Algorithm 1 Bi-ALS-KronF Receiver for Estimating the Symbol Matrices ${\mathbf{S}}^{\left(l\right)}$ and the Channels ${\mathcal{H}}^{\left(SR\right)}$ and ${\mathcal{H}}^{\left(RD\right)}$.

Input: tensors ${\mathcal{X}}^{\left(SR\right)}$, ${\mathcal{X}}^{\left(RD\right)}$, ${\mathcal{G}}^{\left(S\right)}$, ${\mathcal{G}}^{\left(R\right)}$

Output: Estimated symbol matrices and channels

First hop: source–relay

- Step 1: Bi-ALS algorithm

$it=0$

(1) Initialization of ${\mathbf{S}}^{\left(l\right)}\left[0\right]$ with symbols randomly drawn from the alphabet and $s_{11}^{\left(l\right)}=1$, for $l\in [1,L]$.

(2) Update the estimates of ${\mathbf{H}}_{F{M}_S\times M_R}^{\left(SR\right)}$ and $\mathbf{S}$ using Equations (30) and (31) or (32) and (33).

(3) Calculate the error (34) and $\left|err\right[it-1]-err[it\left]\right|.$

- if $\left|err\right[it-1]-err[it\left]\right|\le ϵ$ or $it=$ maximum number of iterations

- stop

- else $it\to it+1$;

(4) Eliminate the scaling ambiguities using Equation (35).

- Step 2: KronF algorithm

(5) Build the rank-one tensor: $\widehat{\widehat{\mathcal{S}}}=\mathrm{reshape}(\widehat{\widehat{\mathbf{S}}},[R_1{N}_1,...,R_L{N}_L])$.

(6) Estimate each vector ${\widehat{\mathbf{s}}}^{\left(l\right)}$ by means of the KronF algorithm recalled in Appendix A, and unvectorize it using Equation (37).

(7) Eliminate the scaling ambiguities using Equation (38).

(8) Project the estimated symbols onto the symbol alphabet.

Second hop: relay–destination

- Step 1: Bi-ALS algorithm

- Apply the stages (1) to (4) of the first hop, using the correspondences (23) and (24).

- Step 2: KronF algorithm

- Apply the stages (5) to (8) of the first hop, using the correspondences (23) and (24).