Energy-based effective length of the impulse response of a recursive filter

A measure for the effective length of the impulse response of a stable recursive digital filter based on accumulated energy is proposed. The new measure finds applications in several fields of digital signal processing, including estimation of the extent of attack transients for filters with dynamically varying inputs, elimination of transients in variable recursive filters, and design and implementation of linear-phase IIR systems. A general definition and a simple algorithm to evaluate it are introduced, and closed-form expressions are derived for first and second-order all-pole filters. The effect of zeros on the effective length is analyzed. An upper bound for the effective length of higher-order filters is derived using results for low-order filters, which is illustrated for classical digital lowpass filters. The use of the measure is demonstrated with examples of implementation of linear-phase IIR systems and estimation of transients in variable IIR filters.