of two steps, are deduced from unfoldings (in green) in Table 13.

The first algorithm is based on the following matrix unfoldings of $\mathcal{X}_{c_2}$ and $\mathcal{X}^{(1)}$ deduced from the coupled CPD-3 models (26) and (19), respectively:

$$ \mathbf{X}_{I_4 I_1 I_2 \times I_3} = (\mathbf{B}^{(2)} \diamond \mathbf{X}_{I_1 I_2 \times R_2}^{(1)})(\mathbf{A}^{(2)})^{T} \quad (72) $$

$$ \mathbf{X}_{I_1 R_2 \times I_2}^{(1)} = (\mathbf{A}^{(1)} \diamond \mathbf{G}^{T})(\mathbf{B}^{(1)})^{T}. \quad (73) $$

These equations are successively exploited as follows. In a first step, the LS estimate of the Khatri-Rao product (KRP) in (72) is calculated, and its factors $(\mathbf{B}^{(2)}, \mathbf{X}_{I_1 I_2 \times R_2}^{(1)})$ are determined using the KRF method. In the second step, after a reshaping of the estimate
$\hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)}$, the LS estimate of the KRP in (73), with $\mathbf{X}_{I_1 R_2 \times I_2}^{(1)}$ replaced with its estimate $\hat{\mathbf{X}}_{I_1 R_2 \times I_2}^{(1)}$ deduced from the reshaping, is calculated, and the KRF method is applied again to estimate the factors $(\mathbf{A}^{(1)}, \mathbf{G}^T)$. The resulting algorithm, namely Closed-Form Algorithm 1, is summarized in Table 18.

<table><caption>**Table 18.** Closed-form algorithms to estimate the parameters of a NCPD-4 model.</caption><thead><tr><th>Models</th><th>Closed-Form Algorithm 1</th><th>Closed-Form Algorithm 2</th></tr></thead><tbody><tr><td rowspan="4">NCPD-4</td><td>$\mathbf{B}^{(2)} \diamond \mathbf{X}_{I_1 I_2 \times R_2}^{(1)} = \mathbf{X}_{I_1 I_2 \times I_3} [(\mathbf{A}^{(2)})^T]^\dagger$</td><td>$\mathbf{A}^{(1)} \diamond \mathbf{X}_{I_3 I_4 \times R_1}^{(2)} = \mathbf{X}_{I_1 I_3 I_4 \times I_2} [(\mathbf{B}^{(1)})^T]^\dagger$</td></tr><tr><td>$\text{KRF} \Rightarrow (\hat{\mathbf{B}}^{(2)}, \hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)})$</td><td>$\text{KRF} \Rightarrow (\hat{\mathbf{A}}^{(1)}, \hat{\mathbf{X}}_{I_3 I_4 \times R_1}^{(2)})$</td></tr><tr><td>Reshaping : $\hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)} \rightarrow \hat{\mathbf{X}}_{I_1 R_2 \times I_2}^{(1)}$</td><td>Reshaping : $\hat{\mathbf{X}}_{I_3 I_4 \times R_1}^{(2)} \rightarrow \hat{\mathbf{X}}_{I_4 R_1 \times I_3}^{(2)}$</td></tr><tr><td>$\mathbf{A}^{(1)} \diamond \mathbf{G}^T = \hat{\mathbf{X}}_{I_1 R_2 \times I_2}^{(1)} [(\mathbf{B}^{(1)})^T]^\dagger$</td><td>$\mathbf{B}^{(2)} \diamond \mathbf{G} = \hat{\mathbf{X}}_{I_4 R_1 \times I_3}^{(2)} [(\mathbf{A}^{(2)})^T]^\dagger$</td></tr><tr><td>$\text{KRF} \Rightarrow (\hat{\mathbf{A}}^{(1)}, \hat{\mathbf{G}})$</td><td>$\text{KRF} \Rightarrow (\hat{\mathbf{B}}^{(2)}, \hat{\mathbf{G}})$</td></tr></tbody></table>

Similarly, a second closed-form solution can be derived from the following coupled matrix unfoldings of $\mathcal{X}_{c_1}$ and $\mathcal{X}^{(2)}$ deduced from the CPD-3 models (25) and (20), respectively:

$$ \mathbf{X}_{I_1 I_3 I_4 \times I_2} = (\mathbf{A}^{(1)} \diamond \mathbf{X}_{I_3 I_4 \times R_1}^{(2)})(\mathbf{B}^{(1)})^T \quad (74) $$

$$ \mathbf{X}_{I_4 R_1 \times I_3}^{(2)} = (\mathbf{B}^{(2)} \diamond \mathbf{G})(\mathbf{A}^{(2)})^T. \quad (75) $$

Following the same approach as for the Closed-Form Algorithm 1, the Closed-Form Algorithm 2 is summarized in Table 18.

Identifiability conditions for the closed-form algorithms are linked with the uniqueness of right inverses of $(\mathbf{B}^{(1)})^T$ and $(\mathbf{A}^{(2)})^T$, which yields the following necessary conditions:

$$ I_2 \geq R_1 \text{ and } I_3 \geq R_2. \quad (76) $$

Comparing the identifiability conditions (76) with (71), one can conclude that the ALS algorithm is less constraining than the closed-form ones.

**Remark 5.** It is worth noting that closed-form algorithms involve the drawback of error propagation for computing the LS estimate of the KRP in the second step due to the use of the estimates $\hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)}$ and $\hat{\mathbf{X}}_{I_3 I_4 \times R_1}^{(2)}$ obtained in the first step. To alleviate error propagation, a solution consists of combining both closed-form algorithms to estimate the factors $(\mathbf{A}^{(1)}, \mathbf{B}^{(2)})$ in parallel. Then, the factor $\mathbf{G}$ is estimated using one of the following formulae deduced from the first and third equations in Table 13:

$$ \hat{\mathbf{G}} = (\hat{\mathbf{A}}^{(1)} \diamond \mathbf{B}^{(1)})^{\dagger} \hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)} \quad \text{or} \quad \hat{\mathbf{G}}^T = (\mathbf{A}^{(2)} \diamond \hat{\mathbf{B}}^{(2)})^{\dagger} \hat{\mathbf{X}}_{I_3 I_4 \times R_1}^{(2)}, \quad (77) $$

where $(\hat{\mathbf{A}}^{(1)}, \hat{\mathbf{X}}_{I_1 I_2 \times R_2}^{(1)})$ and $(\hat{\mathbf{B}}^{(2)}, \hat{\mathbf{X}}_{I_3 I_4 \times R_1}^{(2)})$ are estimates obtained with the closed-form algorithms. The identifiability conditions for these combined closed-form algorithms are given by (76) with additional conditions resulting from the pseudo-inverses to be calculated in (77):

$$ I_1 I_2 \geq R_1 \text{ and } I_3 I_4 \geq R_2. \quad (78) $$

These conditions are necessarily satisfied when conditions (76) are.

### 4.3. NTD-4 Model

In the following, we assume that the core tensors $(\mathcal{G}^{(1)}, \mathcal{G}^{(2)})$ are known, and we describe a three-step ALS-based algorithm and two closed-form algorithms to estimate the factors $(\mathbf{A}^{(1)}, \mathbf{U}, \mathbf{A}^{(2)})$ of the NTD-4 model.
### 4.3.1. ALS Algorithm

As the LS cost functions (61)–(65) for the NCPD-4 model, unfoldings in Table 16 can be used to define three LS criteria to be minimized in order to estimate the matrix factors $(\mathbf{A}^{(1)}, \mathbf{U}, \mathbf{A}^{(2)})$ by means of a three-step ALS-based algorithm. From the first, third, and fifth equations (in blue) in Table 16, we define the following LS cost functions:

$$ \min_{\mathbf{A}^{(2)}} \|\mathbf{X}_{I_3 I_1 I_2 \times I_4} - [\mathbf{I}_{I_3} \otimes (\mathbf{A}_{\mathbf{t}-1}^{(1)} \otimes \mathbf{I}_{I_2}) \mathbf{G}_{R_1 I_2 \times R_2}^{(1)} \mathbf{U}_{t-1}] \mathbf{G}_{I_3 R_3 \times R_4}^{(2)} (\mathbf{A}^{(2)})^T \|_{F}^2 \Rightarrow \mathbf{A}_{\mathbf{t}}^{(2)} \quad (79) $$

$$ \min_{\mathbf{A}^{(1)}} \|\mathbf{X}_{I_2 I_3 I_4 \times I_1} - [\mathbf{I}_{I_2} \otimes (\mathbf{I}_{I_3} \otimes \mathbf{A}_{\mathbf{t}}^{(2)}) \mathbf{G}_{I_3 R_4 \times R_3}^{(2)} \mathbf{U}_{t-1}^{T}] \mathbf{G}_{I_2 R_2 \times R_1}^{(1)} (\mathbf{A}^{(1)})^{T} \|_{F}^{2} \Rightarrow \mathbf{A}_{\mathbf{t}}^{(1)} \quad (80) $$

$$ \min_{\mathbf{U}} \|\mathbf{x}_{I_3 I_4 I_1 I_2} - [(\mathbf{I}_{I_3} \otimes \mathbf{A}_{\mathbf{t}}^{(2)}) \mathbf{G}_{I_3 R_4 \times R_3}^{(2)} \otimes (\mathbf{A}_{\mathbf{t}}^{(1)} \otimes \mathbf{I}_{I_2}) \mathbf{G}_{R_1 I_2 \times R_2}^{(1)}] \mathrm{vec}(\mathbf{U}) \|_{F}^{2} \Rightarrow \mathbf{U}_{t}. \quad (81) $$

Minimizing these LS criteria leads to the following equations for the ALS algorithm:

$$ (\mathbf{A}_{\mathbf{t}}^{(2)})^{T} = [ [\mathbf{I}_{I_3} \otimes (\mathbf{A}_{\mathbf{t}-1}^{(1)} \otimes \mathbf{I}_{I_2}) \mathbf{G}_{R_1 I_2 \times R_2}^{(1)} \mathbf{U}_{t-1}] \mathbf{G}_{I_3 R_3 \times R_4}^{(2)} ]^{\dagger} \mathbf{X}_{I_3 I_1 I_2 \times I_4} \quad (82) $$

$$ (\mathbf{A}_{\mathbf{t}}^{(1)})^{T} = [ [\mathbf{I}_{I_2} \otimes (\mathbf{I}_{I_3} \otimes \mathbf{A}_{\mathbf{t}}^{(2)}) \mathbf{G}_{I_3 R_4 \times R_3}^{(2)} \mathbf{U}_{t-1}^{T}] \mathbf{G}_{I_2 R_2 \times R_1}^{(1)} ]^{\dagger} \mathbf{X}_{I_2 I_3 I_4 \times I_1} \quad (83) $$

$$ \mathrm{vec}(\mathbf{U}_f) = [(\mathbf{I}_{I_3} \otimes \mathbf{A}_{\mathbf{t}}^{(2)}) \mathbf{G}_{I_3 R_4 \times R_3}^{(2)} \otimes (\mathbf{A}_{\mathbf{t}}^{(1)} \otimes \mathbf{I}_{I_2}) \mathbf{G}_{R_1 I_2 \times R_2}^{(1)}]^{\dagger} \mathbf{x}_{I_3 I_4 I_1 I_2}. \quad (84) $$

The uniqueness of the above LS solutions implies the following necessary identifiability conditions in the dimensions:

$$ I_3 I_1 I_2 \geq R_4; \quad I_2 I_3 I_4 \geq R_1; \quad I_3 I_4 I_1 I_2 \geq R_3 R_2. \quad (85) $$

### 4.3.2. Closed-Form Algorithms

Following the same approach as for the NCPD-4 model, we now present two closed-form solutions exploiting the matrix unfoldings (in green) in Table 15 and using the KronF method. From the coupled TD-(2,3) models (37) and (33) of $\mathcal{X}_{c_2}$ and $\mathcal{X}^{(1)}$, we deduce the following matrix unfoldings of the tensors $\mathcal{X}$ and $\mathcal{X}^{(1)}$:

$$ \mathbf{X}_{I_4 I_1 I_2 \times I_3} = (\mathbf{A}^{(2)} \otimes \mathbf{X}_{I_1 I_2 \times R_3}^{(1)}) \mathbf{G}_{R_4 R_3 \times I_3}^{(2)} \quad (86) $$

$$ \mathbf{X}_{I_1 R_3 \times I_2}^{(1)} = (\mathbf{A}^{(1)} \otimes \mathbf{U}^T) \mathbf{G}_{R_1 R_2 \times I_2}^{(1)}. \quad (87) $$

These equations are used to build the first closed-form algorithm as follows: In a first step, the LS method is applied to estimate the Kronecker product (KronP) in Equation (86) as follows:

$$ \mathbf{A}^{(2)} \otimes \mathbf{X}_{I_1 I_2 \times R_3}^{(1)} = \mathbf{X}_{I_4 I_1 I_2 \times I_3} [\mathbf{G}_{R_4 R_3 \times I_3}^{(2)}]^{\dagger}. \quad (88) $$

The factors $(\mathbf{A}^{(2)}, \mathbf{X}_{I_1 I_2 \times R_3}^{(1)})$ are then estimated using the KronF method. In a second step, after a reshaping of the estimate $\hat{\mathbf{X}}_{I_1 I_2 \times R_3}^{(1)}$, the LS method is used to estimate the KronP in Equation (87) as follows:

$$ \mathbf{A}^{(1)} \otimes \mathbf{U}^T = \hat{\mathbf{X}}_{I_1 R_3 \times I_2}^{(1)} (\mathbf{G}_{R_1 R_2 \times I_2}^{(1)})^{\dagger}. \quad (89) $$

The factors $(\mathbf{A}^{(1)}, \mathbf{U})$ are then deduced using the KronF method. This two-step Closed-Form Algorithm 1 is summarized in Table 19.

The second closed-form solution is based on the following matrix unfoldings deduced from the coupled TD-(2,3) models (36) and (34) of $\mathcal{X}_{c_1}$ and $\mathcal{X}^{(2)}$:

$$ \mathbf{X}_{I_1 I_3 I_4 \times I_2} = (\mathbf{A}^{(1)} \otimes \mathbf{X}_{I_3 I_4 \times R_2}^{(2)}) \mathbf{G}_{R_1 R_2 \times I_2}^{(1)} \quad (90) $$

$$ \mathbf{X}_{I_4 R_2 \times I_3}^{(2)} = (\mathbf{A}^{(2)} \otimes \mathbf{U}) \mathbf{G}_{R_4 R_3 \times I_3}^{(2)}, \quad (91) $$
which leads to Closed-Form Algorithm 2, summarized in Table 19.

