Table 1 Iterative p -shift Kalman-like UFIR filtering algorithm

Given:

K, N, p, l = α, …, n, s = α − 1

α = max(m + K, m + 2, m + 2 − p),

Set:

$\mathbf{\Upsilon }={\left({\mathbf{C}}_{s-m}^T{\mathbf{C}}_{s-m}\right)}^{-1}$

${\mathbf{F}}_s={\mathbf{A}}^{s-m+p}\mathbf{\Upsilon }{{\mathbf{A}}^{s-m+p}}^T$

${\bar{\mathbf{x}}}_{s|s-p}={\mathbf{A}}^{s-m+p}\mathbf{\Upsilon }{\mathbf{C}}_{s-m}^T{\mathbf{Y}}_{s-p,m-p}$

Update:

${\mathbf{F}}_l={[{\mathbf{C}}^T\mathbf{C}+{\left(\mathbf{A}{\mathbf{F}}_{l-1}{\mathbf{A}}^T\right)}^{-1}]}^{-1}$

${\bar{\mathbf{x}}}_{l|l-p}=\mathbf{A}{\bar{\mathbf{x}}}_{l-1|l-p-1}$

$+{\mathbf{A}}^p{\mathbf{F}}_l{\mathbf{C}}^T({\mathbf{y}}_{l-p}-\mathbf{C}{\mathbf{A}}^{1-p}{\bar{\mathbf{x}}}_{l-1|l-p-1})$

Remark:

Use ${\bar{\mathbf{x}}}_{l|l-p}$ as the output when l = n