Abstract 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.

中文翻译：

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前向后向过滤和惩罚最小二乘优化：统一框架

摘要 本文提出了一种统一惩罚最小二乘优化（PLSO）和前向后向滤波方案的框架。它提供了一个数学证明，即前向后向滤波（零相位 IIR 滤波器）可以作为 PLSO 的实例呈现。在此结果的基础上，本文提出了一种在时域和频域中设计和实现前向后向滤波和 PLSO 算法的统一方法。还提出了一种新的逐块矩阵公式，用于实现 PLSO 和前向后向滤波算法。本文中提出的方法特别适用于理解时域中零相位滤波器的任务和分析频域中的 PLSO 算法。在本文中，我们展示了时域中零相位数字巴特沃斯滤波器的任务是用脉冲序列拟合信号，并对拟合模型的导数进行惩罚。对于零相位数字切比雪夫滤波器，模型导数的线性组合用于惩罚项。