Singular Value Spectrum Research Articles

Chatter detection in milling based on singular spectrum analysis

Chatter is a frequently encountered problem in metal cutting field which reduces the machining efficiency and surface quality. Therefore, a reliable and robust chatter detection method is necessary to improve the machining performances. In this work, a novel milling chatter detection approach based on singular spectrum analysis (SSA) is proposed. SSA is applied to process the cutting force signal and extract the feature that is closely related to the machining state. The singular value spectrum obtained by SSA is used to describe the energy distribution of the principal modes in the signal. On the basis of frequency domain chatter theory, singular value entropy (SVE) is adopted to evaluate the variation of energy distribution in the signal and the milling chatter is detected accordingly. Milling experiments under different cutting conditions are performed out to verify the effectiveness of the proposed method. Experimental results demonstrate that the proposed method can accurately identify the onset of chatter. This method is simple in operation and fast in calculation, which makes it have great potential for online chatter detection.

Identification of coherent structures in the flow past a NACA0012 airfoil via proper orthogonal decomposition

Proper orthogonal decomposition, POD, is employed to identify coherent structures in the turbulent flow past a NACA0012 airfoil. An analysis of different POD techniques is presented including the standard snapshot method, the spectral POD (SPOD) method, and a Fourier-POD implementation combined with the SPOD. The latter technique is viable once the present turbulent flow has a homogeneous direction along the airfoil span. An assessment of different vector norms and filter functions employed in the POD reconstruction is presented. The evaluation of the several POD techniques allows the investigation of the physics of the current turbulent flow including the observation of coherent structures and their relation to the airfoil noise generation mechanisms. The SPOD technique is applied assuming periodic and non-periodic temporal signals in the construction of the covariance matrix. Although it slightly alters the spectrum of the POD singular values, the periodic SPOD considerably reduces the noise observed in the singular vectors. The application of the Gaussian filtering allows an enhanced control in the response of the SPOD when compared with a square-box filter. The present POD reconstructions employ norms based on kinetic energy and pressure. For both norms, two-dimensional coherent structures are observed along the turbulent boundary layer and wake regions for the first pair of modes. These structures have a spectral content at the same frequency of the tonal noise radiation by the airfoil. The POD analysis also allows the identification of further coherent structures in the present flow. The kinetic energy norm is able to reconstruct low-frequency structures in the flow field for higher POD modes, while the pressure norm reconstructs high-frequency structures. This behavior is related to the physics captured by each POD norm in the present turbulent flow.

An Impedance Boundary Condition EFIE That Is Low-Frequency and Refinement Stable

A discretization of the impedance boundary condition electric field integral equation (IBC-EFIE) is introduced that: 1) yields the correct solution at arbitrarily small frequencies and 2) requires for its solution a number of matrix vector products bounded as the frequency tends to zero and as the mesh density increases. The low frequency stabilization is based on a projector-based discrete Helmholtz splitting, rescaling, and recombination that depends on the low frequency behavior of both the EFIE operator and the surface impedance condition. The dense mesh stabilization is a modification of the perfect electric conductor operator preconditioning approach taking into account the effect on the singular value spectrum of the IBC term.

Noise correction algorithm for nonlinear turbulent shear velocity time series based on energy spectrum of singular values

Noise correction algorithm for nonlinear turbulent shear velocity time series based on energy spectrum of singular values

Contrasting eigenvalue and singular-value spectra for lasing and antilasing in aPT-symmetric periodic structure

It has been proposed and demonstrated that lasing and coherent perfect absorption (CPA or ``antilasing'') coexist in parity-time ($\mathcal{PT}$) symmetric photonic systems. In this work we show that the spectral signature of such a CPA laser displayed by the singular value spectrum of the scattering matrix ($S$) can be orders of magnitude wider than that displayed by the eigenvalue spectrum of $S$. Since the former reflects how strongly light can be absorbed or amplified and the latter announces the spontaneous symmetry breaking of $S$, these contrasting spectral signatures indicate that near perfect absorption and extremely strong amplification can be achieved even in the $\mathcal{PT}$-symmetric phase of $S$, which is known for and defined by its flux-conserving eigenstates. We also show that these contrasting spectral signatures are accompanied by strikingly different sensitivities to disorder and imperfection, suggesting that the eigenvalue spectrum is potentially suitable for sensing and the singular value spectrum for robust switching. A differential light amplifier may also be devised based on these two spectra.

