An optimum filter design for discrete integration

An optimum design for convolutional filters that integrate discrete data is presented here. Such integration filters are rarely found in the literature, although counterparts for discrete differentiation are rather common. The subject filter is first defined in the frequency domain as a band-limited version of the exact integration response. Subsequent inverse Fourier transformation produces an analytical expression in the time domain. Although the resulting digitized impulse response cannot be expressed in closed form, it can be divided into a two-point recursive filter component and a longer, nonrecursive component. The length of the nonrecursive component must be user-selected according to the accuracy desired in the frequency domain; that is, a shorter filter acts as an integrator over a narrower frequency band. The included graph of frequency range as a function of nonrecursive component length allows the user to specify the compromise he desires between accuracy and computational effort.