Parseval's Theorem Research Articles

H ∞ performance analysis and distributed filtering for spatially interconnected systems over specified frequency ranges

ABSTRACT This paper is concerned with performance analysis and distributed filtering for spatially interconnected systems (SISs) over specified frequency ranges, including entire frequency (EF) range and finite frequency (FF) range. The main idea is based on a reformed Parseval's theorem for SISs, in which some novel frequency domain spaces and modified Fourier transformations are formulated to explore the various relationships they have with the time domain spaces. After that, an equivalent frequency domain interpretation for the time domain property of SISs is built, and two numerically verifiable equivalent conditions are formulated to guarantee performance. Furthermore, we investigate the performance of SISs over finite frequency range, which is termed as finite frequency (FF) bounded realness for SISs. Then, the derived results are applied to solve the distributed filtering problem of SISs, and a practical example is finally given to show the viability of the proposed method.

Anomalous strain energy transformation pathways in mechanical metamaterials.

This discussion starts with a mechanics version of Parseval's energy theorem applicable to any discrete lattice material with periodic internal structure: a microtruss, grid, frame, origami or tessellation. It provides a simple relationship between the strain energy volumetric/usual and spectral distributions in the reciprocal space. The spectral energy distribution leads directly to a spectral entropy of lattice deformation (Shannon's type), whose variance with a material coordinate represents the decrease of information about surface loads in the material interior. Spectral entropy is also a basic measure of complexity of mechanical responses of metamaterials to surface and body loads. Considering transformation of the energy volumetric and spectral distributions with a material coordinate pointed away from a surface load, several interesting anomalies are seen even for simple lattice materials, when compared to continuum materials. These anomalies include selective filtering of surface Raleigh waves (sinusoidal pressure patterns), Saint–Venant effect inversion illustrated by energy spectral distribution contours, occurrence of ‘hiding pockets’ of low deformation, and redirection of strain energy maximum away from axis of a concentrated surface load. The latter phenomenon can be significant for impact protection applications of mechanical metamaterials.

Design an anti-windup controller for a PWR power-level control in the presence of control rod speed saturation

Control input saturation can cause severe performance degradation and possible instability in a Pressurized Water Reactor (PWR). Therefore, some form of limit protection is needed. This study proposes a dynamic compensation design problem for a PWR subject to control rod speed saturation. Design of this dynamic compensator is based on minimization of a performance index. First, a nonlinear dynamic feedback controller is designed based on feedback linearization. Then based on Parseval's theorem, the dynamic compensator is design such that the auxiliary control law is the same as that without control rod speed saturation. The proposed control approach is applied to a PWR. Simulation results show that the proposed anti-windup scheme is quite effective in presence of input saturation and it has a remarkable performance for tracking reference trajectory.

АВТОМАТИЧНЕ ВИДАЛЕННЯ ГАУСОВОГО ШУМУ НА ЦИФРОВИХ ЗОБРАЖЕННЯХ ЗА ДОПОМОГОЮ КВАЗІОПТИМАЛЬНОГО ФІЛЬТРА ГАУСА

A mathematical model, method and software are developed for the automatic removal of Gaussian noise in digital images by quasioptimal Gaussian filter. The calculation of Gaussian noise level is performed by a method which based on image filtering and iterative selection of region of interest. As the noise level its standard deviation is used. The useful signal in the image is described by the sum of sinusoids. In the first model the brightness of the useful signal is described by one sinusoid, and in the second model by two mutually perpendicular sinusoids. The used models allow us to take into account distortions of useful signal that arise when filtering images to remove noise. The periods and amplitudes of the sinusoid of the useful signal are calculated based on the radial distribution for the power spectrum of the initial image. The orientation of the brightness of the initial image is expressed in terms of the eccentricity, which is calculated on the basis of the central discrete moments of its power spectrum, transformed in accordance with Parseval's theorem. The parameters of the quasioptimal Gaussian filter are calculated on the basis of the four above-described parameters of the initial image: noise level, eccentricity of the image, period and amplitude of sinusoid of the useful signal. The quasioptimal value of the standard deviation of the Gaussian filter kernel is obtained as the value, at which the standard deviation of brightness of the filtered image from the brightness of useful signal is minimized. When calculating the quasioptimal kernel of the Gaussian filter, simultaneous reduction of the noise level and distortion of the useful signal that occur as a result of low-frequency filtering of the initial image are taken into account. The accuracy of the developed filtering method was verified by removing of Gaussian noise on a set of 100 test images. To estimate the accuracy of the filtering of test images, the root mean square error between the brightness of the filtered and the initial images was used, as well as the peak signal-to-noise ratio for the filtered images. An analysis of the peak signal-to-noise ratio for test filtered images showed that the developed method for removing Gaussian noise is quasioptimal. Software implementation of the developed method of automatic removal of Gaussian noise in digital images is performed in the MATLAB system.