Design and Analysis of a W-Band Metasurface-Based Computational Imaging System

We design and numerically analyze a coherent computational imaging system that utilizes a sparse detector array of planar, frequency-diverse, metasurface antennas designed to operate over the $W$ -band frequency range (75–110 GHz). Each of the metasurface antennas consists of a parallel plate waveguide, into which a center coaxial feed is inserted into the lower plate, launching a cylindrical guided wave. A dense array of metamaterial resonators patterned into the upper plate couples energy from the waveguide to free space radiative modes. The resonance frequency of each element, determined by its specific geometry, can be positioned anywhere within the $W$ -band. The geometry of each element is chosen to produce a resonance frequency selected randomly from the $W$ -band. Since a random subset of elements is resonant at any given frequency, the metasurface antenna forms a sequence of spatially diverse radiation patterns as a function of the excitation frequency. We analyze the metasurface aperture as an imaging system, optimizing key parameters relevant to image quality and resolution, including: aperture size; density and quality factor of the metamaterial resonators; number of detectors and their spatial distribution; bandwidth; and the number of frequency samples. A point-spread function analysis is used to compare the metasurface imager with traditional synthetic aperture radar. The singular value spectrum corresponding to the system transfer function and the mean-square-error associated with reconstructed images are both metrics used to characterize the system performance.

Purification of Rotor Center's Orbit Based on Singular Value Decomposition

Targeting at refinement of rotor center’s orbit, a novel purification method was presented by using singular value decomposition. Firstly, the original vibration signal was reconstructed in phase space by Hankel matrix; then the attractor track matrix was decomposed with singular value decomposition. Secondly, aiming at determine reconstruction order number of singular values, the concept of energy difference spectrum of singular values was defined; then the reconstruction order number was determined according to the peak position of the energy difference spectrum. Finally, the effectiveness of the method was proved by simulation and successful purification of rotor center’s orbits coming from the miniature test-bed and the turbine generator units in the power plant. And the results of comparing the performances of the proposed method to the harmonic window and the ensemble empirical mode decomposition (EEMD) purification method show the proposed method can retain the original signal characteristic effectively and eliminate noises as much as possible. It will provide a new way for rotor center’s orbit purification.

Development of a Method for Selection of Effective Singular Values in Bearing Fault Signal De-Noising

Singular value decomposition (SVD) is a widely used and powerful tool for signal extraction under noise. Noise attenuation relies on the selection of the effective singular value because these values are significant features of the useful signal. Traditional methods of selecting effective singular values (or selecting the useful components to rebuild the faulty signal) consist of seeking the maximum peak of the differential spectrum of singular values. However, owing to the small number of selected effective singular values, these methods lead to excessive de-noised effects. In order to get a more appropriate number of effective singular values, which preserves the components of the original signal as much as possible, this paper used a difference curvature spectrum of incremental singular entropy to determine the number of effective singular values. Then the position was found where the difference of two peaks in the spectrum declines in an infinitely large degree for the first time, and this position was regarded as the boundary of singular values between noise and a useful signal. The experimental results showed that the modified methods could accurately extract the non-stationary bearing faulty signal under real background noise.

Selection of Optimal Decomposition Layer for Thresholding Denoising Using Singular Spectrum Analysis and Wavelet Entropy

To optimize the number of decomposition layers in wavelet threshold denoising for ultrasonic signals, we propose a self-adaptive algorithm of determining the number of decomposition layers based on singular spectrum analysis and wavelet entropy. First the noise-containing signals are decomposed by discrete wavelet transform. The slope of the singular value spectrum for each layer is estimated. Then the wavelet entropy over the signal subinterval is calculated for each layer. Finally the optimal number of decomposition layer is determined by combining the entropy ratio of detail coefficients to original signal and the slope of the singular value spectrum. The performance of the algorithm is evaluated using signal-to-noise ratio (SNR) and the relative error of the peak value (REPV). Experiment shows that the algorithm can self-adaptively determine the optimal number of decomposition layers and filter out the noise contained in the ultrasonic signals. It not only increases the SNR, but also preserves valuable components of the original signal.

Model validity and frequency band selection in operational modal analysis

Experimental modal analysis aims at identifying the modal properties (e.g., natural frequencies, damping ratios, mode shapes) of a structure using vibration measurements. Two basic questions are encountered when operating in the frequency domain: Is there a mode near a particular frequency? If so, how much spectral data near the frequency can be included for modal identification without incurring significant modeling error? For data with high signal-to-noise (s/n) ratios these questions can be addressed using empirical tools such as singular value spectrum. Otherwise they are generally open and can be challenging, e.g., for modes with low s/n ratios or close modes. In this work these questions are addressed using a Bayesian approach. The focus is on operational modal analysis, i.e., with ‘output-only’ ambient data, where identification uncertainty and modeling error can be significant and their control is most demanding. The approach leads to ‘evidence ratios’ quantifying the relative plausibility of competing sets of modeling assumptions. The latter involves modeling the ‘what-if-not’ situation, which is non-trivial but is resolved by systematic consideration of alternative models and using maximum entropy principle. Synthetic and field data are considered to investigate the behavior of evidence ratios and how they should be interpreted in practical applications.

