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Table 1 Summary of the proposed WSFOPDE approach

From: Effective joint DOA-DOD estimation for the coexistence of uncorrelated and coherent signals in massive multi-input multi-output array systems

Require

The received MIMO array data \(\left \{ \mathbf {x}(t) \right \}_{t=1}^{T}\).

Ensure

Estimation of {θk,ϕk}, k=1,2,,K.

Step 1.

\(\hat {\mathbf {R}}_{\mathbf {x}} \leftarrow (1/T){\sum \nolimits }_{t=1}^{T} \mathbf {x}(t)\mathbf {x}^{\mathrm {H}}(t) \);

Step 2.

Us,W← Perform EVD on \(\hat {\mathbf {R}}_{\mathbf {x}}\) in (6);

Step 3.

\(\hat {\mathbf {b}} \leftarrow \) Minimize (21) by MODE algorithm;

Step 4.

\(\left \{\bar {\phi _{i}}\right \}_{i=1}^{K} \leftarrow \) Root polynomial (12) constructed by \(\hat {\mathbf {b}}\);

Step 5.

\(\left \{\bar {\phi _{i}}\right \}_{i=1}^{K_{1}}, \left \{\bar {\phi _{i}}\right \}_{i=K_{1}+1}^{K} \leftarrow \) Discern uncorrelated and coherent DOA via (25);

 

\(\bar {\mathbf {P}}_{\mathbf {D}_{1}\mathbf {D}_{2}} \leftarrow \) Construct oblique projector by (38);

Step 6.

\(\{{\bar \theta }_{k}\}_{k=1}^{K_{1}} \leftarrow \) Perform LS fitting to the auto-paired \({\bar {\mathbf {a}}_{u}({\theta }_{k})}\) by (28) and (33);

Step 7.

\(\bar {\mathbf {Y}}, \bar {\mathbf {U}}_{0}\leftarrow \) Construct virtual MIMO array matrix by (38) and (39), and perform EVD;

Step 8.

\(\left \{{\bar {\theta }}_{k}\right \}_{k=K_{1}+1}^{K} \leftarrow \) Extract auto-paired \(\bar {\mathbf {a}}_{c}({\theta }_{k})\) by (41) and repeat LS fitting.