<table><caption>**Table 19.** Closed-form algorithms to estimate the parameters of the NTD-4 model.</caption><thead><tr><th>Models</th><th>Closed-Form Algorithm 1</th><th>Closed-Form Algorithm 2</th></tr></thead><tbody><tr><td>NTD-4</td><td>A<sup>(2)</sup> ⊗ X<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>3</sub></sub> = X<sub>I<sub>4</sub>I<sub>1</sub>I<sub>2</sub>×I<sub>3</sub></sub>[G<sup>(2)</sup><sub>R<sub>4</sub>R<sub>3</sub>×I<sub>3</sub></sub>]<sup>†</sup><br>KronF ⇒ (Â<sup>(2)</sup>, X̂<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>3</sub></sub>)</td><td>A<sup>(1)</sup> ⊗ X<sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>2</sub></sub> = X<sub>I<sub>1</sub>I<sub>3</sub>I<sub>4</sub>×I<sub>2</sub></sub>[G<sup>(1)</sup><sub>R<sub>1</sub>R<sub>2</sub>×I<sub>2</sub></sub>]<sup>†</sup><br>KronF ⇒ (Â<sup>(1)</sup>, X̂<sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>2</sub></sub>)</td></tr><tr><td>Reshaping</td><td>X̂<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>×R<sub>3</sub></sub> → X̂<sup>(1)</sup><sub>I<sub>1</sub>R<sub>3</sub>×I<sub>2</sub></sub></td><td>Reshaping</td><td>X̂<sup>(2)</sup><sub>I<sub>3</sub>I<sub>4</sub>×R<sub>2</sub></sub> → X̂<sup>(2)</sup><sub>I<sub>4</sub>R<sub>2</sub>×I<sub>3</sub></sub></td></tr><tr><td>A<sup>(1)</sup> ⊗ U<sup>T</sup> = X̂<sup>(1)</sup><sub>I<sub>1</sub>R<sub>3</sub>×I<sub>2</sub></sub>(G<sup>(1)</sup><sub>R<sub>1</sub>R<sub>2</sub>×I<sub>2</sub></sub>)<sup>†</sup><br>KronF ⇒ (Â<sup>(1)</sup>, Û)</td><td>A<sup>(2)</sup> ⊗ U = X̂<sup>(2)</sup><sub>I<sub>4</sub>R<sub>2</sub>×I<sub>3</sub></sub>(G<sup>(2)</sup><sub>R<sub>4</sub>R<sub>3</sub>×I<sub>3</sub></sub>)<sup>†</sup><br>KronF ⇒ (Â<sup>(2)</sup>, Û)</td><td>A<sup>(2)</sup> ⊗ U = X̂<sup>(2)</sup><sub>I<sub>4</sub>R<sub>2</sub>×I<sub>3</sub></sub>(G<sup>(2)</sup><sub>R<sub>4</sub>R<sub>3</sub>×I<sub>3</sub></sub>)<sup>†</sup><br>KronF ⇒ (Â<sup>(2)</sup>, Û)</td></tr></tbody></table>

For these closed-form solutions, necessary identifiability conditions, associated with the right inversion of $G_{R_1 R_2 \times I_2}^{(1)}$ and $G_{R_4 R_3 \times I_3}^{(2)}$, are given by the following:

$$I_2 \geq R_1 R_2 \text{ and } I_3 \geq R_3 R_4. \qquad (92)$$

These conditions are to be compared with the ALS ones (85).

**Remark 6.** As for the NCPD-4 model, the closed-form algorithms in Table 19 involve the drawback of error propagation due to the use, in the second step, of the estimates $\hat{X}_{I_1 I_2 \times R_3}^{(1)}$ and $\hat{X}_{I_3 I_4 \times R_2}^{(2)}$ obtained in the first step. An improved solution consists of combining both closed-form algorithms to estimate the factors $(A^{(1)}, A^{(2)})$ in parallel. Then, the factor $U$ is estimated using one of the following formulae deduced from the expressions of $X_{I_1 I_2 \times R_3}^{(1)}$ and $X_{I_3 I_4 \times R_2}^{(2)}$ given in Table 15:

$$\hat{U} = ((\hat{A}^{(1)} \otimes I_{I_2}) G_{R_1 I_2 \times R_2}^{(1)})^{\dagger} \hat{X}_{I_1 I_2 \times R_3}^{(1)} \text{ or } \hat{U}^{T} = ((I_{I_3} \otimes \hat{A}^{(2)}) G_{I_3 R_4 \times R_3}^{(2)})^{\dagger} \hat{X}_{I_3 I_4 \times R_2}^{(2)}, \quad (93)$$

where $(\hat{A}^{(1)}, \hat{A}^{(2)}, \hat{X}_{I_1 I_2 \times R_3}^{(1)}, \hat{X}_{I_3 I_4 \times R_2}^{(2)})$ are estimates obtained in the first step of the closed-form algorithms. Identifiability conditions for these combined closed-form algorithms are given by (92) with additional conditions for the pseudo-inverses to be calculated in (93):

$$I_1 I_2 \geq R_2 \text{ and } I_3 I_4 \geq R_3. \qquad (94)$$

Note that these conditions are always satisfied when conditions (92) are.

### 4.4. NTD-6 Model

The closed-form algorithms for the NTD-4 model are generalized to the NTD-6 using the correspondences $(I_2, I_3, I_4) \leftrightarrow (I_2 I_3, I_4 I_5, I_6)$, as summarized in Table 20.

<table><caption>**Table 20.** Closed-form algorithms to estimate the parameters of the NTD-6 model.</caption><thead><tr><th>Models</th><th>Closed-Form Algorithm 1</th><th>Closed-Form Algorithm 2</th></tr></thead><tbody><tr><td>NTD-6</td><td>A<sup>(2)</sup> ⊗ X<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>I<sub>3</sub>×R<sub>3</sub></sub> = X<sub>I<sub>6</sub>I<sub>1</sub>I<sub>2</sub>I<sub>3</sub>×I<sub>4</sub>I<sub>5</sub></sub>[G<sup>(2)</sup><sub>R<sub>4</sub>R<sub>3</sub>×I<sub>4</sub>I<sub>5</sub></sub>]<sup>†</sup><br>KronF ⇒ (Â<sup>(2)</sup>, X̂<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>I<sub>3</sub>×R<sub>3</sub></sub>)</td><td>A<sup>(1)</sup> ⊗ X<sup>(2)</sup><sub>I<sub>4</sub>I<sub>5</sub>I<sub>6</sub>×R<sub>2</sub></sub> = X<sub>I<sub>1</sub>I<sub>4</sub>I<sub>5</sub>I<sub>6</sub>×I<sub>2</sub>I<sub>3</sub></sub>[G<sup>(1)</sup><sub>R<sub>1</sub>R<sub>2</sub>×I<sub>2</sub>I<sub>3</sub></sub>]<sup>†</sup><br>KronF ⇒ (Â<sup>(1)</sup>, X̂<sup>(2)</sup><sub>I<sub>4</sub>I<sub>5</sub>I<sub>6</sub>×R<sub>2</sub></sub>)</td></tr><tr><td>Reshaping</td><td>X̂<sup>(1)</sup><sub>I<sub>1</sub>I<sub>2</sub>I<sub>3</sub>×R<sub>3</sub></sub> → X̂<sup>(1)</sup><sub>I<sub>1</sub>R<sub>3</sub>×I<sub>2</sub>I<sub>3</sub></sub></td><td>Reshaping</td><td>X̂<sup>(2)</sup><sub>I<sub>4</sub>I<sub>5</sub>I<sub>6</sub>×R<sub>2</sub></sub> → X̂<sup>(2)</sup><sub>I<sub>6</sub>R<sub>2</sub>×I<sub>4</sub>I<sub>5</sub></sub></td></tr><tr><td>A<sup>(1)</sup> ⊗ U<sup>T</sup> = X̂<sup>(1)</sup><sub>I<sub>1</sub>R<sub>3</sub>×I<sub>2</sub>I<sub>3</sub></sub>(G<sup>(1)</sup><sub>R<sub>1</sub>R<sub>2</sub>×I<sub>2</sub>I<sub>3</sub></sub>)<sup>†</sup><br>KronF ⇒ (Â<sup>(1)</sup>, Û)</td><td>A<sup>(2)</sup> ⊗ U = X̂<sup>(2)</sup><sub>I<sub>6</sub>R<sub>2</sub>×I<sub>4</sub>I<sub>5</sub></sub>(G<sup>(2)</sup><sub>R<sub>4</sub>R<sub>3</sub>×I<sub>4</sub>I<sub>5</sub></sub>)<sup>†</sup><br>KronF ⇒ (Â<sup>(2)</sup>, Û)</td></tr></tbody></table>
and the tensor model of signals received at destination. It also includes the receiver algorithms used for each cooperative system.

<table><caption>**Table 22.** Overview of cooperative systems.</caption><thead><tr><th>Ref.</th><th>OFDM/mmW</th><th>Relay/IRS/UAV</th><th>Coding/Training</th><th>Tensor Models</th><th>Receiver Algorithms</th></tr></thead><tbody><tr><td>[53]</td><td></td><td>Two-hop relay</td><td>Simplified KRST</td><td>CPD-PARATUCK</td><td>ALS + KRF</td></tr><tr><td>[40]</td><td></td><td>Two-hop relay</td><td>Simplified KRST</td><td>NCPD</td><td>ALS</td></tr><tr><td>[54]</td><td></td><td>Two-hop relay</td><td>Simplified KRST</td><td>NCPD</td><td>KRF</td></tr><tr><td>[41]</td><td></td><td>Two-hop relay</td><td>TST</td><td>NTD</td><td>ALS, KronF</td></tr><tr><td>[55]</td><td></td><td>Two-hop relay</td><td>MKRST<br>MKronST</td><td>NCPD</td><td>KRF</td></tr><tr><td>[56]</td><td></td><td>Two-hop relay</td><td>Matrices + Training</td><td>Tucker-2</td><td>KRF + Structured LS</td></tr><tr><td>[57]</td><td></td><td>Two-hop relay</td><td>TST</td><td>Block Tucker-2</td><td>KronF</td></tr><tr><td>[58]</td><td></td><td>Multi-hop relay</td><td>Simplified KRST</td><td>Generalized NCPD</td><td>KRF</td></tr><tr><td>[59]</td><td></td><td>Three-hop relay</td><td>KRST</td><td>NCPD</td><td>ALS + KRF</td></tr><tr><td>[60]</td><td></td><td>Two-hop relay</td><td>Matrices + Training</td><td>CPD</td><td>ALS, MMSE</td></tr><tr><td>[61]</td><td></td><td>Three-hop relay</td><td>TST-CPD</td><td>NTD</td><td>Coupled SVD, ALS</td></tr><tr><td>[42]</td><td></td><td>Two-hop relay</td><td>TST</td><td>Coupled NTD</td><td>KronF</td></tr><tr><td>[62]</td><td></td><td>Three-hop relay</td><td>Matrices + Training</td><td>CPD + structured Tucker</td><td>ALS</td></tr><tr><td>[63]</td><td></td><td>Two-hop relay</td><td>TST</td><td>Block Tucker2-CPD</td><td>ALS, KronF</td></tr><tr><td>[64]</td><td>OFDM/mmW</td><td>Two-hop relay</td><td>Matrices + Training</td><td>Structured CPD</td><td>SS * + ESPRIT</td></tr><tr><td>[65]</td><td>OFDM/mmW</td><td>IRS</td><td>Matrices + Training</td><td>CPD</td><td>Tensor completion</td></tr><tr><td>[66]</td><td>mmW</td><td>Two-hop relay</td><td>Matrices + Training</td><td>CPD</td><td>ALS, KRF</td></tr><tr><td>[67]</td><td>mmW</td><td>One-hop</td><td>Simplified KRST</td><td>NCPD</td><td>SVP *-ALS</td></tr><tr><td>[68,69]</td><td></td><td>IRS</td><td>Training</td><td>CPD</td><td>ALS</td></tr><tr><td>[70]</td><td></td><td>UAV</td><td>Simplified KRST<br>+ Training</td><td>NCPD</td><td>KRF-ALM *</td></tr><tr><td>[71]</td><td></td><td>UAV-IRS</td><td>Simplified KRST</td><td>CPD</td><td>ALS + KRF</td></tr><tr><td>[72]</td><td></td><td>Two-hop relay</td><td>TST + Training</td><td>Tucker-2</td><td>LM * + LMMSE *</td></tr><tr><td>[73]</td><td>OFDM</td><td>Two-hop relay</td><td>TST + simplified TSTF</td><td>Coupled NTD</td><td>KronF</td></tr><tr><td>[74]</td><td>OFDM</td><td>Two-hop relay</td><td>KRSTF</td><td>NCPD</td><td>ALS</td></tr></tbody></table>

Hereafter, we highlight some important characteristics of cooperative systems to motivate the choice of two-hop relay systems taken into account in our comparative study:

* Cooperation scheme: Most cooperative systems use relays for the exchange of information between source and destination nodes. Relay stations are equipped with hardware and signal processing capability that allow signal decoding/coding steps, depending on the relay protocol. Recent works have addressed cooperation schemes using intelligent reflecting surfaces (IRSs), which act as a large number of passive reflecting elements with low energy consumption and limited processing capacity to process the received signals. For a comprehensive presentation of IRS-assisted MIMO communication systems and, more generally, of various applications of IRS-assisted wireless networks, the reader is referred to the following review papers: [75–80]. Note that, unlike the semi-blind receivers proposed in this paper that allow an estimation of individual channels in two-hop relay systems with the use of very few pilot symbols, most existing works regarding IRS-assisted wireless communications have presented supervised solutions using a pilot sequence to estimate the individual transmitter-to-IRS and IRS-to-receiver channels or the cascaded transmitter-IRS receiver.

<footnote>* SS = spatial smoothing; SVP = singular value projection; LM = Levenberg-Marquardt; ALM = Accelerated LM; LMMSE = linear minimum mean-square error.</footnote>
receiver channel, view as a single channel [65,68,69,81–85]. The use of pilot sequences
results in a reduction in transmission rates.

Channel estimation for IRS-assisted MIMO communication systems is a very chal-
lenging task due to the following: (i) the large number of passive IRS elements and,
therefore, of channel coefficients to be estimated; and (ii) the lack of processing capacity
at the IRS.

A lot of methods have been proposed in the literature to solve the channel estimation problem, depending on the channel model, system configuration (SISO/MISO/MIMO, single-/multi-user, single-/multi-IRS, or narrowband/broadband communication), type of receiver (supervised using pilot sequences versus semi-blind), and algorithm (ALS, closed-form, compressed sensing, or deep learning) used for individual or cascaded channel estimation. The reader is invited to consult the survey paper [86] for a more detailed presentation of IRS channel estimation methods.

Another type of cooperative system involves UAV-assisted communications [70,87]. Consult [88] for a survey on civil UAV applications. In the context of 6G wireless networks, from the perspective of connecting everyone and everything, integrated satellite-terrestrial networks [89], also known as integrated satellite terrestrial/aerial networks (IST/ANs) [90], have been the subject of recent studies. Other recent works consider IRS-UAV-assisted wireless communication networks for assisting the communication between a base station and multiple users [91,92] or an Internet of Things (IoT) terminal [71].

In the future, highly digitized world of all connected objects, 6G wireless networks will integrate different functionalities, including sensing, communication, computing, localization, navigation, signal and image processing, and object recognition, combined with artificial intelligence (AI) technology. This is the case of recent integrated (radar) sensing and communication (ISAC) systems [93–95], also known as joint communication and radar/radio sensing (JCAS) systems [96], which aim to improve spectral efficiency while minimizing hardware cost and power consumption.

*   *Relaying protocols:* The two most common relaying protocols are DF and AF, depending on whether the relay decodes the received signals. Although the DF protocol generally offers better performance, most of the relay systems presented in Table 22 use the AF protocol. In [55], the authors compare the system performance obtained with both protocols to jointly and semi-blindly estimate the transmitted symbols and individual channels in a one-way two-hop relay system.

