Linear phase FIR approximation of magnitude response \(|1/\omega|\) for maximal flatness at an arbitrary frequency \(\omega_0\), \(0< \omega_0<\pi\) (Q1970906)

In many signal processing situations, the desired magnitude response of a filter has the form of the rational function \(\text{MR}(\omega)=|1/\omega|\), i.e. the magnitude response of a digital integrator. The aim of this paper is to propose a novel technique to approximate \(\text{MR}(\omega)\) by linear-phase FIR (finite impulse response) structures. The suggested approximations are highly accurate over a band of frequencies centered around \(\omega_0\). The point \(\omega_0\) of the maximal flatness can be chosen in the frequency range \(0<\omega_0< \pi\). The authors derive, by a recursive technique, the exact mathematical formulas for computing the design weights. The proposed methodology is proved to be faster in comparison to other well-known optimization techniques of approximation. Design examples and practical uses illustrate the new approach. The suggested designs are shown to be particularly attractive for processing (integrating) narrowband signals, as they compute small relative errors.