Noise analysis of single stage fractional-order low-pass filter using stochastic and fractional Calculus

Abstract

In this paper, we present the noise analysis of a simple single stage low-pass ¯lter (SSLPF) with the fractional-order capacitor, using stochastic differential equations (SDE). The input noise is considered to be white and various solution statistics of output namely mean, variance, auto-correlation and power spectral density (PSD) are obtained using tools from both stochastic and fractional calculus. We investigate the change in these statistics with the change in the capacitor order. The closed form solutions of the step response of the fractional filter are also provided and it has been found that filters with capacitor order greater than unity have a faster step response but suffer from higher output noise and although, the filters with capacitor order less than unity enjoy advantage of less output noise, but they have a sluggish step response. And hence, an appropriate fractional capacitor can be chosen for the desired circuit behavior. A brief study of more generic class of single stage fractional-order high-pass and all-pass ¯ltering functions has been included. The idea can be extended to more complex and practical fractional order circuits.