Fundamental frequency response bounds of direct-form recursive switched-capacitor filters with capacitance mismatch

A theoretical statistical analysis is developed to investigate the effects of random capacitance matching errors in the frequency response of recursive switched-capacitor filters implemented in direct form. As a result, with appropriate approximations, closed-form solutions for the mean and the standard deviation of the frequency response error are derived. The obtained expressions provide insight into the quantitative influence of capacitance ratio tolerance, numerator and denominator orders, passband and stopband ripples, and edge frequencies, and reveal existing tradeoffs among these parameters, so that the most efficient filter design can be found. The main theoretical results are extensively verified by simulation through Monte Carlo analysis to show the effectiveness of the proposed formulas.