Editing Kalman filter (section)

== Frequency-weighted Kalman filters ==
Pioneering research on the perception of sounds at different frequencies was conducted by Fletcher and Munson in the 1930s. Their work led to a standard way of weighting measured sound levels within investigations of industrial noise and hearing loss. Frequency weightings have since been used within filter and controller designs to manage performance within bands of interest.

Typically, a frequency shaping function is used to weight the average power of the error spectral density in a specified frequency band. Let <math>\mathbf{y} - \hat{\mathbf{y}}</math> denote the output estimation error exhibited by a conventional Kalman filter. Also, let <math>\mathbf{W}</math> denote a causal frequency weighting transfer function. The optimum solution which minimizes the variance of <math>\mathbf{W}\left(\mathbf{y} - \hat{\mathbf{y}}\right)</math> arises by simply constructing <math>\mathbf{W}^{-1} \hat{\mathbf{y}}</math>.

The design of <math>\mathbf{W}</math> remains an open question. One way of proceeding is to identify a system which generates the estimation error and setting <math>\mathbf{W}</math> equal to the inverse of that system.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Iterative Frequency-Weighted Filtering and Smoothing Procedures | journal = IEEE Signal Processing Letters | volume = 21 | issue = 12 | pages = 1467–1470 | date=December 2014 | doi = 10.1109/LSP.2014.2341641 |bibcode = 2014ISPL...21.1467E | s2cid = 13569109 }}</ref> This procedure may be iterated to obtain mean-square error improvement at the cost of increased filter order. The same technique can be applied to smoothers.