Minimum entropy and histogram-pair based JPEG image reversible data hiding

In this paper, based on the block entropy in the spatial domain obtained by summing the squares of all AC coefficients of each 8 × 8 block, a reversible data hiding (RDH) scheme for JPEG images is presented which selects the DCT blocks with minimum entropy for RDH. This scheme applies Parseval theorem and entropy principle to achieve the minimum entropy in the spatial domain. It embeds data into the DCT coefficients based on a histogram–pair RDH scheme using histogram shifting. Moreover, four thresholds including the embedding amplitude threshold, the fluctuation threshold, the lower frequency threshold and the higher frequency threshold, are utilized and adjusted to achieve optimized embedding performance. In particular, the proposed scheme does not embed data into zero-valued DCT coefficients, and can obtain optimal PSNR with relatively small file-size increase. Experimental results have demonstrated that, compared with the prior arts, the quality of proposed marked image is higher, and the file-size increase after RDH is less as well.

On Quasi-Isometry of Threshold-Based Sampling

The problem of isometry for threshold-based sampling such as integrate-and-fire (IF) or send-on-delta (SOD) is addressed. While for uniform sampling the Parseval theorem provides isometry and makes the Euclidean metric canonical, there is no analogy for threshold-based sampling. The relaxation of the isometric postulate to quasi-isometry, however, allows the discovery of the underlying metric structure of threshold-based sampling. This paper characterizes this metric structure making Hermann Weyl's discrepancy measure canonical for threshold-based sampling.

A Novel Nonintrusive Fault Identification for Power Transmission Networks Using Power-Spectrum-Based Hyperbolic S-Transform—Part I: Fault Classification

This paper presents a novel nonintrusive protection scheme for fault classification of power transmission networks in a wide-area measurement system using fault information for decision making. The protection scheme is a noncommunication without global positioning system as it depends completely on locally measured currents for the nonintrusive fault monitoring (NIFM) using the power-spectrum-based hyperbolic S-transform (PS-HST). In this work, the HST is used to extract the high-frequency components of the current signals generated by an electric fault. To effectively select the HST coefficients (HSTCs) representing fault transient signals with increasing performance, a power spectrum of the HSTCs in different scales calculated by Parseval's theorem is proposed in this paper. Finally, back-propagation artificial neural networks and PS-HST are used to identify fault classes in power transmission networks. The proposed method is tested for different breaker on/off conditions by simulations using electromagnetic transients program software. The results obtained have proved that the proposed method is promising and demonstrate a high success rates and reliability for considering different fault resistances and inception times in NIFM applications.

Strain energy spectral density and information content of materials deformation

A broader range of analytical tools can enhance understanding of the unusual mechanical properties of metamaterials and other advanced material systems. Here, we discuss a mechanics analogue of the Parseval's energy theorem that leads to a density of the strain energy in the reciprocal space. It reflects the ways for an elastic medium to translate static deformation patterns between two points in space. A normalized spectral density also provides an information entropy of deformation at those points. Both differential and discrete (numerical) entropies of the Shannon's type are discussed. Spectral entropy is a basic measure of information available in the material interior about surface loads, or a measure of disorder introduced into elastic medium by the deformation. An exact analytical entropy function is derived for an isotropic plane solid under Gauss-distributed and point loads. Approaches to numerical calculation of spectral entropy in computational solid mechanics are also discussed. Energy spectral density and spectral entropy of an elastic continuum is shown to translate, logically, in agreement with the Saint-Venant's principle. However, it also becomes clear that microstructured media may demonstrate anomalous pathways of evolution of the strain energy spectrum, enabling interesting transformation mechanics studies of engineered material systems.

