Abstract
The exact and actual cause of the failure of the
fast-Kalman algorithm due to the generation and propagation of
finite-precision or quantization error is presented. It is
demonstrated that out of all the formulas that constitute this
fast Recursive Least Squares (RLS) scheme only three generate an
amount of finite-precision error that consistently propagates in
the subsequent iterations and eventually makes the algorithm fail
after a certain number of recursions. Moreover, it is shown that
there is a very limited number of specific formulas that transmit
the generated finite-precision error, while there is another
class of formulas that lift or “relax” this error. In addition, a
number of general propositions is presented that allow for the
calculation of the exact number of erroneous digits with which
the various quantities of the fast-Kalman scheme are computed,
including the filter coefficients. On the basis of the previous
analysis a method of stabilization of the fast-Kalman algorithm
is developed and is presented here, a method that allows for the
fast-Kalman algorithm to follow very difficult signals such as music,
speech, environmental noise, and other nonstationary ones.
Finally, a general methodology is pointed out, that allows for
the development of new algorithms which, intrinsically, suffer
far less of finite-precision problems.