Uniformly Damped Binomial Filters: Five-percent Maximum Overshoot Optimal Response Design

In this paper, the five-percent maximum overshoot design of uniformly damped binomial filters (transfer functions) is introduced. First, the Butterworth filter response is represented as a damped binomial filter response. To extend the maximum overshoot response of the second-order Butterworth to higher orders, the binomial theorem is extended to the uniformly damped binomial theorem. It is shown that the five-percent uniformly damped binomial filter is a compromise between the Butterworth filter and the standard binomial filter, with respect to the filter approximation problem in the time and frequency domain. Finally, this paper concludes that in applications of interest, such as step-tracking, where both strong filtering and a fast, smooth transient response with negligible overshoot are desired, the response of the normalized five-percent uniformly damped binomial form is a candidate replacement for both the Butterworth and standard binomial filter forms.