Similarly, for STST, from unfoldings of the TD-(2,3) models (121) and (122), we obtain:

$$
\begin{align}
\hat{\mathbf{H}}^{(\mathrm{SR})} &= \left( (\hat{\mathbf{H}}^{(\mathrm{RD})} \otimes \mathbf{I}_{P_R}) \mathbf{C}_{M_T P_R \times M_R}^{(\mathrm{R})} \right)^{\dagger} \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})} \nonumber \\
&\quad \text{and} \nonumber \\
\left(\hat{\mathbf{H}}^{(\mathrm{SR})}\right)^T &= \left( (\mathbf{I}_{P_S} \otimes \hat{\mathbf{S}}) \mathbf{C}_{P_S R \times M_S}^{(\mathrm{S})} \right)^{\dagger} \hat{\mathbf{X}}_{P_S N \times M_R}^{(\mathrm{R})}. \tag{171}
\end{align}
$$

In Section 7, these alternative solutions will be compared with the closed-form receivers 1.

Necessary identifiability conditions for the algorithms presented in Table 27 are linked with the uniqueness of right inverses of the coding matrices/unfoldings, which gives:

$$P_S \geq M_S, P_R \geq M_R \tag{172}$$

for the KRF receiver of the SKRST system, and

$$P_S \geq M_S R, \quad P_R \geq M_R M_T \tag{173}$$

for the KronF receiver of the STST system.

For the alternative solutions proposed above, the additional steps (170) and (171) to estimate $\mathbf{H}^{(\mathrm{SR})}$ imply the following supplementary identifiability conditions: $M_D P_R \geq M_R$ and $P_S N \geq M_S$. Note that these conditions are always satisfied when (172) and (173) are satisfied.

## 6.3. ALS Receivers for the STST and SKRST Systems

To derive ALS-based receivers, we use the unfoldings (82)–(84) for the STST system and (67), (68) and (70) for the SKRST system, with the correspondences in Table 26, to estimate the symbol matrix and the channels in an alternate and iterative way. Equations for the ALS-based receivers are presented in Table 28.

<table><caption>**Table 28.** Semi-blind receivers.</caption><thead><tr><th colspan="5">Tensor-Based codings—AF protocol</th></tr><tr><th>System</th><th>Unfoldings</th><th>Estimated parameters</th><th>Corresp.</th><th></th></tr></thead><tbody><tr><td rowspan="2">TSTF</td><td>bdiag<sub>f</sub>(X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>I<sub>R</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub>(f)) = [S ⊗ bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub>(f))]bdiag<sub>f</sub>(C<sup>(S)</sup><sub>R<sub>M<sub>S</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub>(f))</td><td></td><td>NGTD-7<br>Equation (52)</td><td></td></tr><tr><td>bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub>(f)) = bdiag<sub>f</sub>(H<sup>(RD)</sup><sub>M<sub>D</sub>×M<sub>T</sub></sub>(f) ⊗ H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub></sub>(f))bdiag<sub>f</sub>(C<sup>(R)</sup><sub>M<sub>T</sub>×P<sub>R</sub>I<sub>R</sub></sub>(f))</td><td></td><td>Equation (49)</td><td></td></tr><tr><td rowspan="3">Receiver<br>KronF</td><td>Estimation steps</td><td></td><td></td><td></td></tr><tr><td>S ⊗ bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub>(f)) = bdiag<sub>f</sub>(X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>I<sub>R</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub>(f))bdiag<sub>f</sub>(C<sup>(S)</sup><sub>P<sub>S</sub>I<sub>S</sub>×RM<sub>S</sub></sub>(f))<sup>*</sup></td><td>Ŝ, bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub>(f))</td><td>Table 21</td><td></td></tr><tr><td>Reshaping bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub>(f)) → bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub>(f))</td><td></td><td>Algorithm 1</td><td></td></tr><tr><td></td><td>bdiag<sub>f</sub>(H<sup>(RD)</sup><sub>M<sub>D</sub>×M<sub>T</sub></sub>(f) ⊗ H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub></sub>(f)) = bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub>(f))bdiag<sub>f</sub>(C<sup>(R)</sup><sub>P<sub>R</sub>I<sub>R</sub>×M<sub>T</sub>M<sub>R</sub></sub>(f))<sup>*</sup></td><td>Ψ̂<sup>(RD)</sup>, Ψ̂<sup>(SR)</sup></td><td></td><td></td></tr><tr><td rowspan="2">System<br>TST</td><td>Unfoldings</td><td>Estimated parameters</td><td>Corresp.</td><td></td></tr><tr><td>X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>I<sub>R</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub> = (S ⊗ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub>)C<sup>(S)</sup><sub>R<sub>M<sub>S</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub></td><td></td><td>NTD-6<br>Equation (118)</td><td></td></tr><tr><td></td><td>H<sup>(SRD)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub> = (H<sup>(RD)</sup> ⊗ H<sup>(SR)</sup>)<sup>T</sup>C<sup>(R)</sup><sub>M<sub>T</sub>M<sub>R</sub></sub>×P<sub>R</sub>I<sub>R</sub></td><td></td><td>Equation (114)</td><td></td></tr><tr><td rowspan="2">Receiver<br>KronF</td><td>Estimation steps</td><td></td><td></td><td></td></tr><tr><td>S ⊗ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub> = X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>I<sub>R</sub></sub>×P<sub>S</sub>I<sub>S</sub></sub>(C<sup>(S)</sup><sub>P<sub>S</sub>I<sub>S</sub>×RM<sub>S</sub></sub>)<sup>*</sup></td><td>Ŝ, Ψ̂<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub></td><td>Table 20</td><td></td></tr><tr><td></td><td>Reshaping Ψ̂<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>I<sub>R</sub>×M<sub>S</sub></sub> → Ψ̂<sup>(SRD)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub></td><td></td><td>Algorithm 1</td><td></td></tr><tr><td></td><td>H<sup>(RD)</sup> ⊗ H<sup>(SR)</sup><sup>T</sup> = Ψ̂<sup>(SRD)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub>I<sub>R</sub></sub>(C<sup>(R)</sup><sub>P<sub>R</sub>I<sub>R</sub>×M<sub>T</sub>M<sub>R</sub></sub>)<sup>*</sup></td><td>Ψ̂<sup>(RD)</sup>, Ψ̂<sup>(SR)</sup></td><td></td><td></td></tr></tbody></table>
<table><caption>Table 28. *Cont.*</caption><thead><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>System<br>STSTF</td><td>Unfoldings</td><td>Estimated parameters</td><td>Corresp.<br>NGTD-5<br>Equation (52)</td></tr><tr><td></td><td>bdiag<sub>f</sub>(X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub>(f)) = [S ⊗ bdiag<sub>f</sub>(H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>(f))]bdiag<sub>f</sub>(C<sup>(S)</sup><sub>R<sub>M<sub>S</sub>×P<sub>S</sub></sub>(f))</td><td></td><td></td></tr><tr><td></td><td>bdiag<sub>f</sub>(H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>(f)) = bdiag<sub>f</sub>(H<sup>(R<sup>D</sup>)</sup><sub>M<sub>D</sub>M<sub>T</sub></sub>(f) ⊗ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>S</sub>×M<sub>R</sub></sub>(f))bdiag<sub>f</sub>(C<sup>(R)</sup><sub>M<sub>T</sub>M<sub>R</sub>×P<sub>R</sub></sub>(f))</td><td></td><td>Equation (49)</td></tr><tr><td>Receiver<br>KronF</td><td>Estimation steps</td><td></td><td></td></tr><tr><td></td><td>S ⊗ bdiag<sub>f</sub>(H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>(f)) = bdiag<sub>f</sub>(X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub>(f))bdiag<sub>f</sub>(C<sup>(S)</sup><sub>P<sub>S</sub>×RM<sub>S</sub></sub>(f))<sup>*</sup></td><td>Š, bdiag<sub>f</sub>(H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>(f))</td><td>-</td></tr><tr><td></td><td>Reshaping bdiag<sub>f</sub>(H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>(f)) → bdiag<sub>f</sub>(H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>(f))</td><td></td><td></td></tr><tr><td></td><td>bdiag<sub>f</sub>(H<sup>(R<sup>D</sup>)</sup><sub>M<sub>D</sub>M<sub>T</sub></sub>(f) ⊗ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>S</sub>×M<sub>R</sub></sub>(f)) = bdiag<sub>f</sub>(H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>(f))bdiag<sub>f</sub>(C<sup>(R)</sup><sub>P<sub>R</sub>×M<sub>T</sub>M<sub>R</sub></sub>(f))<sup>*</sup></td><td>Š&#770;<sup>(R<sup>D</sup>)</sup>, Š&#770;<sup>(S<sup>R</sup>)</sup></td><td></td></tr><tr><td>System<br>STST</td><td>Unfoldings</td><td>Estimated parameters</td><td>Corresp.<br>NTD-4<br>Equation (86)</td></tr><tr><td></td><td>X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub> = (S ⊗ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>)C<sup>(S)</sup><sub>R<sub>M<sub>S</sub>×P<sub>S</sub></sub></sub></td><td></td><td></td></tr><tr><td></td><td>H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub> = (H<sup>(R<sup>D</sup>)</sup> ⊗ H<sup>(S<sup>R</sup>)</sup><sub>R</sub>)C<sup>(R)</sup><sub>M<sub>T</sub>M<sub>R</sub>×P<sub>R</sub></sub></td><td></td><td>Equation (87)</td></tr><tr><td>Receiver<br>KronF</td><td>Estimation steps</td><td></td><td></td></tr><tr><td></td><td>S ⊗ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> = X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub>(C<sup>(S)</sup><sub>P<sub>S</sub>×RM<sub>S</sub></sub>)<sup>*</sup></td><td>Š, H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub></td><td>Table 20</td></tr><tr><td></td><td>Reshaping H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> → H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub></td><td></td><td>Algorithm 1</td></tr><tr><td></td><td>H<sup>(R<sup>D</sup>)</sup> ⊗ H<sup>(S<sup>R</sup>)</sup> = H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>(C<sup>(R)</sup><sub>P<sub>R</sub>×M<sub>T</sub>M<sub>R</sub></sub>)<sup>*</sup></td><td>H&#770;<sup>(R<sup>D</sup>)</sup>, H&#770;<sup>(S<sup>R</sup>)</sup></td><td></td></tr><tr><td>System<br>STST</td><td>Estimation steps</td><td>Estimated parameters</td><td>Corresp.<br>NTD-4<br>Equation (82)</td></tr><tr><td></td><td>(S&#770;<sub>t</sub>)<sup>T</sup> = [(I<sub>P<sub>S</sub></sub> ⊗ (H&#770;<sup>(R<sup>D</sup>)</sup><sub>t-1</sub> ⊗ I<sub>P<sub>R</sub></sub>)C<sup>(R)</sup><sub>M<sub>T</sub>P<sub>R</sub>×M<sub>R</sub></sub> H&#770;<sup>(S<sup>R</sup>)</sup><sub>t-1</sub>)C<sup>(S)</sup><sub>P<sub>S</sub>M<sub>S</sub>×R</sub>]<sup>†</sup>X<sup>(D)</sup><sub>P<sub>S</sub>M<sub>D</sub>P<sub>R</sub>×N</sub></td><td>Š</td><td></td></tr><tr><td>Receiver</td><td>(H&#770;<sup>(R<sup>D</sup>)</sup><sub>t</sub>)<sup>T</sup> = [(I<sub>P<sub>R</sub></sub> ⊗ (I<sub>P<sub>S</sub></sub> ⊗ Š<sub>t</sub>)C<sup>(S)</sup><sub>P<sub>S</sub>R×M<sub>S</sub></sub>(H&#770;<sup>(S<sup>R</sup>)</sup><sub>t-1</sub>)<sup>T</sup>)C<sup>(R)</sup><sub>P<sub>R</sub>M<sub>R</sub>×M<sub>T</sub></sub>]<sup>†</sup>X<sup>(D)</sup><sub>P<sub>R</sub>P<sub>S</sub>N×M<sub>D</sub></sub></td><td>H&#770;<sup>(R<sup>D</sup>)</sup></td><td>Equation (83)</td></tr><tr><td>ALS</td><td>vec(H&#770;<sup>(S<sup>R</sup>)</sup><sub>t</sub>) = [(I<sub>P<sub>S</sub></sub> ⊗ Š<sub>t</sub>) ⊗ (H&#770;<sup>(R<sup>D</sup>)</sup><sub>t</sub> ⊗ I<sub>J</sub>)](C<sup>(S)</sup><sub>P<sub>S</sub>R×M<sub>S</sub></sub> ⊗ C<sup>(R)</sup><sub>M<sub>T</sub>P<sub>R</sub>×M<sub>R</sub></sub>]<sup>†</sup>x<sup>(D)</sup><sub>P<sub>S</sub>NMD<sub>P<sub>R</sub></sub></sub></td><td>H&#770;<sup>(S<sup>R</sup>)</sup></td><td>Equation (84)</td></tr><tr><td colspan="4" style="text-align:center; font-weight:bold;">Matrix-based codings—AF protocol</td></tr><tr><td>System<br>DKRSTF</td><td>Unfoldings</td><td>Estimated parameters</td><td>Corresp.<br>NCPD-5<br>Equation (178)</td></tr><tr><td></td><td>X<sup>(D)</sup><sub>FN<sub>M<sub>S</sub></sub>P<sub>R</sub>×P<sub>S</sub></sub> = (V<sup>(S)</sup><sub>FN×M<sub>S</sub></sub> ◦ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>)C<sup>(S)</sup><sup>T</sup> with V<sup>(S)</sup><sub>FN×M<sub>S</sub></sub> = (A<sup>(S)</sup> ◦ S)W<sup>(S)</sup><sup>T</sup></td><td></td><td></td></tr><tr><td></td><td>H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub> = (B ◦ H<sup>(S<sup>R</sup>)</sup><sub>R</sub>)C<sup>(R)</sup><sup>T</sup> with B = H<sup>(R<sup>D</sup>)</sup>W<sup>(R)</sup></td><td></td><td>Equation (179)</td></tr><tr><td></td><td>V<sup>(S)</sup><sub>F<sub>M<sub>S</sub></sub>N</sub> = (A<sup>(S)</sup> ◦ W<sup>(S)</sup>)S<sup>T</sup></td><td></td><td>Equation (182)</td></tr><tr><td>Receiver<br>KRF</td><td>Estimation steps</td><td></td><td></td></tr><tr><td></td><td>V<sup>(S)</sup><sub>FN×M<sub>S</sub></sub> ◦ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> = X<sup>(D)</sup><sub>FN<sub>M<sub>S</sub></sub>P<sub>R</sub>×P<sub>S</sub></sub>C<sup>(S)</sup><sup>*</sup></td><td>V&#770;<sup>(S)</sup><sub>FN×M<sub>S</sub></sub>, H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub></td><td>Equation (180)</td></tr><tr><td></td><td>Reshaping H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> → H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub> and V&#770;<sup>(S)</sup><sub>FN×M<sub>S</sub></sub> → V&#770;<sup>(S)</sup><sub>F<sub>M<sub>S</sub></sub>N</sub></td><td></td><td></td></tr><tr><td></td><td>B ◦ H<sup>(S<sup>R</sup>)</sup><sub>R</sub> = H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>C<sup>(R)</sup><sup>*</sup></td><td>B&#770;, H&#770;<sup>(S<sup>R</sup>)</sup></td><td>Equation (181)</td></tr><tr><td></td><td>S&#770;<sup>T</sup> = (A<sup>(S)</sup> ◦ W<sup>(S)</sup>) H&#770;<sup>(S)</sup><sub>F<sub>M<sub>S</sub></sub>N</sub> and H&#770;<sup>(R<sup>D</sup>)</sup> = B&#770;(W<sup>(R)</sup>)<sup>H</sup></td><td>S&#770;, H&#770;<sup>(R<sup>D</sup>)</sup></td><td>Equation (183)</td></tr><tr><td>System<br>SKRST</td><td>Unfoldings</td><td>Estimated parameters</td><td>Corresp.<br>NCPD-4<br>Equation (72)</td></tr><tr><td></td><td>X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub> = (S ◦ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub>)C<sup>(S)</sup><sup>T</sup></td><td></td><td></td></tr><tr><td></td><td>H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub> = (H<sup>(R<sup>D</sup>)</sup> ◦ H<sup>(S<sup>R</sup>)</sup><sub>R</sub>)C<sup>(R)</sup><sup>T</sup></td><td></td><td>Equation (73)</td></tr><tr><td>Receiver<br>KRF</td><td>Estimation steps</td><td></td><td></td></tr><tr><td></td><td>S ◦ H<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> = X<sup>(D)</sup><sub>NMD<sub>P<sub>R</sub>×P<sub>S</sub></sub></sub>C<sup>(S)</sup><sup>*</sup></td><td>Š, H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub></td><td>Table 18</td></tr><tr><td></td><td>Reshaping H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> → H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub></td><td></td><td></td></tr><tr><td></td><td>H<sup>(R<sup>D</sup>)</sup> ◦ H<sup>(S<sup>R</sup>)</sup> = H&#770;<sup>(S<sup>R</sup>)</sup><sub>M<sub>D</sub>M<sub>S</sub>×P<sub>R</sub></sub>C<sup>(R)</sup><sup>*</sup></td><td>H&#770;<sup>(R<sup>D</sup>)</sup>, H&#770;<sup>(S<sup>R</sup>)</sup></td><td>Table 18</td></tr></tbody></table>
<table><caption>**Table 28. Cont.**</caption><thead><tr><th>System</th><th>Estimation steps</th><th>Estimated parameters</th><th>Corresp.</th></tr></thead><tbody><tr><td>SKRST</td><td>(<b>Ĥ</b><sub>t</sub><sup>(RD)</sup>)<sup>T</sup> = [<b>C</b><sup>(R)</sup> ◦ (<b>C</b><sup>(S)</sup> ◦ <b>Ŝ</b><sub>t-1</sub>) (<b>Ĥ</b><sub>t-1</sub><sup>(SR)</sup>)<sup>T</sup>]<sup>†</sup> <b>X</b><sub>P<sub>R</sub>P<sub>S</sub>N×M<sub>D</sub></sub><sup>(D)</sup></td><td><b>Ĥ</b><sup>(RD)</sup></td><td>Equation (67)</td></tr><tr><td>Receiver</td><td>(<b>Ŝ</b><sub>t</sub>)<sup>T</sup> = [<b>C</b><sup>(S)</sup> ◦ (<b>Ĥ</b><sub>t</sub><sup>(RD)</sup> ◦ <b>C</b><sup>(R)</sup>) <b>Ĥ</b><sub>t-1</sub><sup>(SR)</sup>]<sup>†</sup> <b>X</b><sub>P<sub>S</sub>M<sub>D</sub>P<sub>R</sub>×N</sub><sup>(D)</sup></td><td><b>Ŝ</b></td><td>Equation (68)</td></tr><tr><td>ALS</td><td>vec(<b>Ĥ</b><sub>t</sub><sup>(SR)</sup>) = [<b>C</b><sup>(S)</sup> ◦ <b>Ŝ</b><sub>t</sub> ⊗ (<b>Ĥ</b><sub>t</sub><sup>(RD)</sup> ◦ <b>C</b><sup>(R)</sup>)]<sup>†</sup> <b>X</b><sub>P<sub>S</sub>NM<sub>D</sub>P<sub>R</sub></sub><sup>(D)</sup></td><td><b>Ĥ</b><sup>(SR)</sup></td><td>Equation (70)</td></tr><tr><td colspan="4" style="text-align:center; font-weight:bold;">Combined codings—DF protocol</td></tr><tr><td>System<br>STST-<br>MSMKron</td><td><b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>R</sub>×P<sub>S</sub></sub> = (<b>Ŝ</b><sup>(q)</sup> ⊗ <b>H</b><sup>(SR)</sup>) <b>C</b><sub>R<sub>1</sub>...R<sub>Q</sub>M<sub>S</sub>×P<sub>S</sub></sub><sup>(S)</sup></td><td>Estimated parameters</td><td>Equation (185)</td></tr><tr><td></td><td><b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>D</sub>×P<sub>R</sub></sub> = (<b>Ŝ</b><sup>(q)</sup> ⊗ <b>H</b><sup>(RD)</sup>) <b>C</b><sub>R<sub>1</sub>...R<sub>Q</sub>M<sub>S</sub>×P<sub>R</sub></sub><sup>(R)</sup></td><td>Estimated parameters</td><td>Equation (186)</td></tr><tr><td>Receiver<br>Multiple<br>KronF</td><td><b>Estimation steps</b><br>First hop: <b>Ŝ</b><sup>(q)</sup> ⊗ <b>H</b><sup>(SR)</sup> = <b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>R</sub>×P<sub>S</sub></sub> (<b>C</b><sub>P<sub>S</sub>×R<sub>1</sub>...R<sub>Q</sub>M<sub>S</sub></sub><sup>(S)</sup>)<sup>*</sup></td><td><b>Ŝ</b><sup>(1)</sup>, ..., <b>Ŝ</b><sup>(Q)</sup>, <b>Ĥ</b><sup>(SR)</sup></td><td>Equation (187)</td></tr><tr><td></td><td>Second hop: <b>Ŝ</b><sup>(q)</sup> ⊗ <b>H</b><sup>(RD)</sup> = <b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>D</sub>×P<sub>R</sub></sub> (<b>C</b><sub>P<sub>R</sub>×R<sub>1</sub>...R<sub>Q</sub>M<sub>S</sub></sub><sup>(R)</sup>)<sup>*</sup></td><td><b>Ŝ</b><sup>(1)</sup>, ..., <b>Ŝ</b><sup>(Q)</sup>, <b>Ĥ</b><sup>(RD)</sup></td><td>Equation (188)</td></tr><tr><td>System<br>SKRST-<br>MSMKR</td><td><b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>R</sub>×P<sub>S</sub></sub> = (<b>Ŝ</b><sup>(q)</sup> ◦ <b>H</b><sup>(SR)</sup>) <b>C</b><sup>(S)</sup><sup>T</sup></td><td>Estimated parameters</td><td>Equation (189)</td></tr><tr><td></td><td><b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>D</sub>×P<sub>R</sub></sub> = (<b>Ŝ</b><sup>(q)</sup> ◦ <b>H</b><sup>(RD)</sup>) <b>C</b><sup>(R)</sup><sup>T</sup></td><td>Estimated parameters</td><td>Equation (190)</td></tr><tr><td>Receiver<br>Multiple<br>KRF</td><td><b>Estimation steps</b><br>First hop: <b>Ŝ</b><sup>(q)</sup> ◦ <b>H</b><sup>(SR)</sup> = <b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>R</sub>×P<sub>S</sub></sub> <b>C</b><sup>(S)</sup><sup>*</sup></td><td><b>Ŝ</b><sup>(1)</sup>, ..., <b>Ŝ</b><sup>(Q)</sup>, <b>Ĥ</b><sup>(SR)</sup></td><td>Equation (191)</td></tr><tr><td></td><td>Second hop: <b>Ŝ</b><sup>(q)</sup> ◦ <b>H</b><sup>(RD)</sup> = <b>X</b><sub>N<sub>1</sub>...N<sub>Q</sub>M<sub>D</sub>×P<sub>R</sub></sub> <b>C</b><sup>(R)</sup><sup>*</sup></td><td><b>Ŝ</b><sup>(1)</sup>, ..., <b>Ŝ</b><sup>(Q)</sup>, <b>Ĥ</b><sup>(RD)</sup></td><td>Equation (192)</td></tr></tbody></table>

