Nonuniform Spectrum Research Articles

The relation between μ-dichotomy spectrum and Sacker-Sell spectrum

We present a relation between μ-dichotomy spectrum and Sacker-Sell spectrum. From this relation, we show that, under some suitable conditions, if the system x′=A(t)x admits a μ-dichotomy, then it admits an exponential dichotomy. Utilizing this result, we present a linearization theorem. Furthermore, we prove that the μ-dichotomy spectrum of the system x′=diag(A1(t),A2(t))x is the union of μ-dichotomy spectra of the subsystems x1′=A1(t)x1 and x2′=A2(t)x2, while this result is wrong if we replace μ-dichotomy spectrum with nonuniform exponential dichotomy spectrum. We also provide a new method to prove the μ-dichotomy spectral theorem.

A new equivalent acceleration-based non-uniform response spectrum method for efficient seismic SSI analysis

A complex multiple-support response spectrum (CMSRS) method with a concise and universal expression is required to estimate the seismic responses of soil–structure interaction (SSI) systems under non-uniform seismic excitations to replace existing complex methods. This paper introduces and defines a new concept called equivalent acceleration of boundary load (EABL) that converts widely used equivalent loads into equivalent accelerations, and further develops an efficient CMSRS-EABL method for the seismic analysis of the SSI system. First, EABL offers three main benefits: it simplifies the expressions without sacrificing precision, extends the applicable scope of viscous-spring artificial boundary (VSAB) to all stress-type artificial boundaries, and presents dynamic equations in a familiar form. Subsequently, the CMSRS–EABL formula was successfully deduced based on the equivalent simplified dynamic equations. This formula significantly simplifies the traditional method by reducing the effective modal participation factors, mean-square-roots, and cross-correlation coefficients by 4, 5 and 15 items, respectively. Finally, a rectangular construction surrounded by a VSAB–soil combination was used as an example to illustrate the convenience of the developed CMSRS–EABL method and verify its precision. The results indicate that the proposed method is characterized by greater simplicity and ingenuity, and it is capable of high precision similar to that of the conventional methods. We believe this method will be of significant benefit in evaluating the anti-seismic performance of deeply buried and long-standing structures.

An efficient and accurate DSRFG method via nonuniform energy spectra discretization

An appropriate set-up of inflow boundary conditions is crucial for accurate prediction of wind effects on the target structures using Large Eddy Simulation (LES). The discretizing and synthesizing random flow generation (DSRFG) method is a commonly used method which however requires considerable computational cost. This paper attempts to improve the efficiency and accuracy of the DSRFG method by modifying the way of discretizing the energy spectra. Since the DSRFG method is originated from the Kraichnan’s method, the grid effects on both methods are analyzed theoretically and examined numerically. Research results show that the Kraichnan’s method is sensitive to the grid effects while the improved DSRFG method is slightly affected. Additionally, the improved DSRFG method is applied to simulate the wind effects on a low-rise building by LES and the results agree well with the NIST aerodynamic database.

Nonuniform μ-dichotomy spectrum and kinematic similarity

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate μ in a large family of growth rates, we consider a notion of spectrum, named nonuniform μ-dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate μ, we describe all possible forms of the nonuniform μ-dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior.

Effect of GeO2 Dopants in FBG Sensor Performance for Temperature and Strain

In this simulation study, the response of fiber Bragg grating (FBG) sensors is investigated and optimized. Uniform and nonuniform FBG spectra with new component are suggested by fine selection with (COMSUL program) and compared theoretically under the effect of several external strain values (0.005, 0.006, 0.007, 0.008, 0.009 and 0.01). These two types operation have been examined by the Optisystem programmer. The measured sensitivity was based on VCSEL laser source with operation wavelengths of 1650, 1600, and 1550 nm via non-uniform and uniform configuration. The achieved sensitivity was found to have different values; 5.7, 2.6, and 1.77, while the highest observed sensitivity value is recorded at a wavelength of 1550 nm. Accordingly, this wavelength was chosen to advance the study. Temperatures of 20, 30, 40, 50, and 60 degrees Celsius were applied. Measured sensitivity between them varied, and satisfied the following functions: sine, Gauss, and Boltzmann indicating altering in sensor responses.

On-chip non-uniformly spaced multi-channel microwave photonic signal processor based on an ultrahigh-Q multimode micro-disk resonator.

