Regularization can be implemented as a rank-based constraint (Oropeza and Sacchi, 2011; Trickett, 2008; Cheng and Sacchi, 2015):
denotes the Frobenius norm.
denotes a matrix constructed from the frequency slice corresponding to .
denotes the noise-free data to be estimated.
denotes the estimated signal.
denotes the Hankel matrix constructed from
, the singular value decomposition of
can be expressed as
are referred to as the left and right singular vector matrices, respectively.
is the singular value matrix. denotes complex tranpose. According to equation 4, the rank of the signal component that is embedded in the Hankel matrix
is assumed to be . The traditional rank-reduction method based on the truncated singular value decomposition (TSVD) (Oropeza and Sacchi, 2011) can be briefly expressed as
which is a solution to equation 4 according to the Eckart-Young-Mirsky theorem (Eckart and Young, 1936).
denotes the denoised signal.
are matrices composed of the left singular vectors in
is the truncated singular value matrix with the first singular values preserved. Although theoretically 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.
in the algorithm workflow denotes an averaging operator along anti-diagonals.