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Table 2 An algorithm for determining the optimal solution \(\{\mathbf {v}^{\star },{\mathbf {Q}_{J}^{\star }}\}\) which maximizes the outage constrained secrecy rate

From: Robust transmit design for secure AF relay networks with imperfect CSI

1. Initialize τ lb =0, \({\tau _{ub}}=P_{s}P_{r}|\mathbf {h}|^{2}/{\sigma _{D}^{2}}\), P s >0, P r >0, P k ≥0, τ=0, τ sz >0 where the step size τ sz is an arbitrarily small positive quantity;

2. while τ∈ [τ lb ,τ ub ]do

3. Calculate \({\bar {g}}(\tau)\) by solving (33) and let \({\bar {f}}(\tau)={\bar {g}}(\tau)\);

4. Let Ï„=Ï„+Ï„ sz ;

5. end while

6. Determine τ ⋆ which maximizes the objective function of (31);

7. Let τ=τ ⋆ and calculate \({\bar {g}}(\tau)\) by solving (33);

8. Calculate \(\{ {{\hat {\mathbf {Q}}}_{\mathbf {v}}},{{\hat {\mathbf {Q}}}_{J}} \}\) by solving (23);

8. Do the matrix decomposition \({\hat {\mathbf {Q}}}_{\mathbf {v}}=\hat {\mathbf {v}}{\hat {\mathbf {v}}}^{H}\);

9. The optimal solution \(\mathbf {v}^{\star }=\hat {\mathbf {v}}\), \({\mathbf {Q}_{J}^{\star }}={{\hat {\mathbf {Q}}}_{J}}\).