An exact equivalence between the discrete- and continuous-time formulations of the Kalman filter

It is shown that if the definition of the covariance of a white noise sequence in discrete-time is derived from the accepted mathematical description for the covariance of a white noise process in continuous-time, compatibility between the discrete- and continuous-time versions of the Kalman filter is complete. Consequently the approach to the limit of the discrete-time filter to obtain its continuous-time equivalent no longer depends on Kalman's non-rigorous argument for dividing the covariance of a white noise sequence by the sampling interval Δt. Such an exact equivalence is essential for comparing the accuray of discrete-time computations with results obtained by numerically integrating the continuous-time filter equations. This approach provides a pragmatic technique for the determination of the most suitable sampling interval for discrete-time Kalman filtering.