Interesting question. The Kalman gain K_n is the bias correction gain that is a matrix. The bandwidth of the Kalman filter is kind of a mistery, because the frequency response function is very complex. It reminiscent of the sinc function, but not exactly. Moreover, the Kalman filter is essentially a LP filter. But you would not want such a LP filter, because there would be no desirable slope at a cutoff frequency wherever it is specified. There may be found an equaivalent bandwidth and its dependence on K_n. But what is a meaning of this, I do not know. May be you have some specific interest here, give us a clue.

According to what I have worked with, the Kalman gain is a quantity that controls the tradeoff between the prediction of future values and the observation of current values. I do not see how that could be related to bandwidth. Maybe in your specific field of operation it may be a different case. You should express your question more.

Thanks a lot, both answers are very helpful, in my case, i used Kalman filter in the GPS tracking loop, so in order to compare it with the traditional one, i have to analyze the equivilant bandwidth of the Kalman filter, you know bandwidth is an important metric in tracking loop, it can decide the noise power. and i was very curious that when i found one paper saying that the bandwidth is proportional to the Kalman gain without mathematical proof, so i proposed the question, actually, till now, i think Yuriy Shmaliy is correct, we should find the equivilant bandwidth of Kalman filter in different applications. after all, there may be no general mathematical expression for it.

Xinhua,

Yes, the ´´bandwidth´´ W of an optimal filter is proportional to the Kalman gain K. An explaination is simple. The lesser K is the more inertial the filter is. So, the W is narrower if K is smaller. In time-invariant linear systems, K is constant and W is constant as well. In time-varying or extended nonlinear models, K is time-varying and W is time-varying.

Is your GPS-based tracking loop designed to work with positioning or time-error correction? Anyway, you may have a trouble with the noise covariance matrices. There is a new Kalman-like filter that ignores these matrices. Our last investigations have shown that it is able to greatly outperform the Kalman filter for nonlinear models in the extended version and if the linear model is harmonic. So, for your problem you may have a more robust and also simpler solution. I can take a look at.

Thanks a lot, Yuriy Shmalily, your answer helps me a lot, especially the simple but meaningful explanation of the relationship between K and 'bandwidth'. by the way, can you show me the reference paper about the Kalman-like filter? thanks a lot.

This is an unbiased FIR filter that is a counterpart to the Kalman filter that has IIR:

Y. S. Shmaliy, An iterative Kalman-like algorithm ignoring noise and initial conditions, IEEE Trans. Signal Processing, vol. 59, no. 6, pp. 2465-2473, Jun. 2011.

Y.S. Shmaliy, Suboptimal FIR filtering of nonlinear models in additive white Gaussian noise, IEEE Trans. Signal Process. , vol. 60, no. 10, pp. 5519-5527, Oct. 2012.

You can find these and other relevant papers on my site here. There are also some fresh results still unpublished which show that such a filter is a strong rival to the Kalman filter. Ask me if you need an assistance.

thanks a lot, Yuriy Shmaliy, you did help me a lot, i will check out all these papers which i think should be very interesting. if i have any proplem, i will let you know. thanks again.

thanks, professor Shmalily, i just checked out your personal site, interesting, i will have an in-depth study on your papers, you know, i am kind of beginner in this field, but my intuition told me that i am really interested in mathematical model of different filters or different applications. but unfortunately, i have to learn more to make up for the shortage of the related knowledge though it may takes much time.

thanks for your help and patience. all the best

Xinhua, there is a fresh paper that may be useful for you to start:

Y. S. Shmaliy, D. Simon, Unbiased FIR state estimation in white Gaussian noise: A review of algorithms, EURASIP J. Advances Signal Process.

It generalizes the unbiased FIR approach and gives a guide to users. Take a look and good luck.

Corrections:

Y. S. Shmaliy, D. Simon, Iterative unbiased FIR state estimation: a review of algorithms, EURASIP J. Advances Signal Process., vol. 113, no. 1, pp. 1-16, 2013.

@Yuri S. Shmaliy,

Dear Professor, do you have slides / lecture notes about your articles?

Thank You.

P.S.

I found something on Super Lectures - a Kalman Like Algorithm with no Requirements for Noise and Initial Conditions.

http://www.superlectures.com/icassp2011/lecture.php?lang=en&id=294