Time-Limited Time-Invariant Wiener Filtering

For the estimation of a signal observed with additive white noise, it is shown that the optimum linear least-squares filter constrained to have its impulse response time-limited to the interval [0,T] satisfies a truncated version of the Wiener-Hopf equation. To solve this equation the covariance for the observed process need only be known for time lags less than T. There is a unique extension of the covariance for lags greater than T, for which the time-limited filter is the optimum Wiener filter; furthermore this same extension is that extension of the covariance for which the optimum Wiener filter gives maximum mean square error, i.e., given limited covariance information we have found the "worst possible" extension of the known information. Parallels are drawn with discrete-time maximum-entropy spectral analysis.