## 6.4. KRF Receiver for the DKRSTF System

Recall the CPD-3 models (139), (140) and (123), with the definition (136), used to design the KRF-based receiver for the DKRSTF system:

$$ \mathcal{X}_{c}^{(D)} = \mathcal{I}_{M_{S}} \times_{1} \mathbf{H}_{M_{D}P_{R} \times M_{S}}^{(SRD)} \times_{2} \mathbf{C}^{(S)} \times_{3} \mathbf{V}_{FN \times M_{S}}^{(S)} \quad (174) $$

$$ \mathcal{H}^{(SRD)} = \mathcal{I}_{M_R} \times_1 \mathbf{B} \times_2 \mathbf{C}^{(R)} \times_3 (\mathbf{H}^{(SR)})^T \quad (175) $$

$$ \mathcal{V}^{(S)} = \mathcal{I}_R \times_1 \mathbf{W}^{(S)} \times_2 \mathbf{A}^{(S)} \times_3 \mathbf{S} \quad (176) $$

$$ \mathbf{B} = \mathbf{H}^{(RD)}\mathbf{W}^{(R)}, \quad (177) $$

where $\mathcal{X}_{c}^{(D)} \in \mathbb{C}^{M_{D}P_{R} \times P_{S} \times FN}$ is a contracted form of $\mathcal{X}^{(D)} \in \mathbb{C}^{M_{D} \times P_{R} \times P_{S} \times F \times N}$, and $\mathbf{H}_{M_{D}P_{R} \times M_{S}}^{(SRD)}$ and $\mathbf{V}_{FN \times M_{S}}^{(S)}$ are matrix unfoldings of the tensors $\mathcal{H}^{(SRD)}$ and $\mathcal{V}^{(S)}$. From the CPD models (174) and (175), we obtain the following tall mode-2 unfoldings:

$$ \mathbf{X}_{FN M_{D} P_{R} \times P_{S}}^{(D)} = (\mathbf{V}_{FN \times M_{S}}^{(S)} \diamond \mathbf{H}_{M_{D} P_{R} \times M_{S}}^{(SRD)}) \mathbf{C}^{(S)T} \quad (178) $$

$$ \mathbf{H}_{M_{D}M_{S} \times P_{R}}^{(SRD)} = (\mathbf{B} \diamond \mathbf{H}^{(SR)T}) \mathbf{C}^{(R)T}. \quad (179) $$

The above unfoldings are alternately exploited in a two-step KRF-based algorithm. In the first step, the LS estimate of the KRP in (178) is calculated, and its factors are estimated using the KRF method:

$$ \mathbf{V}_{FN \times M_S}^{(S)} \diamond \mathbf{H}_{M_D P_R \times M_S}^{(SRD)} = \mathbf{X}_{FN M_D P_R \times P_S}^{(D)} (\mathbf{C}^{(S)T})^{\dagger} \underset{\Rightarrow}{\stackrel{\text{KRF}}{\rightleftharpoons}} (\hat{\mathbf{V}}_{FN \times M_S}^{(S)}, \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(SRD)}). \quad (180) $$
In the second step, the estimate $\hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})}$ is reshaped into $\hat{\mathbf{H}}_{M_D M_S \times P_R}^{(\mathrm{SRD})}$, and then the LS estimate of the KRP in (179) is calculated. The KRF method is applied again to estimate its factors:

$$ \mathbf{B} \diamond \mathbf{H}^{(\mathrm{SR})} = \hat{\mathbf{H}}_{M_D M_S \times P_R}^{(\mathrm{SRD})} (\mathbf{C}^{(R)T})^{\dagger} \underset{\mathrm{KRF}}{\Rightarrow} (\hat{\mathbf{B}}, \hat{\mathbf{H}}^{(\mathrm{SR})}). \quad (181) $$

Finally, after reshaping $\hat{\mathbf{V}}_{F_N \times M_S}^{(S)}$ into $\hat{\mathbf{V}}_{F_M \times N'}^{(S)}$, LS estimates of $\mathbf{S}$ and $\mathbf{H}^{(\mathrm{RD})}$ are obtained using the following unfolding deduced from the CPD-3 model (176):

$$ \mathbf{V}_{F_M \times N}^{(S)} = (\mathbf{A}^{(S)} \diamond \mathbf{W}^{(S)}) \mathbf{S}^T, \quad (182) $$

and the definition (177) of $\mathbf{B}$, which gives:

$$ \hat{\mathbf{S}}^T = (\mathbf{A}^{(S)} \diamond \mathbf{W}^{(S)})^{\dagger} \hat{\mathbf{V}}_{F_M \times N}^{(S)} \quad \text{and} \quad \hat{\mathbf{H}}^{(\mathrm{RD})} = \hat{\mathbf{B}}(\mathbf{W}^{(R)})^{\dagger}. \quad (183) $$