Nonrelativistic Banks-Casher relation and random matrix theory for multicomponent fermionic superfluids

We apply QCD-inspired techniques to study nonrelativistic N-component degenerate fermions with attractive interactions. By analyzing the singular-value spectrum of the fermion matrix in the Lagrangian, we derive several exact relations that characterize the spontaneous symmetry breaking U(1)xSU(N)$\to$Sp(N) through bifermion condensates. These are nonrelativistic analogues of the Banks-Casher relation and the Smilga-Stern relation in QCD. Non-local order parameters are also introduced and their spectral representations are derived, from which a nontrivial constraint on the phase diagram is obtained. The effective theory of soft collective excitations is derived and its equivalence to random matrix theory is demonstrated in the epsilon-regime. We numerically confirm the above analytical predictions in Monte Carlo simulations.

Growing Echo-State Network With Multiple Subreservoirs.

An echo-state network (ESN) is an effective alternative to gradient methods for training recurrent neural network. However, it is difficult to determine the structure (mainly the reservoir) of the ESN to match with the given application. In this paper, a growing ESN (GESN) is proposed to design the size and topology of the reservoir automatically. First, the GESN makes use of the block matrix theory to add hidden units to the existing reservoir group by group, which leads to a GESN with multiple subreservoirs. Second, every subreservoir weight matrix in the GESN is created with a predefined singular value spectrum, which ensures the echo-sate property of the ESN without posterior scaling of the weights. Third, during the growth of the network, the output weights of the GESN are updated in an incremental way. Moreover, the convergence of the GESN is proved. Finally, the GESN is tested on some artificial and real-world time-series benchmarks. Simulation results show that the proposed GESN has better prediction performance and faster leaning speed than some ESNs with fixed sizes and topologies.

Estimates of ocean forecast error covariance derived from Hessian Singular Vectors

Experience in numerical weather prediction suggests that singular value decomposition (SVD) of a forecast can yield useful a priori information about the growth of forecast errors. It has been shown formally that SVD using the inverse of the expected analysis error covariance matrix to define the norm at initial time yields the Empirical Orthogonal Functions (EOFs) of the forecast error covariance matrix at the final time. Because of their connection to the 2nd derivative of the cost function in 4-dimensional variational (4D-Var) data assimilation, the initial time singular vectors defined in this way are often referred to as the Hessian Singular Vectors (HSVs). In the present study, estimates of ocean forecast errors and forecast error covariance were computed using SVD applied to a baroclinically unstable temperature front in a re-entrant channel using the Regional Ocean Modeling System (ROMS). An identical twin approach was used in which a truth run of the model was sampled to generate synthetic hydrographic observations that were then assimilated into the same model started from an incorrect initial condition using 4D-Var. The 4D-Var system was run sequentially, and forecasts were initialized from each ocean analysis. SVD was performed on the resulting forecasts to compute the HSVs and corresponding EOFs of the expected forecast error covariance matrix. In this study, a reduced rank approximation of the inverse expected analysis error covariance matrix was used to compute the HSVs and EOFs based on the Lanczos vectors computed during the 4D-Var minimization of the cost function. This has the advantage that the entire spectrum of HSVs and EOFs in the reduced space can be computed. The associated singular value spectrum is found to yield consistent and reliable estimates of forecast error variance in the space spanned by the EOFs. In addition, at long forecast lead times the resulting HSVs and companion EOFs are able to capture many features of the actual realized forecast error at the largest scales. Forecast error growth via the HSVs was found to be significantly influenced by the non-normal character of the underlying forecast circulation, and is accompanied by a forward energy cascade, suggesting that forecast errors could be effectively controlled by reducing the error at the largest scales in the forecast initial conditions. A predictive relation for the amplitude of the basin integrated forecast error in terms of the mean aspect ratio of the forecast error hyperellipse (quantified in terms of the mean eccentricity) was also identified which could prove useful for predicting the level of forecast error a priori. All of these findings were found to be insensitive to the configuration of the 4D-Var data assimilation system and the resolution of the observing network.

Exact Mapping from Singular-Value Spectrum of Fractal Images to Entanglement Spectrum of One-Dimensional Quantum Systems

We examine the snapshot entropy of general fractal images defined by their singular values. Remarkably, the singular values for a large class of fractals are in exact correspondence with the entanglement spectrum of free fermions in one dimension. These fermions allow for a holographic interpretation of the logarithmic scaling of the snapshot entropy, which is in agreement with the Calabrese-Cardy formula. However, the coarse-grained entropy exhibits a linear scaling due to the degeneracy of the spectrum, in contrast with the logarithmic scaling behavior in one-dimensional quantum near-critical systems.

