Table 1 The SVD-USSD algorithm

Step 1: set k=0, initialize modulation codes {x(n,m)},

n=0,1,...,N−1,m=0,1,...,M−1 using randomly generate codes;

Step 2:

for n=0:N-1

Step 2.1: solve\(\widetilde {\mathcal {P}1}_{n}^{(p)}\) by \(\rho _{n}(p) = \frac {1}{N_{p}} \mathbf {a}_{p}^{H} \mathbf {b}_{p}\);

construct the matrixG_n;

Step 2.2: solve\(\widetilde {\mathcal {P}2}_{n}\);

for fixedX_n, perform SVD operation on \(\mathbf {X}_{n} \mathbf {G}_{n}^{-1/2} = \mathbf {U}_{1} {\Sigma } \mathbf {U}_{2}^{H}\) ;

calculate the matrix\(\mathbf {U} = \mathbf {U}_{1} \mathbf {U}_{2}^{H}\);

calculate the corresponding matrix\(\mathbf {V}= \mathbf {U} \mathbf {G}_{n}^{1/2}\);

for fixedV, calculate the modulation code by

\(x(n,m) = e^{j \phi _{n,m}}, \quad \phi _{n,m} = \text {arg} \left (\sum _{m=0}^{M-1} v_{m} \right)\);

end

Step 3: set k=k+1, repeat the Step (2) until a certain stop criterion,

e.g. \(\sum _{n=0}^{N-1} \sum _{m=0}^{M-1} \| x(n,m)^{(k+1)} -x(n,m)^{(k)} \|^{2} \leq \epsilon \),

where ε is a predefined value.