*   *Coding schemes:* Various coding schemes are used in the cooperative systems reported in Table 22. However, much of the works propose cooperative systems employing KRST and TST codings. In the present paper, we consider two coding families: tensor-based (e.g., TST and TSTF, and simplified versions denoted as STST and STSTF) and matrix-based (e.g., SKRST and DKRSTF) codings employed at the source and the relay. Two combined codings, denoted as SKRST-MSMKR and STST-MSMKron, are also considered.

*   *Communication channels:* Two types of channels are most often considered, namely flat-fading and frequency-selective fading channels. In the latter case, the channels, which depend on frequency, are represented by means of third-order tensors, implying more channel coefficients to be estimated.

*   *Receiver algorithms:* Parameter estimation algorithms can be classified into two main categories: iterative and closed-form algorithms. The works cited in Table 22 mainly use ALS-based algorithms and closed-form solutions based on KRF and KronF methods, described respectively in Appendices B and C. These algorithms will be applied in Section 6 to devise semi-blind receivers for the joint estimation of symbol matrices and channel coefficients for all considered relay systems.

5.2. *Overview of Two-Hop Relay Systems*

Relay systems can be classified according to the following characteristics:
* One-way/two-way;

* AF/DF relaying protocol;

* Two-hop/multihop;

* Half-duplex/full-duplex (the relay can then transmit and receive simultaneously in both directions, during the same time slot).

In Sections 5.3–5.5, we provide an overview of one-way half-duplex two-hop relay systems, applying several coding schemes that take into account different types of diversity, which results in different tensor models for the received signals. The codings considered are the following: tensor space-time-frequency (TSTF), tensor space-time (TST), simplified TSTF (STSTF), simplified TST (STST), simplified Khatri–Rao space-time (SKRST), and double Khatri–Rao space-time-frequency (DKRSTF), all using the AF protocol.

Additionally, to provide a comparison with the DF protocol, we consider STST and SKRST combined with MSMKron and MSMKR codings, as proposed in [97,98] in the context of point-to-point systems.

The relay systems considered can be classified according to different choices concerning the coding scheme, relaying protocol, tensor model, and receiver algorithm. Figure 13 illustrates this classification according to the coding structure, mentioning the order of the tensor of encoded signals to be transmitted and specifying the tensor model for the signals received at the destination node. It is noteworthy that certain codings are particular cases of more generic ones, resulting from simple restrictions in terms of diversities taken into account, which induces simplified tensor models.

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Classification of relay systems according to the coding scheme and tensor model.</figcaption></figure>

Let us introduce the general framework of a one-way, two-hop relay system, as shown in Figure 14, which is composed of source (S), relay (R), and destination (D) nodes, assumed to be equipped with multiple antennas: $M_S$ transmit and $M_D$ receive antennas at the source and destination, respectively, while the relay has $M_R$ receive and $M_T$ transmit antennas. Note that, in this paper, the direct link between the source and the destination is assumed to be unavailable. Table 23 summarizes the definitions of the design parameters and the matrices and tensors used to describe all the relay systems.

<figure><img src="image_16.png" /><figcaption>**Figure 14.** One-way, two-hop cooperative system.</figcaption></figure>
<table><caption>**Table 23.** Design parameters and system matrices and tensors.</caption><thead><tr><th>Design Parameters</th><th>Definitions</th><th>Dimensions</th><th>Codings</th></tr></thead><tbody><tr><td>M<sub>S</sub>, M<sub>T</sub></td><td>Numbers of transmit antennas at the source and relay nodes</td><td>N × R</td><td>TSTF, TST, STSTF, STST, DKRSTF, STST-MSMKron</td></tr><tr><td>M<sub>R</sub>, M<sub>D</sub></td><td>Numbers of receive antennas at the relay and destination nodes</td><td>N × M<sub>S</sub></td><td>SKRST, SKRST-MSMKR</td></tr><tr><td>N</td><td>Number of symbols per data stream</td><td>M<sub>R</sub> × F × M<sub>S</sub></td><td>TSTF, STSTF</td></tr><tr><td>R</td><td>Number of data streams</td><td>M<sub>D</sub> × F × M<sub>T</sub></td><td>TSTF, STSTF</td></tr><tr><td>F</td><td>Number of subcarriers</td><td>M<sub>R</sub> × M<sub>S</sub></td><td>TST, STST, SKRST, DKRSTF, STST-MSMKron, SKRST-MSMKR</td></tr><tr><td>P<sub>S</sub>, P<sub>R</sub></td><td>Time-spreading lengths at source and relay</td><td>M<sub>D</sub> × M<sub>T</sub></td><td>TST, STST, DKRSTF</td></tr><tr><td>I<sub>S</sub>, I<sub>R</sub></td><td>Numbers of chips at source and relay</td><td>M<sub>D</sub> × M<sub>R</sub></td><td>SKRST</td></tr><tr><th>Matrices/Tensors</th><th>Definitions</th><th>Dimensions</th><th>Codings</th></tr><tr><td>S</td><td>Symbol matrix</td><td>N × R</td><td>TSTF, TST, STSTF, STST, DKRSTF, STST-MSMKron</td></tr><tr><td>S</td><td>Symbol matrix</td><td>N × M<sub>S</sub></td><td>SKRST, SKRST-MSMKR</td></tr><tr><td>H<sup>(SR)</sup></td><td>Source-relay channel tensor</td><td>M<sub>R</sub> × F × M<sub>S</sub></td><td>TSTF, STSTF</td></tr><tr><td>H<sup>(RD)</sup></td><td>Relay-destination channel tensor</td><td>M<sub>D</sub> × F × M<sub>T</sub></td><td>TSTF, STSTF</td></tr><tr><td>H<sup>(SR)</sup></td><td>Source-relay channel matrix</td><td>M<sub>R</sub> × M<sub>S</sub></td><td>TST, STST, SKRST, DKRSTF, STST-MSMKron, SKRST-MSMKR</td></tr><tr><td>H<sup>(RD)</sup></td><td>Relay-destination channel matrix</td><td>M<sub>D</sub> × M<sub>T</sub></td><td>TST, STST, DKRSTF</td></tr><tr><td>H<sup>(RD)</sup></td><td>Relay-destination channel matrix</td><td>M<sub>D</sub> × M<sub>R</sub></td><td>SKRST</td></tr><tr><td>H<sup>(RD)</sup></td><td>Relay-destination channel matrix</td><td>M<sub>D</sub> × M<sub>S</sub></td><td>STST-MSMKron, SKRST-MSMKR</td></tr><tr><td>C<sup>(S)</sup></td><td>Source-coding tensor</td><td>M<sub>S</sub> × F × P<sub>S</sub> × J<sub>S</sub> × R</td><td>TSTF</td></tr><tr><td>C<sup>(S)</sup></td><td>Source-coding tensor</td><td>M<sub>S</sub> × F × P<sub>S</sub> × R</td><td>STSTF</td></tr><tr><td>C<sup>(S)</sup></td><td>Source-coding tensor</td><td>M<sub>S</sub> × P<sub>S</sub> × J<sub>S</sub> × R</td><td>TST</td></tr><tr><td>C<sup>(S)</sup></td><td>Source-coding tensor</td><td>M<sub>S</sub> × P<sub>S</sub> × R</td><td>STST</td></tr><tr><td>C<sup>(S)</sup></td><td>Source-coding tensor</td><td>M<sub>S</sub> × P<sub>S</sub> × R<sub>1</sub> × ... × R<sub>Q</sub></td><td>STST-MSMKron</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay-coding tensor</td><td>M<sub>T</sub> × F × P<sub>R</sub> × J<sub>R</sub> × M<sub>R</sub></td><td>TSTF</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay-coding tensor</td><td>M<sub>T</sub> × F × P<sub>R</sub> × M<sub>R</sub></td><td>STSTF</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay-coding tensor</td><td>M<sub>T</sub> × P<sub>R</sub> × J<sub>R</sub> × M<sub>R</sub></td><td>TST</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay-coding tensor</td><td>M<sub>T</sub> × P<sub>R</sub> × M<sub>R</sub></td><td>STST</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay-coding tensor</td><td>M<sub>S</sub> × P<sub>R</sub> × R<sub>1</sub> × ... × R<sub>Q</sub></td><td>STST-MSMKron</td></tr><tr><td>C<sup>(S)</sup></td><td>Source space-time coding matrix</td><td>P<sub>S</sub> × M<sub>S</sub></td><td>SKRST, DKRSTF, SKRST-MSMKR</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay space-time coding matrix</td><td>P<sub>R</sub> × M<sub>R</sub></td><td>SKRST, DKRSTF</td></tr><tr><td>C<sup>(R)</sup></td><td>Relay space-time coding matrix</td><td>P<sub>R</sub> × M<sub>S</sub></td><td>SKRST-MSMKR</td></tr><tr><td>W<sup>(S)</sup></td><td>Source space-coding matrix</td><td>M<sub>S</sub> × R</td><td>DKRSTF</td></tr><tr><td>W<sup>(R)</sup></td><td>Relay space-coding matrix</td><td>M<sub>T</sub> × M<sub>R</sub></td><td>DKRSTF</td></tr><tr><td>A<sup>(S)</sup></td><td>Source frequency-coding matrix</td><td>F × R</td><td>DKRSTF</td></tr></tbody></table>

In the following, we consider three classes of coding: tensor-based codings in Section 5.3, matrix-based codings in Section 5.4, and tensor- and matrix-based codings combined with MSMKron and MSMKR codings, respectively, in Section 5.5. With the first two classes of coding, the AF relaying protocol will be used, while the DF protocol will be considered in the case of combined codings.

### 5.3. AF Two-Hop Relay Systems Using Tensor-Based Codings

High-order tensors have been used to define tensor coding schemes, as proposed for the first time with the TST coding for CDMA systems [37] and the TSTF coding for OFDM systems [38], in the context of point-to-point wireless communications, in 2012 and 2014, respectively. Let us start our survey by presenting the most general two-hop system using fifth-order TSTF codings at both the source and relay, from which simpler systems will be straightforwardly derived. To alleviate the presentation, we consider the noiseless case. The noise will be introduced in the experimental study, in Section 7.

#### 5.3.1. TSTF Coding

Fifth-order TSTF codings $C^{(S)} \in \mathbb{C}^{M_S \times F \times P_S \times J_S \times R}$ and $C^{(R)} \in \mathbb{C}^{M_T \times F \times P_T \times J_T \times M_T}$ are employed at the source and relay, respectively, with the AF protocol at the relay. The channels between the source and the relay (SR), and between the relay and the destination (RD), are assumed to be frequency-selective fading and quasic-static, i.e., time-invariant during the transmission, composed of $P_S$ and $P_T$ time slots at the source and relay, respectively. They are represented by third-order tensors $H^{(SR)} \in \mathbb{C}^{M_R \times F \times M_S}$ and $H^{(RD)} \in \mathbb{C}^{M_D \times F \times M_R}$.
Let us consider the first hop, detailing the coding at the source and the transmission from the source to the relay. The symbol matrix $\mathbf{S} \in \mathbb{C}^{N \times R}$ to be transmitted contains $R$ data streams, each one composed of $N$ information symbols. At the source, the $n$th symbols $s_{n,r}$ of the $R$ data streams are linearly combined and duplicated in the space-frequency-time-chip domains, owing to the coding tensor $\mathcal{C}^{(S)}$, via the following equation, which defines the fifth-order tensor $\mathcal{U}^{(S)} \in \mathbb{C}^{M_S \times F \times P_S \times J_S \times N}$ of coded signals:

$$ \mathcal{U}^{(S)} = \mathcal{C}^{(S)} \times_5 \mathbf{S} \leftrightarrow u_{m_S f, p_S, j_S, n}^{(S)} = \sum_{r=1}^{R} c_{m_S f, p_S, j_S, r}^{(S)} s_{n, r}. \quad (98) $$

This coding is associated with a transmission during $P_S$ time slots composed of $N$ symbol periods each, a time spreading composed of $J_S$ chips, and a repetition over $F$ subcarriers.

The coded signals are transmitted via the $M_S$ transmit antennas from the source to the relay through the transmission channel $\mathcal{H}^{(SR)}$. The $m_R$th antenna of the relay receives during the $j_S$th chip period of the $n$th symbol period of the $p_S$th time slot, and associated with the $f$th subcarrier, the signal given by the following:

$$ \mathcal{X}^{(R)} = \mathcal{U}^{(S)} \times_1^3 \mathcal{H}^{(SR)} = \mathcal{H}^{(SR)} \times_3^1 \mathcal{U}^{(S)} \leftrightarrow \mathcal{X}_{m_R f, p_S, j_S, n}^{(R)} = \sum_{m_S=1}^{M_S} h_{m_R f, m_S}^{(SR)} u_{m_S f, p_S, j_S, n}^{(S)}. \quad (99) $$

Combining Equations (98) and (99) allows for rewriting the fifth-order tensor $\mathcal{X}^{(R)} \in \mathbb{C}^{M_R \times F \times P_S \times J_S \times N}$ of signals received at the relay as follows:

$$ \mathcal{X}^{(R)} = \mathcal{H}^{(SR)} \times_3^1 \mathcal{C}^{(S)} \times_5 \mathbf{S} \leftrightarrow \mathcal{X}_{m_R f, p_S, j_S, n}^{(R)} = \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_R f, m_S}^{(SR)} c_{m_S f, p_S, j_S, r}^{(S)} s_{n, r}, \quad (100) $$

or equivalently:

$$ \mathcal{X}^{(R)} = \mathcal{C}^{(S)} \times_1^3 \mathcal{H}^{(SR)} \times_5 \mathbf{S}, \quad (101) $$

which is a GTD-(2,5) model whose core tensor is the source-coding tensor $\mathcal{C}^{(S)}$.

