Heisenberg Uncertainty Principle for n-Dimen-sional Linear Canonical Transforms

The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical trans-forms domain.

A Novel Method for Forecasting Freak Wave Occurrence in A Short Crested Coastal Sea

In order to forecast the distribution of crest amplitudes and the occurrence of freak waves in a short crested coastal sea,a novel transformed linear simulation method is initially proposed in this paper.A Hermite transformation model expressed as a monotonic cubic polynomial serves as the foundation for the novel simulation technique.The wave crest amplitude exceedance probabilities of two sea states-one with a directional wave spectrum based on the measured wave elevation data at the Yura coast and the other with a typical directional JONSWAP wave spectrum-have been predicted using the novel simulation method that has been proposed.The likelihood that a particular critical wave crest amplitude will be exceeded is directly correlated with the probability that freak waves will occur.It is shown that the novel simulation approach suggested can provide predictions that are more precise than those obtained from the Rayleigh crest amplitude distribution model,the Jahns and Wheeler crest amplitude distribution model,or the conventional linear simulation method.This study also demonstrated that the nonlinear simulation method is less effective than the novel simulation method in terms of efficiency.

A Multi-Gap Multi-Channel Gas Switch for the Linear Transformer Driver

A multi-gap and multi-channel gas switch with convexo-convex discal planet electrodes was designed and investigated. Eight gaps are formed in series by a trigger electrode, six intermediate electrodes and two high voltage electrodes with a uniform gap length of 5 ram. The self breakdown and triggered breakdown performance of the switch are reported. Both the delay time and jitter decrease with the increase in the trigger voltage, switching coefficient and the decrease in the trigger isolating resistor. The delay time of the switch is about 40 ns, and the jitter is less than 2 ns when charged with 4-85 kV and triggered by a voltage pule of -75 kV. The inductance of the switch is about 30 nH.

Numerical Analysis of the Output-Pulse Shaping Capability of Linear Transformer Drivers

Output-pulse shaping capability of a linear transformer driver （LTD） module under different conditions is studied, by conducting the whole circuit model simulation by using the PSPICE code. Results indicate that a higher impedance profile of the internal transmission line would lead to a wider adjustment range for the output current rise time and a narrower adjustment range for the current peak. The number of cavities in series has a positive effect on the output- pulse shaping capability of LTD. Such an improvement in the output-pulse shaping capability can primarily be ascribed to the increment in the axial electric length of LTD. For a triggering time interval longer than the time taken by a pulse to propagate through the length of one cavity, the output parameters of LTD could be improved significantly. The present insulating capability of gas switches and other elements in the LTD cavities may only tolerate a slightly longer deviation in the triggering time interval. It is feasible for the LTD module to reduce the output current rise time, though it is not useful to improve the peak power effectively.

Influences of Switching Jitter on the Operational Performances of Linear Transformer Drivers-Based Drivers

A whole circuit model of a linear transformer drivers （LTD） module composed of 60 cavities in series was developed in the software PSPICE to study the influence of switching jitter on the operational performances of LTDs. In the model, each brick in each cavity is capable of operating with jitter in its switch. Additionally, the manner of triggering cables entering into cavities was considered. The performances of the LTD module operating with three typical cavity-triggering sequences were simulated and the simulation results indicate that switching jitter affects slightly the peak and starting time of the output current pulse. However, the enhancement in switching jitter would significantly lengthen the rise time of the output current pulse. Without considering other factors, a jitter lower than 10 ns may be necessary for the switches in the LTD module to provide output current parameters with an acceptable deviation.

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the polar coordinate form of quaternion-valued signals,the UP of the two-sided quaternion linear canonical transform(QLCT)is strengthened in terms of covariance.The condition giving rise to the equal relation of the derived result is obtained as well.The novel UP with covariance can be regarded as one in a tighter form related to the QLCT.It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.

An Improved Robust Model Predictive Control Approach to Systems with Linear Fractional Transformation Perturbations

In this paper, a robust model predictive control approach is proposed for a class of uncertain systems with time-varying, linear fractional transformation perturbations. By adopting a sequence of feedback control laws instead of a single one, the control performance can be improved and the region of attraction can be enlarged compared with the existing model predictive control （MPC） approaches. Moreover, a synthesis approach of MPC is developed to achieve high performance with lower on-line computational burden. The effectiveness of the proposed approach is verified by simulation examples.

