A comparison of Δ coefficients and the δ parameterization, part I: Coefficient accuracy

The multinotch was introduced in [1] as a way to parameterize digital filters so as to preserve numerical fidelity of the filter while providing precalculation to reduce computational latency. Still, as the sample rate got large relative to the frequencies being filtered, the biquad coefficients in the multinotch got sensitive. A coefficient adjustment called Δ coefficients introduced in [2] worked extremely well on coefficient sensitivity, but did not address potential signal overflow problems. The δ parameterization [3], [4] is another method of adjusting digital filter coefficients to compensate for relatively high sample rates. The δ operator also has the advantage of generating a differential form of the filter [5], which - for high sample rates - approximates the internal behavior of a cascade of analog biquad filters. This paper will compare the coefficient accuracy of biquad filters that are parameterized via fixed point δ operators with those parameterized via fixed point Δ coefficients. Follow on comparisons of internal signal growth issues will be studied in [6].