Table 4 Two-stage fixed-interval TV OUFIR smoothing algorithm

Given:

K, N=constant, q, m=n−N+1⩾0,

s=m+K−1, m+K⩽l⩽n.

Set:

${\widehat{\mathbf{x}}}_s$ by (75) and G _s by (76).

Update:

${\mathbf{G}}_l={\mathbf{H}}_l^T{\mathbf{H}}_l+{\left({\mathbf{F}}_l{\mathbf{G}}_{l-1}{\mathbf{F}}_l^T\right)}^{-1}{]}^{-1}$,

${\widehat{\mathbf{x}}}_l={\mathbf{F}}_l{\widehat{\mathbf{x}}}_{l-1}+{\mathbf{G}}_l{\mathbf{H}}_l^T({\mathbf{z}}_l-{\mathbf{H}}_l{\mathbf{F}}_l{\widehat{\mathbf{x}}}_{l-1})$.

Use ${\widehat{\mathbf{x}}}_n$ when l=n and compute

${\widehat{\mathbf{x}}}_{n-q}={\bar{\mathcal{B}}}_{n,m}\left(q\right){\left({\mathcal{F}}_{n,0}^{m+1}\right)}^{-1}{\widehat{\mathbf{x}}}_n$.

Instruction:

This algorithm is valid for any n⩾N−1. The fixed

interval of M=N=N _opt points is from time

index m to n.