New Digital Infinite Impulse Response Filters on Atomic Function ha(x)

Abstract

Digital infinite impulse response low pass filters on atomic functions $\text{h}_{a}$(x) are first proposed. Due to its compactly support and the presence of interval where it is constant, atomic function $\text{h}_{a}(x)$ can play the role of an ideal low pass filter magnitude response. Since $\text{h}_{a}(x)$ is an infinitely differentiable function its spectrum decay is plenty fast. The synthesis of new infinite impulse response filters is performed by using analogue filter prototypes. Herewith the recently designed construction method for analogue low pass filter with magnitude response approximating function $\text{h}_{a}(x)$ is applied. This method is based on representation of truncated entire function using Cauchy integral formula. To carry out the approximation the partial Fourier sums of $\text{h}_{a}(x)$ function are used in the formula. Rational fractions which approximate atomic function $\text{h}_{a}(x)$ can be found by replacement of integral with Riemann sum. Then, the problem of integration contour optimization need to be solved. The obtained rational functions have the convenient form of representation with sum of partial fractions. After receiving the nonnegative rational fraction which defines the frequency response of analogue filter the transfer function of digital infinite impulse response filter is obtained using standard transforms. New digital filters are stable since the poles of their transfer functions are situated inside the unit circle. The magnitude responses of new digital infinite impulse response filters approximates the infinitely smooth atomic function $\text{h}_{a}(x)$. This property makes the new low pass filters substantially different from classical low pass infinite impulse response filters on Butterworth, Chebyshev and elliptical analogue prototypes which magnitude responses with order increase are approaching to discontinuous rectangular function. Due to the shape of new filters magnitude responses their impulse responses are fast decaying functions.