A frequency-domain approach to interpolation from a nonuniform grid

We introduce a linear time-varying (LTV) system framework for the modeling and design of linear algorithms for interpolating band-limited signals from nonuniformly spaced samples. The LTV model characterizes the interpolation process in the frequency domain via the notion of bifrequency transmission function (BFTF). The BFTF provides a convenient means of assessing interpolator quality because it conveys how frequency components in the input are mapped to the frequency axis of the interpolated output. We show plots of BFTFs for several common linear interpolators, with an emphasis on a minimum-mean-squared-error method due to J.L. Yen. We prove that Yen's algorithm can be obtained as a special case of an optimal frequency-domain design of its BFTF using a weighted least-squares criterion. This design procedure is then generalized by using an unevenly weighted least-squares error measure that can incorporate knowledge of the approximate spectral shape of the original analog signal, to reduce interpolation error. Block interpolators are considered for situations where the number of signal samples is large or infinite, and methods are described for avoiding the matrix inversion to compute the interpolation. Finally, the performances of all interpolators studied in this paper are compared via simulation.