Table 6 Fixed-point TV UFIR smoothing algorithm

Given:

K, v=constant⩾0, q=n−v, n⩾K.

Set:

${\widehat{\mathbf{x}}}_{K-1}$ by (75) and G _K−1 by (76) for m=0.

Update:

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

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

Compute ${\widehat{\mathbf{x}}}_{n-q}$ for n>v as follows:

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