Optimum estimation of nonstationary Gaussian signals in noise

This paper treats the problem of estimating a signal
s(t)
in the presence of noise
n(t)
where
s(t)
and
n(t)
are independent nonstationary Gaussian processes. Specifically, we present the maximum likelihood and the minimum mean-square estimates of
s(t)
for each
t, T_{1} \leq t \leq T_{2}
, by observing the entire signal-plus-noise waveform
x(\cdot)
during the interval
[T_{1}, T_{2}]
. Under the condition that the signal cannot be detected perfectly in the presence of noise, we explicilyty prove that both estimates, denoted by
ŝ(t)
, are given by
E\{s(t)|x(\cdot)\}
, the conditional expectation of
s(t)
given
x(\cdot)
. With the use of simultaneously orthogonal expansions of
x(t)
and
n(t)
, we further obtain explicit expressions in the form of infinite series for
ŝ(t)
and
\epsilon(t) \equiv E|s(t) - ŝ(t)|^{2}
, the minimum mean-square error. Moreover, if the integral equation
\sum_{j=0}^{p} \int H̃_{j}(s, u) \frac{\delta^{j}}{\delta u^{j}}X(u, t) du = S(s, t)
admits a formal solution
\{H̃_{j}(s,t)\}
, then
ŝ(t)
and
\epsilon(t)
have the closed-form expressions
ŝ(t) = \sum_{j=0}^{p} \int H̃_{j} (t,u)x^{j}(u)du,
\epsilon(t) = \sum_{j=0}^{p} \int H̃_{j}(t,u)\frac{\delta^{j}}{\delta u^{j}}N(u,t)du,
where
S(s,t), N(s,t)
and
X(s,t)
are the covariances of
s(t), n(t)
and
x(t)
, respectively, and
p
is the largest integer for which the
2p
th partial derivatives of the covariances are continuous. The last result is a generalization of the classical Wiener filtering theory for stationary processes, and it is valid without the condition of imperfect detection if existence of the formal solution is assumed instead. Finally, we exhibit a general solution of the integral equation in the case where both
s(t)
and
n(t)
have rational power spectra.