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 forwardbackward 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 forwardbackward 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.
Smoothness priors is a well-known and most commonly used method in the analysis of stochastic processes making it very useful in the field of stochastic signal processing. It is particularly suited for smoothing the noisy data and detrending the time-series signals. The method is based on an optimization problem where the n-th order derivative of the signal enters as a constraint. When the method is designed in discrete time domain, the backward difference rule is used to perform differential-to-difference conversion. Moreover, the solution depends on a smoothness trade-off parameter. An efficient algorithm for the trade-off parameter selection remains an important and challenging issue. In this paper, first, we propose a closed-form expression for the tradeoff parameter. The closed-form expression resulted from a frequency domain interpretation of the smoothness priors procedure. The trade-off parameter determines the amount of frequency components that the procedure allows to pass. We show that the trade-off parameter is related to the arbitrary choice of cutoff frequency. Second, we introduce a new way to the design and implementation of smoothness priors using bilinear transformation method. Frequency analysis and experiments on both synthetic and real world signals with different levels of noise demonstrate that bilinear transform is indeed more effective for smoothness priors implementation when compared with the traditional ones, i.e., the backward difference rule.
The V-index, an ECG marker quantifying spatial heterogeneity of ventricular repolarization, significantly improves the accuracy and sensitivity of the ECG for the diagnosis of NSTEMI and independently predicts mortality during follow-up.
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