Optical Diagnostics of Multi-Gap Gas Switches for Linear Transformer Drivers

The trigger characteristics of a multi-gap gas switch with double insulating layers,a square-groove electrode supporter and a UV pre-ionizing structure are investigated aided by a high sensitivity fiber-bundle array detector, a UV fiber detector, and a framing camera, in addition to standard electrical diagnostics. The fiber-bundle-array detector is used to track the turn-on sequence of each electrode gap at a timing precision of 0.6 ns. Each fiber bundle, including five fibers with different azimuth angles, aims at the whole emitting area of each electrode gap and is fed to a photomultiplier tube. The UV fiber detector with a spectrum response of 260-320 nm,including a fused-quartz fiber of 200 μm in diameter and a solar-blinded photomultiplier tube, is adopted to study the effect of UV pre-ionizing on trigger characteristics. The framing camera,with a capacity of 4 frames per shot and an exposure time of 5 ns, is employed to capture the evolution of channel arcs. Based on the turn-on light signal of each electrode gap, the breakdown delay is divided into statistical delay and formative delay. A decrease in both of them, a smaller switch jitter and more channel arcs are observed with lower gas pressure. An increase in trigger voltage can reduce the statistical delay and its jitter, while higher trigger voltage has a relatively small influence on the formative delay and the number of channel arcs. With the UV pre-ionizing structure at 0.24 MPa gas pressure and 60 kV trigger voltage, the statistical delay and its jitter can be reduced by 1.8 ns and 0.67 ns, while the formative delay and its jitter can only be reduced by 0.5 ns and 0.25 ns.

THE BLUE AND MINQUE IN GAUSS-MARKOFF MODEL WITH LINEAR TRANSFORMATION OF THE OBSERVABLE VARIABLES

For a singular linear model A = （y, Xβ, σ2 V） and its transformed model MF = （Fy, FXβ, σ 2FVF＇）, where V is nonnegative definite and X can be rank-deficient, the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala （1981） are strengthened and consequences in Puntanen and Nurhonen （1992）, and Puntanen （1996） are extended.

Research Progress on Discretization of Linear Canonical Transform

Linear canonical transformation(LCT)is a generalization of the Fourier transform and fractional Fourier transform.The recent research has shown that the LCT is widely used in signal processing and applied mathematics,and the discretization of the LCT becomes vital for the applic-ations of LCT.Based on the development of discretization LCT,a review of important research progress and current situation is presented,which can help researchers to further understand the discretization of LCT and can promote its engineering application.Meanwhile,the connection among different discretization algorithms and the future research are given.

Diffraction separation and imaging based on double sparse transforms

Reflection imaging results generally reveal large-scale continuous geological information,and it is difficult to identify small-scale geological bodies such as breakpoints,pinch points,small fault blocks,caves,and fractures,etc.Diffraction imaging is an important method to identify small-scale geological bodies and it has higher resolution than reflection imaging.In the common-offset domain,reflections are mostly expressed as smooth linear events,whereas diffractions are characterized by hyperbolic events.This paper proposes a diffraction extraction method based on double sparse transforms.The linear events can be sparsely expressed by the high-resolution linear Radon transform,and the curved events can be sparsely expressed by the Curvelet transform.A sparse inversion model is built and the alternating direction method is used to solve the inversion model.Simulation data and field data experimental results proved that the diffractions extraction method based on double sparse transforms can effectively improve the imaging quality of faults and other small-scale geological bodies.

A DECOMPOSITION DEPENDED ON DIRECTIONS FOR NONSINGULAR LINEAR TRANSFORMATION

In this paper. we give a decomposition depending on p(1≤p≤n-2) orthonormaldirections assigned for nonsingular linear transformation F on a n-dimension (n≥3)Euclidean space En, and then prove foal there exist q(q=n-p) quasi-Principaldirections.for F depending on the preceding p orthonormal directions. As applicance ofthe preceding result, we derive that there exist at least two orthonormal principaldirections of strain in arbitrary plane of body which is in homogeneous deformation,and strain energy density is.function of 5 real numbers under arbitrary quasi-principalbase.for the preceding nonsingular linear transformation.