Necessary identifiability conditions are linked to the uniqueness of LS estimates in (180)–(183), i.e., full column-rank conditions for $\mathbf{C}^{(S)}$, $\mathbf{C}^{(R)}$, $\mathbf{A}^{(S)} \diamond \mathbf{W}^{(S)}$, and $(\mathbf{W}^{(R)})^T$, which gives:

$$ P_S \geq M_S, P_R \geq M_R, F_M S \geq R, M_R \geq M_T. \quad (184) $$

## 6.5. KronF Receiver for the STST-MSMKron System

Since the DF protocol is employed for this system, the signals received at the relay are processed to estimate the symbol matrices before re-encoding and forwarding to the destination. From the TD-($Q+2$) model (153), we deduce the following tall mode-2 unfolding of the tensor $\mathcal{X}^{(R)} \in \mathbb{C}^{M_R \times P_S \times N_1 \times \cdots \times N_Q}$ of signals received at the relay:

$$ \mathbf{X}_{N_1 \cdots N_Q M_R \times P_S}^{(R)} = \left( \bigotimes_{q=1}^{Q} \mathbf{S}^{(q)} \otimes \mathbf{H}^{(SR)} \right) \mathbf{C}_{R_1 \cdots R_Q M_S \times P_S}^{(S)}. \quad (185) $$

Analogously, from the TD model (154), we deduce the following tall mode-2 unfolding of the tensor $\mathcal{X}^{(D)} \in \mathbb{C}^{M_D \times P_R \times N_1 \times \cdots \times N_Q}$:

$$ \mathbf{X}_{N_1 \cdots N_Q M_D \times P_R}^{(D)} = \left( \bigotimes_{q=1}^{Q} \hat{\mathbf{S}}^{(q)} \otimes \mathbf{H}^{(RD)} \right) \mathbf{C}_{R_1 \cdots R_Q M_S \times P_R}^{(R)}. \quad (186) $$

Equations (185) and (186) are exploited by a KronF-based receiver, at the relay and the destination, to estimate the symbol matrices $\hat{\mathbf{S}}^{(q)}$ and $\hat{\hat{\mathbf{S}}}^{(q)}$, $q \in \langle Q \rangle$, simultaneously with the channel matrices $\hat{\mathbf{H}}^{(SR)}$ and $\hat{\mathbf{H}}^{(RD)}$, using the THOSVD algorithm:

$$ \bigotimes_{q=1}^{Q} \mathbf{S}^{(q)} \otimes \mathbf{H}^{(SR)} = \mathbf{X}_{N_1 \cdots N_Q M_R \times P_S}^{(R)} (\mathbf{C}_{R_1 \cdots R_Q M_S \times P_S}^{(S)})^{\dagger} \underset{\text{KronF}}{\Rightarrow} (\hat{\mathbf{S}}^{(1)}, \dots, \hat{\mathbf{S}}^{(Q)}, \hat{\mathbf{H}}^{(SR)}), \quad (187) $$

$$ \bigotimes_{q=1}^{Q} \hat{\mathbf{S}}^{(q)} \otimes \mathbf{H}^{(RD)} = \mathbf{X}_{N_1 \cdots N_Q M_D \times P_R}^{(D)} (\mathbf{C}_{R_1 \cdots R_Q M_S \times P_R}^{(R)})^{\dagger} \underset{\text{KronF}}{\Rightarrow} (\hat{\hat{\mathbf{S}}}^{(1)}, \dots, \hat{\hat{\mathbf{S}}}^{(Q)}, \hat{\mathbf{H}}^{(RD)}). \quad (188) $$

Note that the symbol matrices estimated at the relay and the destination are denoted $\hat{\mathbf{S}}^{(q)}$ and $\hat{\hat{\mathbf{S}}}^{(q)}$, respectively. It is worth mentioning that, with the DF protocol, the destination node estimates only the RD channel, while the SR channel is estimated at the relay.

## 6.6. KRF Receiver for the SKRST-MSMKR System

Similarly to the STST-MSMKron system, the SKRST-MSMKR one employs the DF protocol, and the proposed KRF receiver follows the same steps as previously described for
the KronF receiver. Thus, from the CPD-(Q + 2) models (160) and (161), we deduce the following tall mode-2 unfoldings of the tensors X (R) and X (D):

$$
\mathbf{X}_{N_1 \cdots N_Q M_R \times P_S}^{(R)} = \left( \underset{q=1}{\overset{Q}{\diamond}} \mathbf{S}^{(q)} \diamond \mathbf{H}^{(SR)} \right) \mathbf{C}^{(S)T}, \qquad (189)
$$

$$
\mathbf{X}_{N_1 \cdots N_Q M_D \times P_R}^{(D)} = \left( \underset{q=1}{\overset{Q}{\diamond}} \hat{\mathbf{S}}^{(q)} \diamond \mathbf{H}^{(RD)} \right) \mathbf{C}^{(R)T}. \qquad (190)
$$

with $\mathbf{C}^{(S)} \in \mathbb{C}^{P_S \times M_S}$ and $\mathbf{C}^{(R)} \in \mathbb{C}^{P_R \times M_R}$. Note that these equations imply $M_S = M_T$,
which means that the source and the relay must have the same number of transmit antennas.

Equations (189) and (190) are exploited by a KRF-based receiver, at the relay and
the destination, to estimate the symbol matrices $\hat{\mathbf{S}}^{(q)}$ and $\hat{\hat{\mathbf{S}}}^{(q)}$, $q \in \langle Q \rangle$, with the channel
matrices $\hat{\mathbf{H}}^{(SR)}$ and $\hat{\hat{\mathbf{H}}}^{(RD)}$, using the THOSVD algorithm:

$$
\underset{q=1}{\overset{Q}{\diamond}} \mathbf{S}^{(q)} \diamond \mathbf{H}^{(SR)} = \mathbf{X}_{N_1 \cdots N_Q M_R \times P_S}^{(R)} (\mathbf{C}^{(S)T})^{\dagger} \underset{\Rightarrow}{\mathbf{KRF}} (\hat{\mathbf{S}}^{(1)}, \dots, \hat{\mathbf{S}}^{(Q)}, \hat{\mathbf{H}}^{(SR)}), \quad (191)
$$

$$
\underset{q=1}{\overset{Q}{\diamond}} \hat{\mathbf{S}}^{(q)} \diamond \mathbf{H}^{(RD)} = \mathbf{X}_{N_1 \cdots N_Q M_D \times P_R}^{(D)} (\mathbf{C}^{(R)T})^{\dagger} \underset{\Rightarrow}{\mathbf{KRF}} (\hat{\hat{\mathbf{S}}}^{(1)}, \dots, \hat{\hat{\mathbf{S}}}^{(Q)}, \hat{\hat{\mathbf{H}}}^{(RD)}). \quad (192)
$$

As for the STST-MSMKron system, the destination node estimates only the RD channel, while the SR channel is estimated at the relay.

**Remark 12.** The necessary identifiability conditions for the KronF and KRF receivers (187)–(188) and (191)–(192) are linked to the uniqueness of LS estimates in these equations, which respectively gives:

$$
P_S \geq M_S \prod_{q=1}^{Q} R_q, \quad P_R \geq M_T \prod_{q=1}^{Q} R_q, \qquad (193)
$$

and

$$
P_S \geq M_S, \quad P_R \geq M_S. \tag{194}
$$

From the conditions (193) and (194), we conclude that the SKRST-MSMKR system is
less constraining for choosing the time-spreading lengths than the STST-MSMKron system.

The unfoldings used to derive the proposed semi-blind receivers for all considered
relay systems, as well as the factors estimated at each step, are summarized in Table 28.
The correspondences with equations and algorithms associated with the generic nested
tensor models are also given.

6.7. Zero-Forcing Receivers for the STST and SKRST Systems

In Section 7, the proposed ALS and closed-form receivers of the STST and SKRST
systems will be compared to zero-forcing (ZF) receivers which assume a perfect knowledge
of communication channels at the destination node. These receivers can be directly derived
from the unfoldings of the received signals tensor used to estimate the symbol matrix with
the ALS receivers.

Thus, by exploiting the correspondences in Table 26, the ZF receiver for the STST
system is deduced from (82) as follows:

$$
\hat{\mathbf{S}}^T = \left[ (\mathbf{I}_{P_S} \otimes (\mathbf{H}^{(RD)} \otimes \mathbf{I}_{P_R})) \mathbf{C}_{M_T P_R \times M_R}^{(R)} \mathbf{H}^{(SR)} \right] \mathbf{C}_{P_S M_S \times R}^{(S)} \mathbf{X}_{P_S M_D P_R \times N}^{(D)} \quad (195)
$$

Similarly, the ZF receiver for the SKRST system, deduced from (68), is given by:

$$
\hat{\mathbf{S}}^T = \left[ \mathbf{C}^{(S)} \diamond (\mathbf{H}^{(RD)} \diamond \mathbf{C}^{(R)}) \mathbf{H}^{(SR)} \right]^\dagger \mathbf{X}_{P_S M_D P_R \times N}^{(D)} \quad (196)
$$
The SER obtained with ZF receivers serves as a reference to evaluate the proposed semi-blind receivers. ZF receivers are also used to analyze the impact of the choice of some design parameters on the SER performance in the ideal situation where the channels are perfectly known.

## 6.8. Ambiguity Relations, Identifiability Conditions, and Transmission Rates

Factors estimated using KRF and KronF methods are obtained up to scalar ambiguities, as discussed in Appendices B and C. To eliminate these ambiguities, some a priori knowledge is required.

For KronF receivers, the knowledge of only one element of one of the matrices involved in each KronP is enough. Thus, for the KronF receiver of the TSTF system, based on KronPs in (165) and (167), we assume the knowledge of the elements $s_{1,1}$ and $h_{1,f,1}^{(RD)}$ of $\mathbf{S}$ and $\mathbf{H}_{f.}^{(RD)}$, for $f \in \langle F \rangle$. This a priori knowledge is used to deduce the scaling factors allowing to eliminate the scaling ambiguities as follows:

$$ 
\begin{aligned}
\lambda_{\mathbf{S}} &= s_{1,1} / \hat{s}_{1,1}; & \lambda_{\mathbf{H}_{f.}^{(RD)}} &= h_{1,f,1}^{(RD)} / \hat{h}_{1,f,1}^{(RD)}, \quad \text{for } f \in \langle F \rangle, \\
\hat{\mathbf{S}} &\leftarrow \lambda_{\mathbf{S}} \hat{\mathbf{S}}; & \hat{\mathbf{H}}_{\mathrm{FM}_{D}P_{R}I_{R} \times F M_{S}}^{(\mathrm{SRD})} &\leftarrow \lambda_{\mathbf{S}}^{-1} \hat{\mathbf{H}}_{\mathrm{FM}_{D}P_{R}I_{R} \times F M_{S}}^{(\mathrm{SRD})}; \\
\hat{\mathbf{H}}_{f.}^{(\mathrm{RD})} &\leftarrow \lambda_{\mathbf{H}_{f.}^{(\mathrm{RD})}} \hat{\mathbf{H}}_{f.}^{(\mathrm{RD})}, & \hat{\mathbf{H}}_{f.}^{(\mathrm{SR})} &\leftarrow \lambda_{\mathbf{H}_{f.}^{(\mathrm{RD})}}^{-1} \hat{\mathbf{H}}_{f.}^{(\mathrm{SR})},
\end{aligned}
\quad (197) $$

with $\mathbf{H}_{FM_{D}P_{R}I_{R} \times F M_{S}}^{(\mathrm{SRD})} = \mathrm{bdiag}_{f}(\mathbf{H}_{M_{D}P_{R}I_{R} \times M_{S}}^{(\mathrm{SRD})}(f))$. The same procedure is applied to the receivers of the TST, STSTF, and STST systems, noting that the TST and STST systems do not exploit frequency diversity, and therefore, $F = 1$ in these cases.