Multi-channel microwave photonic (MWP) signal processing can simultaneously perform different task operations on multiple signals carried by multiple wavelengths, which holds great potential for ultrafast signal processing and characterization in a wavelength-division-multiplexed (WDM) network. As emerging telecommunication services create more data, an elastic optical network, which has a flexible and non-uniform spectrum channel spacing, is an alternative architecture to meet the ever-increasing data transfer need. Here, for the multi-channel ultra-fast signal processing in the elastic optical network, we propose and demonstrate an on-chip non-uniformly spaced multi-channel microwave photonic signal processor based on an ultrahigh-Q multimode micro-disk resonator (MDR). In the proposed signal processor, an MDR supporting multiple different order whispering-gallery modes (WGMs) with an ultrahigh Q-factor is specifically designed. Benefiting from the large and different free spectral ranges (FSRs) provided by the different order WGMs, a non-uniformly spaced multi-channel microwave photonic signal processor is realized, and various processing functions are experimentally demonstrated including bandpass filtering with a narrow passband of 103 MHz, a rejection ratio of 22.3 dB and a frequency tuning range from 1 to 30 GHz, multiple frequency measurement with a frequency measurement range from 1 to 30 GHz, a frequency resolution better than 200 MHz and a measurement accuracy of 91.3 MHz, and phase shifting with a phase tuning range from -170°∼160°, an operational bandwidth of 26 GHz from 6 GHz to 32 GHz and a small power variation of 0.43 dB. Thanks to the coexistence of different order WGMs supported by the MDR, non-uniformly spaced multi-channel signal processing is enabled with the key advantages including a broad operation bandwidth, an ultra-narrow frequency selectivity, and a large phase tuning range with a small power variation. The proposed signal processor is promising to be widely used in future elastic optical networks with flexible spectrum grids.

Non-uniform angular spectrum method in a complex medium based on iteration.

The traditional angular spectrum method has an inherent problem that the region of diffraction propagation should be homogeneous. However, in some cases, the medium of the diffraction propagation region is inhomogeneous. In this Letter, based on iteration we proposed the non-uniform angular spectrum method for diffraction propagation calculation in a complex medium. By phase pre-processing in the spatial domain and diffraction calculation in the spatial frequency domain, the diffraction propagation problem of the light field in a complex medium is solved. Theoretical formulation and numerical examples as well as experimental investigation are presented to confirm the validity of the proposed method. The advantages of this method include faster computation, smaller memory requirement, and the ability to compute a larger area compared with the finite element method as well as the ability to compute the non-paraxial case compared with the standard fast Fourier transform beam propagation method.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

Robust Spectral Peaks Detection in Vibration and Acoustic Signals

This paper brings a practical solution to the problem of spectral peak detection in non-uniform spectra. It applies a robust probabilistic approach that fits the histogram of trimmed spectral data with a truncated Gamma distribution. The estimated distribution parameters are used to derive a threshold through a hypothesis test on the presence of peaks. The proposed approach gains its robustness from the formulation of the no-peak distribution, while no knowledge is available about the amount of peaks in spectral data. The authors propose a pre-processing step to cope with a non-uniform spectrum. The proposed methodology is validated on both simulated and experimental vibration and acoustic signals.

Blocked Azimuth Spectrum Reconstruction Algorithm for Onboard Real-Time Dual-Channel SAR Imaging

Dual-channel synthetic aperture radar (SAR) is a widely used technology to achieve high-resolution and wide-swath imaging in current SAR system. Due to the fact of nonuniform sampling and spectrum ambiguity, the Doppler spectrum must be reconstructed before focusing. However, the traditional spectrum reconstruction algorithms require large onboard storage memory and computational resources and hence, make it hard to realize real-time processing. To solve the problem, this letter proposes a blocked spectrum reconstruction algorithm for onboard real-time imaging. Different from the traditional ones, the new method updates parameters block by block in azimuth, and hence, reducing the storage and computational resources at the cost of sacrificing the accuracy of the spectrum reconstruction to some extent. Fortunately, the accuracy loss is acceptable under properly set focusing depth, i.e., azimuth block width. The simulation and real data experiments verify the proposed approach.

