Estimation of the Average Density of a Nonuniform Poisson Process

The average density
\lambda (t)
of a Poisson process is estimated by a nonstationary shot noise process
s(t)
. The process
s(t)
is generated by a time-varying system whose impulse response
h (t, \alpha)
is so determined as to minimize the mean-square (ms) estimation error. It is shown that, under realistic assumptions,
h (t, \alpha )
is a truncated parabola in the variable α . The analysis is extended to two-dimensional processes. The resulting smoothing kernel is an elliptic paraboloid.