Performance Analysis and Optimization of Finite Impulse Response Filters Using Allan Variance

The design of filters seeks a separation of noise from a desired signal, and the boundary between both is a tradeoff that is a fundamental topic in signal theory. In the presence of signals wherein noise properties have time-varying components, this tradeoff is particularly challenging to optimize. The Mean Squared Error (MSE) is a standard performance metric for evaluating the performance of filters. For signals with non-white noise characteristics - which encompass nearly all real-world signals - the calculation of MSE typically requires repeated analysis across multiple experiments. Prior work by the authors introduced and extended Allan VARiance (AVAR) methods, which analyze variances within increasing data windows, to optimize Moving Average (MA) filters. That work suggested an equivalence between the time-consuming iterative process of using the MSE for filter optimization versus an analysis of the area under an AVAR curve, which can be calculated in one step. This paper extends the use of the AVAR area method for selecting an optimal Finite Impulse Response (FIR) filter, where optimality is defined as the filter that minimizes the MSE between desired and filtered signals. Prior results are further extended to illustrate that the discrete integration of the AVAR curve yields a performance index that, in one step, generates the MSE-optimal filter for input with drift (random walk) corrupted by white noise. AVAR is compared against the MSE to show that both the performance indices give nearly equivalent optimal FIR filter designs. This AVAR FIR filter optimization is achieved with only one iteration versus hundreds of iterations to optimize filters using MSE calculations.