量子力学、香农采样定理与数理统计学的数学联系

对于原子的基本结构，目前物理学认为电子围绕原子核运动，其不同轨道之间通常是分离的。对波函数、坍塌的解释有不同的观点。从信号处理的角度看：不妨假设电子围绕原子核做连续的复杂运动；利用奈奎斯特-香农采样定理（Nyquist-Shannon sampling theorem），可以把高速运动的微观粒子的连续行为的长期观察结果，采样成不连续的概率方式，即量子力学。“标准”的量子力学建立时（1926年前后），数学里的概率论和数理统计学还没有发展成熟。数学工具的不足，是引起波函数与坍塌解释不同看法的数学原因。

Generalizing the sampling theorem for a frequency-time domain to sample signals under the conditions of a priori uncertainty

The radio monitoring of radiation and interference with electronic means is characterized by the issue related to the structural-parametric a priori uncertainty about the type and parameters of the ensemble of signals by radio-emitting sources. Given this, it is a relevant task to devise a technique for the mathematical notation of signals in order to implement their processing, overcoming their a priori uncertainty in terms of form and parameters. A given problem has been solved by the method of generalization and proof for the finite signals of the Whittaker-Kotelnikov-Shannon sampling theorem (WKS) in the frequency-time domain. The result of proving it is a new discrete frequency-temporal description of an arbitrary finite signal in the form of expansion into a double series on the orthogonal functions such as sinx/x, or rectangular Woodward strobe functions, with an explicit form of the phase-frequency-temporal modulation function. The properties of the sampling theorem in the frequency-time domain have been substantiated. These properties establish that the basis of the frequency-time representation is orthogonal, the accuracy of approximation by the basic functions sinx/x and rectangular Woodward strobe functions are the same, and correspond to the accuracy of the UCS theorem approximation, while the number of reference points of an arbitrary, limited in the width of the spectrum and duration, signal, now taken by frequency and time, is determined by the signal base. The devised description of signals in the frequency-time domain has been experimentally investigated using the detection-recovery of continuous, simple pulse, and linear-frequency-modulated (LFM) radio signals. The constructive nature of the resulting description has been confirmed, which is important and useful when devising methods, procedures, and algorithms for processing signals under the conditions of structural-parametric a priori uncertainty.

Overview of Compressed Sensing: Sensing Model, Reconstruction Algorithm, and Its Applications

With the development of intelligent networks such as the Internet of Things, network scales are becoming increasingly larger, and network environments increasingly complex, which brings a great challenge to network communication. The issues of energy-saving, transmission efficiency, and security were gradually highlighted. Compressed sensing (CS) helps to simultaneously solve those three problems in the communication of intelligent networks. In CS, fewer samples are required to reconstruct sparse or compressible signals, which breaks the restrict condition of a traditional Nyquist–Shannon sampling theorem. Here, we give an overview of recent CS studies, along the issues of sensing models, reconstruction algorithms, and their applications. First, we introduce several common sensing methods for CS, like sparse dictionary sensing, block-compressed sensing, and chaotic compressed sensing. We also present several state-of-the-art reconstruction algorithms of CS, including the convex optimization, greedy, and Bayesian algorithms. Lastly, we offer recommendation for broad CS applications, such as data compression, image processing, cryptography, and the reconstruction of complex networks. We discuss works related to CS technology and some CS essentials.

An MCM-Enhanced Compressive Sensing for Weak Fault Feature Extraction of Rolling Element Bearings under Variable Speeds

The compressive sensing (CS) theory provides a new slight to the big-data problem led by the Shannon sampling theorem in rolling element bearings condition monitoring, where the measurement matrix of CS tends to be designed by the random matrix (RM) to preserve the integrity of signal roughly. However, when the signal to be analyzed is infected with strong noise, not only does the signal become insufficiently sparse, but the randomness of the measurement matrix will bring down the sensing efficiency, resulting in the loss of fault feature. Thus, a sensing-enhanced CS scheme based on a series of modes after VMD decomposition is proposed under this paper. The core of this scheme is as follows: (1) the principal mode of VMD with better sparsity replaces the raw signal for compressive sensing; (2) all these modes contain the time-frequency characteristics of the raw signal; (3) a new measurement matrix called mode-circulant matrix (MCM) is defined by circulating the mode matrix, and when the amount of samples is shrunk, the sensing efficiency can be enhanced greatly. Besides, considering the fault signal of rolling bearings under variable speed, there is a need to use order tracking to overcome the nonstationarity of the signal before applying CS theory. The analysis results of simulation and experiment prove that the VMD- and MCM-based CS can successfully extract the weak fault feature of rolling bearings with operating speed changing.

Frequency Aliasing-Based Spatial-Wavenumber Filter for Online Damage Monitoring

The spatial-wavenumber filter method can extract the specific mode of the Lamb wave, thereby distinguishing the incident wave and the damage reflection wave. This method has been widely studied for damage imaging. However, the diameter of piezoelectric transducer (PZT) sensor limits the spatial sampling wavenumber of the linear PZT sensor array, which limits the application of this method because of the Nyquist–Shannon sampling theorem. Therefore, the wavenumber filtering range of spatial-wavenumber filter should be less than half of the spatial sampling wavenumber. In this paper, a frequency aliasing based spatial-wavenumber filter for online damage monitoring is proposed. In this method, the wavenumber filtering range is extended to the spatial sampling wavenumber, and two wavenumber results will be calculated as for the frequency aliasing. Subsequently, the wavenumber of the received Lamb wave signal can be obtained according to the average arrival time difference between the two adjacent sensors in the linear PZT sensor array. Finally, the damage is localized using the spatial-wavenumber filter and cruciform PZT sensor array. This method was validated on an epoxy laminate plate. The maximum damage localization errors are less than 2 cm. It is indicated that this method can extend the spatial-wavenumber filtering range to the spatial sampling wavenumber and the application of spatial-wavenumber filter-based online damage monitoring.

A Two-Stage Compression Method for the Fault Detection of Roller Bearings

Data measurement of roller bearings condition monitoring is carried out based on the Shannon sampling theorem, resulting in massive amounts of redundant information, which will lead to a big-data problem increasing the difficulty of roller bearing fault diagnosis. To overcome the aforementioned shortcoming, a two-stage compressed fault detection strategy is proposed in this study. First, a sliding window is utilized to divide the original signals into several segments and a selected symptom parameter is employed to represent each segment, through which a symptom parameter wave can be obtained and the raw vibration signals are compressed to a certain level with the faulty information remaining. Second, a fault detection scheme based on the compressed sensing is applied to extract the fault features, which can compress the symptom parameter wave thoroughly with a random matrix called the measurement matrix. The experimental results validate the effectiveness of the proposed method and the comparison of the three selected symptom parameters is also presented in this paper.