I was studying Direction of Arrival (DOA) and I ran into an article that indicated that the "desired covariance matrix is Hermitian Toeplitz and owns the low-rank property, the covariance matrix of the interpolated ULA can be recovered by solving the minimization problem"
- Note that R is the covariance matrix of received samples.
$$ min_{\bar{R_v} \in C^{\|V^+\|\times \|{V^+}\|} } rank(\bar{R_v}) $$ subject to:
$$ \bar{R_v} = \bar{R^H_v} ;<\bar{R_v}>_{n1,n2} = <{R_v}>_{n1,n2} ; n1,n2 \in V^+ , n1-n2 \in D $$
where H denotes the Hermitian notation and D, is a different set (the smaller set) and V is the virtual set of ULA (the more extensive set to be interpolated).
The constraints above can ensure that $$\bar{R_v}$$ is a Hermitian Toeplitz matrix, and the known correlation information on D is entirely contained in $$R_V$$.
Clearly, the minimization problem above is a nonconvex NP-hard problem. By using the nuclear norm minimization, can be reformulated as:
$$ min_{\bar{R_v} \in C^{\|V^+\|\times \|{V^+}\|} } \|(\bar{R_v}) \|_{*} $$ subject to:
$$ \bar{R_v} = \bar{R^H_v} ;<\bar{R_v}>_{n1,n2} = <{R_v}>_{n1,n2} ; n1,n2 \in V^+ , n1-n2 \in D $$
I'm seeking the reason why I can interpolate the Covariance matrix using that minimization problem!? Is there any reference on this matter?