# fixed.forgettingFactorInverse

Compute the inverse of the forgetting factor required for streaming input data

## Description

## Examples

## Input Arguments

## Output Arguments

## Algorithms

In real-time applications, such as when data is streaming continuously from a radar array
[1], the QR decomposition is often computed continuously as each new row of data arrives. In
these systems, the previously computed upper-triangular matrix, *R*, is
updated and weighted by forgetting factor *ɑ*, where 0 < *ɑ* < 1. This computation treats the matrix *A* as if it is
infinitely tall. The series of transformations is as follows.

$$\begin{array}{l}{R}_{0}=\mathrm{zeros}\left(n,n\right)\\ \left[\begin{array}{c}{R}_{0}\\ A\left(1,:\right)\end{array}\right]\to \left[\begin{array}{c}{R}_{1}\\ 0\end{array}\right]\\ \left[\begin{array}{c}\alpha {R}_{1}\\ A\left(2,:\right)\end{array}\right]\to \left[\begin{array}{c}{R}_{2}\\ 0\end{array}\right]\\ \vdots \\ \left[\begin{array}{c}\alpha {R}_{k}\\ A\left(k,:\right)\end{array}\right]\to \left[\begin{array}{c}{R}_{k+1}\\ 0\end{array}\right]\end{array}$$

Without the forgetting factor *ɑ*, the values of *R*
would grow without bound.

With the forgetting factor, the gain in *R* is

$$g=\sqrt{\frac{1}{2}{\displaystyle {\int}_{0}^{\infty}{\alpha}^{x}dx}}=\sqrt{\frac{-1}{2\mathrm{log}\left(\alpha \right)}}.$$

The gain of computing *R* without a forgetting factor from an
*m*-by-*n* matrix *A* is $$\sqrt{m}$$. Therefore,

$$\begin{array}{l}\sqrt{m}=\sqrt{\frac{-1}{2\mathrm{log}\left(\alpha \right)}}\\ m=\frac{-1}{2\mathrm{log}\left(\alpha \right)}\\ \alpha ={e}^{-1/\left(2m\right)}.\end{array}$$

## References

[1] Rader, C.M. "VLSI Systolic Arrays
for Adaptive Nulling." *IEEE Signal Processing Magazine* (July 1996):
29-49.

## See Also

### Functions

### Blocks

- Real Partial-Systolic Q-less QR Decomposition with Forgetting Factor | Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor | Real Partial-Systolic Matrix Solve Using Q-less QR Decomposition with Forgetting Factor | Complex Partial-Systolic Matrix Solve Using Q-less QR Decomposition with Forgetting Factor

**Introduced in R2021b**