We now follow the same reasoning for the second hop. Due to the AF protocol, the signals received at the relay are re-encoded using the coding tensor $\mathcal{C}^{(R)} \in \mathbb{C}^{M_T \times F \times P_R \times J_R \times M_R}$ to give the seventh-order tensor $\mathcal{U}^{(R)} \in \mathbb{C}^{M_T \times F \times P_R \times J_R \times P_S \times J_S \times N}$ of coded signals:

$$ \mathcal{U}^{(R)} = \mathcal{C}^{(R)} \times_5^1 \mathcal{X}^{(R)} \leftrightarrow u_{m_T f, p_R, j_R, m_S, j_S, n}^{(R)} = \sum_{m_R=1}^{M_R} c_{m_T f, p_R, j_R, m_R}^{(R)} \mathcal{X}_{m_R f, p_S, j_S, n}^{(R)} \quad (102) $$

These coded signals are transmitted via the relay to the destination through the channel $\mathcal{H}^{(RD)}$, giving the seventh-order tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times F \times P_R \times J_R \times P_S \times J_S \times N}$ of signals received at the destination:

$$ \mathcal{X}^{(D)} = \mathcal{H}^{(RD)} \times_3^1 \mathcal{U}^{(R)} \leftrightarrow \mathcal{X}_{m_D f, p_R, j_R, m_S, j_S, n}^{(D)} = \sum_{m_T=1}^{M_T} h_{m_D f, m_T}^{(RD)} u_{m_T f, p_R, j_R, m_S, j_S, n}^{(R)} \quad (103) $$

Replacing $\mathcal{U}^{(R)}$, $\mathcal{X}^{(R)}$ and $\mathcal{U}^{(S)}$ with their expressions (102), (99) and (98) gives the following:

$$ \mathcal{X}^{(D)} = \mathcal{H}^{(RD)} \times_3^1 \mathcal{C}^{(R)} \times_5^1 \mathcal{X}^{(R)} \quad (104) $$

$$ = \mathcal{H}^{(RD)} \times_3^1 \mathcal{C}^{(R)} \times_5^1 \mathcal{H}^{(SR)} \times_5^1 \mathcal{U}^{(S)} \quad (105) $$

$$ = \mathcal{H}^{(RD)} \times_3^1 \mathcal{C}^{(R)} \times_5^1 \mathcal{H}^{(SR)} \times_5^1 \mathcal{C}^{(S)} \times_7 \mathbf{S} \quad (106) $$
or in scalar form:

$$x_{m_D, f, p_R, j_R, p_S, j_S, n}^{(D)} = \sum_{m_T=1}^{M_T} \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_D, f, m_T}^{(RD)} c_{m_T, f, p_R, j_R, m_R}^{(R)} \eta_{m_R, f, m_S}^{(SR)} c_{m_S, f, p_S, j_S, r}^{(S)} s_{n, r}. \quad (107)$$

Note that, in Equation (106), the operation $\times_3^1$ and the first operation $\times_5^1$ correspond to contractions along the modes $m_T$ and $m_R$, respectively, while the second operation $\times_5^1$ and the mode-7 product correspond to contractions along the modes $m_S$ and $r$, respectively, as shown in Equation (107). From this equation, we conclude that the tensor $\mathcal{X}^{(D)}$ satisfies the Tucker train (TT) model $[\mathcal{H}^{(RD)}, \mathcal{C}^{(R)}, \mathcal{H}^{(SR)}, \mathcal{C}^{(S)}, \mathbf{S}; M_T, M_R, M_S, R]$, as represented by means of the block diagram in Figure 15.

<figure><img src="image_17.png" /><figcaption>Figure 15. Tucker train model of a two-hop relay system using TSTF codings.</figcaption></figure>

Comparing Equation (106) with the equation of the NGTD-7 model presented in Table 17, we deduce the following correspondences:

$$(\mathcal{H}^{(RD)}, \mathcal{C}^{(R)}, \mathcal{H}^{(SR)}, \mathcal{C}^{(S)}, \mathbf{S}) \leftrightarrow (\mathcal{A}^{(1)}, \mathcal{G}^{(1)}, \mathcal{U}, \mathcal{G}^{(2)}, \mathbf{A}^{(2)}) \quad (108)$$

$$(M_D, F, P_R, J_R, P_S, J_S, N, M_T, M_R, M_S, R) \leftrightarrow (I_1, I_2, I_3, I_4, I_5, I_6, I_7, R_1, R_2, R_3, R_4). \quad (109)$$

Now, let us define the effective channel tensor $\mathcal{H}^{(SRD)} \in \mathbb{C}^{M_D \times F \times P_R \times J_R \times M_S}$ between the source and the destination, as follows:

$$\mathcal{H}^{(SRD)} = \mathcal{H}^{(RD)} \times_3^1 \mathcal{C}^{(R)} \times_5^1 \mathcal{H}^{(SR)} = \mathcal{C}^{(R)} \times_1^3 \mathcal{H}^{(RD)} \times_5^1 \mathcal{H}^{(SR)}. \quad (110)$$

This tensor satisfies a GTD-(2,5) model whose core tensor is the relay-coding tensor $\mathcal{C}^{(R)}$. The expressions (105) and (106) of $\mathcal{X}^{(D)}$ can then be rewritten in terms of the effective channel as follows:

$$\mathcal{X}^{(D)} = \mathcal{H}^{(SRD)} \times_5^1 \mathcal{U}^{(S)} \quad (111)$$

$$= \mathcal{H}^{(SRD)} \times_5^1 \mathcal{C}^{(S)} \times_7 \mathbf{S} = \mathcal{C}^{(S)} \times_1^5 \mathcal{H}^{(SRD)} \times_7 \mathbf{S}. \quad (112)$$

From Equation (111), we conclude that the TT structure of the tensor $\mathcal{X}^{(D)}$ can be interpreted as the contraction of the tensors $\mathcal{H}^{(SRD)}$ and $\mathcal{U}^{(S)}$ along the mode $m_S$, as illustrated in Figure 15.

Moreover, a comparison with the NGTD-7 model reveals that $\mathcal{X}^{(D)}$ is the nesting of the tensor $\mathcal{H}^{(SRD)}$, corresponding to the component $\mathcal{X}^{(1)}$ defined in (45), with the tensor $\mathcal{X}^{(R)}$ defined in (101) and to be associated with the component $\mathcal{X}^{(2)}$ in (46), in red and blue in Equations (106) and (107), respectively, sharing the tensor factor $\mathcal{H}^{(SR)}$ (in green).

In Section 6, under the assumption that coding tensors are known at the destination, the coupled GTD-(2,5) models (112) and (110) will be exploited to design a KronF-based semi-blind receiver composed of two stages; in the first one, the symbol matrix and the effective channel tensor are jointly estimated, applying the KronF algorithm to the model (112),
and in the second one, individual channel tensors ($\mathcal{H}^{(RD)}$, $\mathcal{H}^{(SR)}$) are estimated using the KronF algorithm applied to model (110) of the effective channel estimated in the previous stage. See Table 28.

The TSTF system presented above is the most comprehensive relay system utilizing tensor codings that combine signal diversities across space (antennas), frequency (sub-carriers), time (time-spreading lengths and symbol periods), and code (chips) domains, corresponding to the indices ($m_D, f, p_R, j_R, p_S, j_S, n$) of the received signals' tensor $\mathcal{X}^{(D)}$. This system extends the relay system proposed in [41], which does not take frequency and chip diversities into account, inducing third-order instead of fifth-order coding tensors, and it considers flat-fading channels, leading to a NTD-4 model for the fourth-order tensor of signals received at the destination instead of a NGTD-7 model for the TSTF system. The coding in [41], denoted as STST, as well as the tensor-based codings TST and STSTF, can be derived from the TSTF coding by applying certain simplifications, as detailed hereafter.

### 5.3.2. TST Coding

With TST coding, we consider flat fading channels $\mathbf{H}^{(SR)} \in \mathbb{C}^{M_D \times M_S}$ and $\mathbf{H}^{(RD)} \in \mathbb{C}^{M_D \times M_T}$, and we assume that both the source and relay transmit signals using only one subcarrier. This implies removing frequency diversity from coding tensors, resulting in fourth-order tensors $\mathcal{C}^{(S)} \in \mathbb{C}^{M_S \times P_S \times J_S \times R}$ and $\mathcal{C}^{(R)} \in \mathbb{C}^{M_T \times P_R \times J_R \times M_R}$. The signals received at the destination then form a sixth-order tensor, $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times J_R \times P_S \times J_S \times N}$, directly deduced from Equation (107), which is simplified as follows:

$$x_{m_D, p_R, j_R, p_S, j_S, n}^{(D)} = \sum_{m_T=1}^{M_T} \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_D, m_T, m_R}^{(RD)} c_{m_T, p_R, j_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} c_{m_S, p_S, j_S, r}^{(S)} s_{n, r}. \quad (113)$$

The tensor $\mathcal{X}^{(D)}$ satisfies a NTD-6 model, such as that introduced in Table 17, corresponding to the GTTD-(2,4,2,4,2) model $[\mathbf{H}^{(RD)}, \mathcal{C}^{(R)}, \mathbf{H}^{(SR)}, \mathcal{C}^{(S)}, \mathbf{S}; M_T, M_R, M_S, R]$, which can be viewed as the nesting of the following two TD-(2,4) models deduced from Equations (110) and (101), sharing the common matrix factor $\mathbf{H}^{(SR)}$:

$$\mathcal{H}^{(SRD)} = \mathcal{C}^{(R)} \times_1 \mathbf{H}^{(RD)} \times_4 (\mathbf{H}^{(SR)})^T \in \mathbb{C}^{M_D \times P_R \times J_R \times M_S} \quad (114)$$

$$\mathcal{X}^{(R)} = \mathcal{C}^{(S)} \times_1 \mathbf{H}^{(SR)} \times_4 \mathbf{S} \in \mathbb{C}^{M_R \times P_S \times J_S \times N}. \quad (115)$$

$\mathcal{H}^{(SRD)}$ and $\mathcal{X}^{(R)}$ are the tensors of the effective channel and of signals received at the relay, respectively. Figure 16 depicts a block diagram of the TST system, highlighting its nesting structure. Equations (114) and (115) are analogous to Equations (43) and (44) with the following correspondences:

$$(\mathbf{H}^{(RD)}, \mathcal{C}^{(R)}, \mathbf{H}^{(SR)}, \mathcal{C}^{(S)}, \mathbf{S}) \leftrightarrow (\mathbf{A}^{(1)}, \mathcal{G}^{(1)}, \mathbf{U}, \mathcal{G}^{(2)}, \mathbf{A}^{(2)}) \quad (116)$$

$$(M_D, P_R, J_R, P_S, J_S, N, M_T, M_R, M_S, R) \leftrightarrow (I_1, I_2, I_3, I_4, I_5, I_6, R_1, R_2, R_3, R_4). \quad (117)$$

<figure><img src="image_18.png" /><figcaption>Figure 16. Tucker train model of a two-hop relay system using TST codings.</figcaption></figure>
Moreover, Equation (112) becomes the following:

$$ \mathcal{X}^{(D)} = \mathcal{C}^{(S)} \times_{1}^{4} \mathcal{H}^{(SRD)} \times_{6} \mathbf{S}. \qquad (118) $$

In the next section, the coupled TD-(2,4) models (114) and (118) will be used to design a KronF-based semi-blind receiver. See Table 28.

### 5.3.3. STSTF and STST Codings

Simplified versions of the TSTF and TST codings, denoted as STSTF and STST, are now introduced. These coding schemes do not consider chip diversity for STSTF, and both chip and frequency diversities for STST, leading to fourth-order coding tensors $\mathcal{C}^{(S)} \in \mathbb{C}^{M_S \times F \times P_S \times R}$, $\mathcal{C}^{(R)} \in \mathbb{C}^{M_T \times F \times P_R \times M_R}$, and third-order tensors $\mathcal{C}^{(S)} \in \mathbb{C}^{M_S \times P_S \times R}$, $\mathcal{C}^{(R)} \in \mathbb{C}^{M_T \times P_R \times M_R}$, respectively. With these simplifications, we can directly infer the signals received at the destination from Equations (107) and (113):

$$ x_{m_D, f, p_R, p_S, n}^{(D)} = \sum_{m_T=1}^{M_T} \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_D, f, m_T}^{(RD)} c_{m_T, f, p_R, m_R}^{(R)} h_{m_R, f, m_S}^{(SR)} c_{m_S, f, p_S, r}^{(S)} s_{n, r} \qquad (119) $$

for the STSTF system, and

$$ x_{m_D, p_R, p_S, n}^{(D)} = \sum_{m_T=1}^{M_T} \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_D, m_T}^{(RD)} c_{m_T, p_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} c_{m_S, p_S, r}^{(S)} s_{n, r} \qquad (120) $$

for the STST one. The received signals in (119) and (120) define fifth- and fourth-order tensors that satisfy, respectively, the NGTD-5 and NTD-4 models, such as those introduced in Sections 3.3.3 and 3.3.2, respectively.

**Remark 8.** Note that, for STST, the tensors $\mathcal{H}^{(SRD)}$ and $\mathcal{X}^{(R)}$ are modeled by means of the following TD-(2,3) models, directly deduced from (114) and (115):

$$ \mathcal{H}^{(SRD)} = \mathcal{C}^{(R)} \times_{1} \mathbf{H}^{(RD)} \times_{3} (\mathbf{H}^{(SR)})^{T} \in \mathbb{C}^{M_D \times P_R \times M_S} \qquad (121) $$

$$ \mathcal{X}^{(R)} = \mathcal{C}^{(S)} \times_{1} \mathbf{H}^{(SR)} \times_{3} \mathbf{S} \in \mathbb{C}^{M_R \times P_S \times N}. \qquad (122) $$

In summary, the signals received at the destination for a TSTF system form a seventh-order tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times F \times P_R \times J_R \times P_S \times J_S \times N}$, satisfying a NGTD-7 model, i.e., a generalized Tucker train decomposition, GTTD-(3,5,3,5,2). The TST, STSTF, and STST systems are particular cases of TSTF, with certain signal diversities omitted (frequency for TST, chip for STSTF, and both for STST). These systems satisfy Tucker train models previously defined as GTTD-(2,4,2,4,2), GTTD-(3,4,3,4,2), and TTD-(2,3,2,3,2), corresponding to NTD-6, NGTD-5, and NTD-4 models, respectively. Note that the STST system corresponds to the relay system proposed in [41], while the other three are new.

### 5.4. AF Two-Hop Relay Systems Using Matrix-Based Codings

Coding schemes based on Khatri–Rao products provide a less complex coding structure than tensor codings. Following the same approach as in Section 5.3.1, let us start by describing the general DKRSTF coding based on a double Khatri–Rao product to introduce signal diversities in the space, time, and frequency domains.

#### 5.4.1. DKRSTF Coding

This coding was introduced in [39] as an OFDM extension of the Khatri–Rao ST coding (KRST) defined in [99], in the context of a point-to-point communication system. Here, we propose using DKRSTF in a two-hop relay system. As introduced in Table 23, the coding matrices $\mathcal{C}^{(S)} \in \mathbb{C}^{P_S \times M_S}$ and $\mathcal{C}^{(R)} \in \mathbb{C}^{P_R \times M_R}$ spread the transmitted symbols across $P_S$ and $P_R$ time blocks, while the matrices $\mathbf{A}^{(S)} \in \mathbb{C}^{F \times R}$, $\mathbf{W}^{(S)} \in \mathbb{C}^{M_S \times R}$ and $\mathbf{W}^{(R)} \in \mathbb{C}^{M_T \times M_R}$ are
employed to add frequency and space diversities. Table 24 summarizes the equations of the two-hop system using DKRSTF coding.

