Abstract

Resolving the fine-scale details of a signal from coarse-scale measurements is a classical problem in signal processing. This problem is usually formulated in terms of extrapolation in frequency, i.e., as extrapolating the Fourier transform of the signal from a set of low-frequencies to a larger set. An approach to perform extrapolation in frequency is to use a multiplier, or a filter, that minimizes a suitable approximation error metric over a known collection of signals. However, one of the drawbacks of this approach is that this multiplier is not able to exploit the relations between the signals in the collection. In this work, we propose a formulation that is translation-invariant, finding both the optimal multipliers and the optimal centering for the signals in the collection. A consequence of our formulation is that the optimal centering does not correspond to a usual choice such as the center of mass. We perform numerical experiments supporting our claims.