Worst-Case Uncertainty Bounds in Covariance Interpolation

In this work, we establish and compute worst case bounds in covariance interpolation for continuous-time stationary stochastic processes, a problem that appears in applications such as broad-band direction-of-arrival estimation and optimal sensor placement. More specifically, for two such stochastic processes whose covariance functions agree on a finite discrete set of time-lags, we would like to compute the maximal possible discrepancy of the covariance functions for real-valued time-lags outside this discrete grid. In array processing, this difference quantifies the inherent uncertainty pertaining to the discrete sampling of space determined by the array geometry. Computing this uncertainty corresponds to solving an infinite-dimensional non-convex problem. However, we herein prove that the maximal objective value may be bounded from above by a finite-dimensional convex optimization problem, allowing for efficient computation by standard methods. Furthermore, we empirically observe that for the case of signals whose spectra are supported on an interval, this upper bound is sharp, i.e., provides an exact quantification of the covariance uncertainty.