A New Formulation for Design of Digital IIR Band Stop Filters

This paper introduces a new general mathematical method for calculating the poles and zeros of a digital filter when transforming the poles and zeros of an analog filter in the s-domain into the poles and zeros of a digital filter in the zdomain by using inverse bilinear z-transformation with prewarping frequency technique. This paper also analyzes the poles and zeros of the digital Butterworth filters and indicates where they are distributed in the geometric curve in the zdomain, which allows visualization of the pole-zero locations and the stability of the desired digital filter design. In addition, the center frequency and the notch frequency are discussed in the designing of the digital Butterworth band stop. From that, the relationships among the lower frequency, the upper frequency and the notch frequency are derived and used as a new method to design a digital Butterworth band stop filter. In a special case, in which the notch frequency is equal to onequarter of the sampling frequency fs, the pole-zero transfer functions of the digital Butterworth filters are also investigated and it can be used to design a half-band filter in multi-rate digital signal processing.