Fourier-Like Multipliers and Applications for Integral Operators

Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operators, and study their boundedness and Schatten class properties. We show that the wavelet multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator. Moreover we prove that a signal which is almost time and almost bandlimited can be approximated by its projection on the span of the first eigenfunctions of the phase space restriction operator, corresponding to the largest eigenvalues which are close to one.

Superunitary Representations of Heisenberg Supergroups

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

Gyrator wavelet transform

The gyrator transform is a linear canonical transform, which generates the rotation of an optical signal in position-spatial frequency planes. Gyrator wavelet transform is a relatively newer optical information processing tool obtained by combining the gyrator transform with the wavelet transform. This combination provides multi-resolution analysis of an image which is twisted in spatial frequency planes. The proposed tool satisfies basic algebraic properties, such as the linearity property and Parseval's theorem. Considering the usefulness of this tool, here a study of features, applications, and implementation of the gyrator wavelet transform is presented. This work studies the features of the gyrator wavelet transform, which can find a role in different applications such as edge enhancement, image encryption, image hiding, and watermarking.

Multiscale energy reallocation during low-frequency steady-state brain response.

Traditional task-evoked brain activations are based on detection and estimation of signal change from the mean signal. By contrast, the low-frequency steady-state brain response (lfSSBR) reflects frequency-tagging activity at the fundamental frequency of the task presentation and its harmonics. Compared to the activity at these resonant frequencies, brain responses at nonresonant frequencies are largely unknown. Additionally, because the lfSSBR is defined by power change, we hypothesize using Parseval's theorem that the power change reflects brain signal variability rather than the change of mean signal. Using a face recognition task, we observed power increase at the fundamental frequency (0.05 Hz) and two harmonics (0.1 and 0.15 Hz) and power decrease within the infra-slow frequency band (<0.1 Hz), suggesting a multifrequency energy reallocation. The consistency of power and variability was demonstrated by the high correlation (r > .955) of their spatial distribution and brain-behavior relationship at all frequency bands. Additionally, the reallocation of finite energy was observed across various brain regions and frequency bands, forming a particular spatiotemporal pattern. Overall, results from this study strongly suggest that frequency-specific power and variability may measure the same underlying brain activity and that these results may shed light on different mechanisms between lfSSBR and brain activation, and spatiotemporal characteristics of energy reallocation induced by cognitive tasks.

Deep searches for broadband extended gravitational-wave emission bursts by heterogeneous computing

We present a heterogeneous search algorithm for broadband extended gravitational-wave emission (BEGE), expected from gamma-ray bursts and energetic core-collapse supernovae. It searches the $(f,\dot{f})$-plane for long duration bursts by inner engines slowly exhausting their energy reservoir by matched filtering on a {\em Graphics Processor Unit} (GPU) over a template bank of millions of one-second duration chirps. Parseval's Theorem is used to predict the standard deviation $\sigma$ of filter output, taking advantage of near-Gaussian noise in LIGO S6 data over 350-2000 Hz. Tails exceeding a mulitple of $\sigma$ are communicated back to a {\em Central Processing Unit} (CPU). This algorithm attains about 65\% efficiency overall, normalized to the Fast Fourier Transform (FFT). At about one million correlations per second over data segments of 16 s duration $(N=2^{16}$ samples), better than real-time analysis is achieved on a cluster of about a dozen GPUs. We demonstrate its application to the capture of high frequency hardware LIGO injections. This algorithm serves as a starting point for deep all-sky searches in both archive data and real-time analysis in current observational runs.

Application of energies of optimal frequency bands for fault diagnosis based on modified distance function

Low-dimensional relevant feature sets are ideal to avoid extra data mining for classification. The current work investigates the feasibility of utilizing energies of vibration signals in optimal frequency bands as features for machine fault diagnosis application. Energies in different frequency bands were derived based on Parseval's theorem. The optimal feature sets were extracted by optimization of the related frequency bands using genetic algorithm and a Modified distance function (MDF). The frequency bands and the number of bands were optimized based on the MDF. The MDF is designed to a) maximize the distance between centers of classes, b) minimize the dispersion of features in each class separately, and c) minimize dimension of extracted feature sets. The experimental signals in two different gearboxes were used to demonstrate the efficiency of the presented technique. The results show the effectiveness of the presented technique in gear fault diagnosis application.

