In this paper we propose a novel adaptive nonparametric regularization (NPR) method for solving optimization problems with linear constraint. The regular parameter in NPR algorithm is adaptively updated. We prove that the NPR is convergent and provide an early stop method (ESM) with a termination criterion to handle the perturbation problem which is unsolved in other methods such as the augmented Lagrange method (ALM). We then introduce a new differential operator which can absolutely annihilate the amplitude-modulated and frequency-modulated (AM-FM) signals (AM-FM operator, AFO). The proposed operator is a precise operator and thus can obtain a more accurate solution in operator based signal demodulation and separation problems. We apply the NPR algorithm in signal demodulation and separation based on the AFO and propose signal demodulation (NPR-AFOSD) and separation (NPR-AFOSS) algorithms. The experimental results on both synthetic AM-FM signals and the real-life data demonstrate that the proposed demodulation and separation methods are more effective and robust than the state-of-the-art methods.

在本文中，我们提出了一种新颖的自适应非参数正则化（NPR）方法，用于解决具有线性约束的优化问题。NPR算法中的常规参数会自适应更新。我们证明了NPR是收敛的，并提供了一种具有终止准则的提前停止方法（ESM）来处理摄动问题，而该问题是其他方法（如增强拉格朗日方法（ALM））无法解决的。然后，我们引入了一种新的差分运算符，它可以完全消除振幅调制和频率调制（AM-FM）信号（AM-FM运算符，AFO）。所提出的算子是精确算子，因此可以在基于算子的信号解调和分离问题中获得更精确的解决方案。我们将NPR算法应用于基于AFO的信号解调和分离，并提出信号解调（NPR-AFOSD）和分离（NPR-AFOSS）算法。对合成AM-FM信号和实际数据的实验结果表明，所提出的解调和分离方法比最新技术更有效，更稳健。