For KRF receivers, a priori knowledge of one row of one of the matrices involved in each KRP is required. For instance, in the case of the DKRSTF system based on KRPs in (180) and (181), the first rows of $\mathbf{V}_{FN \times M_{S}}^{(S)}$ and $\mathbf{B}$ can be calculated using the first rows of $\mathbf{S}$ and $\mathbf{H}^{(RD)}$, respectively, from the unfolding (123) of $\mathcal{V}^{(S)}$ and the definition (177) of $\mathbf{B}$, as follows:

$$ [\mathbf{V}_{FN \times M_{S}}^{(S)}]_{1,\cdot} = (\mathbf{a}_{1,\cdot}^{(S)} \diamond \mathbf{s}_{1,\cdot}) \mathbf{W}^{(S)T} \quad \text{and} \quad \mathbf{b}_{1,\cdot} = \mathbf{h}_{1,\cdot}^{(RD)} \mathbf{W}^{(R)}. \quad (198) $$

The ambiguity relations are then given by the following:

$$ 
\begin{aligned}
\mathbf{\Lambda}_{\mathbf{V}^{(S)}} &= \mathrm{diag}([\mathbf{V}_{FN \times M_S}^{(S)}]_{1,\cdot}) [\mathrm{diag}([\hat{\mathbf{V}}_{FN \times M_S}^{(S)}]_{1,\cdot})]^{-1}; & \mathbf{\Lambda}_{\mathbf{B}} &= \mathrm{diag}(\mathbf{b}_{1,\cdot}) [\mathrm{diag}(\hat{\mathbf{b}}_{1,\cdot})]^{-1} \\
\hat{\mathbf{V}}_{FN \times M_S}^{(S)} &\leftarrow \hat{\mathbf{V}}_{FN \times M_S}^{(S)} \mathbf{\Lambda}_{\mathbf{V}^{(S)}}; & \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})} &\leftarrow \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})} \mathbf{\Lambda}_{\mathbf{V}^{(S)}}^{-1} \\
\hat{\mathbf{B}} &\leftarrow \hat{\mathbf{B}} \mathbf{\Lambda}_{\mathbf{B}}; & \hat{\mathbf{H}}^{(\mathrm{SR})} &\leftarrow \hat{\mathbf{H}}^{(\mathrm{SR})} \mathbf{\Lambda}_{\mathbf{B}}^{-1}.
\end{aligned}
\quad (199) $$

For the KRF receiver of the SKRST system, we have the following ambiguity relations:

$$ 
\begin{aligned}
\mathbf{\Lambda}_{\mathbf{S}} &= \mathrm{diag}(\mathbf{s}_{1,\cdot}) [\mathrm{diag}(\hat{\mathbf{s}}_{1,\cdot})]^{-1}; & \mathbf{\Lambda}_{\mathbf{H}^{(RD)}} &= \mathrm{diag}(\hat{\mathbf{h}}_{1,\cdot}^{(RD)}) [\mathrm{diag}(\hat{\mathbf{h}}_{1,\cdot}^{(RD)})]^{-1}, \\
\hat{\mathbf{S}} &\leftarrow \hat{\mathbf{S}} \mathbf{\Lambda}_{\mathbf{S}}; & \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})} &\leftarrow \hat{\mathbf{H}}_{M_D P_R \times M_S}^{(\mathrm{SRD})} \mathbf{\Lambda}_{\mathbf{S}}^{-1}, \\
\hat{\mathbf{H}}^{(\mathrm{RD})} &\leftarrow \hat{\mathbf{H}}^{(\mathrm{RD})} \mathbf{\Lambda}_{\mathbf{H}^{(RD)}}; & \hat{\mathbf{H}}^{(\mathrm{SR})} &\leftarrow \hat{\mathbf{H}}^{(\mathrm{SR})} \mathbf{\Lambda}_{\mathbf{H}^{(RD)}}^{-1}.
\end{aligned}
\quad (200) $$

For combined codings based on multiple KronPs and KRPs, the closed-form receivers are described by Equations (187) and (188) for the STST-MSMKron system and (191)–(192) for the SKRST-MSMKR one. Due to the DF protocol, each estimation step at the relay and destination requires a priori knowledge of one element or one row of each symbol matrix to remove ambiguities. In this case, the channels $\mathbf{H}^{(SR)}$ and $\mathbf{H}^{(RD)}$ are estimated blindly, that is to say without a priori knowledge on $\mathbf{H}^{(RD)}$, unlike the AF protocol.
Finally, for the ALS receivers of the SKRST and STST systems, the procedure to eliminate the ambiguities results from the discussion in Sections 3.3.1 and 3.3.2 on the uniqueness of the NCDP-4 and NTD-4 models. The fact that the coding matrices/tensors are assumed to be perfectly known at the destination ensures the uniqueness of the tensor models and avoids permutation ambiguity in estimated factor matrices. Only scaling ambiguities have to be removed. By exploiting the ambiguity relations (32) and (42) and the correspondences in Table 26, we deduce the following ambiguity relations for the ALS receivers of the SKRST and STST systems:

$$
\text{NCPD-4: } \Lambda_S = \operatorname{diag}(\mathbf{s}_{1,\cdot}) [\operatorname{diag}(\hat{\mathbf{s}}_{1,\cdot})]^{-1} \quad \text{and} \quad \Lambda_{\mathbf{H}^{(RD)}} = \operatorname{diag}(\mathbf{h}_{1,\cdot}^{(RD)}) [\operatorname{diag}(\hat{\mathbf{h}}_{1,\cdot}^{(RD)})]^{-1}, \\
\hat{\mathbf{S}} \leftarrow \hat{\mathbf{S}}_{\infty} \Lambda_S; \quad \hat{\mathbf{H}}^{(RD)} \leftarrow \hat{\mathbf{H}}_{\infty}^{(RD)} \Lambda_{\mathbf{H}^{(RD)}}; \quad \hat{\mathbf{H}}^{(SR)} \leftarrow \Lambda_{\mathbf{H}^{(RD)}}^{-1} \hat{\mathbf{H}}_{\infty}^{(SR)} \Lambda_S^{-1}, \qquad (201)
$$

$$
\text{NTD-4: } \lambda_S = s_{1,1} / \hat{s}_{1,1} \quad \text{and} \quad \lambda_{\mathbf{H}^{(RD)}} = h_{1,1}^{(RD)} / \hat{h}_{1,1}^{(RD)}, \\
\hat{\mathbf{S}} \leftarrow \lambda_S \hat{\mathbf{S}}_{\infty}; \quad \hat{\mathbf{H}}^{(RD)} \leftarrow \lambda_{\mathbf{H}^{(RD)}} \hat{\mathbf{H}}_{\infty}^{(RD)}; \quad \hat{\mathbf{H}}^{(SR)} \leftarrow \lambda_S^{-1} \lambda_{\mathbf{H}^{(RD)}}^{-1} \hat{\mathbf{H}}_{\infty}^{(SR)}, \quad (202)
$$

where $(\hat{\mathbf{S}}_{\infty}, \hat{\mathbf{H}}_{\infty}^{(RD)}, \hat{\mathbf{H}}_{\infty}^{(SR)})$ are estimates at convergence. In practice, the a priori information on the RD channel, necessary to remove ambiguities, can be obtained by applying a supervised procedure, that is to say a pilot symbol or a pilot sequence sent via the relay to the destination in the cases of KronF and KRF receivers, respectively. Several examples of this procedure can be found in the literature in the context of relay systems [40–43,55,58,60,100].

In Section 4, necessary conditions have been established for identifiability with ALS and closed-form estimation algorithms. By exploiting the correspondences in Table 26, identifiability conditions are derived in terms of design parameters for the proposed semi-blind receivers. For the KRF and KronF receivers of the DKRSTF, STST-MSMKron, and SKRST-MSMKR systems, identifiability conditions are given in (184), (193) and (194).

Table 29 summarizes the identifiability conditions for each proposed semi-blind receiver, as well as the transmission rate for each relay system. An analysis of the results in this table allows us to draw some preliminary conclusions with a view of choosing the design parameters in order to obtain the best trade-off between different performance criteria:

* ALS-based receivers are less constraining than closed-form ones. The matrices to be inverted in the ALS steps induce greater flexibility in the choice of design parameters compared to closed-form receivers which impose more restrictive conditions on these parameters.

* Matrix-based coding schemes induce fewer restrictions than tensor codings, i.e., softer constraints for the choice of design parameters.

* Exploiting chip diversity in the encoding process leads to identifiability conditions that are less constraining without degrading transmission rate.

<table><caption>**Table 29.** Identifiability conditions in terms of design parameters.</caption><thead><tr><th>Systems/Receivers</th><th>Necessary Conditions</th><th>Corresp.</th><th>Transmission Rates</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/KronF</td><td>P<sub>R</sub>J<sub>R</sub> ≥ M<sub>T</sub>M<sub>R</sub>, P<sub>S</sub>J<sub>S</sub> ≥ M<sub>S</sub>R</td><td>(96)</td><td></td></tr><tr><td>TST/KronF</td><td>P<sub>R</sub>J<sub>R</sub> ≥ M<sub>T</sub>M<sub>R</sub>, P<sub>S</sub>J<sub>S</sub> ≥ M<sub>S</sub>R</td><td>(95)</td><td></td></tr><tr><td>STSTF/KronF</td><td>P<sub>R</sub> ≥ M<sub>T</sub>M<sub>R</sub>, P<sub>S</sub> ≥ M<sub>S</sub>R</td><td>(97)</td><td>(NR − 1)/NP<sub>S</sub>(P<sub>R</sub> + 1)</td></tr><tr><td>STST/KronF</td><td>P<sub>R</sub> ≥ M<sub>T</sub>M<sub>R</sub>, P<sub>S</sub> ≥ M<sub>S</sub>R</td><td>(173)</td><td></td></tr><tr><td>STST/ALS</td><td>P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ R, P<sub>R</sub>P<sub>S</sub>N ≥ M<sub>T</sub>, P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ M<sub>R</sub>M<sub>S</sub></td><td>(85)</td><td></td></tr><tr><td>STST/ZF</td><td>P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ R</td><td>(85) *</td><td></td></tr><tr><td colspan="4" style="text-align: center; font-weight: bold;">Matrix-based codings—AF protocol</td></tr><tr><td>DKRSTF/KRF</td><td>P<sub>S</sub> ≥ M<sub>S</sub>, P<sub>R</sub> ≥ M<sub>R</sub>, FM<sub>S</sub> ≥ R, M<sub>R</sub> ≥ M<sub>T</sub></td><td>(184)</td><td>(N − 1)R/NP<sub>S</sub>(P<sub>R</sub> + 1)</td></tr><tr><td>SKRST/KRF</td><td>P<sub>S</sub> ≥ M<sub>S</sub>, P<sub>R</sub> ≥ M<sub>R</sub></td><td>(172)</td><td></td></tr><tr><td>SKRST/ALS</td><td>P<sub>R</sub>P<sub>S</sub>N ≥ M<sub>R</sub>, P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ M<sub>S</sub>, P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ M<sub>R</sub>M<sub>S</sub></td><td>(71)</td><td>(N − 1)M<sub>S</sub>/NP<sub>S</sub>(P<sub>R</sub> + 1)</td></tr><tr><td>SKRST/ZF</td><td>P<sub>R</sub>P<sub>S</sub>M<sub>D</sub> ≥ M<sub>S</sub></td><td>(71) *</td><td></td></tr></tbody></table>
<table><caption>Table 29. *Cont.*</caption><thead><tr><th>Systems/Receivers</th><th>Necessary Conditions</th><th>Corresp.</th><th>Transmission Rates</th></tr></thead><tbody><tr><td></td><td colspan="3">Combined codings—DF protocol</td></tr><tr><td>STST-MSMKron/KronF</td><td>P<sub>S</sub> ≥ R<sub>1</sub> · · · R<sub>Q</sub>M<sub>S</sub>, P<sub>R</sub> ≥ R<sub>1</sub> · · · R<sub>Q</sub>M<sub>S</sub></td><td>(193)</td><td>(∑<sub>q</sub>N<sub>q</sub>R<sub>q</sub> - Q) / ∏<sub>q</sub>N<sub>q</sub>P<sub>S</sub>(P<sub>R</sub> + 1)</td></tr><tr><td>SKRST-MSMKR/KRF</td><td>P<sub>S</sub> ≥ M<sub>S</sub>, P<sub>R</sub> ≥ M<sub>S</sub></td><td>(194)</td><td>(∑<sub>q</sub>N<sub>q</sub> - Q)M<sub>S</sub> / ∏<sub>q</sub>N<sub>q</sub>P<sub>S</sub>(P<sub>R</sub> + 1)</td></tr></tbody></table>

## 7. Simulation Results

In this section, we present Monte Carlo simulation results to illustrate the performance of the proposed MIMO relay systems and associated semi-blind receivers.

For each system, information symbols are randomly drawn from a unit energy 4-QAM constellation. Channel coefficients are simulated as independent identically distributed (i.i.d.) complex random variables, following a Gaussian distribution with zero-mean and unit variance. The coding matrices/tensors are generated in such a way that each matrix or matrix unfolding to be inverted in the closed-form receiver algorithms is an orthonormal truncated discrete Fourier transform (DFT) matrix in order to simplify the computation of pseudo-inverses, as discussed in Remark 11.

All receiving antennas are assumed to be subject to an additive white Gaussian noise (AWGN), so the noisy received signal tensors are simulated as follows:

$$ \mathcal{X}_{n}^{(R)} = \mathcal{X}^{(R)} + \sqrt{N_0} \mathcal{N}^{(R)} \quad (203) $$

$$ \mathcal{X}_{n}^{(D)} = \mathcal{X}^{(D)} + \sqrt{N_0} \mathcal{N}^{(D)}, \quad (204) $$

where the noise-free tensors $\mathcal{X}^{(R)}$ and $\mathcal{X}^{(D)}$ are defined in Sections 5 and 6, $\mathcal{N}^{(R)}$ and $\mathcal{N}^{(D)}$ represent unit energy noise tensors at the relay and the destination, respectively, and $N_0$ is the noise spectral density. At each Monte Carlo run, $N_0$ is calculated according to the desired signal-to-noise ratio (SNR) value, varying between -10 and 20 dB, and determined as follows:

$$ \text{SNR} = 10 \log_{10} \left( \frac{\| \mathcal{X}^{(D)} \|_{F}^{2}}{N_{0}} \right). \quad (205) $$

The criteria used to evaluate the proposed receivers are the SER and NMSEs of the estimated SR and RD channels and of the received signals tensor $\mathcal{X}^{(D)}$, reconstructed using the parameters estimated at each Monte Carlo run. The corresponding curves are plotted versus the SNR, and the design parameter values are indicated at the top of each figure.

The SER is computed after the projection of the estimated symbols onto the symbol alphabet. The NMSE of estimated channels and reconstructed signals is defined as follows:

$$ \begin{aligned} \text{NMSE}(\mathcal{H}) &= 10 \log_{10} \left( \frac{1}{\text{MC}} \sum_{mc=1}^{\text{MC}} \frac{\| \mathcal{H}_{mc} - \hat{\mathcal{H}}_{mc} \|_{F}^{2}}{\| \mathcal{H}_{mc} \|_{F}^{2}} \right) \\ \text{NMSE}(\mathcal{X}^{(D)}) &= 10 \log_{10} \left( \frac{1}{\text{MC}} \sum_{mc=1}^{\text{MC}} \frac{\| \mathcal{X}_{mc}^{(D)} - \hat{\mathcal{X}}_{mc}^{(D)} \|_{F}^{2}}{\| \mathcal{X}_{mc}^{(D)} \|_{F}^{2}} \right), \end{aligned} \quad (206) $$

where MC denotes the number of Monte Carlo runs. $\mathcal{H}_{mc}$ and $\hat{\mathcal{H}}_{mc}$ represent the channel $\mathcal{H}$ simulated and estimated at the $mc$-th Monte Carlo run, respectively. $\mathcal{X}_{mc}^{(D)}$ and $\hat{\mathcal{X}}_{mc}^{(D)}$ are defined similarly. The performance criteria are averaged over $10^4$ Monte Carlo runs for various system configurations. In addition to the SER and NMSE curves, an assessment of the computational complexity of each receiver is also measured based on the computation time.

