Table 1 Procedure of deriving the compression matrix in each long-term period with the PCA method at the MS

Initialization

S:

Number of the short-term period T _s in every

long-term period T _l.

M:

Number of the dominating eigenvectors chosen

to form the compression matrix.

Procedures

1)

Channel estimation: perform channel

estimation continuously in T _l ⁽ⁿ⁾ to obtain

S high-dimensional channel vectors

\({\textbf {h}_{k}^{\left ({n,1} \right)},\textbf {h}_{k}^{\left ({n,2} \right)} \ldots, \textbf {h}_{k}^{\left ({n,S} \right)}}\).

2)

Computation of covariance matrix: \({\tilde {\mathbf {H}}^{\left (n \right)} =}\)

\({{\left [ {{{\left ({\textbf {h}_{k}^{\left ({n,1} \right)}} \right)}^{H}},{{\left ({\textbf {h}_{k}^{\left ({n,2} \right)}} \right)}^{H}}\ldots,{{\left ({\textbf {h}_{k}^{\left ({n,S} \right)}} \right)}^{H}}} \right ]^{H}}}\)

\({\in {\mathbb {C}^{S \times {N_{\mathrm {t}}}}}}\) and compute its covariance matrix.

3)

Eigen-decomposition: perform eigen-

decomposition on the above covariance matrix

\({\text {Cov}\left ({{{\tilde {\mathbf {H}}}^{\left (n \right)}},{{\tilde {\mathbf {H}}}^{\left (n \right)}}} \right) = {\textbf {U}^{\left (n \right)}}{\textbf {D}^{\left (n \right)}}{\left ({{\textbf {U}^{\left (n \right)}}} \right)^{H}}}\).

4)

Formation of compression matrix: choose M

dominating eigenvectors to form compression

matrix \({{\bar {\mathbf {U}}^{\left (n \right)}} \in {\mathbb {C}^{{N_{\mathrm {t}}} \times M}}}\).