<table><caption>**Table 24.** Two-hop system with DKRSTF codings.</caption><thead><tr><th>Coded and Received Signals</th><th>Symbols/Codings</th><th>Channels</th><th>Encode/Received Signals</th><th>Dimensions</th></tr></thead><tbody><tr><td rowspan="2">Signals coded at source</td><td rowspan="2">S ∈ C<sup>N×R</sup><br>A<sup>(S)</sup>, C<sup>(S)</sup>, W<sup>(S)</sup></td><td rowspan="2">First hop</td><td>V<sub>FN×M<sub>S</sub></sub><sup>(S)</sup> = (A<sup>(S)</sup> ♦ S)W<sup>(S)</sup><sup>T</sup></td><td>M<sub>S</sub> × F × N</td></tr><tr><td>U<sub>P<sub>S</sub>FN×M<sub>S</sub></sub><sup>(S)</sup> = C<sup>(S)</sup> ♦ V<sub>FN×M<sub>S</sub></sub><sup>(S)</sup></td><td>M<sub>S</sub> × P<sub>S</sub> × F × N</td></tr><tr><td>Signals received at relay</td><td></td><td>H<sup>(SR)</sup></td><td>X<sub>M<sub>R</sub>×P<sub>S</sub>FN</sub><sup>(R)</sup> = H<sup>(SR)</sup>U<sub>M<sub>S</sub>×P<sub>S</sub>FN</sub><sup>(S)</sup></td><td>M<sub>R</sub> × P<sub>S</sub> × F × N</td></tr><tr><td>Signals coded at relay</td><td>C<sup>(R)</sup>, W<sup>(R)</sup></td><td>Second hop</td><td>U<sub>P<sub>R</sub>P<sub>S</sub>FN×M<sub>T</sub></sub><sup>(R)</sup> = (C<sup>(R)</sup> ♦ X<sub>P<sub>S</sub>FN×M<sub>R</sub></sub><sup>(R)</sup>)W<sup>(R)</sup><sup>T</sup></td><td>M<sub>T</sub> × P<sub>R</sub> × P<sub>S</sub> × F × N</td></tr><tr><td>Signals received at destination</td><td></td><td>H<sup>(RD)</sup></td><td>X<sub>M<sub>D</sub>×P<sub>R</sub>P<sub>S</sub>FN</sub><sup>(D)</sup> = H<sup>(RD)</sup>U<sub>M<sub>T</sub>×P<sub>R</sub>P<sub>S</sub>FN</sub><sup>(R)</sup></td><td>M<sub>D</sub> × P<sub>R</sub> × P<sub>S</sub> × F × N</td></tr></tbody></table>

The first Khatri–Rao product provides space-frequency pre-coded signals at the source, forming the third-order tensor $V^{(S)} \in C^{M_S \times F \times N}$, such as:

$$V_{FN \times M_S}^{(S)} = (A^{(S)} \diamond S)W^{(S)} \Leftrightarrow V^{(S)} = \mathcal{I}_R \times_1 W^{(S)} \times_2 A^{(S)} \times_3 S. \quad (123)$$

This tensor $V^{(S)}$ satisfies the CPD model, $[W^{(S)}, A^{(S)}, S; R]$, which can be written in scalar notation as follows:

$$v_{m_S, f, n}^{(S)} = \sum_{r=1}^{R} w_{m_S, r} a_{f, r}^{(S)} s_{n, r}. \quad (124)$$

Then, the pre-coded signals are spread over $P_S$ blocks using a second Khatri–Rao ST coding matrix $C^{(S)}$ to give the fourth-order tensor $U^{(S)} \in C^{M_S \times P_S \times F \times N}$ of space–time-frequency encoded signals, such as:

$$U_{P_S FN \times M_S}^{(S)} = C^{(S)} \diamond V_{FN \times M_S}^{(S)} \Leftrightarrow U_c^{(S)} = \mathcal{I}_{M_S} \times_1 I_{M_S} \times_2 C^{(S)} \times_3 V_{FN \times M_S}^{(S)}, \quad (125)$$

where $U_c^{(S)} \in C^{M_S \times P_S \times FN}$ denotes a contracted form of $U^{(S)}$, resulting from a combination of the modes $(f, n)$. In scalar form, we have:

$$u_{m_S, p_S, f, n}^{(S)} = c_{p_S, m_S}^{(S)} v_{m_S, f, n}^{(S)}. \quad (126)$$

After space-time-frequency coding at the source and transmission through the channel $H^{(SR)} \in C^{M_R \times M_S}$, the signals received at the relay form a fourth-order tensor $X^{(R)} \in C^{M_R \times P_S \times F \times N}$ given by:

$$X^{(R)} = U^{(S)} \times_1 H^{(SR)} \Leftrightarrow x_{m_R, p_S, f, n}^{(R)} = \sum_{m_S=1}^{M_S} h_{m_R, m_S}^{(SR)} u_{m_S, p_S, f, n}^{(S)} \quad (127)$$

Replacing $U^{(S)}$ with its expression (125) gives the following contracted form $X_c^{(R)} \in C^{M_R \times P_S \times FN}$ of $X^{(R)}$:

$$X_c^{(R)} = U_c^{(S)} \times_1 H^{(SR)} = \mathcal{I}_{M_S} \times_1 H^{(SR)} \times_2 C^{(S)} \times_3 V_{FN \times M_S}^{(S)}, \quad (128)$$

or, in scalar form,

$$x_{m_R, p_S, f, n}^{(R)} = \sum_{m_S=1}^{M_S} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} v_{m_S, f, n}^{(S)}. \quad (129)$$
Combining (129) with (124) gives:

$$x_{m_R, p_S, f, n}^{(R)} = \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} w_{m_S, r}^{(S)} a_{f, r}^{(S)} s_{n, r}. \quad (130)$$

Equations (123) and (128) are to be compared to Equations (20) and (25) associated with an NCDP-4 model. From this comparison, we conclude that the tensor $\mathcal{X}^{(R)}$ satisfies the NCDP-4 model $\llbracket \mathbf{H}^{(SR)}, \mathbf{C}^{(S)}, \mathbf{W}^{(S)}, \mathbf{A}^{(S)}, \mathbf{S}; M_S, R \rrbracket$ with the following correspondences:

$$(\mathbf{H}^{(SR)}, \mathbf{C}^{(S)}, \mathbf{W}^{(S)}, \mathbf{A}^{(S)}, \mathbf{S}) \leftrightarrow (\mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)}) \quad (131)$$

$$(M_R, P_S, F, N, M_S, R) \leftrightarrow (I_1, I_2, I_3, I_4, R_1, R_2). \quad (132)$$

Due to the AF protocol, the signals received at the relay are encoded using a Khatri-Rao space-time coding with matrices $\mathbf{W}^{(R)}$ and $\mathbf{C}^{(R)}$ to give the fifth-order tensor $\mathcal{U}^{(R)} \in \mathbb{C}^{M_T \times P_R \times P_S \times F \times N}$ of coded signals defined by the following equation:

$$\mathbf{U}_{P_R P_S F N \times M_T}^{(R)} = (\mathbf{C}^{(R)} \diamond \mathbf{X}_{P_S F N \times M_R}^{(R)}) \mathbf{W}^{(R)T} \leftrightarrow \mathcal{U}_c^{(R)} = \mathcal{I}_{M_R} \times_1 \mathbf{W}^{(R)} \times_2 \mathbf{C}^{(R)} \times_3 \mathbf{X}_{P_S F N \times M_R}^{(R)}, \quad (133)$$

where $\mathcal{U}_c^{(R)} \in \mathbb{C}^{M_T \times P_R \times P_S F N}$ denotes a contracted form of $\mathcal{U}^{(R)}$, resulting from a combination of the modes $(p_S, f, n)$. After transmission through the channel $\mathbf{H}^{(RD)} \in \mathbb{C}^{M_D \times M_T}$, the signals received at the destination form a fifth-order tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times P_S \times F \times N}$ whose a contracted form is given by:

$$\mathcal{X}_c^{(D)} = \mathcal{U}_c^{(R)} \times_1 \mathbf{H}^{(RD)} \leftrightarrow \mathbf{X}_{M_D \times P_R P_S F N}^{(D)} = \mathbf{H}^{(RD)} \mathbf{U}_{M_T \times P_R P_S F N}^{(R)}. \quad (134)$$

Combining Equations (134) and (133) gives:

$$\mathcal{X}_c^{(D)} = \mathcal{I}_{M_R} \times_1 (\mathbf{H}^{(RD)} \mathbf{W}^{(R)}) \times_2 \mathbf{C}^{(R)} \times_3 \mathbf{X}_{P_S F N \times M_R}^{(R)}. \quad (135)$$

When $M_R > M_T$, the term $\mathbf{H}^{(RD)}\mathbf{W}^{(R)} \in \mathbb{C}^{M_D \times M_R}$ can be interpreted as a virtual antenna array at the destination due to a virtual increase in the number of transmit antennas at the relay, thanks to the space coding matrix $\mathbf{W}^{(R)}$. Note that a necessary identifiability condition (184) with the KRF receiver for the DKRSTF system is $M_R \ge M_T$.

Let us define the virtual channel as follows:

$$\mathbf{B} \triangleq \mathbf{H}^{(RD)} \mathbf{W}^{(R)} \leftrightarrow b_{m_D, m_R} = \sum_{m_T=1}^{M_T} h_{m_D, m_T}^{(RD)} w_{m_T, m_R}^{(R)}. \quad (136)$$

Then, Equation (135) defines the CPD-3 model $\llbracket \mathbf{B}, \mathbf{C}^{(R)}, \mathbf{X}_{P_S F N \times M_R}^{(R)}; M_R \rrbracket$. From this equation, we conclude that the tensor $\mathcal{X}^{(D)}$ satisfies a NCPD-5 model, corresponding to the cascade of the CPD-3 model (135) with the NCPD-4 model (130) of the tensor $\mathcal{X}^{(R)}$, which is itself the cascade of two CPD-3 models, as illustrated in the block diagram in Figure 17.

<figure><img src="image_19.png" /><figcaption>Figure 17. NCPD-5 model for the DKRSTF system as a cascade of three CPD-3 models.</figcaption></figure>
When the CPD-3 model (135) and the expression (129) of $x_{m_R, f, p_S, n}^{(R)}$ are taken into account, the tensor of signals received at the destination can also be written in scalar form as follows:

$$x_{m_D, p_R, p_S, f, n}^{(D)} = \sum_{m_R=1}^{M_R} b_{m_D, m_R} c_{p_R, m_R}^{(R)} x_{m_R, p_S, f, n}^{(R)} \quad (137)$$

$$= \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} b_{m_D, m_R} c_{p_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} v_{m_S, f, n}^{(S)} \quad (138)$$

$$= \sum_{m_S=1}^{M_S} h_{m_D, p_R, m_S}^{(SRD)} c_{p_S, m_S}^{(S)} v_{m_S, f, n}^{(S)} \quad (139)$$

where $\mathcal{H}^{(SRD)} \in \mathbb{C}^{M_D \times P_R \times M_S}$ is the effective channel between the source and the destination, defined as follows:

$$h_{m_D, p_R, m_S}^{(SRD)} = \sum_{m_R=1}^{M_R} b_{m_D, m_R} c_{p_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} \quad (140)$$

and $v_{m_S, f, n}^{(S)}$ is defined in (124). In Section 6.4, the three CPD-3 models (123), (139), and (140), with the definition (136) of $\mathbf{B}$, will be exploited to devise a KRF-based, closed-form receiver, allowing for the joint estimation of the three unknown matrices $(\mathbf{S}, \mathbf{H}^{(SR)}, \mathbf{H}^{(RD)})$.

**Remark 9.** From Equations (124), (136), and (138), the signals received at the destination can be written as follows:

$$x_{m_D, p_R, p_S, f, n}^{(D)} = \sum_{m_T=1}^{M_T} \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} \sum_{r=1}^{R} h_{m_D, m_T}^{(RD)} w_{m_T, m_R}^{(R)} c_{p_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} w_{m_S, r}^{(S)} a_{f, r}^{(S)} s_{n, r}. \quad (141)$$

Comparing this equation with (119), we deduce that the DKRSTF system can be viewed as a particular case of STSTF when considering flat-fading channels, with the following choices for the coding tensors:

$$c_{m_S, f, p_S, r}^{(S)} = c_{p_S, m_S}^{(S)} w_{m_S, r}^{(S)} a_{f, r}^{(S)} \quad \text{and} \quad c_{m_T, f, p_R, m_R}^{(R)} = w_{m_T, m_R}^{(R)} c_{p_R, m_R}^{(R)}. \quad (142)$$

Note that, in the case of DKRSTF, the frequency diversity is omitted from the coding at the relay.

### 5.4.2. SKRST Coding

Now, we consider a simplified version of KRST coding, denoted as SKRST, introduced in [40,53] for two-hop relay systems. At the source, the coding consists of a simple Khatri-Rao product using the coding matrix $\mathbf{C}^{(S)} \in \mathbb{C}^{P_S \times M_S}$, with $\mathbf{S} \in \mathbb{C}^{N \times M_S}$, resulting in the following encoded signals tensor $\mathcal{U}^{(S)} \in \mathbb{C}^{M_S \times P_S \times N}$:

$$\mathcal{U}^{(S)} = \mathcal{I}_{M_S} \times_1 \mathbf{I}_{M_S} \times_2 \mathbf{C}^{(S)} \times_3 \mathbf{S} \leftrightarrow \mathbf{U}_{P_S N \times M_S}^{(S)} = \mathbf{C}^{(S)} \diamond \mathbf{S}, \quad (143)$$

or, in scalar form,

$$u_{m_S, p_S, n}^{(S)} = c_{p_S, m_S}^{(S)} s_{n, m_S}. \quad (144)$$

Without the frequency-space coding matrices $(\mathbf{A}^{(S)}, \mathbf{W}^{(S)}, \mathbf{W}^{(R)})$, Equations (130), (137) and (135) of the signals received at the relay and the destination are then simplified as follows:

$$x_{m_R, p_S, n}^{(R)} = \sum_{m_S=1}^{M_S} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} s_{n, m_S} \leftrightarrow \mathcal{X}^{(R)} = \mathcal{I}_{M_S} \times_1 \mathbf{H}^{(SR)} \times_2 \mathbf{C}^{(S)} \times_3 \mathbf{S} \quad (145)$$
$$x_{m_D, p_R, p_S, n}^{(D)} = \sum_{m_R=1}^{M_R} h_{m_D, m_R}^{(RD)} c_{p_R, m_R}^{(R)} x_{m_R, p_S, n}^{(R)} \longleftrightarrow \mathcal{X}_c^{(D)} = \mathcal{I}_{M_R} \times_1 \mathbf{H}^{(RD)} \times_2 \mathbf{C}^{(R)} \times_3 \mathbf{X}_{P_S N \times M_R}^{(R)} \quad (146)$$

with $\mathcal{X}_c^{(D)} \in \mathbb{C}^{M_D \times P_R \times P_S N}$. Note that this simplification implies a constraint on the relay, such that the numbers of transmit and receive antennas must be equal ($M_T = M_R$). The coupled Equations (145) and (146), to be compared to Equations (20) and (25), define a NCPD-4 model for the fourth-order tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times P_S N}$, which can be written in the following scalar form:

$$x_{m_D, p_R, p_S, n}^{(D)} = \sum_{m_R=1}^{M_R} \sum_{m_S=1}^{M_S} h_{m_D, m_R}^{(RD)} c_{p_R, m_R}^{(R)} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} s_{n, m_S}, \quad (147)$$

with the following correspondences:

$$(\mathbf{H}^{(RD)}, \mathbf{C}^{(R)}, \mathbf{H}^{(SR)}, \mathbf{C}^{(S)}, \mathbf{S}) \longleftrightarrow (\mathbf{A}^{(1)}, \mathbf{B}^{(1)}, \mathbf{G}, \mathbf{A}^{(2)}, \mathbf{B}^{(2)}) \quad (148)$$

$$(M_D, P_R, P_S, N, M_R, M_S) \longleftrightarrow (I_1, I_2, I_3, I_4, R_1, R_2). \quad (149)$$

Let us define the effective channel tensor $\mathcal{H}^{(SRD)} \in \mathbb{C}^{M_D \times P_R \times M_S}$ as follows:

$$\mathcal{H}^{(SRD)} = \mathcal{I}_{M_R} \times_1 \mathbf{H}^{(RD)} \times_2 \mathbf{C}^{(R)} \times_3 (\mathbf{H}^{(SR)})^T, \quad (150)$$

which satisfies the CPD-3 model $[\mathbf{H}^{(RD)}, \mathbf{C}^{(R)}, \mathbf{H}^{(SR)}]^T; M_R]$ and corresponds to Equation (19). Noting that the tensor $\mathcal{X}^{(R)}$ of signals received at the relay, defined in (145), satisfies the CPD-3 model $[\mathbf{H}^{(SR)}, \mathbf{C}^{(S)}, \mathbf{S}; M_S]$ sharing the common factor $\mathbf{H}^{(SR)}$ with $\mathcal{H}^{(SRD)}$, we conclude that the NCPD-4 model of $\mathcal{X}^{(D)}$ is associated with the nesting of the CPD-3 models of $\mathcal{H}^{(SRD)}$ and $\mathcal{X}^{(R)}$, as illustrated with the block diagram in Figure 18. The correspondences (148) and (149) will be employed to design ALS- and KRF-based receivers for the SKRST system. See Table 28.