To perform a comprehensive evaluation of the considered systems, we follow the methodology summarized in Figure 19, consisting of four main comparisons concerning receiver algorithms, design parameters, relaying protocols, and coding schemes. The STST

<footnote>* Only the condition related to the symbol matrix estimation is considered.</footnote>
and SKRST codings were selected as reference systems, except for in the last set of simulations, where all coding schemes are compared.

<figure><img src="image_21.png" /><figcaption>Figure 19. Plan of simulations for performance comparison.</figcaption></figure>

## 7.1. Comparison of Receiver Algorithms

The first simulations aimed to compare the performance of the closed-form (KronF/KRF) and ALS algorithms used in the semi-blind receivers presented in Table 28 for the STST and SKRST systems. A comparison was also conducted with the corresponding ZF receivers.

In the case of ALS receivers, the convergence is decided when the difference between two successive normalized reconstruction errors becomes smaller than a predefined threshold $\eta = 10^{-5}$, i.e., $|\epsilon_t - \epsilon_{t-1}| \le \eta$ with:

$$ \epsilon_t = \frac{\|\mathcal{X}_n^{(D)} - \hat{\mathcal{X}}_t^{(D)}\|_F^2}{\|\mathcal{X}_n^{(D)}\|_F^2}. \qquad (207) $$

Figure 20 shows the SER versus SNR for the STST and SKRST systems. As expected, the best SER is obtained with ZF receivers due to a perfect knowledge of the channels at the destination. Slightly better performance is observed when ALS receivers are employed, compared to closed-form receivers. However, this improvement strongly depends on the choice of the threshold for convergence. A lower threshold improves the performance but at the cost of a higher computation time.

From the results in Figure 20, we conclude that the STST system outperforms the SKRST system in terms of SER. For example, for a SER of $10^{-3}$, there is a gap of about 8 dB between the STST and SKRST performances. This is due to the use of tensor codings that provide increased diversity compared to matrix codings. Analyzing the signals encoded at the source, in Table 25, we conclude that, for STST, each symbol $s_{n,r}$ is repeated $M_S P_S$ times, while, for SKRST, each symbol $s_{n,m_S}$ is repeated only $P_S$ times. On the other hand, it is worth noting that the STST system requires a higher computation time with quasi the same number of iterations for convergence compared to the SKRST system, as illustrated in Figure 21a,b.

Figure 21a compares the computation times (in seconds) versus SNR, and Figure 21b shows the normalized reconstruction error $\epsilon_t$ versus the number of iterations for four different SNR values. As expected, closed-form and ZF receivers have constant computation times versus SNR, the shortest being obtained with ZF receivers since they only involve the estimation of symbols. Due to the knowledge of coding matrices and tensors, the three-step ALS receivers converge in very few iterations (three iterations for high SNR and
up to six iterations for low SNR). Nevertheless, closed-form receivers require much less calculation than ALS-based ones, the cheapest being the KRF receiver for SKRST.

<figure><img src="image_22.png" /><figcaption>**Figure 20.** SER comparison with different receivers for STST and SKRST.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 21.** Comparison of (a) computation time for ZF, KronF/KRF, and ALS receivers and (b) number of iterations for convergence of ALS receivers for STST and SKRST.</figcaption></figure>

Figure 22 shows the NMSE of the estimated channels $H^{(SR)}$ and $H^{(RD)}$ versus SNR for the ALS and KronF/KRF receivers of the STST and SKRST systems. For both channels, the performance of ALS receivers is very close to that of closed-form receivers. We note that the alternative method proposed in Section 6.2, denoted as KRF2 and KronF2 in Figure 22, which combines both closed-form algorithms in parallel, allows for a significant improvement of the estimate of $H^{(RD)}$.

From this first set of simulation results, we can conclude that ALS and closed-form receivers lead to a gap of approximately 3 dB for a SER of $10^{-2}$, compared to ZF receivers. Additionally, ALS receivers slightly outperform closed-form receivers, penalized by error propagation, at the cost of a higher computation time.
<figure><img src="image_24.png" /><figcaption>**Figure 22.** NMSE of estimated channels with the KronF/KRF and ALS receivers for STST and SKRST: (a) $\hat{H}^{(SR)}$ and (b) $\hat{H}^{(RD)}$.</figcaption></figure>

## 7.2. Impact of Design Parameters and Relay Protocol

The following simulations aimed to evaluate the impact of certain design parameters on the SER performance of STST and SKRST systems using ZF receivers. More precisely, we evaluate the impact of time-spreading lengths and numbers of antennas while keeping the other design parameters fixed. The impact of the number $Q$ of symbol matrices in combined codings STST-MSMKron and SKRST-MSMKR is also evaluated.

### 7.2.1. Impact of Time-Spreading Lengths: $P_S$ and $P_R$

Figure 23 shows the SER versus SNR for different values of time-spreading lengths at the source and relay nodes. We compare the following configurations: ($P_S, P_R$) = (4,4), (4,8), and (8,4). From the simulation results, we deduce that an increase in time-spreading lengths, whether at the source or the relay, leads to a gain in diversity and, therefore, to an improvement in the SER. A comparison of the configurations ($P_S, P_R$) = (4,8) and (8,4) shows that the SER improves more by increasing $P_S$, rather than $P_R$. This is because $P_S$ modifies the system diversity at both the relay and destination, while $P_R$ acts only at the destination. On the other hand, from Table 29, we conclude that an increase in $P_S$ degrades the transmission rate more than an increase in $P_R$.

<figure><img src="image_25.png" /><figcaption>**Figure 23.** Impact of time-spreading lengths with ZF receivers of STST and SKRST.</figcaption></figure>

The above conclusions regarding the impact of time-spreading lengths on SER performance are valid for both the STST and SKRST systems, with, however, a greater impact for STST, whose SER vanishes for an SNR greater than 5 dB.
7.2.2. Impact of Numbers of Antennas: $M_S, M_R, M_T$, and $M_D$

Figure 24 shows the SER versus SNR for different configurations of the number of antennas at the source ($M_S$), the relay ($M_R, M_T$), and the destination ($M_D$). Due to the constraint imposed on the SKRST system, we assume that $M_R = M_T$ for the following configurations: ($M_S, M_R, M_T, M_D$) = (2,2,2,2), (4,2,2,2), (2,4,4,2), and (2,2,2,4), compared using the ZF receivers of STST and SKRST systems.

The (2,2,2,2) configuration serves as a reference to evaluate the impact of an increase in the number of antennas at the different nodes.

With the SKRST system (Figure 24a), increasing the number $M_S$ of antennas at the source leads to a degradation of the SER for all SNR values. This is due to the dimension of the symbol matrix depending on $M_S$ and, therefore, to an increase in the number of symbols to be estimated when $M_S$ is increased without changing the diversity of the system. With the (2,4,4,2) and (2,2,2,4) configurations, we observe some improvement in performance compared to the (2,2,2,2) configuration. In particular, an increase in the number $M_R$ of receive antennas at the relay allows a more significant improvement in the SER than an increase in the number $M_D$ of receiving antennas at the destination because $M_R$ is directly linked to the spatial diversity provided via the coding matrix $C^{(R)}$ at the relay, which is not the case for $M_D$.

For the STST system (Figure 24b), any increase in the numbers of antennas improves the SER. The (2,4,4,2) configuration provides the most significant improvement because $M_R$ and $M_T$ are both involved in the dimension of the coding tensor $C^{(R)}$ at the relay. Comparing the (4,2,2,2) and (2,2,2,4) configurations, we observe that both give nearly the same SER, with a slight advantage when increasing the number $M_S$ of antennas at the source.

<figure><img src="image_26.png" /><figcaption>Figure 24. Impact of numbers of antennas with ZF receivers of (a) SKRST and (b) STST.</figcaption></figure>

<figure><img src="image_27.png" /></figure>

In Figure 25, we compare the DKRSTF, STSTF, and TSTF systems for the following configurations: ($M_T, M_R$) = (2, 2) and (2,4). As discussed in Section 5.4.1, the configuration $M_R > M_T$ is associated with a virtual array at the destination for the DKRSTF system due to space coding at the relay. Note that all three systems exploit frequency diversity, with frequency-selective channels for the STSTF and TSTF systems, while flat-fading channels are assumed for the DKRSTF system. As expected, from the results on Figure 25, we conclude that increasing the number $M_R$ of receive antennas at the relay allows for an improvement of the SER, with the best improvement obtained using DKRSTF. For example, for a SER of $10^{-2}$, the DKRSTF system achieves a performance gain of approximately 8 dB between the two configurations, while the STSTF and TSTF systems provide a maximum
gain of 2 dB. Note also that the ($M_T, M_R$) = (2, 4) configuration with DKRSTF gives the same SER as the ($M_T, M_R$) = (2, 2) configuration with STSTF because the doubling of $M_R$ compensates for the lack of frequency coding at the relay in the case of DKRSTF, while $F = 2$ with STSTF. See the equivalence relationships (142) between the two codings at the relay. The TSTF system provides the best SER, thanks to a greater diversity provided via tensor coding.

<figure><img src="image_28.png" /><figcaption>**Figure 25.** SER comparison for the DKRSTF, STSTF, and TSTF systems with $M_R \ge M_T$.</figcaption></figure>

### 7.2.3. Impact of the Number *Q* of Symbol Matrices in Combined Codings

We now evaluate the impact of the number *Q* of symbol matrices in the MSMKron and MSMKR codings by varying the numbers $N_q$ of data streams for $q \in \langle Q \rangle$, so as to have the same number of symbols to estimate for each value of *Q*.

Figure 26 shows the SER versus SNR obtained with the STST-MSMKron and SKRST-MSMKR systems for the following configurations: ($N_q, Q) \in \{(6,1), (3,2), (2,3)\}, q \in \langle Q \rangle$. From the simulation results, we conclude that increasing *Q* improves the SER, with the best performance obtained for *Q* = 3, leading to an SNR gain of 3 dB with STST-MSMKron and 4 dB with SKRST-MSMKR, for a SER of $10^{-2}$, compared to the case of *Q* = 1. This improvement is due to the fact that each symbol is repeated $N_q^{Q-1}$ times with the SKRST-MSMKR coding and ($N_q R_q$)$^{Q-1}$ times with the STST-MSMKron coding, which induces a greater repetition of each transmitted symbol when *Q* is increased, thanks to the mutual space-time spreading of information symbols provided via the multiple KRPs and KronPs of symbol matrices. The best SER is obtained with the STST-MSMKron coding.

<figure><img src="image_29.png" /><figcaption>**Figure 26.** Impact of the number *Q* of symbol matrices in combined codings with ZF receivers.</figcaption></figure>
### 7.2.4. Impact of Relaying Protocol: AF and DF

In this section, we compare the impact of the AF and DF protocols for the STST and SKRST systems. With the DF protocol, the KronF and KRF receivers are directly deduced from the ones proposed for the combined codings by fixing $Q = 1$. See Sections 6.5 and 6.6.

Figures 27 and 28 respectively depict the SER and NMSE of estimated channels versus SNR. From Figure 27, we observe a clear advantage of using the DF protocol at the relay to improve the SER at the destination. Indeed, estimating the symbols at the relay before their re-encoding and then their transmission to the destination makes it possible to avoid the amplification of the noise contained in the signals received at the relay and directly coded in the case of the AF protocol. Moreover, it is obvious that, whatever protocol is used, the STST system achieves a better SER than the SKRST system. This corroborates the greater efficiency of tensor coding compared to matrix coding.

<figure><img src="image_30.png" /><figcaption>**Figure 27.** Impact of AF/DF protocols on SER performance of STST and SKRST.</figcaption></figure>

From Figures 28a,b, we observe that, for both protocols, the NMSEs of estimated channels obtained with the STST and SKRST systems are nearly identical, although the SKRST system uses more a priori information on the RD channel (a row instead of a single element of $\mathbf{H}^{(\mathrm{RD})}$ with STST). This additional information compensates for the consideration of less diversity with SKRST than with STST.

<figure><img src="image_31.png" /><figcaption>**Figure 28.** Impact of AF/DF protocols on NMSE of estimated channels for STST and SKRST: (a) $\hat{\mathbf{H}}^{(\mathrm{SR})}$ and (b) $\hat{\mathbf{H}}^{(\mathrm{RD})}$.</figcaption></figure>
Moreover, regarding the DF protocol, we observe that the SR and RD channels, which play a symmetrical role in terms of estimation, present the same NMSE. Note that this estimation is then perfectly blind in the sense that the a priori information needed to remove ambiguities only concerns the transmitted symbols.

As expected, and for the same reason as for the SER, from Figure 28a, we conclude that the SR channel is better estimated with the DF protocol than with AF, while, from Figure 28b, we draw the conclusion that the NMSE of the RD channel is smaller with AF than with DF, thanks to a priori information on the RD channel used with AF.

### 7.3. Comparison of Coding Schemes

In this section, we carry out a comparison of all the relay systems considered in this paper, in terms of SER (Figure 29), the NMSE of estimated channels (Figure 30), the NMSE of reconstructed signals, and the computation time (Figure 31), versus SNR. The design parameters are chosen such that the transmission rate is approximately the same for all simulated systems.

<figure><img src="image_32.png" /><figcaption>Figure 29. SER comparison for all considered relay systems.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Figure 30. NMSE of estimated channels for all considered relay systems: (a) $\hat{H}^{(SR)}$ and (b) $\hat{H}^{(RD)}$.</figcaption></figure>

In Figure 29, we observe the best performance when tensor-based codings (STST-MSMKron, TSTF, TST, STSTF, and STST) are used. The best SER is obtained with the combined STST-MSMKron coding, thanks to the multiple Kronecker product of symbol
matrices that induces mutual diversity in the transmitted signals and to the use of the DF protocol.

