Synthesis of a Complex RⁱCR Filter with Flat Passband Characteristics

Abstract

In this paper, a synthesis of a RⁱCR filter with flat passband characteristics is proposed. Because the RⁱCR filter include no inductors, we can reduce the required elements when we realize the filter using OTA's and C's. The transfer function T_c of the proposed circuit is obtained through the frequency transformation based on the special case of BLLT. Its prototype transfer function T_b is also a complex transfer function and has multiple transmission zeros placed at ω=-1 and 1. The kernel function of T_b is obtained by an analytical method not by numerical calculations depending on computers as used in many of the conventional methods. The proposed filter can be designed through a graphical method based on the relationship among ω_c/b_w (selectivity), ω_c (center frequency) and A_min (minimum attenuation). The proposed circuit can be composed of 2n+4 OTAs and 2n capacitors. As an example, a 5th-order RⁱCR filter is designed. The proposed filter is compared with the complex Butterworth filter from the view point of the frequency characteristics and the number of the required elements. It is shown that the proposed RⁱCR filter has maximally flat response in the passband.