Kalman filter/smoother-based design and implementation of digital IIR filters

Recently, a unified framework was proposed for forward-backward filtering and penalized least-squares optimization. It was shown that forward-backward filtering can be presented as instances of penalized least-squares optimization. In other words, the output of a zero-phase digital infinite-impulse response (IIR) filter can be computed by solving a constrained optimization problem, in which the weight controlling the constraint is directly related to cutoff frequencies with closed-form equations. It was also shown that a zero-phase digital IIR filter can be formed as an optimal smoothing Wiener filter for a random process obtained from an autoregressive (AR) or AR-moving average (ARMA) model driven by input (innovation) noise in presence of an observation noise. In this paper, the problem of zero-phase digital IIR filtering is re-examined using Kalman filter/smoother. The paper shows that every zero-phase digital IIR filter can be viewed as a special case of an optimal smoothing Wiener filter. Based on the fact that the formulations of the optimum filter by Wiener and Kalman are equivalent in steady state, we present a Kalman filter/smoother framework to the design and implementation of digital IIR filters. As an example, the zero-phase digital Butterworth filter is designed using Kalman smoother and compared with the traditional design (forward filtering and backward smoothing) method.