<figure><img src="image_34.png" /><figcaption>**Figure 31.** Comparison of considered relay systems in terms of (a) NMSE of reconstructed received signals and (b) computation time.</figcaption></figure>

For codings with the AF protocol, a classification can be made according to a continuous improvement in SER performance: (1) STST, (2) STSTF, (3) TST, and (4) TSTF, which corresponds to an increasing diversity associated with an increasing order (from four to seven) of the tensor $\mathcal{X}^{(D)}$ of signals received at the destination.

Comparing the SERs obtained with TST and STSTF codings allows us to conclude that exploiting chip diversity at both the source and relay nodes ($J_S$ and $J_R$ dimensions of $\mathcal{X}^{(D)}$) is more beneficial than exploiting frequency diversity, which introduces only one dimension ($F$) to $\mathcal{X}^{(D)}$.

Regarding matrix-based codings, we draw a similar conclusion for the improvement in SER when increasing signal diversity, leading to the following classification (from worst to best): (1) SKRST, (2) DKRSTF, and (3) SKRST-MSMKR.

Concerning channel estimation, Figures 30a and 30b respectively plot the NMSE of the estimated SR and RD channel matrices/tensors for all the systems considered.

To facilitate the comparative analysis of simulation results in these figures, Table 30 provides the numerical values of the NMSE of estimated channels for an SNR of 0 dB. This table also presents the values of the NMSE of reconstructed signals, shown in Figure 31a.

<table><caption>**Table 30.** Values of NMSE of estimated channels $H^{(SR)}$ and $H^{(RD)}$ and reconstructed signals $\mathcal{X}^{(D)}$ for SNR = 0 dB.</caption><thead><tr><th>Systems/Receivers</th><th>H<sup>(SR)</sup>/H<sup>(SR)</sup></th><th>NMSE<br>H<sup>(RD)</sup>/H<sup>(RD)</sup></th><th>χ<sup>(D)</sup></th></tr></thead><tbody><tr><td>TSTF/KronF</td><td>-12.33</td><td>-22.08</td><td>-12.09</td></tr><tr><td>TST/KronF</td><td>-11.27</td><td>-23.81</td><td>-10.41</td></tr><tr><td>STSTF/KronF</td><td>-8.33</td><td>-16.23</td><td>-8.33</td></tr><tr><td>STST/KronF</td><td>-7.52</td><td>-18.02</td><td>-6.67</td></tr><tr><td>DKRSTF/KRF</td><td>-7.48</td><td>-20.98</td><td>-6.61</td></tr><tr><td>SKRST/KRF</td><td>-6.73</td><td>-18.04</td><td>-5.76</td></tr><tr><td>STST-MSMKron/KronF</td><td>-8.05</td><td>-8.06</td><td>-6.91</td></tr><tr><td>SKRST-MSMKR/KRF</td><td>-9.96</td><td>-9.90</td><td>-8.99</td></tr></tbody></table>

Note that the SR channel is estimated without a priori knowledge, unlike the RD channel for which a coefficient or a row is assumed to be known a priori to remove
ambiguities, which explains a better estimation of the RD channel, except with the STST-MSMKron and SKRST-MSMKR systems for which the use of the DF protocol avoids any a priori knowledge on the channels.

As expected, the best NMSE results are achieved with codings that provide greater diversity. For instance, when comparing the TSTF and TST systems with their simplified versions, STSTF and STST, it is evident that TSTF and TST deliver superior performance.

For the same reason, tensor-based coding systems outperform matrix-based coding systems. It is also important to highlight the significant improvement in RD channel estimation observed with KRF-based receivers (for SKRST and DKRSTF systems) compared to KronF-based receivers (for STST and STSTF systems). This improvement is due to the a priori knowledge of an entire RD channel row required to remove ambiguities with KRF-based receivers, whereas KronF-based receivers only require a priori knowledge of a single channel coefficient, as detailed in Equations (200) and (197), respectively.

Finally, Figure 31 compares the performance of all relay systems in terms of the NMSE of reconstructed received signals and computation time. It is worth noting that the reconstruction of the signals received at the destination using symbol and channel estimates provides a meaningful measure to compare the effectiveness of the proposed semi-blind receivers.

In Figure 31a, we observe that the best performance is obtained with the TSTF system, with a SNR gain of about 2 dB compared to the TST system which gives the second best performance, for any SNR value. This better performance is achieved at the cost of a higher computation time, as shown in Figure 31b. In general, as already mentioned, we observe that the lowest performance is obtained with matrix codings that offer less diversity than tensor codings, inducing a lower computational complexity and, therefore, a lower computation time.

Comparing the results in Figure 31b and Table A1 in Appendix D makes the difference in computational cost even more evident. For instance, when comparing STST and SKRST systems which both generate fourth-order tensors for received signals, we observe that the complexity of the receiver STST/KronF is $O(NM^2PR) + O(M^4)$, while for SKRST/KRF, it is $O(NM^2P) + O(M^3)$, which corroborates the higher computational cost of STST/KronF, as shown in Figure 31b. A similar conclusion can be drawn for the comparison of STSTF/KronF and DKRSTF/KRF receivers. For the STST-MSMKron/KronF receiver, the complexity increases with a multiplicative factor $R^Q$ which induces an exponential growth of the complexity as the number Q of symbol matrices increases in the coding scheme. This highlights the trade-off between computational cost and SER performance that improves as Q increases.

It should be noted that the computational costs depend more on the number of antennas than on other design parameters, with, in particular, $M^4$ and $M^5$ terms for the complexities of the KronF and ALS algorithms, respectively. This can become problematic in the case of massive MIMO systems involving a large number of antennas. For such systems, optimizing the computational cost will be crucial to avoid possible real-time difficulties.

Comparing the results in Figure 31a and Table 30 with those in Figure 29, we observe that the NMSE of reconstructed signals follows the same improvement as the SER when the system diversity is increased with the AF protocol, which is not the case of combined codings due to a degradation in channel estimation with the DF protocol. Regarding the robustness to noise, Table 31 provides the SNR thresholds from which the systems achieve the desired performance in terms of SER or NMSE of the reconstructed signals, deduced, respectively, from Figures 29 and 31a. Two different values of SER and NMSE are considered. For example, for a fixed SER of $10^{-2}$, the STST-MSMKron system is the most robust to noise, achieving the desired SER performance from an SNR level of -5 dB. On the other hand, for the NMSE of the reconstructed signals set to -10 dB, the TSTF system is the most robust, with the required NMSE performance obtained from an SNR level of -2 dB. Table 31 highlights the systems with the best (green) and least (red) robustness to noise relative to the SNR levels required to achieve the desired performance. This analysis
provides key information for selecting the system based on the desired balance between performance and noise tolerance. Note that the absence of an SNR threshold in Table 31 means that the corresponding performance is achieved for any SNR value greater than the largest value for which the SER is non-zero. For example, with the TST/KronF receiver, for any SNR value greater than 0 dB, the SER is zero.

<table><caption>**Table 31.** SNR thresholds for desired SER and NMSE (of reconstructed signals).</caption><thead><tr><th></th><th>TSTF<br>KronF</th><th>TST<br>KronF</th><th>STSTF<br>KronF</th><th>STST<br>KronF</th><th>DKRSTF<br>KRF</th><th>SKRST<br>KRF</th><th>STST-MSMKron<br>KronF</th><th>SKRST-MSMKR<br>KRF</th></tr></thead><tbody><tr><td rowspan="2">SER</td><td>10<sup>-2</sup></td><td>-4 dB</td><td>-1 dB</td><td>0 dB</td><td>2 dB</td><td>4 dB</td><td>6 dB</td><td>-5 dB</td></tr><tr><td>10<sup>-3</sup></td><td>-2 dB</td><td>-</td><td>-</td><td>4 dB</td><td>9 dB</td><td>12 dB</td><td>5 dB</td></tr><tr><td rowspan="2">NMSE</td><td>-10 dB</td><td>-2 dB</td><td>0 dB</td><td>1 dB</td><td>3 dB</td><td>4 dB</td><td>3 dB</td><td>1 dB</td></tr><tr><td>-20 dB</td><td>7 dB</td><td>9 dB</td><td>11 dB</td><td>12 dB</td><td>13 dB</td><td>12 dB</td><td>11 dB</td></tr></tbody></table>

To complement the previous comparison and help the user achieve the best trade-off between performance and complexity, taking into account the constraints imposed by the necessary identifiability conditions (NIC) and the required a priori knowledge (AK), Table 32 summarizes the characteristics of each system and the performance of associated semi-blind receivers, with the following criteria: (i) diversities, (ii) assumption of the channels (flat fading (FF) versus frequency selective fading (FSF)), and (iii) performance in terms of the NIC, AK, SER, NMSE of estimated channels, and computation time (CT). Performance is compared using positive signs (+) for advantages and negative signs (−) for disadvantages.

<table><caption>**Table 32.** Comparison of systems' characteristics and receivers' performance.</caption><thead><tr><th rowspan="2">Systems/Receivers</th><th colspan="4">Diversities*</th><th colspan="2">Channels</th><th rowspan="2">NIC</th><th rowspan="2">AK</th><th colspan="3">Performance</th></tr><tr><th>M</th><th>P</th><th>J</th><th>F</th><th>FF</th><th>FSF</th><th>SER</th><th>H<sup>(SR)</sup></th><th>H<sup>(RD)</sup></th><th>CT</th></tr></thead><tbody><tr><td>TSTF/KronF</td><td>+</td><td>+</td><td>+</td><td>+</td><td></td><td>+</td><td>--</td><td>--</td><td>+++</td><td>+++</td><td>+++</td><td>---</td></tr><tr><td>TST/KronF</td><td>+</td><td>+</td><td>+</td><td></td><td>+</td><td></td><td>--</td><td>--</td><td>++</td><td>+++</td><td>+++</td><td>---</td></tr><tr><td>STSTF/KronF</td><td>+</td><td>+</td><td></td><td>+</td><td></td><td>+</td><td>---</td><td>--</td><td>++</td><td>++</td><td>++</td><td>--</td></tr><tr><td>STST/KronF</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>---</td><td>--</td><td>++</td><td>+</td><td>++</td><td>--</td></tr><tr><td>STST/ALS</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>--</td><td>--</td><td>++</td><td>+</td><td>++</td><td>---</td></tr><tr><td>DKRSTF/KRF</td><td>+</td><td>+</td><td></td><td>+</td><td>+</td><td></td><td>--</td><td>---</td><td>+</td><td>+</td><td>+++</td><td>--</td></tr><tr><td>SKRST/KRF</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>--</td><td>---</td><td>+</td><td>+</td><td>++</td><td>--</td></tr><tr><td>SKRST/ALS</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>--</td><td>---</td><td>+</td><td>+</td><td>++</td><td>---</td></tr><tr><td>STST-MSMKron/KronF</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>---</td><td>--</td><td>+++</td><td>+</td><td>+</td><td>--</td></tr><tr><td>SKRST-MSMKR/KRF</td><td>+</td><td>+</td><td></td><td></td><td>+</td><td></td><td>---</td><td>--</td><td>++</td><td>++</td><td>+</td><td>--</td></tr></tbody></table>

From Table 32, we can draw the following conclusions:

* **Diversity:** All coding schemes used at both the source and the relay include space-time (ST) coding. TST coding incorporates additional chip diversity, while STSTF and DKRSTF take into account frequency diversity. TSTF is the most comprehensive coding simultaneously exploiting the space-time-chip-frequency diversities (*M*, *P*, *J*, *F*), resulting in the best performance.

* **Channels:** Systems based on TSTF and STSTF codings consider frequency-selective fading channels, i.e., third-order channel tensors, while all other systems assume flat fading channels, i.e., channel matrices. It is worth noting that, despite a larger number of channel coefficients to be estimated, the best SER is obtained with TSTF coding due to the exploitation of all diversities, at the cost of a higher computation time.

* **Necessary identifiability conditions:** These conditions are related to the uniqueness of the pseudo-inverses of coding matrices or matrix unfoldings of coding tensors, depending on both the coding and the receiver type (ALS/KronF/KRF). ALS receivers are the

<footnote>* M = antennas, P = time-spreading, J = chip, F = frequency.</footnote>
least restrictive, i.e., the most flexible in the choice of design parameters. TSTF and TST are less restrictive than their simplified versions (STSTF and STST), thanks to the incorporation of chip diversity. Matrix codings entail fewer constraints than tensor codings. Note that frequency diversity has no impact on the identifiability conditions.

*   *A priori knowledge:* With matrix codings (SKRST and DKRSTF) leading to CP models for received signals, associated semi-blind receivers use the KRF algorithm, which requires knowledge of an entire row of symbol and RD-channel matrices to remove ambiguities. On the other hand, systems based on tensor codings are modeled using Tucker or generalized Tucker models for which the semi-blind receivers use the KronF algorithm, which requires knowledge of only one element of the symbol and RD-channel matrices. With the DF protocol, only a few pilot symbols are needed, with receivers performing blind channel estimation in the sense that no a priori channel information is needed. In particular, the STST-MSMKron system requires knowledge of only one pilot symbol in each symbol matrix.

*   *SER and NMSE of estimated channels:* These performances have already been commented in detail previously. It is worth highlighting that the TSTF and STST-MSMKron systems provide the best performance in terms of symbol estimation. Regarding channel estimation, the performance of the TSTF and TST systems are particularly remarkable.

*   *Computation time:* As expected, iterative receivers based on the ALS algorithm require the highest computation times due to their iterative nature, which leads to a refinement in the estimation of unknown parameters. In general, the computation time reflects the amount of diversity taken into account. Higher diversity leads to higher-order tensors, which induce higher computation times. Matrix coding systems perform the best in terms of computational complexity.

In summary, the numerical simulations performed reveal a clear hierarchy in the performances of relay systems, depending on diversity management (via the choice of the coding scheme), relaying protocol and channel characteristics, which induce a more or less complex receiver and guide the choice of coding in order to satisfy a desired trade-off for a given application. Systems incorporating frequency and/or chip diversity present superior overall performance with less restrictive identifiability conditions using chip diversity and less a priori knowledge with KronF type receivers. Matrix-based codings are more suitable for low complexity requirements, but for better performance, tensor codings such as TST, TSTF, or STST-MSMKron are more appropriate, inducing higher computational costs. Moreover, these three coding schemes offer the best balance between SER and NMSE, in addition to exhibiting greater robustness to noise for different SNR levels.