Efficient Bit Loading Algorithm for OFDM-NOMA Systems With BER Constraints

This paper proposes a novel bit loading algorithm to multicarrier non-orthogonal multiple access (NOMA) systems to maximize the system total throughput while satisfying the users’ individual quality of service (QoS) constraints. Although bit loading is generally a non-deterministic polynomial-time (NP)-hard problem, even for orthogonal multiple access (OMA), the mutual interference between the users, and the dependence of power coefficients and modulation orders are additional challenges that add substantial complexity to the optimization problem. Therefore, in this paper, we propose an efficient bit loading algorithm for multicarrier NOMA systems and compare the complexity and throughput with a corresponding OMA system. The obtained results show that for the case of NOMA, bit loading provides an additional degree of freedom that allows non-uniform spectrum sharing among the users. In particular scenarios, the system may experience hybrid modes of operation, where the system switches between NOMA and OMA. The obtained numerical results show that NOMA can outperform OMA by 100% in terms of spectral efficiency, for a two-user scenario. The complexity of the loading process for NOMA is noticeably higher than OMA, which is due to the high complexity associated with the bit error rate (BER) computation in the case of NOMA.

In-Situ Evaluation of the Pitch of a Reflective-Type Scale Grating by Using a Mode-Locked Femtosecond Laser

Major modifications are made to the setup and signal processing of the method of in-situ measurement of the pitch of a diffraction grating based on the angles of diffraction of the diffracted optical frequency comb laser emanated from the grating. In the method, the improvement of the uncertainty of in-situ pitch measurement can be expected since every mode in the diffracted optical frequency comb laser can be utilized. Instead of employing a Fabry-Pérot etalon for the separation of the neighboring modes in the group of the diffracted laser beams, the weight-of-mass method is introduced in the method to detect the light wavelength in the Littrow configuration. An attempt is also made to reduce the influence of the non-uniform spectrum of the optical comb laser employed in the setup through normalization operation. In addition, an optical alignment technique with the employment of a retroreflector is introduced for the precise alignment of optical components in the setup. Furthermore, a mathematical model of the pitch measurement by the proposed method is established, and theoretical analysis on the uncertainty of pitch measurement is carried out based on the guide to the expression of uncertainty in measurement (GUM).

Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

Abstract For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

EDFA gain flattening optimization with long period fiber gratings in WDM system

Abstract This paper highlights a proposed optimized gain flatness technique for nonuniform gain spectrum of the erbium-doped fiber amplifier (EDFA) by introduction of long period fiber gratings (LPFG). In this analysis, EDFA gain spectrum has been evaluated between 1525 and 1600 nm with −20 dBm input signal power. Attenuation peaks of LPFG are optimized with a grating period of 240 µm and grating length of 30000 µm in the wavelength range of 1.1–1.8 μm, to compensate the peak gain of EDFA. Results have shown that EDFA peak gain of 35.94 dB is achieved at 1532.89 nm wavelength. This 35.94 dB peak gain is flattened up to 2.65 dB using long period fiber gratings. Also, less than ±0.3 dB gain flatness is achieved between 1528 and 1560 nm wavelength. The proposed less complex technique can be used to modify the grating parameters during fabrication in order to produce the loss peak at desired wavelength, which is efficient to flatten EDFA gain peak.

Non-uniform adaptive angular spectrum method and its application to neural network assisted coherent beam combining

Increasing the number of laser beams that can be coherently combined requires accurate and fast algorithms for compensating phase and alignment errors. The paper proposes to use a Fully Connected Artificial Neural Network (FCANN) to correct the beam positioning perturbations by evaluating the beam shifts and tilts from two images taken at slightly different locations. Then, since it is practically impossible to have a large enough experimental dataset to train the neural network, this approach required developing an accurate and fast simulation method to evaluate the beam propagation in arbitrary directions, overcoming the limitations occurring when the computation must be repeated a large number of times. The numerical approach is a variant of the Angular Spectrum (AS) method, called Non Uniform ADaptive Angular Spectrum (NUADAS) method, which relies on the combination of non-uniform and adaptive Fourier transform algorithms to allow the computation of an arbitrary field distribution in a plane that is shifted and tilted with respect to the source. The parallel implementation of the NUADAS method is discussed and the numerical and experimental validations are presented. Then, an FCANN is trained using the synthetic dataset generated with the NUADAS method and the results are discussed, demonstrating the viability of the proposed approach not only for coherent beam combing, but also in other beam alignment applications.

