The truncation error in the application of sampling series to electromagnetic problems

Truncation error interpolation methods used in the evaluation and representation of scattered fields are discussed. Three main points are considered: the choice of the sampling point lattice: the choice of the type of sampling expansion: and the determination of the number of terms of the sampling series that must be retained to insure a negligible truncation error. The aim is to provide a clear reference frame for the applications which allow the optimal choice for the reconstruction algorithm. The cases of bounded and square integrable band-limited functions are separately dealt with in order to exploit the properties of each method. A general analysis of the multidimensional truncation error is carried out. No particular assumption on the sampling expansion is made, except for that concerning the use of a rectangular sampling lattice. Simple and effective expressions relating the multidimensional error to the corresponding one-dimensional bounds are derived. The one-dimensional error is then analyzed in detail with reference to the most relevant case of central interpolation. By very simple methods, error bounds for all the sampling expansions explicitly considered in the literature are derived in a systematic way. Expressions providing the values of the sampling rate and number of retained samples which minimize the overall computer time and memory requirement are derived. Some numerical examples are presented and briefly discussed