Approximating Infinite Impulse Response Filters with Finite Impulse Response Filters Using Allan Variance

This work presents the application of Allan VARiance (AVAR) to determine the order of an FIR filter whose output approximates that of an IIR filter. AVAR methods are typically used to analyze the variance of static windowed averages of data; prior recent work by the authors extended AVAR methods to include optimization of moving average filter designs, and more recently optimization of finite impulse response (FIR) filters. One of the main advantages of FIR over IIR filters is that the output of FIR depends only on the data size equal to one more than the filter order, whereas the data size influencing the output of the IIR filter increases with the length of data. A consequence of this is that the AVAR optimization of IIR filters remains an unsolved problem, but one that is solvable if IIR filters can be well approximated by FIR filter designs. In this work, the similarity between FIR and IIR filters is quantified using the normalized distance between the AVAR curves of error. This approximation is demonstrated through time-domain results of a signal with both low- and high-frequency noise contributions, namely drift (random walk) input corrupted by white noise. The results show that the AVAR-equivalent FIR filter output is similar to that of the IIR filter output if one chooses a sufficiently high FIR filter order. In addition, an iterative algorithm is presented to quickly estimate the necessary order of an FIR filter that is AVAR-equivalent to an IIR filter.