<figure><img src="image_20.png" /><figcaption>Figure 18. NCPD-4 model for the SKRST system.</figcaption></figure>

## 5.5. DF Two-Hop Relay Systems Using STST-MSMKron and SKRST-MSMKR Codings

In the preceding two sections, as well as in most of the systems presented in Table 22, the AF protocol was employed. A significant improvement in the symbol error rate (SER) can be obtained with the DF protocol at the cost of additional computational complexity at the relay. In [55], the authors propose two-hop relay systems using multiple Khatri-Rao product-based space-time (MKRST) and multiple Kronecker product-based space-time (MKronST) codings at the source and relay, with a comparison of AF, DF, and estimate-forward (EF) protocols.
In this section, we consider two-hop relay systems with a DF protocol, using STST and SKRST codings, respectively, combined with multiple symbol matrices Kronecker and Khatri–Rao products, which simplify the MKronST and MKRST codings in eliminating the pre-coding matrix. These combined codings, respectively denoted as STST-MSMKron and SKRST-MSMKR, were originally proposed in [97,98], in the context of point-to-point systems. Here, we extend their use to a cooperative scenario.

Let us consider Q symbol matrices $\mathbf{S}^{(q)}$, with $q \in \langle Q \rangle$, to be transmitted via the source. With MSMKron and MSMKR codings, each symbol $s_{i,j}^{(q)}$ of a given symbol matrix $\mathbf{S}^{(q)}$ is duplicated via Kronecker (Kron) and Khatri–Rao (KR) products with other symbol matrices $\mathbf{S}^{(q')}$, where $q' \neq q$. These multiple Kron and KR products induce a mutual spreading of transmitted symbols, thus providing additional diversity. Below, we detail these two systems.

### 5.5.1. STST-MSMKron Coding

For the STST-MSMKron coding, the symbol matrices $\mathbf{S}^{(q)} \in \mathbb{C}^{N_q \times R_q}$ are encoded at the source using the $(Q+2)$-order ST coding tensor $\mathcal{C}^{(S)} \in \mathbb{C}^{M_S \times P_S \times R_1 \times \cdots \times R_Q}$, leading to the following $(Q+2)$-order tensor $\mathcal{U}^{(S)} \in \mathbb{C}^{M_S \times P_S \times N_1 \times \cdots \times N_Q}$ of signals to be transmitted:

$$ \mathcal{U}^{(S)} = \mathcal{C}^{(S)} \times_1 \mathbf{I}_{M_S} \times_2 \mathbf{I}_{P_S} \times_3 \mathbf{S}^{(1)} \times_4 \cdots \times_{Q+2} \mathbf{S}^{(Q)}, \quad (151) $$

which can be represented in a compact form as $\mathcal{U}_c^{(S)} = \mathcal{C}^{(S)} \times_3 \mathbf{S} \in \mathbb{C}^{M_S \times P_S \times N}$, with $\mathbf{S} = \bigotimes_{q=1}^Q \mathbf{S}^{(q)} \in \mathbb{C}^{N \times R}$, $N = \prod_{q=1}^Q N_q$ and $R = \prod_{q=1}^Q R_q$. In a scalar format, the coded signals transmitted via the $m_S$th antenna, during the $p_S$th time slot, are given by:

$$ u_{m_S, p_S, n_1, \dots, n_Q}^{(S)} = \sum_{r_1=1}^{R_1} \cdots \sum_{r_Q=1}^{R_Q} c_{m_S, p_S, r_1, \dots, r_Q}^{(S)} \prod_{q=1}^{Q} s_{n_q, r_q}^{(q)}. \quad (152) $$

After transmission through the channel $\mathbf{H}^{(SR)} \in \mathbb{C}^{M_R \times M_S}$, the signals received at the relay form the $(Q+2)$-order tensor $\mathcal{X}^{(R)} \in \mathbb{C}^{M_R \times P_S \times N_1 \times \cdots \times N_Q}$ given by:

$$ \begin{aligned} \mathcal{X}^{(R)} &= \mathcal{U}^{(S)} \times_1 \mathbf{H}^{(SR)} \\ &= \mathcal{C}^{(S)} \times_1 \mathbf{H}^{(SR)} \times_2 \mathbf{I}_{P_S} \times_3 \mathbf{S}^{(1)} \times_4 \cdots \times_{Q+2} \mathbf{S}^{(Q)}. \end{aligned} \quad (153) $$

Note that the mode-2 product with identity matrix $\mathbf{I}_{P_S}$ can be interpreted as repeating the transmission of the same symbols during $P_S$ time slots. The DF protocol, used at the relay, involves estimating the information symbols and then re-encoding the estimated symbols using the tensor $\mathcal{C}^{(R)} \in \mathbb{C}^{M_S \times P_R \times R_1 \times \cdots \times R_Q}$ before transmission to the destination. After transmission through the channel $\mathbf{H}^{(RD)} \in \mathbb{C}^{M_D \times M_S}$ of the signals coded at the relay, the tensor structure (153) is repeated at the destination for the tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times N_1 \times \cdots \times N_Q}$, given by:

$$ \mathcal{X}^{(D)} = \mathcal{C}^{(R)} \times_1 \mathbf{H}^{(RD)} \times_3 \hat{\mathbf{S}}^{(1)} \times_4 \cdots \times_{Q+2} \hat{\mathbf{S}}^{(Q)}, \quad (154) $$

where $\hat{\mathbf{S}}^{(q)}$ denotes the estimate of $\mathbf{S}^{(q)}$ obtained via the decoding process at the relay.

Due to the DF protocol, the equations describing the signals received at the relay and the destination are similar, and they are written in scalar form as follows:

$$ x_{m_R, p_S, n_1, \dots, n_Q}^{(R)} = \sum_{m_S=1}^{M_S} \sum_{r_1=1}^{R_1} \cdots \sum_{r_Q=1}^{R_Q} h_{m_R, m_S}^{(SR)} c_{m_S, p_S, r_1, \dots, r_Q}^{(S)} \prod_{q=1}^{Q} s_{n_q, r_q}^{(q)}, \quad (155) $$
$$x_{m_D, p_R, n_1, \dots, n_Q}^{(D)} = \sum_{m_S=1}^{M_S} \sum_{r_1=1}^{R_1} \dots \sum_{r_Q=1}^{R_Q} h_{m_D, m_S}^{(RD)} c_{m_S, p_R, r_1, \dots, r_Q}^{(R)} \prod_{q=1}^{Q} \hat{s}_{n_{\bar{q}}, r_q}^{(q)}. \quad (156)$$

### 5.5.2. SKRST-MSMKR Coding

For the SKRST-MSMKR coding, the symbol matrices $\mathbf{S}^{(q)} \in \mathbb{C}^{N_q \times M_S}$ are encoded at the source using the coding matrix $\mathbf{C}^{(S)} \in \mathbb{C}^{P_S \times M_S}$, giving a $(Q+2)$-order tensor $\mathcal{U}^{(S)} \in \mathbb{C}^{M_S \times P_S \times N_1 \times \dots \times N_Q}$ for the encoded signals, defined by means of the following matrix unfolding:

$$\mathbf{U}_{P_S N_1 \cdots N_Q \times M_S}^{(S)} = \mathbf{C}^{(S)} \diamond \mathbf{S}^{(1)} \diamond \cdots \diamond \mathbf{S}^{(Q)} \leftrightarrow \mathcal{U}^{(S)} = \mathcal{I}_{M_S} \times_1 \mathbf{I}_{M_S} \times_2 \mathbf{C}^{(S)} \times_3 \mathbf{S}^{(1)} \cdots \times_{Q+2} \mathbf{S}^{(Q)}, \quad (157)$$

which can be compactly written as $\mathbf{U}_{P_S N \times M_S}^{(S)} = \mathbf{C}^{(S)} \diamond \mathbf{S}$, with $\mathbf{S} = \bigcirc_{q=1}^{Q} \mathbf{S}^{(q)} \in \mathbb{C}^{N \times M_S}$ and $N = \prod_{q=1}^{Q} N_q$. The coded signals to be transmitted by the $m_S$th antenna, during the $p_S$th time slot, are given by:

$$u_{m_S, p_S, n_1, \dots, n_Q}^{(S)} = c_{p_S, m_S}^{(S)} \prod_{q=1}^{Q} s_{n_{\bar{q}}, m_S}^{(q)}. \quad (158)$$

After transmission through the channel $\mathbf{H}^{(SR)}$, the signals received at the relay form a $(Q+2)$-order tensor $\mathcal{X}^{(R)} \in \mathbb{C}^{M_R \times P_S \times N_1 \times \dots \times N_Q}$ given by:

$$\mathcal{X}^{(R)} = \mathcal{U}^{(S)} \times_1 \mathbf{H}^{(SR)} \leftrightarrow \mathbf{X}_{M_R \times P_S N_1 \cdots N_Q}^{(R)} = \mathbf{H}^{(SR)} \mathbf{U}_{M_S \times P_S N_1 \cdots N_Q}^{(S)}. \quad (159)$$

Replacing $\mathbf{U}_{M_S \times P_S N_1 \cdots N_Q}^{(S)}$ with its expression (157) gives:

$$\mathbf{X}_{M_R \times P_S N_1 \cdots N_Q}^{(R)} = \mathbf{H}^{(SR)} (\mathbf{C}^{(S)} \diamond \mathbf{S}^{(1)} \diamond \cdots \diamond \mathbf{S}^{(Q)})^T. \quad (160)$$

The tensor $\mathcal{X}^{(R)}$ satisfies the CPD-$(Q+2)$ model $\llbracket \mathbf{H}^{(SR)}, \mathbf{C}^{(S)}, \mathbf{S}^{(1)}, \dots, \mathbf{S}^{(Q)}, M_S \rrbracket$. With the DF protocol, the symbol matrices are estimated at the relay and then re-encoded with the coding matrix $\mathbf{C}^{(R)} \in \mathbb{C}^{P_R \times M_S}$ before transmission to the destination. The tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times N_1 \times \dots \times N_Q}$ of signals received at the destination satisfies the CPD-$(Q+2)$ model $\llbracket \mathbf{H}^{(RD)}, \mathbf{C}^{(R)}, \hat{\mathbf{S}}^{(1)}, \dots, \hat{\mathbf{S}}^{(Q)}, M_S \rrbracket$, given by:

$$\mathbf{X}_{M_D \times P_R N_1 \cdots N_Q}^{(D)} = \mathbf{H}^{(RD)} (\mathbf{C}^{(R)} \diamond \hat{\mathbf{S}}^{(1)} \diamond \cdots \diamond \hat{\mathbf{S}}^{(Q)})^T. \quad (161)$$

Note that this equation implies that $M_T = M_S$. The signals received at the relay and the destination are given in scalar format as follows:

$$x_{m_R, p_S, n_1, \dots, n_Q}^{(R)} = \sum_{m_S=1}^{M_S} h_{m_R, m_S}^{(SR)} c_{p_S, m_S}^{(S)} \prod_{q=1}^{Q} s_{n_{\bar{q}}, m_S}^{(q)} \quad (162)$$

$$x_{m_D, p_R, n_1, \dots, n_Q}^{(D)} = \sum_{m_S=1}^{M_S} h_{m_D, m_S}^{(RD)} c_{p_R, m_S}^{(R)} \prod_{q=1}^{Q} \hat{s}_{n_{\bar{q}}, m_S}^{(q)}. \quad (163)$$

It is worth mentioning that STST and SKSRT systems with a DF protocol can be deduced easily from STST-MSMKron and SKRST-MSMKR systems by considering only one matrix symbol, i.e., $Q = 1$. This configuration will be considered in the experimental study, in Section 7.

Table 25 summarizes the tensors of signals encoded at the source and of signals received at the relay and the destination for each two-hop relay system considered in this overview, unifying their tensor representation and mentioning the tensor model of $\mathcal{X}^{(D)}$.
**Remark 10.** Analyzing the nested models in Table 25 leads to the following conclusions:

* In the case of tensor-based codings, the nested models of signals received at the relay and the destination result from the nesting of two TD-(2,4) and two TD-(2,3) models for the TST and STST systems, respectively, and two GTD-(2,5) and two GTD-(2,4) models for the TSTF and STSTF systems, respectively, with coding tensors as core tensors.

* In the case of matrix-based codings, the nested models are cascades of three and two CPD-3 models for the DKRSTF and SKRST systems, respectively, with a coding matrix as factor of each CPD-3 model.

* In the case of combined codings, with the DF protocol, the signals received at relay and destination form (Q + 2)-order tensor models which satisfy a CPD model with a coding matrix as a factor for the SKRST-MSMKR system, and a TD model with a coding tensor as a core tensor for the STST-MSMKron system.

Capitalizing on these tensor models and assuming the coding matrices/tensors are known at relay and destination allow closed-form, semi-blind receivers to be devised for all the considered relay systems, as shown in the next section. Moreover, knowledge of coding matrices/tensors guarantees the uniqueness of the tensor models. In Section 7, extensive Monte Carlo simulation results will be presented to compare the performance of the different relay systems and associated semi-blind receivers.

