A Comparison of the $\delta$ Parameterization and the $\tau$ Parameterization

This paper continues the analysis of biquad structures - discretized at high sample rates relative to the dynamic frequencies of the biquad - by introducing the τ parameterization as an alternative to the venerable δ parameterization [1], [2]. The τ parameterization has slightly improved coefficient accuracy as the scale factor between sample frequency and biquad dynamic frequencies falls below the typical several orders of magnitude assumed in the use of the δ parameterization. The use of the τ parameterization, which is based on a Trapezoidal Rule equivalent, provides an intuitively appealing result. In the mid-range sample rates, there is a potential for significant decrease in controller latency. Furthermore, the use of a biquad based structure such as the MultiNotch [3] or Biquad State Space [4], [5] allows us to seriously consider doing away with “one size fits all” discretization, applying τ or δ parameterized biquads for low frequency dynamics and traditional discrete biquads for high frequency dynamics. Finally, a restructuring of the fixed point calculations of the τ
^-1
integrator allows for improved signal fidelity while maintaining the signal growth immunity of the δ parameterization [6].