Analytical SVD for reconstruction of the radiating current in inverse EM scattering

In this paper, an analytical reconstruction of the radiating current for the inverse scattering problem with a line measurement configuration is presented. Here, a generalized continuous singular value decomposition (SVD) of the scattering integral operator is extracted by means of the spectral representation of Green’s function. The proposed formulation can be used in analyzing image resolution and reconstruction of the radiating currents. In addition, the singular value spectrum relation with radiating/non-radiating modes is studied. Study of the singular spectrum gives the opportunity to investigate the growing behavior of evanescent modes and the sub-wavelength focusing property in double negative slab. The singular value spectrum plays an important role in the truncation process of analytical SVD. By virtue of the proposed analytical method, the equivalent current reconstruction can be done in a very short computation time. In addition, the singular system description is based on medium Green’s function and it is applicable to 2-D inverse scattering problems.

Feature extraction for rolling bearing fault diagnosis by electrostatic monitoring sensors

There are mainly two problems with the current feature extraction methods used in the electrostatic monitoring of rolling bearings, which affect their abilities to identify early faults: (1) since noises are mixed in the electrostatic signals, it is difficult to extract weak early fault features; (2) traditional time and frequency domain features have limited ability to provide a quantitative indicator of degradation state. With regard to these two problems, a new feature extraction method for rolling bearing fault diagnosis by electrostatic monitoring sensors is proposed in this paper. First, the spectrum interpolation is adopted to suppress the power-frequency interference in the electrostatic signal. Then the resultant signal is used to construct Hankel matrix, the number of useful components is automatically selected based on the difference spectrum of singular values, after that the signal is reconstructed to remove background noises and random pulses. Finally, the permutation entropy of the denoised signal is calculated and smoothed using the exponential weighted moving average method, which is used to be a quantitative indicator of bearing performance state. The simulation and experimental results show that the proposed method can effectively remove noises and significantly bring forward the time when early faults are detected.

Enhancing microwave tomography in a circular metallic chamber by an inhomogeneous background

Microwave tomography is enhanced by using an inhomogeneous background. In this paper, the measurement region is located in a circular perfect electrical conductor (PEC) chamber where a known object is placed inside the imaging domain as an inhomogeneous background. This can not only make use of the prior information about the background, but also increase the equivalent radiation source for the target detection. The Green function of a circular PEC chamber with inhomogeneous background is obtained using the method of moments. Based on the Green functions for both homogeneous and inhomogeneous background in circular PEC chamber, the properties of the radiation operators are analyzed by comparing the condition numbers and the singular value spectra. Simulations are carried out in homogeneous, lossless inhomogeneous and lossy inhomogeneous backgrounds respectively, and the relative errors are discussed. The results show that using inhomogeneous background can improve the convergence rate and accuracy, and the lossy inhomogeneous background produces better results than the lossless one. In addition, it can enhance the inversion results without changing the microwave tomography system, which can be used in the medical imaging and industrial nondestructive detection.

An optimal layered inhomogeneous background used in microwave tomography system in metallic chamber

An optimal layered inhomogeneous background which can be used in an embedded microwave tomography system is proposed. The method is based on a new evaluation method of integral radiation operator with respect to an configuration and optimal methods such as simulated annealing method. First, the integral radiation operator is calculated using the finite element method. Then, a kind of metric which can be used to evaluate the operator is proposed. The metric contains information about the whole singular value spectrum of a integral radiation operator. A set of synthetic researches is performed to show the correlation between the metric and inversion error. The method can evaluate an integral radiation operator using a number, and it can be used in optimal process easily as the cost function. Simulated annealing method is employed to obtain the permittivity of each layer in the optimal layered inhomogeneous background. Finally, synthetic researches are employed both on simple target and complex target to test the optimal layered inhomogeneous background. The results show that the optimal layered inhomogeneous background can expedite the convergence process and more accurate inversion results can be obtained.

Coded apertures for x-ray scatter imaging

We examine coding strategies for coded aperture scatter imagers. Scatter imaging enables tomography of compact regions from snapshot measurements. We present coded aperture designs for pencil and fan beam geometries, and compare their singular value spectra with that of the Radon transform and selected volume tomography. We show that under dose constraints scatter imaging improves conditioning over alternative techniques, and that specially designed coded apertures enable snapshot 1D and 2D tomography.

Measuring large optical transmission matrices of disordered media.

We report a measurement of the large optical transmission matrix (TM) of a complex turbid medium. The TM is acquired using polarization-sensitive, full-field interferometric microscopy equipped with a rotating galvanometer mirror. It is represented with respect to input and output bases of optical modes, which correspond to plane wave components of the respective illumination and transmitted waves. The modes are sampled so finely in angular spectrum space that their number exceeds the total number of resolvable modes for the illuminated area of the sample. As such, we investigate the singular value spectrum of the TM in order to detect evidence of open transmission channels, predicted by random-matrix theory. Our results comport with theoretical expectations, given the experimental limitations of the system. We consider the impact of these limitations on the usefulness of transmission matrices in optical measurements.