During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.

Variational Mode Decomposition

Harmonics estimation in emerging power system: Key issues and challenges

Estimation theory with applications to communications and control

A nonlinear filter is proposed for estimating a complex sinusoidal signal and its parameters (frequency, amplitude, and phase) from measurements corrupted by white noise. This filter is derived by applying an extended complex Kalman filter (ECKF) to a nonlinear stochastic system whose state variables are a function of its frequency and a sample of an original signal, and then, proof of the stability is given in the case of a single complex sinusoid. Simulations demonstrate that the proposed nonlinear filter is effective as a method for estimating a single complex sinusoid and its frequency under a low signal-to-noise ratio (SNR). In addition, the effect of the initial condition in the filter on frequency estimation is also discussed.

A nonlinear filter for estimating a sinusoidal signal and its parameters in white noise: on the case of a single sinusoid

The paper deals with developing and testing frequency-adaptive PMU algorithms with wider linearity range than specified in IEEE Std C37.118-1. This goal is achieved by means of three different concepts encompassing robust state-of-the-art design approaches: 1) FIR bandpass filtering, 2) extended Kalman filtering (EKF), and 3) discrete Fourier transform (DFT) demodulation with FIR low-pass smoothing. While FIR-based PMUs are linear phase with no overshoot in either phase or amplitude step responses, the adaptive EKF PMU is more computer-intensive but allows for a reduced group delay and better out-of-band interference rejection at the cost of a phase step response with overshoot. Frequency measurement performances of the various PMUs are assessed in detail. It turned out that FIR PMUs are best for meeting Std C37-118-1 metrics but they are outperformed by EKF under changing harmonics. Test results on three recent commercial PMU models further confirm that PMU algorithms meeting standard C37-118-1 can behave quite differently under dynamic conditions.

Wide Frequency Range Adaptive Phasor and Frequency PMU Algorithms

The problem of estimating time-varying harmonic components of a signal measured in noise is considered. The approach used is via state estimation. Two methods are proposed, one involving pole-placement of a state observer, the other using quadratic optimization techniques. The result is the development of a new class of filters, akin to recursive frequency-sampling filters, for inclusion in a parallel bank to produce sliding harmonic estimates. Kalman filtering theory is applied to effect the good performance in noise, and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate. These filters can be seen as generalizing the DFT.

P. Parker

Papers

A Kalman filtering approach to short-time Fourier analysis