<table><caption>**Table 25.** Tensors of encoded and received signals for two-hop relay systems.</caption><thead><tr><th>Systems</th><th>Tensor Writing</th><th>Scalar Writing</th><th>Models of X<sup>(D)</sup></th></tr></thead><tbody><tr><td colspan="4" style="text-align:center; font-weight:bold;">Tensor-based codings-AF protocol</td></tr><tr><td>TSTF</td><td>U<sup>(S)</sup> = C<sup>(S)</sup> ×<sub>5</sub> S ∈ C<sup>M<sub>S</sub>×F×P<sub>S</sub>×I<sub>S</sub>×N</sup><br/>X<sup>(R)</sup> = C<sup>(S)</sup> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>5</sub> S ∈ C<sup>M<sub>R</sub>×F×P<sub>S</sub>×I<sub>S</sub>×N</sup><br/>X<sup>(D)</sup> = C<sup>(R)</sup> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>5</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×F×P<sub>R</sub>×I<sub>R</sub>×P<sub>S</sub>×I<sub>S</sub>×N</sup></td><td>u<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>r</sub> c<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,I<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,f,m<sub>S</sub></sub>c<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,I<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,f,P<sub>R</sub>,I<sub>R</sub>,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>m<sub>T</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,f,m<sub>T</sub></sub>c<sup>(R)</sup><sub>m<sub>T</sub>,f,P<sub>R</sub>,I<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,I<sub>S</sub>,n</sub></td><td>NGTD-7</td></tr><tr><td>TST</td><td>U<sup>(S)</sup> = C<sup>(S)</sup> ×<sub>4</sub> S ∈ C<sup>M<sub>S</sub>×P<sub>S</sub>×I<sub>S</sub>×N</sup><br/>X<sup>(R)</sup> = C<sup>(S)</sup> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>4</sub> S ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×I<sub>S</sub>×N</sup><br/>X<sup>(D)</sup> = C<sup>(R)</sup> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>4</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×F×P<sub>R</sub>×I<sub>R</sub>×P<sub>S</sub>×I<sub>S</sub>×N</sup></td><td>u<sup>(S)</sup><sub>m<sub>S</sub>,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>r</sub> c<sup>(S)</sup><sub>m<sub>S</sub>,P<sub>S</sub>,I<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,f,m<sub>S</sub></sub>c<sup>(S)</sup><sub>m<sub>S</sub>,P<sub>S</sub>,I<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,P<sub>R</sub>,I<sub>R</sub>,P<sub>S</sub>,I<sub>S</sub>,n</sub> = ∑<sub>m<sub>T</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,f,m<sub>T</sub></sub>c<sup>(R)</sup><sub>m<sub>T</sub>,f,P<sub>R</sub>,I<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,I<sub>S</sub>,n</sub></td><td>NTD-6</td></tr><tr><td>STSTF</td><td>U<sup>(S)</sup> = C<sup>(S)</sup> ×<sub>4</sub> S ∈ C<sup>M<sub>S</sub>×F×P<sub>S</sub>×N</sup><br/>X<sup>(R)</sup> = C<sup>(S)</sup> ×<sub>3</sub> H<sup>(SR)</sup> ×<sub>4</sub> S ∈ C<sup>M<sub>R</sub>×F×P<sub>S</sub>×N</sup><br/>X<sup>(D)</sup> = C<sup>(R)</sup> ×<sub>3</sub> H<sup>(RD)</sup> ×<sub>4</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×F×P<sub>R</sub>×P<sub>S</sub>×N</sup></td><td>u<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,n</sub> = ∑<sub>r</sub> c<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,f,m<sub>S</sub></sub>c<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,f,P<sub>R</sub>,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>T</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,f,m<sub>T</sub></sub>c<sup>(R)</sup><sub>m<sub>T</sub>,f,P<sub>R</sub>,I<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,I<sub>S</sub>,n</sub></td><td>NGTD-5</td></tr><tr><td>STST</td><td>U<sup>(S)</sup> = C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>S</sub>×P<sub>S</sub>×N</sup><br/>X<sup>(R)</sup> = C<sup>(S)</sup> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×N</sup><br/>X<sup>(D)</sup> = C<sup>(R)</sup> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>3</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×P<sub>R</sub>×P<sub>S</sub>×N</sup></td><td>u<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,n</sub> = ∑<sub>r</sub> c<sup>(S)</sup><sub>m<sub>S</sub>,f,P<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,m<sub>S</sub></sub>c<sup>(S)</sup><sub>m<sub>S</sub>,P<sub>S</sub>,r</sub>s<sub>n,r</sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,P<sub>R</sub>,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>T</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,m<sub>T</sub></sub>c<sup>(R)</sup><sub>m<sub>T</sub>,P<sub>R</sub>,I<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,f,P<sub>S</sub>,n</sub></td><td>NTD-4</td></tr><tr><td colspan="4" style="text-align:center; font-weight:bold;">Matrix-based codings-AF protocol</td></tr><tr><td>DKRSTF</td><td>V<sup>(S)</sup> = I<sub>R</sub> ×<sub>1</sub> W<sup>(S)</sup> ×<sub>2</sub> A<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>S</sub>×F×N</sup><br/>X<sup>(R)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>2</sub> C<sup>(S)</sup> ×<sub>3</sub> V<sup>(S)</sup><sub>FN</sub> ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×FN</sup><br/>X<sup>(D)</sup> = I<sub>M<sub>R</sub></sub> ×<sub>1</sub> H<sup>(RD)</sup> W<sup>(R)</sup> ×<sub>2</sub> C<sup>(R)</sup> ×<sub>3</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×P<sub>R</sub>×P<sub>S</sub>FN</sup></td><td>v<sup>(S)</sup><sub>m<sub>S</sub>,f,n</sub> = ∑<sub>r</sub> w<sup>(S)</sup><sub>m<sub>S</sub>,r</sub>a<sup>(S)</sup><sub>f,r</sub>s<sub>n,r</sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,f,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,m<sub>S</sub></sub>c<sup>(S)</sup><sub>P<sub>S</sub>,m<sub>S</sub></sub>v<sub>m<sub>S</sub>,f,n</sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,P<sub>R</sub>,P<sub>S</sub>,f,n</sub> = ∑<sub>m<sub>R</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,m<sub>T</sub></sub>w<sup>(R)</sup><sub>m<sub>T</sub>,m<sub>R</sub></sub>c<sup>(R)</sup><sub>P<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,f,n</sub></td><td>NCPD-5</td></tr><tr><td>SKRST</td><td>U<sup>(S)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>2</sub> C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>S</sub>×P<sub>S</sub>×N</sup><br/>X<sup>(R)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>2</sub> C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×N</sup><br/>X<sup>(D)</sup> = I<sub>M<sub>R</sub></sub> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>2</sub> C<sup>(R)</sup> ×<sub>3</sub> X<sup>(R)</sup> ∈ C<sup>M<sub>D</sub>×P<sub>R</sub>×P<sub>S</sub>×N</sup></td><td>u<sup>(S)</sup><sub>m<sub>S</sub>,P<sub>S</sub>,n</sub> = c<sup>(S)</sup><sub>P<sub>S</sub>,m<sub>S</sub></sub>s<sub>n,m<sub>S</sub></sub><br/>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>S</sub></sub> h<sup>(SR)</sup><sub>m<sub>R</sub>,m<sub>S</sub></sub>c<sup>(S)</sup><sub>P<sub>S</sub>,m<sub>S</sub></sub>s<sub>n,m<sub>S</sub></sub><br/>x<sup>(D)</sup><sub>m<sub>D</sub>,P<sub>R</sub>,P<sub>S</sub>,n</sub> = ∑<sub>m<sub>T</sub></sub> h<sup>(RD)</sup><sub>m<sub>D</sub>,m<sub>T</sub></sub>c<sup>(R)</sup><sub>P<sub>R</sub>,m<sub>R</sub></sub>x<sup>(R)</sup><sub>m<sub>R</sub>,P<sub>S</sub>,n</sub></td><td>NCPD-4</td></tr></tbody></table>
<table><caption>Table 25. *Cont.*</caption><thead><tr><th>Systems</th><th>Tensor Writing</th><th>Scalar Writing</th><th>Models of X<sup>(D)</sup></th></tr></thead><tbody><tr><td colspan="4" style="text-align:center; font-weight:bold;">Combined codings-DF protocol</td></tr><tr><td>STST-MSMKron</td><td>U<sub>c</sub><sup>(S)</sup> = C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>S</sub>×P<sub>S</sub>×N<sub>1</sub>...N<sub>Q</sub></sup></td><td>u<sub>m<sub>S</sub>,p<sub>S</sub>,n<sub>1</sub>...n<sub>Q</sub></sub> = &sum;<sub>r<sub>1</sub></sub> &middot;&middot;&middot; &sum;<sub>Q</sub> c<sub>m<sub>S</sub>,p<sub>S</sub>,r<sub>1</sub>...r<sub>Q</sub></sub><sup>(S)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,r<sub>q</sub></sub><sup>(q)</sup></td><td></td></tr><tr><td>S = <sup>Q</sup>&#x20D7;<sub>q=1</sub> S<sup>(q)</sup></td><td>X<sub>c</sub><sup>(R)</sup> = C<sup>(S)</sup> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×N<sub>1</sub>...N<sub>Q</sub></sup><br>X<sub>c</sub><sup>(D)</sup> = C<sup>(R)</sup> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>3</sub> S&#x20D7; ∈ C<sup>M<sub>D</sub>×P<sub>R</sub>×N<sub>1</sub>...N<sub>Q</sub></sup></td><td>x<sub>m<sub>R</sub>,p<sub>S</sub>,n<sub>1</sub>...n<sub>Q</sub></sub><sup>(R)</sup> = &sum;<sub>m<sub>S</sub></sub> &middot;&middot;&middot; &sum;<sub>r<sub>1</sub></sub> h<sub>m<sub>R</sub>,m<sub>S</sub></sub><sup>(SR)</sup> c<sub>m<sub>S</sub>,p<sub>S</sub>,r<sub>1</sub>...r<sub>Q</sub></sub><sup>(S)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,r<sub>q</sub></sub><sup>(q)</sup><br>x<sub>m<sub>D</sub>,p<sub>R</sub>,n<sub>1</sub>...n<sub>Q</sub></sub><sup>(D)</sup> = &sum;<sub>m<sub>S</sub></sub> &middot;&middot;&middot; &sum;<sub>r<sub>1</sub></sub> h<sub>m<sub>D</sub>,m<sub>S</sub></sub><sup>(RD)</sup> c<sub>m<sub>S</sub>,p<sub>R</sub>,r<sub>1</sub>...r<sub>Q</sub></sub><sup>(R)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,r<sub>q</sub></sub><sup>(q)</sup></td><td>TD-(Q + 2)</td></tr><tr><td>SKRST-MSMKR</td><td>U<sub>c</sub><sup>(S)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>2</sub> C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>S</sub>×P<sub>S</sub>×N<sub>1</sub>...N<sub>Q</sub></sup></td><td>u<sub>m<sub>S</sub>,p<sub>S</sub>,n<sub>1</sub>...n<sub>Q</sub></sub> = c<sub>p<sub>S</sub>,m<sub>S</sub></sub><sup>(S)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,m<sub>S</sub></sub><sup>(q)</sup></td><td></td></tr><tr><td>S = <sup>Q</sup>&#x20D7;<sub>q=1</sub> S<sup>(q)</sup></td><td>X<sub>c</sub><sup>(R)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>1</sub> H<sup>(SR)</sup> ×<sub>2</sub> C<sup>(S)</sup> ×<sub>3</sub> S ∈ C<sup>M<sub>R</sub>×P<sub>S</sub>×N<sub>1</sub>...N<sub>Q</sub></sup><br>X<sub>c</sub><sup>(D)</sup> = I<sub>M<sub>S</sub></sub> ×<sub>1</sub> H<sup>(RD)</sup> ×<sub>2</sub> C<sup>(R)</sup> ×<sub>3</sub> S&#x20D7; ∈ C<sup>M<sub>D</sub>×P<sub>R</sub>×N<sub>1</sub>...N<sub>Q</sub></sup></td><td>x<sub>m<sub>R</sub>,p<sub>S</sub>,n<sub>1</sub>...n<sub>Q</sub></sub><sup>(R)</sup> = &sum;<sub>m<sub>S</sub></sub> h<sub>m<sub>R</sub>,m<sub>S</sub></sub><sup>(SR)</sup> c<sub>p<sub>S</sub>,m<sub>S</sub></sub><sup>(S)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,m<sub>S</sub></sub><sup>(q)</sup><br>x<sub>m<sub>D</sub>,p<sub>R</sub>,n<sub>1</sub>...n<sub>Q</sub></sub><sup>(D)</sup> = &sum;<sub>m<sub>S</sub></sub> h<sub>m<sub>D</sub>,m<sub>S</sub></sub><sup>(RD)</sup> c<sub>p<sub>R</sub>,m<sub>S</sub></sub><sup>(R)</sup> &middot;&middot;&middot; &sum;<sub>q=1</sub><sup>Q</sup> s<sub>n<sub>q</sub>,m<sub>S</sub></sub><sup>(q)</sup></td><td>CPD-(Q + 2)</td></tr></tbody></table>

## 6. Semi-Blind Receivers

In the following, we assume that the coding matrices and tensors used at the source and relay are known at the destination. In Section 4, two families of algorithms were presented to estimate unknown parameters in various NCPD and NTD models, namely ALS and closed-form algorithms. In this section, these algorithms are used to design the following semi-blind receivers for relay systems modeled by means of nested tensor models:

- KronF receiver for the TSTF system (Table 21, for a NGTD-7 model);

- KronF receiver for the TST system (Table 20, for a NTD-6 model);

- KronF receiver for the STSTF system (Remark 7, for a NGTD-5 model);

- KronF receiver for the STST system (Table 19, for a NTD-4 model);

- ALS receiver for the STST system (Equations (82)–(84), for a NTD-4 model);

- KRF receiver for the SKRST system (Table 18, for a NCPD-4 model);

- ALS receiver for the SKRST system (Equations (67), (68) and (70), for a NCPD-4 model).

In Table 26, we provide the correspondences between the tensors $X^{(D)}$ of signals received at the destination in systems TSTF/TST/STSTF/STST/SKRST with the generic nested tensor models described in Section 3.

