Forward-backward filtering and penalized least-Squares optimization: A Unified framework

The paper proposes a framework for unification of the penalized least-squares optimization (PLSO) and forward-backward filtering scheme. It provides a mathematical proof that forward-backward filtering (zero-phase IIR filters) can be presented as instances of PLSO. On the basis of this result, the paper then represents a unifying approach to the design and implementation of forward-backward filtering and PLSO algorithms in the time and frequency domain. A new block-wise matrix formulation is also presented for implementing the PLSO and forward-backward filtering algorithms. The approach presented in this paper is particularly suited for understanding the task of zero-phase filters in the time domain and analyzing PLSO algorithms in the frequency domain. In this paper, we show that the task of a zero-phase digital Butterworth filter in the time domain is to fit the signal with impulse train and penalties on the derivatives of the fitted model. For a zero-phase digital Chebyshev filter, a linear combination of derivatives of the model is used in the penalty term.