Minimum uncertainty filters for pulses

Abstract

The objective of this paper is to calculate filters with a minimum uncertainty, the product of filter length and bandwidth. The method is applicable to producing minimum uncertainty filters with time or frequency domain constraints on the filter. The calculus of variations is used to derive the conditions that minimize a filter’s uncertainty. The general solution is a linear combination of Hermite functions, where the Hermite functions are summed from low to high order until the filter’s constraints are met. Filters constrained to have zero amplitude at zero hertz have an uncertainty at least three times greater than expected from the uncertainty principle, and the minimum uncertainty filter is a first derivative Gaussian. For the previous filter, the minimum uncertainty high cut filter is a Gaussian function of frequency, but the minimum uncertainty low cut filter is a linear function of frequency.