Optimum estimation of nonstationary Gaussian signals in noise
Toshihito Kadota · IEEE Transactions on Information Theory · 1969
This paper treats the problem of estimating a signals(t)in the presence of noisen(t)wheres(t)andn(t)are independent nonstationary Gaussian processes. Specifically, we present the maximum likelihood and the minimum mean-square estimates ofs(t)for eacht, T_{1} \leq t \leq T_{2}, by observing the entire signal-plus-noise waveformx(\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 byŝ(t), are given byE\{s(t)|x(\cdot)\}, the conditional expectation ofs(t)givenx(\cdot). With the use of simultaneously orthogonal expansions ofx(t)andn(t), we further obtain explicit expressions in the form of infinite series forŝ(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)\}, thenŝ(t)and\epsilon(t)have the closed-form expressionsŝ(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,whereS(s,t), N(s,t)andX(s,t)are the covariances ofs(t), n(t)andx(t), respectively, andpis the largest integer for which the2pth 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 boths(t)andn(t)have rational power spectra.