Streaming orthogonal prediction filter in $t$ -$x$ domain for random noise attenuation

Appendix A: Direct derivation of the Sherman-Morrison Formula

The Sherman-Morrison formula can be derived directly by solving the linear problem as

$$(\mathbf{A}-\mathbf{u}\mathbf{v})\mathbf{x}=\mathbf{b},$$ (22)

where $\mathbf{u}$ is a column vector and $\mathbf{v}$ is a row vector. Assuming $\mathbf{A}^{-1}$ is already known, we pre-multiply equation A-1 by $\mathbf{A}^{-1}$ and denote $\mathbf{A}^{-1}\mathbf{u}=\mathbf{z}$ and $\mathbf{A}^{-1}\mathbf{b}=\mathbf{y}$ to give

$$\mathbf{x}-\mathbf{z}\mathbf{v}\mathbf{x}=\mathbf{y}.$$ (23)

Because $\mathbf{v}\mathbf{x}$ is a scalar quantity, we can denote it as $\alpha$ . To find $\alpha$ , we pre-multiply the equation A-2 by $\mathbf{v}$ to give

$$\alpha-\mathbf{v}\mathbf{z}\alpha=\mathbf{v}\mathbf{y}.$$ (24)

Since $\mathbf{v}\mathbf{z}$ and $\mathbf{v}\mathbf{y}$ in the equation A-3 are scalars, one can easily solve for $\alpha$ to get

$$\alpha=\frac{\mathbf{v}\mathbf{y}}{1-\mathbf{v}\mathbf{z}}.$$ (25)

Thus the solution can be written as

$$\mathbf{x} = \mathbf{y}+\alpha\mathbf{z}$$ (26)

$$= \mathbf{A}^{-1}\mathbf{b}+\mathbf{A}^{-1}\mathbf{u} (1-\mathbf{v}\mathbf{A}^{-1}\mathbf{u})^{-1} \mathbf{v}\mathbf{A}^{-1}\mathbf{b}$$ (27)

$$= [\mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{u} (1-\mathbf{v}\mathbf{A}^{-1}\mathbf{u})^{-1} \mathbf{v}\mathbf{A}^{-1}]\mathbf{b}.$$ (28)

We then obtain the Sherman-Morrison formula

$$(\mathbf{A}-\mathbf{u}\mathbf{v})^{-1}=\mathbf{A}^{-1}+ \mathbf{A... ...athbf{u}(1- \mathbf{v}\mathbf{A}^{-1}\mathbf{u})^{-1}\mathbf{v}\mathbf{A}^{-1}.$$ (29)

Streaming orthogonal prediction filter in $t$ -$x$ domain for random noise attenuation