Method and Algorithm for Synthesis of Controlled Digital Low-Pass Butterworth Filters on the Example of A 4th Order Filter

The article presents a method for synthesizing controlled digital recursive low-pass filters with an infinite impulse response and its modification for Butterworth filters. Controllability refers to the explicit dependence of the filter coefficients on the cutoff frequency. The technique is based on a bilinear transformation of the transfer function of an analog low-pass filter prototype and frequency transformation of the amplitude-frequency characteristics of the resulting digital filter. The main idea of the technique is that for an analog prototype filter with a cutoff frequency of 1 rad/s, the parameters of the transfer function of biquadrate or bilinear links having a frequency dimension will be numerically equal to the correction coefficients of similar parameters of a controlled filter with an arbitrary cutofffrequency. As an example, the case of synthesis of a fourth-order Butterworth digital filter is considered. In this case, the transfer function of a filter of arbitrary order is represented as a cascade connection of links of order II, if the filter is of even order (in the case of odd order, one cascaded link of I order is added). The algorithm is implemented in the Mathcad environment. As an example, a digital recursive low-pass Butterworth filter 4
^th
order is calculated.