## 8. Conclusions and Perspectives

In this paper, we have first provided an in-depth overview of tensor models, with a detailed description of nested tensor decompositions which are very useful to represent cooperative communication systems. Initially proposed for fourth-order tensors, nested decompositions have been generalized to higher-order tensors using graph-based representations and a novel interpretation in terms of cascading tensor models. Some new nested models have been introduced to represent relay systems using tensor codings. Several parameter estimation methods have been described, with a particular focus on closed-form algorithms based on KRF and KronF methods when some factors of nested models are a priori known, as is the case with coding matrices/tensors in the context of relay systems.

In a second part, a thorough survey of cooperative communication systems, and more particularly of one-way two-hop MIMO relay systems, was carried out using various coding strategies, with the aim of showing how these strategies impact the tensor model of signals received at the destination node. Capitalizing on this tensor model, two classes of semi-blind receivers were proposed to jointly estimate the transmitted information symbols and the individual communication channels: iterative receivers based on ALS and closed-form ones using KRF and KronF methods, which consist of rank-one matrix or tensor approximations. Extensive Monte Carlo simulation results were presented to
analyze and compare the impact of the coding strategy, of the choice of design parameters, of the relay protocol and of the type of receiver on the system performance in terms of SER, channel estimate NMSE, and computation time. Simulation results show that the best SER performance is achieved with the most general TSTF tensor coding at the price of higher computational complexity.

Several extensions of this work can be considered, such as multi-user and multi-relay scenarios, two-way relaying case, and power and resource allocation, as well as three-dimensional (3D) polarized channels that combine dual-polarized antenna arrays with double-directional channels and time-varying mmWave channels characterized by an inherent sparse and low-rank structure that allows for the employment of compressed sensing methods. In terms of cooperative communications, combinations of relay-, IRS-, and UAV-aided communication systems are particularly interesting to study for future 6G wireless networks. We believe the tensor codings and nested tensor models highlighted in this paper for two-hop relay systems can be very useful in designing such new combined assisted communication networks.

The development of ISAC systems opens the way for new research problems for which sensing and cooperative communication systems are combined in a general framework allowing for estimates of target physical parameters, e.g., angles of arrival/departure (AoA/AoD), time delays, and Doppler shift, at a sensing base station, and communication channels and information symbols at an user equipment. The nested tensor models studied in this paper are also very attractive for modeling such ISAC systems.

**Funding:** This work was supported in part by the CAPES-COFECUB bilateral research project number Ma 985-23. The work of Danilo S. Rocha was supported in part by CAPES—Finance Code 001.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

## Appendix A. List of Acronyms

<table><thead><tr><th>Acronyms</th><th>Definitions</th><th>References</th></tr></thead><tbody><tr><td colspan="3" style="text-align: center; font-weight: bold;">Tensor models</td></tr><tr><td>TD</td><td>Tucker decomposition</td><td>[7]</td></tr><tr><td>TTD</td><td>Tensor train decomposition</td><td>[44]</td></tr><tr><td>NTD</td><td>Nested TD</td><td>[41]</td></tr><tr><td>CNTD</td><td>Coupled NTD</td><td>[42]</td></tr><tr><td>DCNTD</td><td>Doubly coupled NTD</td><td>[43]</td></tr><tr><td>GTD</td><td>Generalized TD</td><td>[38]</td></tr><tr><td>NGTD</td><td>Nested GTD</td><td>—</td></tr><tr><td>CPD</td><td>Canonical polyadic decomposition</td><td>[34]</td></tr><tr><td>NCPD</td><td>Nested CPD</td><td>[39,40]</td></tr><tr><td>CCPD</td><td>Coupled CPD</td><td>[101,102]</td></tr><tr><td>CNCPD</td><td>Coupled NCPD</td><td>—</td></tr><tr><td>DCNCPD</td><td>Doubly coupled NCPD</td><td>[43]</td></tr><tr><td>PARALIND</td><td>PARAFAC with linearly dependent loadings</td><td>[103]</td></tr><tr><td>CONFAC</td><td>Constrained factor decomposition</td><td>[35,36]</td></tr><tr><td>BTD</td><td>Block-term decomposition</td><td>[104]</td></tr><tr><td>PARAFAC</td><td>Parallel factors decomposition</td><td>[8,47]</td></tr><tr><td>PARATUCK</td><td>PARAFAC-Tucker decomposition</td><td>[51–53]</td></tr><tr><td>CNTD-CPD</td><td>Coupled nested TD-CPD</td><td>[97]</td></tr></tbody></table>
<table><thead><tr><th>Acronyms</th><th>Definitions</th><th>References</th></tr></thead><tbody><tr><td colspan="3"><strong>Codings</strong></td></tr><tr><td>STST</td><td>Simplified tensor space-time</td><td>[41]</td></tr><tr><td>STSTF</td><td>Simplified tensor space-time-frequency</td><td>-</td></tr><tr><td>TST</td><td>Tensor space-time</td><td>[37]</td></tr><tr><td>TSTF</td><td>Tensor space-time frequency</td><td>[38]</td></tr><tr><td>SKRST</td><td>Simplified Khatri-Rao space-time</td><td>[40,53]</td></tr><tr><td>DKRSTF</td><td>Double Khatri-Rao space-time frequency</td><td>[39]</td></tr><tr><td>MSMKR</td><td>Multiple symbol matrices Khatri-Rao (product)</td><td>[98]</td></tr><tr><td>MSMKron</td><td>Multiple symbol matrices Kronecker (product)</td><td>[97]</td></tr><tr><td colspan="3"><strong>Algorithms</strong></td></tr><tr><td>ALS</td><td>Alternating least squares</td><td></td></tr><tr><td>KRF</td><td>Khatri-Rao factorization</td><td></td></tr><tr><td>KronF</td><td>Kronecker factorization</td><td></td></tr><tr><td>ZF</td><td>Zero-forcing</td><td></td></tr><tr><td colspan="3"><strong>Other acronyms</strong></td></tr><tr><td>AF</td><td>Amplify-and-forward</td><td></td></tr><tr><td>DF</td><td>Decode-and-forward</td><td></td></tr><tr><td>SISO</td><td>Single-input single-output</td><td></td></tr><tr><td>MISO</td><td>Multiple-input single-output</td><td></td></tr><tr><td>MIMO</td><td>Multiple-input multiple-output</td><td></td></tr><tr><td>CDMA</td><td>Code division multiple access</td><td></td></tr><tr><td>OFDM</td><td>Orthogonal frequency-division multiplexing</td><td></td></tr><tr><td>IRS</td><td>Intelligent reflecting surface</td><td></td></tr><tr><td>UAV</td><td>Unmanned aerial vehicular</td><td></td></tr><tr><td>SER</td><td>Symbol error rate</td><td></td></tr><tr><td>NMSE</td><td>Normalized mean square error</td><td></td></tr></tbody></table>

## Appendix B. KRF Method

In this Appendix, we present the Khatri-Rao factorization method, denoted as KRF, to estimate the matrix factors of a Khatri-Rao product, abbreviated as KRP, based on the minimization of an LS criterion, first proposed in [49].

Let us consider the case of the noisy KRP: $Y = A \diamond B + E$ of two matrices $A = [a_1, \dots, a_R] \in \mathbb{K}^{I \times R}$ and $B = [b_1, \dots, b_R] \in \mathbb{K}^{I \times R}$, where $E$ is an additive noise matrix due to modeling error and measurement noise, with the following LS criterion to be minimized:

$$ \min_{A,B} \|Y - A \diamond B\|_F^2. \qquad (A1) $$

By vectorizing $Y$, this criterion can be rewritten as follows:

$$ \min_{A,B} \|\mathrm{vec}(Y) - \mathrm{vec}(A \diamond B)\|_2^2 = \min_{\substack{a_r, b_r \\ r \in \langle R \rangle}} \sum_{r=1}^{R} \|y_r - a_r \diamond b_r\|_2^2. \qquad (A2) $$

Since each term in this sum can be minimized separately, the column vectors $a_r \in \mathbb{K}^I$ and $b_r \in \mathbb{K}^J$ are estimated by minimizing $\min_{a_r, b_r} \|y_r - a_r \diamond b_r\|_2^2$.

The idea behind the KRF method is to form a rank-one matrix by unvectorizing each KRP $y_r$ for $r \in \langle R \rangle$, as

$$ Y_r \triangleq \mathrm{unvec}(y_r) = \mathrm{unvec}(a_r \diamond b_r) = b_r a_r^T = b_r \circ a_r \in \mathbb{K}^{J \times I} \qquad (A3) $$

and to calculate the rank-one approximation of this matrix using the reduced SVD algorithm. The vectors $(a_r, b_r)$ are determined via a simple identification of the formed rank-one matrix
with the left and right singular vectors associated with the largest singular value. The LS criterion is then replaced with a rank-one matrix approximation:

$$ \min_{\mathbf{a}_r, \mathbf{b}_r} \|\mathbf{Y}_r - \mathbf{b}_r \mathbf{a}_r^T\|_F^2 = \min_{\mathbf{a}_r, \mathbf{b}_r} \|\mathbf{Y}_r - \mathbf{b}_r \circ \mathbf{a}_r\|_F^2, \quad r \in \langle R \rangle, \quad (A4) $$

where $\circ$ denotes the outer product. The vectors $\mathbf{a}_r$ and $\mathbf{b}_r$ are estimated by computing the rank-one reduced SVD of $\mathbf{Y}_r$:

$$ \mathbf{Y}_r = \sigma_r^{(1)} \mathbf{u}_r^{(1)} (\mathbf{v}_r^{(1)})^H, \quad (A5) $$

where $\sigma_r^{(1)}$ denotes the largest singular value, and $\mathbf{u}_r^{(1)}$ and $\mathbf{v}_r^{(1)}$ are the left and right singular vectors associated with $\sigma_r^{(1)}$, respectively. Identifying the reduced SVD (A5) with $\mathbf{b}_r \mathbf{a}_r^T$ leads to the following estimates [49]:

$$ \hat{\mathbf{a}}_r = \sqrt{\sigma_r^{(1)} \mathbf{v}_r^{(1)*}}, \quad \hat{\mathbf{b}}_r = \sqrt{\sigma_r^{(1)} \mathbf{u}_r^{(1)}}. \quad (A6) $$

Note:

* The $R$ columns of $\mathbf{A}$ and $\mathbf{B}$ can be estimated in parallel by computing the rank-one approximation of $R$ matrices;

* Each vector, $\mathbf{a}_r$ and $\mathbf{b}_r$, $r \in \langle R \rangle$, is only estimated up to a scaling factor, since $(\lambda_r \mathbf{a}_r) \diamond (\frac{1}{\lambda_r} \mathbf{b}_r) = \mathbf{a}_r \diamond \mathbf{b}_r$ for every $\lambda_r \in \mathbb{K}$. To eliminate this scaling ambiguity, we need to know one component of one of these two vectors. For example, if we assume that the first component $(\mathbf{a}_r)_1$ is equal to 1, then $\lambda_r = \frac{1}{(\hat{\mathbf{a}}_r)_1}$, and the ambiguity is removed as follows:

$$ \hat{\mathbf{a}}_r = \frac{1}{(\hat{\mathbf{a}}_r)_1} \hat{\mathbf{a}}_r, \quad \hat{\mathbf{b}}_r = (\hat{\mathbf{a}}_r)_1 \hat{\mathbf{b}}_r, \quad (A7) $$

where $\hat{\mathbf{a}}_r$ and $\hat{\mathbf{b}}_r$ are defined in (A6). We can conclude that the factors $\mathbf{A}$ and $\mathbf{B}$ of a Khatri-Rao product are estimated without scaling ambiguity if we have a priori knowledge of one row of $\mathbf{A}$ or $\mathbf{B}$. This follows from the fact that $\mathbf{A}\mathbf{\Lambda} \diamond \mathbf{B}\mathbf{\Lambda}^{-1} = \mathbf{A} \diamond \mathbf{B}$ for any non-singular matrix $\mathbf{\Lambda} = \mathrm{diag}(\lambda_1, \dots, \lambda_R) \in \mathbb{K}^{R \times R}$.

We now present a generalization of the KRF method to the case of a multiple KRP [33]:

$$ \min_{\mathbf{A}^{(n)}, n \in \langle N \rangle} \|\mathbf{Y} - \underset{n=1}{\overset{N}{\diamond}} \mathbf{A}^{(n)}\|_F^2, \quad (A8) $$

with $\mathbf{A}^{(n)} = [\mathbf{a}_1^{(n)}, \dots, \mathbf{a}_R^{(n)}] \in \mathbb{K}^{I_n \times R}$, $n \in \langle N \rangle$. This generalization is based on the rank-one approximation of $R$ tensors of order $N$ constructed from the columns of the factors $\mathbf{A}^{(n)}$. Let us consider the column $\mathbf{y}_r = \underset{n=1}{\overset{N}{\diamond}} \mathbf{a}_r^{(n)} \in \mathbb{K}^{I_1 \cdots I_N}$, for $r \in \langle R \rangle$, and the associated rank-one tensor $\mathcal{X}_r = \underset{n=1}{\circ} \mathbf{a}_r^{(n)} \in \mathbb{K}^{I_1 \cdots I_N}$ defined by the transformation $\mathbf{y}_r \to \mathcal{X}_r$ such as:

$$ (\mathcal{X}_r)_{i_1 \cdots i_N} = (\mathbf{y}_r)_{\overline{i_1 \cdots i_N}}, \quad (A9) $$

where

$$ \overline{i_1 \cdots i_N} \triangleq i_N + \sum_{n=1}^{N-1} (i_n - 1) \prod_{k=n+1}^{N} I_k. \quad (A10) $$

As for the case of the KRP of two vectors, the LS criterion (A8) is replaced with:

$$ \min_{\mathbf{a}_r^{(n)}, n \in \langle N \rangle} \|\mathbf{y}_r - \underset{n=1}{\overset{N}{\diamond}} \mathbf{a}_r^{(n)}\|_F^2 = \min_{\mathbf{a}_r^{(n)}, n \in \langle N \rangle} \|\mathcal{X}_r - \underset{n=1}{\circ} \mathbf{a}_r^{(n)}\|_F^2, \quad r \in \langle R \rangle. \quad (A11) $$

The columns $\mathbf{a}_r^{(n)}$ are then determined in calculating the rank-one approximation of the tensor $\mathcal{X}_r$ by means of the truncated higher-order SVD (THOSVD) algorithm [48]. Like
for a Khatri–Rao product of two vectors, the R columns of the factors $\mathbf{A}^{(n)}$ can be estimated in parallel.

