Solution of a linear mean square estimation problem when process statistics are undefined

The problem of estimating a signal
x(t)
from linear measurements
z(t)=x(t)+w(t)
is considered. The signal
w(t)
is white noise with known statistics, while
x(t)
is an unknown nonrandom signal drawn from a set of admissible waveforms having bounded second derivatives for which statistics are undefined. Problems of this type arise naturally in the radar tracking of an evasive vehicle under the control of an intelligent adversary. Such problems are also important in trajectory estimation on a missile test range, since a priori statistics for a malfunctioning missile cannot by nature be obtained. The problem is formulated as a minimax estimation problem in which the object is to find that filter design which minimizes the maximun value of estimation error taken over all admissible signal waveforms
\ddot{x}| \leq\alpha
. An analytic solution is obtained for design of the optimal filter when measurements extend into the infinite past. The impulse response function is exhibited, and its performance is compared with that obtained using more usual estimation techniques. It is discovered that the impulse response function of the optimal filter has a finite time history terminating at a critical observation lag time:
t= T_{max}
.