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Table 1 The joint beamforming and power control update algorithm

From: Joint beamforming and power control algorithm for cognitive MIMO broadcast channels via game theory

1. Initial data: beamforming vectors \({\hat {\mathbf {u}}_{k}}\), power control p k , channel matrices H k , and target SINRs \(\gamma _{k}^{*}\) for SUk, noise covariance matrix W, constants β, μ and tolerance ε.

2. Whitening the received signal at the SUk, using the rank of SUk-transformed MIMO channel matrix \({\rho _{k}} = \text {rank}\left ({{{\tilde {\mathbf {H}}}_{k}}} \right)\) to calculate \(\hat {\mathbf {\Lambda }}_{k}^{- 2}\), calculating S k (n) and the eigenvector \({\overset {\frown }{\mathbf {u}}_{k}}\) corresponding to the minimum eigenvalue of S k . At the same time, using (32) to update \({\hat {\mathbf {u}}_{k}}\).

3. If ρ k <N t , adding zeros to obtain N-dimension matrix \({\bar {\mathbf {u}}_{k}}\), and then using \({{\mathbf {u}}_{k}} = {{\mathbf {V}}_{k}}{\bar {\mathbf {u}}_{k}}\) to obtain original beamforming vectors.

4. Using (33) to update power.

5. If satisfy |J k (n+1)−J k (n)|≤ε, where ε (ε>0) is also a small-valued positive number in practice to represent an infinitely small quantity, iteration continues. Otherwise, go to step 2.

6. Iteration termination if the optimal condition (24) is true, then stop, an optimal Nash equilibrium has been reached. Otherwise, go to step 2. Where checking the optimal condition (24) ensures the optimal Nash equilibrium is reached and the algorithm does not stop in a suboptimal fixed point.