Noting that $\bigcirc_{n=1}^{N} \lambda_n \mathbf{a}_r^{(n)} = \bigcirc_{n=1}^{N} \mathbf{a}_r^{(n)}$ for any $\lambda_n$ such that $\prod_{n=1}^{N} \lambda_n = 1$, one concludes that each estimated column vector $\mathbf{a}_r^{(n)}$ is unique up to a scalar ambiguity. To eliminate these ambiguities, one component of $N-1$ columns among the $N$ vectors $\mathbf{a}_r^{(n)}, n \in \langle N \rangle$, must be known.

## Appendix C. KronF Method

We now consider the Kronecker factorization method, denoted as KronF, to estimate the matrix factors of a multiple Kronecker product, denoted as KronP, in minimizing the following LS cost function:

$$ \min_{\mathbf{A}^{(n)}, n \in \langle N \rangle} \|\mathbf{Y} - \bigotimes_{n=1}^{N} \mathbf{A}^{(n)}\|_{F}^{2} \qquad (A12) $$

with $\mathbf{A}^{(n)} \in \mathbb{K}^{I_n \times J_n}, n \in \langle N \rangle$.

Inspired by the approach proposed in [105] to solve the problem of approximating a given matrix by a simple Kronecker product using a SVD-based rank-one matrix approximation, a generalization to a multiple Kronecker product is obtained in constructing a rank-one tensor from $\mathbf{Y}$, defined as the outer product of vectorized forms $\text{vec}(\mathbf{A}^{(n)}) \in \mathbb{K}^{J_n I_n}$ of the matrices $\mathbf{A}^{(n)}$:

$$ \mathcal{X} = \bigcirc_{n=1}^{N} \text{vec}(\mathbf{A}^{(n)}) \in \mathbb{K}^{I_1 I_1 \times I_2 I_2 \times \cdots \times I_N I_N}. \qquad (A13) $$

The problem of estimating the factors $\mathbf{A}^{(n)}$ can be solved in the sense of minimizing the following LS criterion [33]:

$$ \min_{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}} \|\mathbf{Y} - \bigotimes_{n=1}^{N} \mathbf{A}^{(n)}\|_{F}^{2} = \min_{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}} \|\mathcal{X} - \bigcirc_{n=1}^{N} \text{vec}(\mathbf{A}^{(n)})\|_{F}^{2}. \qquad (A14) $$

This minimization amounts to finding the best rank-one approximation for the tensor $\mathcal{X}$ using the THOSVD method. This approximation gives an estimate of the vectorized forms $\text{vec}(\mathbf{A}^{(n)})$, from which it is easy to deduce an estimate of the factors $\mathbf{A}^{(n)}$ via a simple unvectorization operation. Since, for $\tilde{\mathbf{A}}^{(n)} = \lambda_n \mathbf{A}^{(n)}, n \in \langle N \rangle$, we have $\bigcirc_{n=1}^{N} \text{vec}(\tilde{\mathbf{A}}^{(n)}) = \bigcirc_{n=1}^{N} \text{vec}(\mathbf{A}^{(n)})$ if $\prod_{n=1}^{N} \lambda_n = 1$, we deduce that each estimate $\text{vec}(\mathbf{A}^{(n)})$ is subject to a scalar scaling ambiguity $\lambda_n$ requiring knowledge of one component to eliminate this ambiguity.

## Appendix D. Complexity of the Receiver Algorithms

The computational complexity of the proposed semi-blind receivers is evaluated using Big-O notation, considering the most computationally expensive operations. For KronF/KRF-based methods, the dominant cost comes from SVD-based rank-1 approximations, while for ALS methods, it is associated with pseudo-inverse computations. The computational complexities of the proposed receivers are summarized in Table A1, assuming $P_S = P_R = P$, $J_S = J_R = J$, $M_S = M_R = M_T = M_D = M$, $N_1 = \dots = N_Q = N$, and $R_1 = \dots = R_Q = R$.

It is important to note that, for ALS receivers, the complexity is evaluated for a single iteration. The total computational cost is consequently dependent on the number of iterations required for convergence.

In this analysis, we consider that the computational complexity of the KronF and KRF algorithms, when applied to an $I_1 \times I_2$ matrix, is $O(I_1 I_2)$, corresponding to the computation of a rank-1 SVD. For the ALS algorithms, the complexity of the pseudo-inverse calculation is $O(I_1 I_2^2)$ when $I_1 \ge I_2$, or $O(I_1^2 I_2)$ when $I_1 \le I_2$.
<table><caption>**Table A1.** Complexity of the receivers.</caption><thead><tr><th>System/Receiver</th><th>Matrix Dimension</th><th>Complexity</th></tr></thead><tbody><tr><td colspan="3"><strong>Tensor-based codings-AF protocol</strong></td></tr><tr><td>TSTF/KronF</td><td>S ⊗ bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>J<sub>R</sub>×M<sub>S</sub>(f)) ⇒ F(NM<sub>D</sub>P<sub>R</sub>I<sub>R</sub> × RM<sub>S</sub>)<br>bdiag<sub>f</sub>(H<sup>(RD)</sup><sub>M<sub>D</sub>×M<sub>T</sub>(f) × H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub>(f)</sub>) ⇒ F(M<sub>D</sub>M<sub>S</sub> × M<sub>T</sub>M<sub>R</sub>)</td><td>O(FNM<sup>2</sup>PJR)<br>O(FM<sup>4</sup>)</td></tr><tr><td>TST/KronF</td><td>S ⊗ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>J<sub>R</sub>×M<sub>S</sub><sup>T</sup></sub> ⇒ NM<sub>D</sub>P<sub>R</sub>J<sub>R</sub> × RM<sub>S</sub><br>H<sup>(RD)</sup> ⊗ H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub><sup>T</sup></sub> ⇒ M<sub>D</sub>M<sub>S</sub> × M<sub>T</sub>M<sub>R</sub></td><td>O(NM<sup>2</sup>PJR)<br>O(M<sup>4</sup>)</td></tr><tr><td>STSTF/KronF</td><td>S ⊗ bdiag<sub>f</sub>(H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub>(f)) ⇒ F(NM<sub>D</sub>P<sub>R</sub> × RM<sub>S</sub>)<br>bdiag<sub>f</sub>(H<sup>(RD)</sup><sub>M<sub>D</sub>×M<sub>T</sub>(f) × H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub>(f)</sub>) ⇒ F(M<sub>D</sub>M<sub>S</sub> × M<sub>T</sub>M<sub>R</sub>)</td><td>O(FNM<sup>2</sup>PR)<br>O(FM<sup>4</sup>)</td></tr><tr><td>STST/KronF</td><td>S ⊗ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub><sup>T</sup></sub> ⇒ NM<sub>D</sub>P<sub>R</sub> × RM<sub>S</sub><br>H<sup>(RD)</sup> ⊗ H<sup>(SR)</sup><sub>M<sub>S</sub>×M<sub>R</sub><sup>T</sup></sub> ⇒ M<sub>D</sub>M<sub>S</sub> × M<sub>T</sub>M<sub>R</sub></td><td>O(NM<sup>2</sup>PR)<br>O(M<sup>4</sup>)</td></tr><tr><td>STST/ALS</td><td>[(I<sub>P<sub>S</sub></sub> ⊗ (Ĥ<sup>(RD)</sup><sub>t−1</sub> ⊗ I<sub>P<sub>R</sub></sub>)C<sup>(R)</sup><sub>M<sub>T</sub>P<sub>R</sub>×M<sub>R</sub></sub>Ĥ<sup>(SR)</sup><sub>t−1</sub>)C<sup>(S)</sup><sub>P<sub>S</sub>M<sub>S</sub>×R</sub>]<sup>†</sup> ⇒ P<sub>S</sub>M<sub>D</sub>P<sub>R</sub> × R<br>[(I<sub>P<sub>R</sub></sub> ⊗ (I<sub>P<sub>S</sub></sub> ⊗ Ŵ<sub>t</sub>)C<sup>(S)</sup><sub>P<sub>S</sub>R×M<sub>S</sub></sub>(Ĥ<sup>(SR)</sup><sub>t−1</sub>)<sup>T</sup>)C<sup>(R)</sup><sub>P<sub>R</sub>M<sub>R</sub>×M<sub>T</sub></sub>]<sup>†</sup> ⇒ P<sub>R</sub>P<sub>S</sub>N × M<sub>T</sub><br>[(I<sub>P<sub>S</sub></sub> ⊗ Ŵ<sub>t</sub>) ⊗ (Ĥ<sup>(RD)</sup><sub>t</sub> ⊗ I<sub>J<sub>T</sub></sub>)(C<sup>(S)</sup><sub>P<sub>S</sub>R×M<sub>S</sub></sub> ⊗ C<sup>(R)</sup><sub>M<sub>T</sub>P<sub>R</sub>×M<sub>R</sub></sub>)]<sup>†</sup> ⇒ P<sub>S</sub>NM<sub>D</sub>P<sub>R</sub> × M<sub>R</sub>M<sub>S</sub></td><td>O(MP<sup>2</sup>R<sup>2</sup>)<br>O(NM<sup>2</sup>P<sup>2</sup>)<br>O(NM<sup>5</sup>P<sup>2</sup>)</td></tr><tr><td colspan="3"><strong>Matrix-based codings-AF protocol</strong></td></tr><tr><td>DKRSTF/KRF</td><td>V<sup>(S)</sup><sub>FN×M<sub>S</sub></sub> ♦ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub></sub> ⇒ FNM<sub>D</sub>P<sub>R</sub> × M<sub>S</sub><br>B ♦ H<sup>(SR)</sup><sub>M<sub>D</sub>M<sub>S</sub>×M<sub>R</sub><sup>T</sup></sub> ⇒ M<sub>D</sub>M<sub>S</sub> × M<sub>R</sub></td><td>O(FNM<sup>2</sup>P)<br>O(M<sup>3</sup>)</td></tr><tr><td>SKRST/KRF</td><td>S ♦ H<sup>(SRD)</sup><sub>M<sub>D</sub>P<sub>R</sub>×M<sub>S</sub><sup>T</sup></sub> ⇒ NM<sub>D</sub>P<sub>R</sub> × M<sub>S</sub><br>H<sup>(RD)</sup> ♦ H<sup>(SR)</sup><sub>M<sub>D</sub>M<sub>S</sub>×M<sub>R</sub><sup>T</sup></sub> ⇒ M<sub>D</sub>M<sub>S</sub> × M<sub>R</sub></td><td>O(NM<sup>2</sup>P)<br>O(M<sup>3</sup>)</td></tr><tr><td>SKRST/ALS</td><td>[C<sup>(R)</sup> ♦ (C<sup>(S)</sup> ♦ Ŵ<sub>t−1</sub>) (Ĥ<sup>(SR)</sup><sub>t−1</sub>)<sup>T</sup>]<sup>†</sup> ⇒ P<sub>R</sub>P<sub>S</sub>N × M<sub>R</sub><br>[C<sup>(S)</sup> ♦ (Ĥ<sup>(RD)</sup><sub>t</sub> ♦ C<sup>(R)</sup>) Ĥ<sup>(SR)</sup><sub>t−1</sub>]<sup>†</sup> ⇒ P<sub>S</sub>M<sub>D</sub>P<sub>R</sub> × M<sub>S</sub><br>[(C<sup>(S)</sup> ♦ Ŵ<sub>t</sub>) ⊗ (Ĥ<sup>(RD)</sup><sub>t</sub> ♦ C<sup>(R)</sup>)]<sup>†</sup> ⇒ P<sub>S</sub>NM<sub>D</sub>P<sub>R</sub> × M<sub>S</sub>M<sub>R</sub></td><td>O(NM<sup>2</sup>P<sup>2</sup>)<br>O(M<sup>3</sup>P<sup>2</sup>)<br>O(NM<sup>5</sup>P<sup>2</sup>)</td></tr><tr><td colspan="3"><strong>Combined codings-DF protocol</strong></td></tr><tr><td>STST-MSMKron/KronF</td><td>Q<sub>q=1</sub> S<sup>(q)</sup> ⊗ H<sup>(SR)</sup> ⇒ N<sub>1</sub> ... N<sub>Q</sub>M<sub>R</sub> × R<sub>1</sub> ... R<sub>Q</sub>M<sub>S</sub><br>Q<sub>q=1</sub> Ŵ<sup>(q)</sup> ⊗ H<sup>(RD)</sup> ⇒ N<sub>1</sub> ... N<sub>Q</sub>M<sub>D</sub> × R<sub>1</sub> ... R<sub>Q</sub>M<sub>S</sub></td><td>O(N<sup>Q</sup>M<sup>2</sup>R<sup>Q</sup>)<br>O(N<sup>Q</sup>M<sup>2</sup>R<sup>Q</sup>)</td></tr><tr><td>SKRST-MSMKR/KRF</td><td>Q<sub>q=1</sub> S<sup>(q)</sup> ♦ H<sup>(SR)</sup> ⇒ N<sub>1</sub> ... N<sub>Q</sub>M<sub>R</sub> × M<sub>S</sub><br>Q<sub>q=1</sub> Ŵ<sup>(q)</sup> ♦ H<sup>(RD)</sup> ⇒ N<sub>1</sub> ... N<sub>Q</sub>M<sub>D</sub> × M<sub>S</sub></td><td>O(N<sup>Q</sup>M<sup>2</sup>)<br>O(N<sup>Q</sup>M<sup>2</sup>)</td></tr></tbody></table>

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