I have a Kalman filter (discrete, linear, time invariant) that works nicely for the application. However, I have to compress all the parameters and the innovations. Ideally I would like to have integer coefficients in order to minimize the number of bits. So first I truncate the parameters to 4 decimal places before compression, hoping the filter will not suffer much from the truncation, but the filter diverges. Does anyone know how to do this type of compression without degrading the filter? Any help will be appreciated.
If you are currently using double-precision floating point, you might initially try switching to a single-precision representation. If single (32bit) floating point is still too large for your application, then the next thing to try would be an appropriate 16bit fixed-point representation.
The choice of where you fix the decimal place will depend on your precision requirements. 4 decimal places requires devoting 10 of the 16 bits to the fractional component. The range of valid numbers you can then represent in your filter ought to be [-32, 31.9990] in increments of (approximately) 0.0001. If you require larger or smaller values than these, 16 bit fixed point may not be suitable for your application.
Arguably, the compression benefits of integer representations are not generally worth the implementation complexity on modern systems.
In general filters are very sensitive to coefficients. Perhaps use of
paraunitary QMF lattice network implementation is a solution cf P.P. Vaidyanathan
Multirate Systems and Filter Banks .P307 quote "Such robustness to quantization is however not offered by direct implementation of the filters" unquote..
Cheers
Truncation of parameters of the system makes system unstable.
Simplest is you can try with Rounding or going with alternative structures for implementation of kalman filter which are insensitive to truncation like (lattice )
What you mention seems to be related to the result of roundoff errors. That may depend on the implementation of Kalman filter you are using. I may suggest to check chapters 6, and especially 7 of Kalman Filtering Theory and Practice by M.S Grewal and A.P. Andrews ( I use 1993 edition). Rescaling of some of the variables may remedy the situation – but you will have to carefully analyse the system to find a way through...
Another thing - what about using steady state gains - if it works there will be no divergence issue. There is also a gain scheduling approach ( based on several gain- matrices instead of the one at infinity
Have you considered covariance factorization ("square-root filtering") to reduce the effect of roundoff errors? G. J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, 1977. Grewal and Andrews, section 6.4.
C.A.:
Thanks for your answer. The KF is a steady-state filter, so we don't need to compute covariances or gains. The filter is a result of doing 2-D system identification on an image. The end result is a 2-D KF in steady-state form (space-invariant). The roundoff accumulation comes from truncating the coefficients to 4 decimal places. I need to compress the parameter matrices, the gains, and the innovations. I'm looking for the most compact way of doing this. I have to try MATLAB"s fixed point toolbox.
Jose
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