Rank-reduction method

Regularization can be implemented as a rank-based constraint (Oropeza and Sacchi, 2011; Trickett, 2008; Cheng and Sacchi, 2015):

$$\begin{split} \hat{\mathbf{X}} &= \arg \min_{\mathbf{x}} \par... ...to}&\quad \text{rank} (\mathcal{H}(\mathbf{X})) =N, \end{split}$$ (4)

where $\parallel\cdot\parallel_F$ denotes the Frobenius norm. $\mathbf{D}$ denotes a matrix constructed from the frequency slice corresponding to $w_0$. $\mathbf{X}$ denotes the noise-free data to be estimated. $\hat{\mathbf{X}}$ denotes the estimated signal. $\mathcal{H}(\mathbf{X})$ denotes the Hankel matrix constructed from $\mathbf{X}$. Let $\mathbf{M}=\mathcal{H}(\mathbf{D})$, the singular value decomposition of $\mathbf{M}$ can be expressed as

$$\mathbf{M} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H,$$ (5)

where $\mathbf{U}$ and $\mathbf{V}$ are referred to as the left and right singular vector matrices, respectively. $\boldsymbol{\Sigma}$ is the singular value matrix. $(\cdot)^H$ denotes complex tranpose. According to equation 4, the rank of the signal component that is embedded in the Hankel matrix $\mathbf{M}$ is assumed to be $N$. The traditional rank-reduction method based on the truncated singular value decomposition (TSVD) (Oropeza and Sacchi, 2011) can be briefly expressed as

$$\hat{\mathbf{S}} = \mathbf{U}_N\boldsymbol{\Sigma}_N\mathbf{V}_N^H,$$ (6)

which is a solution to equation 4 according to the Eckart-Young-Mirsky theorem (Eckart and Young, 1936). $\hat{\mathbf{S}}$ denotes the denoised signal. $\mathbf{U}_N$ and $\mathbf{V}_N$ are matrices composed of the left $N$ singular vectors in $\mathbf{U}$ and $\mathbf{V}$, respectively. $\boldsymbol{\Sigma}_N$ is the truncated singular value matrix with the first $N$ singular values preserved. Although theoretically $N$ equals to the number of distinct dipping components, it is practically defined as a relatively large number considering data complexity, otherwise signal energy can be damaged. The algorithm workflow for the traditional rank-reduction method (RR) is outlined in Algorithm 1. $\mathcal{A}$ in the algorithm workflow denotes an averaging operator along anti-diagonals.