<table><caption>Table 26. Correspondences between $X^{(D)}$ and generic nested tensor models.</caption><thead><tr><th colspan="14" style="text-align:center; font-weight:bold;">Tensor-Based Codings</th></tr><tr><th>Models</th><th>X<sup>(1)</sup></th><th>A<sup>(1)</sup>/A<sup>(1)</sup></th><th>G<sup>(1)</sup></th><th>U/U</th><th>G<sup>(2)</sup></th><th>A<sup>(2)</sup></th><th>I<sub>1</sub></th><th>I<sub>2</sub></th><th>I<sub>3</sub></th><th>I<sub>4</sub></th><th>I<sub>5</sub></th><th>I<sub>6</sub></th><th>I<sub>7</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>R<sub>4</sub></th></tr></thead><tbody><tr><td>TSTF/NGTD-7</td><td>H<sup>(SRD)</sup></td><td>H<sup>(RD)</sup></td><td>C<sup>(R)</sup></td><td>H<sup>(SR)</sup></td><td>C<sup>(S)</sup></td><td>S</td><td>M<sub>D</sub></td><td>F</td><td>P<sub>R</sub></td><td>J<sub>R</sub></td><td>P<sub>S</sub></td><td>J<sub>S</sub></td><td>N</td><td>M<sub>T</sub></td><td>M<sub>R</sub></td><td>M<sub>S</sub></td><td>R</td></tr><tr><td>TST/NTD-6</td><td>H<sup>(SRD)</sup></td><td>H<sup>(RD)</sup></td><td>C<sup>(R)</sup></td><td>H<sup>(SR)</sup></td><td>C<sup>(S)</sup></td><td>S</td><td>M<sub>D</sub></td><td>P<sub>R</sub></td><td>J<sub>R</sub></td><td>P<sub>S</sub></td><td>J<sub>S</sub></td><td>N</td><td>-</td><td>M<sub>T</sub></td><td>M<sub>R</sub></td><td>M<sub>S</sub></td><td>R</td></tr><tr><td>STSTF/NGTD-5</td><td>H<sup>(SRD)</sup></td><td>H<sup>(RD)</sup></td><td>C<sup>(R)</sup></td><td>H<sup>(SR)</sup></td><td>C<sup>(S)</sup></td><td>S</td><td>M<sub>D</sub></td><td>F</td><td>P<sub>R</sub></td><td>P<sub>S</sub></td><td>N</td><td>-</td><td>-</td><td>M<sub>T</sub></td><td>M<sub>R</sub></td><td>M<sub>S</sub></td><td>R</td></tr><tr><td>STST/NTD-4</td><td>H<sup>(SRD)</sup></td><td>H<sup>(RD)</sup></td><td>C<sup>(R)</sup></td><td>H<sup>(SR)</sup></td><td>C<sup>(S)</sup></td><td>S</td><td>M<sub>D</sub></td><td>P<sub>R</sub></td><td>P<sub>S</sub></td><td>N</td><td>-</td><td>-</td><td>-</td><td>M<sub>T</sub></td><td>M<sub>R</sub></td><td>M<sub>S</sub></td><td>R</td></tr></tbody><thead><tr><th colspan="14" style="text-align:center; font-weight:bold;">Matrix-Based Codings</th></tr><tr><th>Models</th><th>X<sup>(1)</sup></th><th>A<sup>(1)</sup></th><th>B<sup>(1)</sup></th><th>G</th><th>A<sup>(2)</sup></th><th>B<sup>(2)</sup></th><th>I<sub>1</sub></th><th>I<sub>2</sub></th><th>I<sub>3</sub></th><th>I<sub>4</sub></th><th>-</th><th>-</th><th>-</th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>-</th><th>-</th></tr></thead><tbody><tr><td>SKRST/NCPD-4</td><td>H<sup>(SRD)</sup></td><td>H<sup>(RD)</sup></td><td>C<sup>(R)</sup></td><td>H<sup>(SR)</sup></td><td>C<sup>(S)</sup></td><td>S</td><td>M<sub>D</sub></td><td>P<sub>R</sub></td><td>P<sub>S</sub></td><td>N</td><td>-</td><td>-</td><td>-</td><td>M<sub>R</sub></td><td>M<sub>S</sub></td><td>-</td><td>-</td></tr></tbody></table>

Note that the DKRSTF, STST-MSMKron, and SKRST-MSMKR systems that satisfy specific tensor models will be considered separately in Sections 6.4, 6.5 and 6.6, respectively. To facilitate the understanding of the receivers presented in a unified way, in Table 28, we first detail the development of the KronF receiver for the TSTF system represented by a NGTD-7 model, which is the most general one. An analysis of the parametric complexity of the proposed receivers is presented in Appendix D.
## 6.1. KronF Receiver for the TSTF System

In Table 21, two closed-form algorithms derived from two different ways of combining the matrix unfoldings (49)–(52) of the NGTD-7 model are proposed. Each algorithm is composed of two steps conditioning the order of estimation of the unknown factors ($\mathcal{A}^{(1)}$, $\mathcal{U}$, $\mathcal{A}^{(2)}$). In the context of a relay system, the closed-form algorithm 1 allows for first estimating the matrix factor $\mathbf{A}^{(2)}$, corresponding to the symbol matrix $\mathbf{S}$, while algorithm 2 first estimates the tensor factor $\mathcal{A}^{(1)}$ which corresponds to the channel tensor $\mathcal{H}^{(RD)}$. Note that these two closed-form algorithms can be used in parallel in order to eliminate propagation errors in the estimation of these two factors. In the next subsection, such a strategy will be exploited to devise closed-form receivers for the STST and SKRST systems. Now, let us detail the two-step KronF receiver for the TSTF system, derived from the coupled GTD-(2,5) models (112) and (110).

Using the correspondences in Table 26, Equation (52) gives the following block-diagonal matrix unfolding of $\mathcal{X}^{(D)}$ deduced from the GTD-(2,5) model (112):

$$ \text{bdiag}_f(\mathbf{X}_{NM_D P_R J_R \times P_S J_S}^{(D)}(f)) = [\mathbf{S} \otimes \text{bdiag}_f(\mathbf{H}_{M_D P_R J_R \times M_S}^{(\text{SRD})}(f))] \text{bdiag}_f(\mathbf{C}_{RM_S \times P_S J_S}^{(S)}(f)). \quad (164) $$

The LS estimate of the KronP in this equation is given by:

$$ \mathbf{S} \otimes \text{bdiag}_f(\mathbf{H}_{M_D P_R J_R \times M_S}^{(\text{SRD})}(f)) = \text{bdiag}_f(\mathbf{X}_{NM_D P_R J_R \times P_S J_S}^{(D)}(f)) \text{bdiag}_f(\mathbf{C}_{RM_S \times P_S J_S}^{(S)}(f))^{\dagger}. \quad (165) $$

The factors $\mathbf{S}$ and $\mathcal{H}^{(\text{SRD})}$ are then estimated by applying the KronF method.

After reshaping the estimate $\text{bdiag}_f(\hat{\mathbf{H}}_{M_D P_R J_R \times M_S}^{(\text{SRD})}(f))$ into $\text{bdiag}_f(\hat{\mathbf{H}}_{M_D M_S \times P_R J_R}^{(\text{SRD})}(f))$, in a second step, we use the following unfolding deduced from the GTD-(2,5) model (110) of $\mathcal{H}^{(\text{SRD})}$ to estimate the individual channels:

$$ \text{bdiag}_f(\mathbf{H}_{M_D M_S \times P_R J_R}^{(\text{SRD})}(f)) = \text{bdiag}_f(\mathbf{H}_{M_D \times M_T}^{(\text{RD})}(f) \otimes \mathbf{H}_{M_S \times M_R}^{(\text{SR})}(f)) \text{bdiag}_f(\mathbf{C}_{M_T M_R \times P_R J_R}^{(R)}(f)), \quad (166) $$

which corresponds to the unfolding (49), leading to the following LS estimate of the KronP:

$$ \text{bdiag}_f(\mathbf{H}_{M_D \times M_T}^{(\text{RD})}(f) \otimes \mathbf{H}_{M_S \times M_R}^{(\text{SR})}(f)) = \text{bdiag}_f(\hat{\mathbf{H}}_{M_D M_S \times P_R J_R}^{(\text{SRD})}(f)) \text{bdiag}_f(\mathbf{C}_{M_T M_R \times P_R J_R}^{(R)}(f))^{\dagger}. \quad (167) $$

The KronF method is applied once again to obtain estimates of the individual channel tensors $\mathcal{H}^{(\text{SR})}$ and $\mathcal{H}^{(\text{RD})}$.

KronF receivers for other tensor-based coding systems can easily be derived using the same approach, with simplified unfoldings. For instance, in the case of the TST system, where frequency diversity is not considered, the simplification involves fixing $F = 1$. Similarly, for the STSTF system which does not consider chip diversity, the simplification results from choosing $J_S = J_R = 1$.

For each two-step KronF receiver, the procedure consists of (i) jointly estimating $(\mathbf{S}, \mathcal{H}^{(\text{SRD})})$ using the expression of the received signal tensor $\mathcal{X}^{(D)}$ in terms of the effective channel and (ii) estimating the individual channels using the estimated effective channel. Table 28 presents the unfoldings used for each system to derive KronF receivers.

**Remark 11.** The matrix Moore-Penrose pseudo-inverses to be calculated with the proposed closed-form receivers can be simplified when choosing orthonormal matrices. For instance, let us consider the case of a column-orthonormal matrix, $\mathbf{A} \in \mathbb{C}^{M \times N}$, i.e., $\mathbf{A}^H \mathbf{A} = \mathbf{I}_N$, which implies $r(\mathbf{A}) = N$ and $M \ge N$. Its left-inverse is then given by $\mathbf{A}^\dagger = \mathbf{A}^H$. Similarly, if $\mathbf{A}$ is row-orthonormal, i.e., $\mathbf{A}\mathbf{A}^H = \mathbf{I}_M$, implying $r(\mathbf{A}) = M$ and $N \ge M$, its right-inverse is also given by $\mathbf{A}^\dagger = \mathbf{A}^H$. In the case of the pseudo-inversion of $\mathbf{A}^T$, we have $(\mathbf{A}^T)^\dagger = \mathbf{A}^*$ if $\mathbf{A}^T$ is column- or row-orthonormal.

In Table 28, this property is used to simplify the calculation of right-inverses of matrix unfoldings of coding tensors, chosen row-orthonormal by construction.
The same simplification is valid for the left-inverse of the KRP and KronP of two column-orthonormal matrices. Indeed, for $\mathbf{A} \in \mathbb{C}^{M \times N}$, $\mathbf{B} \in \mathbb{C}^{P \times N}$ and $\mathbf{C} \in \mathbb{C}^{P \times Q}$, we have:

$$ (\mathbf{A} \diamond \mathbf{B})^H (\mathbf{A} \diamond \mathbf{B}) = \mathbf{A}^H \mathbf{A} \odot \mathbf{B}^H \mathbf{B} = \mathbf{I}_N \odot \mathbf{I}_N = \mathbf{I}_N \qquad (168) $$

$$ (\mathbf{A} \otimes \mathbf{C})^H (\mathbf{A} \otimes \mathbf{C}) = \mathbf{A}^H \mathbf{A} \otimes \mathbf{C}^H \mathbf{C} = \mathbf{I}_N \otimes \mathbf{I}_Q = \mathbf{I}_{NQ}, \qquad (169) $$

which implies $(\mathbf{A} \diamond \mathbf{B})^\dagger = (\mathbf{A} \diamond \mathbf{B})^H$ and $(\mathbf{A} \otimes \mathbf{C})^\dagger = (\mathbf{A} \otimes \mathbf{C})^H$. This property is used to simplify the calculation of the left-inverse of the KRP $(\mathbf{A}^{(S)} \diamond \mathbf{W}^{(S)})$ of coding matrices in the KRF receiver of the DKRSTF system.

## 6.2. KronF Receiver for the STST System and KRF Receiver for the SKRST System

Closed-form receivers for the STST and SKRST systems are derived from Tables 19 and 18, respectively, by exploiting the correspondences in Table 26, with $(\mathcal{H}^{(SRD)}, \mathcal{X}^{(R)})$ playing the role of $(\mathcal{X}^{(1)}, \mathcal{X}^{(2)})$. This gives the two closed-form receivers summarized in Table 27.

Closed-form receiver 1 will be taken as the standard algorithm, allowing for the estimation of the symbol matrix in the first step and thus avoiding error propagation in the information symbol recovery. As $\hat{\mathbf{H}}^{(SR)}$ and $\hat{\mathbf{H}}^{(RD)}$ are calculated, in the second step, from the estimate $\hat{\mathbf{H}}_{M_D M_S \times P_R}^{(SRD)}$, this introduces error propagation in channel estimates.

<table><caption>Table 27. Closed-form receiver for SKRST and STST systems.</caption><thead><tr><th>System/Receiver</th><th>Closed-Form Receiver 1</th><th>Closed-Form Receiver 2</th></tr></thead><tbody><tr><td>SKRST/KRF</td><td>S<sub>SKRST/KRF</sub> = X<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> = [C<sup>(S)</sup><sub>M<sub>D</sub>P<sub>R</sub> &times; P<sub>S</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KRF &rarr; (S&#770;, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)<br><br>Reshaping : H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> &rarr; H&#770;<sub>M<sub>D</sub>M<sub>S</sub> &times; P<sub>R</sub></sub><sup>(SRD)</sup></td><td>H<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup> = X<sub>M<sub>D</sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> = [C<sup>(R)</sup><sub>M<sub>D</sub>P<sub>S</sub>N &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KRF &rarr; (H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup>, X&#770;<sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup>)<br><br>Reshaping : X&#770;<sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> &rarr; X&#770;<sub>NM<sub>R</sub> &times; P<sub>S</sub></sub><sup>(R)</sup></td></tr><tr><td>STST/KronF</td><td>S<sub>STST/KronF</sub> = X<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> = [C<sup>(S)</sup><sub>M<sub>D</sub>P<sub>R</sub> &times; P<sub>S</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (S&#770;, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)<br><br>Reshaping : H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> &rarr; H&#770;<sub>M<sub>D</sub>M<sub>S</sub> &times; P<sub>R</sub></sub><sup>(SRD)</sup></td><td>H<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup> = X<sub>M<sub>D</sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> = [C<sup>(R)</sup><sub>M<sub>D</sub>P<sub>S</sub>N &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup>, X&#770;<sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup>)<br><br>Reshaping : X&#770;<sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> &rarr; X&#770;<sub>NM<sub>R</sub> &times; P<sub>S</sub></sub><sup>(R)</sup></td></tr><tr><td></td><td>H<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup> &otimes; H<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> =<br>H&#770;<sub>M<sub>D</sub>M<sub>S</sub> &times; P<sub>R</sub></sub><sup>(SRD)</sup> = [C<sup>(R)</sup><sub>M<sub>T</sub>M<sub>R</sub> &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup>, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)<br><br>H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> &otimes; H<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup> =<br>H&#770;<sub>M<sub>D</sub>M<sub>S</sub> &times; P<sub>R</sub></sub><sup>(SRD)</sup> = [C<sup>(S)</sup><sub>M<sub>T</sub>M<sub>R</sub> &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(RD)</sup>, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)</td><td>S<sub>STST/KronF</sub> = X&#770;<sub>M<sub>D</sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> = [C<sup>(S)</sup><sub>M<sub>D</sub>P<sub>S</sub>N &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (S&#770;, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)<br><br>S<sub>STST/KronF</sub> = X&#770;<sub>M<sub>D</sub>P<sub>S</sub>N &times; M<sub>R</sub></sub><sup>(R)</sup> = [C<sup>(S)</sup><sub>M<sub>D</sub>P<sub>S</sub>N &times; P<sub>R</sub></sub>]<sup>T</sup><sup>&dagger;</sup><br>KronF &rarr; (S&#770;, H&#770;<sub>M<sub>D</sub>P<sub>R</sub> &times; M<sub>S</sub></sub><sup>(SRD)</sup>)</td></tr></tbody></table>

To eliminate this error propagation in the estimation of $\mathbf{H}^{(RD)}$, an alternative solution consists of using both closed-form algorithms in parallel.

The channel $\mathbf{H}^{(SR)}$ can then be estimated either from the estimate $\hat{\mathbf{H}}_{M_D P_R \times M_S}^{(SRD)}$ obtained with Algorithm 1, or the estimate $\hat{\mathbf{X}}_{P_S N \times M_R}^{(R)}$ obtained with Algorithm 2. For SKRST, using mode-3 and mode-1 unfoldings of the CPD-3 models (150) of $\mathcal{H}^{(SRD)}$ and (145) of $\mathcal{X}^{(R)}$, respectively, we derive the following two solutions to estimate $\mathbf{H}^{(SR)}$:

$$ \hat{\mathbf{H}}^{(SR)} = (\hat{\mathbf{H}}^{(RD)} \diamond \mathbf{C}^{(R)})^{\dagger} \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(SRD)} \quad \text{and} \quad (\hat{\mathbf{H}}^{(SR)})^{T} = (\mathbf{C}^{(S)} \diamond \hat{\mathbf{S}})^{\dagger} \hat{\mathbf{X}}_{P_S N \times M_R}^{(R)} \qquad (170) $$