Appendix B: Damped TSVD formula

We first reformulate equation A-7 as

$$\mathbf{S} = \mathbf{M}^{(4)} -\mathbf{U}_1^S(\mathbf{U}_1^S)^H\mathbf{N}.$$ (25)

Inserting equation 8 into equation B-1, we can further derive

$$\begin{split} \mathbf{S} = \mathbf{U}_1^{M^{(4)}}\Sigma_1^{M^... ...)}})^H -\mathbf{U}_1^S(\mathbf{U}_1^S)^H\mathbf{N}. \end{split}$$ (26)

The SVD of $\mathbf{M}^{(4)}$ can be expressed as

$$\mathbf{M}^{(4)} = [\mathbf{U}_1^{M^{(4)}}\quad \mathbf{U}_2^{M^{... ...c} (\mathbf{V}_1^{M^{(4)}})^H\\ (\mathbf{V}_2^{M^{(4)}})^H \end{array}\right].$$ (27)

Because equations B-3 and A-6 are both SVDs of $\mathbf{M}^{(4)}$, we let

$$\mathbf{U}_1^S =\mathbf{U}_1^{M^{(4)}},$$ (28)

$$\Sigma_1 =\Sigma_1^{M^{(4)}},$$ (29)

$$(\mathbf{N}^H\mathbf{U}_1^S+\mathbf{V}_1^S\Sigma_1^S)(\Sigma_1)^{-1} =\mathbf{V}_1^{M^{(4)}}.$$ (30)

Inserting equations B-4 and B-6 into equation B-2, we can derive:

$$\mathbf{S} =\mathbf{U}_1^{M^{(4)}}\left\{\Sigma_1^{M^{(4)}}(\math... ...\Sigma_1(\mathbf{V}_1^{M^{(4)}})^H-\Sigma_1^S(\mathbf{V}_1^S)^H\right]\right\}.$$ (31)

For simplification, we assume that there exist such $\mathbf{A}$ and $\mathbf{B}$ that $\mathbf{V}_1^S=\mathbf{V}_1^{M^{(4)}}\mathbf{A}$ and $\Sigma_1= \Sigma_1^{M^{(4)}}\mathbf{B}$. $\mathbf{A}$ is a square matrix and $\mathbf{B}$ is a diagonal matrix. Then we can simplify $\mathbf{S}$ as:

$$\mathbf{S} = \mathbf{U}_1^{M^{(4)}}\Sigma_1^{M^{(4)}}\mathbf{T}\left(\mathbf{V}_1^{M^{(4)}}\right)^H.$$ (32)

$$\mathbf{T} = \mathbf{I} - \mathbf{B}\left(\mathbf{I}-(\Sigma_1)^{-1}\Sigma_1^S\mathbf{A}^H\right),$$ (33)

where $\mathbf{I}$ is a unit matrix and here we name $\mathbf{T}$ the damping operator.

From equation B-5, $\mathbf{B}=\mathbf{I}$. From equation B-6, it is straightforward to derive

$$\begin{split} \mathbf{V}_1^S&=(\mathbf{V}_1^{M^{(4)}}-\mathbf... ...hbf{U}_1^S(\Sigma_1)^{-1})\Sigma_1(\Sigma_1^S)^{-1} \end{split}$$ (34)

where $(\mathbf{V}_1^{M^{(4)}})^{o}$ satisfies that $\parallel\mathbf{I}-\mathbf{V}_1^{M^{(4)}}(\mathbf{V}_1^{M^{(4)}})^{o} \parallel\rightarrow 0$. Considering $\mathbf{V}_1^S=\mathbf{V}_1^{M^{(4)}}\mathbf{A}$, then the following relation holds

$$\begin{split} \mathbf{A}&\approx (\mathbf{I}-(\mathbf{V}_1^{M... ...\\ &=(\mathbf{I}-\Gamma)\Sigma_1(\Sigma_1^S)^{-1}, \end{split}$$ (35)

where $\Gamma=(\mathbf{V}_1^{M^{(4)}})^{o}\mathbf{N}^H\mathbf{U}_1^S(\Sigma_1)^{-1}$.

Inserting equation B-11 and $\mathbf{B}=\mathbf{I}$ into equation B-9, we can obtain a simplified formula:

$$\mathbf{T} = \mathbf{I}-\Gamma.$$ (36)

Combing equations B-8 and B-12, we can conclude that the true signal is a damped version of the previous TSVD method (equation 8), with the damping operator defined by equation B-12. Right now, there is still one unknown parameter needed to be defined: $\Gamma$. Although we have a potential selection $\Gamma=(\mathbf{V}_1^{M^{(4)}})^{o}\mathbf{N}^H\mathbf{U}_1^S(\Sigma_1)^{-1}$, as defined during the derivation of $\mathbf{A}$, we cannot calculate it because we do not know $\mathbf{N}$ and $\mathbf{U}_1^S$. A very pleasant denoising performance can be obtained when $\Gamma$ is chosen as

$$\Gamma \approx \hat{\delta}^N\left(\Sigma_1^{M^{(4)}}\right)^{-N},$$ (37)

where $\hat{\delta}$ denotes the maximum element of $\Sigma_2^{M^{(4)}}$ and $N$ denotes the damping factor. We use such approximation because of three reasons. (1) $\hat{\delta}$ reflects the energy of random noise and $\Sigma_1^{M^{(4)}}$ contains the information of signal. (2) Because the diagonal elements of $\Sigma^{M^{(4)}}$ are in a descending order, $\hat{\delta}$ is certainly smaller than every diagonal element of $\Sigma_1^{M^{(4)}}$, and $\hat{\delta}/\delta_1<\hat{\delta}/\delta_2<\cdots<\hat{\delta}/\delta_K$, where $\delta_i$ denotes $i$th diagonal entry in $\Sigma_1^{M^{(4)}}$. (3) $\hat{\delta}$ is zero in the zero random noise situation. Besides, we introduce the parameter $N$ to control the strength of damping operator, the greater the $N$, the weaker the damping, and the damped MSSA reverts to the basic MSSA when $N\to\infty$.

Combining equations B-8, B-12, and B-13, we conclude the approximation of $\mathbf{S}$ as:

$$\mathbf{S} = \mathbf{U}_1^{M^{(4)}} \Sigma_1^{M^{(4)}}\mathbf{T}(\mathbf{V}_1^{M^{(4)}})^H,$$ (38)

$$\mathbf{T} =\mathbf{I}-(\Sigma_1^{M^{(4)}})^{-N}\hat{\delta}^N.$$ (39)