Assume we have one block Hankel matrix as follows $\mathscr{H}\{\mathbf{X}\}=\left[\begin{array}{cccc}{\mathcal{H}\left\{\mathbf{x}_{1}\right\}} & {\mathcal{H}\left\{\mathbf{x}_{2}\right\}} & {\cdots} & {\mathcal{H}\left\{\mathbf{x}_{2 n+1}\right\}} \\ {\mathcal{H}\left\{\mathbf{x}_{2}\right\}} & {\mathcal{H}\left\{\mathbf{x}_{3}\right\}} & {\cdots} & {\mathcal{H}\left\{\mathbf{x}_{2 n+2}\right\}} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {\mathcal{H}\left\{\mathbf{x}_{N-2 n}\right\}} & {\mathcal{H}\left\{\mathbf{x}_{N-2 n+1}\right\}} & {\cdots} & {\mathcal{H}\left\{\mathbf{x}_{N}\right\}}\end{array}\right]$
where $\mathcal{H}\{\mathbf{x}_i\}=\left[\begin{array}{cccc}{\mathbf{x}_{i}(1)} & {\mathbf{x}_{i}(2)} & {\cdots} & {\mathbf{x}_{i}(2 m+1)} \\ {\mathbf{x}_{i}(2)} & {\mathbf{x}_{i}(3)} & {\cdots} & {\mathbf{x}_{i}(2 m+2)} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {\mathbf{x}_{i}(M-2 m)} & {\mathbf{x}_{i}(M-2 m+1)} & {\cdots} & {\mathbf{x}_{i}(M)}\end{array}\right]$ is a low rank matrix, i.e., $RANK(\mathcal{H}\{\mathbf{x}_i\})=r<min(M-2m, 2m+1)$. Can we prove $\mathscr{H}\{\mathbf{x}\}$ is also low rank?