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

Let G be a separable locally compact group with type-I left regular representation, Gˆ its dual and A(G) its Fourier algebra. We prove an analogue of Parseval's theorem and that the mappingT⟶u(x):=∫GˆTr[T(π)π(x)−1]dμ(π) is an isometric isomorphism of Banach spaces from L1(Gˆ) onto A(G).

Development of Contactless Sand Production Test Instrument

Background:Sand production is an important factor affecting reservoir exploitation. Excessive sand production affects the oil well life, causes the damage to the oil recovery device and brings serious hidden danger in the process of safety production.Objective:This paper presents a sand content computation model for the crude oil, which uses the power of sand production and Parseval theorem, develops a system to monitor the sand content.Method:The main problem of designing the mechanical structure and matching layer for sand sensor is introduced. Furthermore a contactless Doppler ultrasonic method is used to develop sand amount detection system‚ finally the system measurement accuracy is calibrated by the developed calibration system.Conclusion:The sand production computation model can describe the relationship between sand monitor output and sand amount; compute the oil sand carrying ratio. The sand sensor is installed on the outside of pipe, not directly contacted with the sand particles. Measurement accuracy of the sand production is higher than the same type instrument.

Finite-Frequency Model Reduction of Takagi–Sugeno Fuzzy Systems

This paper considers the model-reduction problem for continuous-time Takagi–Sugeno (T–S) fuzzy systems. Different from existing full-frequency methods, a finite-frequency model-reduction method is proposed in this paper. The proposed method can get a better approximation performance when input signals belong to a finite-frequency domain. To this end, a finite-frequency $H_\infty$ performance index is first defined. Then, a sufficient finite-frequency performance analysis condition is derived by the aid of Parseval's theorem and quadratic functions. Based on this condition and projection lemma, three model-reduction algorithms for T–S fuzzy systems with input signals in low-frequency, middle-frequency, and high-frequency domain are obtained, respectively. Finally, an example is given to illustrate the effectiveness of the proposed method.

Nobel approach in signifying damage for a bellows type flexible joint

The durability of an exhaust system, particularly for a flexible joint, has been addressed in terms of damage augmented by energy spectral densities. Movement of a flexible joint in motion is calculated by solving forward kinematics of the Stewart platform, converting displacement signals obtained by a set of potentiometers into a Cartesian pose. Those time series are then projected onto the uni-directional field thus defined separately, i.e., damage for the application of the Minor rule and energy for power spectrum densities. Parseval's theorem allows us to bridge two fields so that one can access not only information on fatigue properties, but also on effect of aging in the presence of shock and impact.

Fault detection for singular switched linear systems with multiple time-varying delay in finite frequency domain

ABSTRACTThis paper deals with the problem of the fault detection (FD) for continuous-time singular switched linear systems with multiple time-varying delay. In this paper, the actuator fault is considered. Besides, the systems faults and unknown disturbances are assumed in known frequency domains. Some finite frequency performance indices are initially introduced to design the switched FD filters which ensure that the filtering augmented systems under switching signal with average dwell time are exponentially admissible and guarantee the fault input sensitivity and disturbance robustness. By developing generalised Kalman–Yakubovic–Popov lemma and using Parseval's theorem and Fourier transform, finite frequency delay-dependent sufficient conditions for the existence of such a filter which can guarantee the finite-frequency H− and H∞ performance are derived and formulated in terms of linear matrix inequalities. Four examples are provided to illustrate the effectiveness of the proposed finite frequency method.

Fault detection for stochastic parameter-varying Markovian jump systems with application to networked control systems

This paper investigates the fault detection (FD) problem for networked control systems (NCS). The random delays from the sensor to the controller (S-C) and from the controller to the actuator (C-A) are modeled as a Markov chain. Whilst considering the data dropout phenomenon happens unavoidably and the measurement drops intermittently, C-A data dropout and S-C data dropout are coped with by introducing two independent Bernoulli processes. Based on the above two points, a stochastic parameter-varying Markovian jump system (SPVMJS) is constructed. By combining discrete Parseval theorem and strict S-procedure, a novel finite frequency approach is proposed to design an H−/H∞ fault detection filter for the SPVMJSs. Finally, a simulation example on the networked control system is presented to illustrate the effectiveness of the proposed method.