Smooth linearization under nonuniform hyperbolicity

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

A Novel Automatic White Balance Method for Color Constancy Under Different Color Temperatures

The spectrum distribution of the light source between scenes is usually uniform in all scenes lit by a given light source per the color constancy assumption. Nonuniform spectrum distribution results in pictures with different color temperatures in different lighting conditions. With the conventional automated white balance method, the results under different degrees of color cast are inconsistent. To resolve this, a new automated white balance algorithm is proposed to achieve uniformity in images with color cast caused by color temperature variations. The proposed method features the selection of near-white points through automated regression and the dynamic adjustment of the color components’ weights. Finally, white balance is implemented according to the selected near-white points with various percentages. The proposed method can effectively select the appropriate white reference point of the image scene under different color temperatures, so that the adjusted results of the method under different color temperatures are close. The first major contribution of this study is the minimization of differences of continual color temperature images with different color temperatures and the achievement of favorable visual effects. The second contribution is the superior outcomes yielded by the proposed method in the visualization and quantification of data compared with those from the existing methods. The experimental results demonstrate that this method yields favorable results in the improvement of experiments with continual different color temperatures. The images at various color temperatures are adjusted to approach real-world color temperatures. In addition, this method provides superior color constancy preservation to other methods.

Modeling of Active Fiber Loop Ring-Down Spectroscopy Considering Non-Linear Effects of EDFA Gain

Non-linear effects of erbium-doped fiber amplifier (EDFA) gain have been significant factors in deteriorating the performance of active fiber loop ring-down spectroscopy (FLRDS). The non-linear effects involve two aspects: the gain saturation and the non-uniformity of the gain spectrum. The gain saturation of EDFA induces pulse-to-pulse gain fluctuations, which causes ring-down decays to deviate from the exponential law. The non-uniform gain spectrum introduces measurement errors in the active FLRDS systems operating in wavelength-tuning or wavelength-switching modes. A theoretical model of active FLRDS that integrates both non-linear gain effects is presented in this paper. It can be used for qualitative and quantitative analysis of the measurement errors introduced by the non-linear gain effects on the active FLRDS. The model provides theoretical guidance to the performance prediction and optimization of the active FLRDS systems.

Performance investigation of hybrid optical amplifiers for high-speed optical networks

The development of lightwave systems required large capacity communication networks in today’s high-tech era. To fulfill these challenges required proper amplification of the signals. Hybrid optical amplifier made a remarkable change in present optical networks in order to amplify the signals for better performance of a system. In this paper, we have proposed different hybrid optical amplifier configurations [such as semiconductor optical amplifier (SOA)-erbium-doped fiber amplifier (EDFA), EDFA-EDFA, Raman-EDFA] at channel spacing of 0.8 nm for 8 × 10 Gbps for wavelength division multiplexed optical networks. The performance is examined on the basis of quality factor (dB), maximum output power (dBm), minimum bit error rate (BER), and minimum eye closure by varying the transmission distance from 20 to 180 km. It is demonstrated that Raman-EDFA configuration provides better results for the small distances up to 100 km as compared to EDFA-EDFA and SOA-EDFA configurations. The variation in Q factor for Raman-EDFA, EDFA-EDFA, and SOA-EDFA is 33.35–13.56 dB, 32.30–10.95 dB, and 18.52–25.03 dB, respectively. When distance is increased from 100 to 180 km, SOA-EDFA provides better results than Raman-EDFA in terms of BER, but it shows non-uniform gain spectrum.

Fast reconstruction of non-uniform sampling multidimensional NMR spectroscopy via a deep neural network

Multidimensional nuclear magnetic resonance (NMR) spectroscopy is used to examine the chemical structures of the studied systems. Unfortunately, the application of NMR spectra is limited by their long acquisition time, especially for 3D, 4D, and higher dimensional spectra. Non-uniform sampling (NUS) has been widely recognized as a powerful tool to reduce the NMR experimental time. But the quality of NUS spectra depends on appropriate reconstruction algorithms. As an effective data processing method, deep learning has been widely used in many fields in recent years. In this work, a deep learning-based strategy for fast reconstruction of non-uniform sampling NMR spectra is proposed. In our experiments, the proposed deep neural network has better performance in removing artifacts and preserving weak peaks than typical convolutional neural networks of U-Net and DenseNet. Besides, a novel approach of generating training data is utilized to reduce the computational burden of neural networks, and thus training our network can be easier and faster than previous deep learning-based works. Compared with the two currently available methods, SMILE and hmsIST, our strategy can provide comparable reconstruction quality in terms of peak intensities and the fidelity of peak shape. The reconstruction time of our methods is also comparable to or faster than the two methods, especially for 3D spectra.