Appendix B: Derivation for equations 11, 14 and 15

We first reformulate equation 10 as

$$\mathbf{S} = \tilde{\mathbf{M}} -\mathbf{U}_1^S(\mathbf{U}_1^S)^H\mathbf{N}.$$ (29)

Inserting equation 6 into equation 29, we can further derive

$$\begin{split} \mathbf{S} = \mathbf{U}_1^M\Sigma_1^M(\mathbf{V}_1^M)^H -\mathbf{U}_1^S(\mathbf{U}_1^S)^H\mathbf{N}. \end{split}$$ (30)

Because equations 5 and 9 are both SVDs of $\mathbf{M}$, we let

$$\mathbf{U}_1^S=\mathbf{U}_1^M,$$ (31)

and

$$\Sigma_1=\Sigma_1^M,$$ (32)

and

$$(\mathbf{N}^H\mathbf{U}_1^S+\mathbf{V}_1^S\Sigma_1^S)(\Sigma_1)^{-1}=\mathbf{V}_1^M.$$ (33)

Considering $\Sigma_1= \Sigma_1^M\mathbf{B}$,

$$\mathbf{B}=\mathbf{I}.$$ (34)

From equation 33,

$$\begin{split} \mathbf{V}_1^S&=(\mathbf{V}_1^M-\mathbf{N}^H\ma... ...a_1(\Sigma_1^S)^{-1}\\ &=\mathbf{V}_1^M\mathbf{A}, \end{split}$$ (35)

where $(\mathbf{V}_1^M)^{o}$ satisfies that $\parallel\mathbf{I}-\mathbf{V}_1^M(\mathbf{V}_1^M)^{o} \parallel\rightarrow 0$.

Considering $\mathbf{V}_1^S=\mathbf{V}_1^M\mathbf{A}$,

$$\begin{split} \mathbf{A}&\approx (\mathbf{I}-(\mathbf{V}_1^M)... ...\\ &=(\mathbf{I}-\Gamma)\Sigma_1(\Sigma_1^S)^{-1}, \end{split}$$ (36)

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

Inserting equations 31 and 33 into equation 30, we can obtain:

$$\begin{split} \mathbf{S} &= \mathbf{U}_1^M\Sigma_1^M(\mathbf{... ...1^M)^H-\Sigma_1^S(\mathbf{V}_1^S)^H\right]\right\}. \end{split}$$ (37)