Resampling in frequency domain, a method for interpolation of time series

Resampling is a process often used in geophysical applications. Especially in the field of seismic and radar data processing it is applied mainly to time series. In most cases the resampling process influences the contents of time series both, in the time domain and in the frequency domain. If a time series is resampled to a larger sampling interval the frequency spectrum will vary since the Nyquist frequency ifN) decreases and the frequency components higher then the new Nyquist frequency fold into the area below. Vice versa, by resampling to a smaller sampling interval, fNy is increasing and usually an unknown spectrum, depending on the used method will be added. Resampling is performed by interpolation. The simplest way is to use a linear function. On the other hand polynomial functions or cubic or rational splines lead to more satisfying results, but all of these methods influence the frequency spectrum. If the interpolation is performed in the frequency domain, the spectrum of the resampled data set can be controlled in such a way that it is unchanged. The principle of this method is based on the fact that the sampling interval of a time series in the time domain automatically defines the Nyquist frequency. But vice versa, in the frequency domain full control over the sampling interval is given by alteration Of/Ny' This work presents the above method and demonstrates that a discrete time series can be transformed into a continuous time series by moving/Ny to infinity and that it is very easy for that reason to resample to larger or smaller sampling intervals which are not integral multiples of the original sampling interval. The problem of aliasing is also discussed and practical examples for application of this method to GPR data are mentioned.