Semi-Blind Receivers for Two-Hop MIMO Relay Systems with a Combined TSTF-MSMKron Coding

. 2023 Jun 27;23(13):5963. doi: 10.3390/s23135963

Algorithm 2 THOSVD Receiver for Estimating the Symbol Matrices

S^{(l)}

and the Channels

H^{(S R)}

and

H^{(R D)}

Input: tensors

X^{(S R)}

X^{(R D)}

G^{(S)}

G^{(R)}

Output: Estimated symbol matrices and channels

First hop: source–relay

(1) Calculate the LS estimate

Z^{(S R)}

defined in (39).

(2) Build the rank-one tensor

Z^{(S R)}

of size

R_{1} N_{1} \times \dots \times R_{L} N_{L} \times F M_{S} M_{R}

from

Z^{(S R)}

(3) Compute the SVD of each mode-n unfolding of

Z^{(S R)}

, and calculate the estimates

{\hat{s}}^{(l)} = vec ({\hat{S}}^{(l)})

and

{\hat{h}}^{(S R)} = vec ({\hat{H}}_{M_{R} \times F M_{S}}^{(S R)})

as the first left singular vector of each mode-n unfolding.

(4) Unvectorize

{\hat{s}}^{(l)}

and

{\hat{h}}^{(S R)}

to obtain the estimates

{\hat{\hat{S}}}^{(l)}

and

{\hat{\hat{H}}}_{M_{R} \times F M_{S}}^{(S R)}

(5) Eliminate the scaling ambiguities.

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

Second hop: relay–destination

- Apply the stages (1) to (6) of the first hop, using the correspondences (23) and (24).