Stability and Accuracy of Discrete-Time High Pass Filters with Application to Geophone Deconvolution

Abstract

Low frequency noise in measured sensor data is amplified when integrated. In the integration of measured acceleration data to displacement, such low frequency noise can lead to significant drift errors. In non-real-time applications, time domain and frequency domain detrending methods can be employed to remove bias and drift errors. For real-time applications, recursive high-pass digital filters, such as Butterworth filters, are computationally simple to implement. This research focuses on developing a discrete time state-space model to simultaneously filter out low frequency noise, deconvolve, and integrate voltage measurements from a geophone sensor. A circuit model for the sensor was chosen. Forward and inverse dynamical systems describing the circuit were derived utilizing the theory of linear time-invariant systems. The stability and accuracy of Butterworth filter design using the bilinear transformation method can be affected by the filter order and cut-off frequency. This research reveals the root cause of the numerical instability of high-pass digital Butterworth filters having low cutoff frequencies (less than half a percent of the sampling frequency) and high filter orders (greater than 6). These instabilities arise when filter coefficients are computed from discrete time poles and can be avoided by converting a continuous-time state-space model for the filter to discrete time via a matrix exponential. The method is demonstrated using measured geophone data.