A Class of Finite Memory Interpolation Filters

Sufficient conditions are given for an interpolation filter to have an impulse response that vanishes outside a finite interval of the time axis, that is to have a finite memory. These conditions are that the transfer function be of the form , where is proper, rational, and has poles limited to the strip ; and where is a polynomial. The filters are included in this class, and these are characterized by the fact that their effect is to interpolate an -order polynomial in each interval through past and future points. The interpolation filters described can be used to derive digital filters that approximate an arbitrary linear timeinvariant continuous-time operator. It is shown that in the case of integration, the filters lead to well-known Lagrangian integration formulas.