Speech Encryption in Linear Canonical Transform Domain Based on Chaotic Dynamic Modulation

In order to transmit the speech information safely in the channel,a new speech encryp-tion algorithm in linear canonical transform(LCT)domain based on dynamic modulation of chaot-ic system is proposed.The algorithm first uses a chaotic system to obtain the number of sampling points of the grouped encrypted signal.Then three chaotic systems are used to modulate the corres-ponding parameters of the LCT,and each group of transform parameters corresponds to a group of encrypted signals.Thus,each group of signals is transformed by LCT with different parameters.Fi-nally,chaotic encryption is performed on the LCT domain spectrum of each group of signals,to realize the overall encryption of the speech signal.The experimental results show that the proposed algorithm is extremely sensitive to the keys and has a larger key space.Compared with the original signal,the waveform and LCT domain spectrum of obtained encrypted signal are distributed more uniformly and have less correlation,which can realize the safe transmission of speech signals.

LMI_based robust iterative learning controller design for discrete linear uncertain systems

This paper addresses the design problem of robust iterative learning controllers for a class of linear discrete-time systems with norm-bounded parameter uncertainties. An iterative learning algorithm with current cycle feedback is proposed to achieve both robust convergence and robust stability. The synthesis problem of the proposed iterative learmng control （ILC） system is reformulated as a γ-suboptimal H-infinity control problem via the linear fractional transformation （LFT）. A sufficient condition for the convergence of the ILC algorithm is presented in terms of linear matrix inequalities （LMIs）. Furthermore, the linear wansfer operators of the ILC algorithm with high convergence speed are obtained by using existing convex optimization techniques. The simulation results demonstrate the effectiveness of the proposed method.

Predicting Nonlinear Wave Trough Distributions Utilizing A Transformed Linear Simulation Method

This paper first proposes a new approach for predicting the nonlinear wave trough distributions by utilizing a transformed linear simulation method. The linear simulation method is transformed based on a Hermite transformation model where the transformation is chosen to be a monotonic cubic polynomial and calibrated such that the first four moments of the transformed model match the moments of the true process. The proposed new approach is applied for calculating the wave trough distributions of a nonlinear sea state with the surface elevation data measured at the coast of Yura in the Japan Sea, and its accuracy and efficiency are convincingly validated by comparisons with the results from two theoretical distribution models, from a linear simulation model and a secondorder nonlinear simulation model. Finally, it is further demonstrated in this paper that the new approach can be applied to all the situations characterized by similar nondimensional spectrum.

Nonlinear symbolic LFT model for UAV

A nonlinear modeling framework is presented for an oceanographic unmanned aerial vehicle （UAV） by using symbolic modeling and linear fractional transformation （LFT） techniques . Consequently, an exact nonlinear symbolic LFT model of the UAV is derived in a standard M-A form where M represents the nominal, known, part of the system and A contains the time-varying, uncertain and nonlinear components. The advantages of the proposed modeling approach are that： it not only provides an ideal starting point to obtain various final design-oriented models through subse- quent assumptions and simplifications, but also it facilitates the control system analysis with models of different levels of fidelity/complexity. Furthermore, a linearized symbolic LFT model of the UAV is proposed based on the LFT differentiation, which is amenable directly to a sophisticated linear ro- bust control strategy such as μ synthesis/analysis. Both of the derived LFT models are validated with the original nonlinear model in time domain. Simulation results show the effectiveness of the pro- posed algorithm.

Line-element based nonlinear adaptive piecewise compensating correction for LVDT sensors

In order to solve the linear variable differential transformer （LVDT） displacement sensor nonlinearity of overall range and extend its working range, a novel line-element based adaptively seg- menting method for piecewise compensating correction was proposed. According to the mechanical structure of LVDT, the output equation was calculated, and then the theoretic nonlinear source of output was analyzed. By the proposed line-element adaptive segmentation method, the nonlinear output of LVDT was divided into linear and nonlinear regions with a given threshold. Then the com- pensating correction function was designed for nonlinear parts employing polynomial regression tech- nique. The simulation of LVDT validates the feasibility of proposed scheme, and the results of cali- bration and testing experiments fully prove that the proposed method has higher accuracy than the state-of-art correction algorithms.

ON THE DECOMPOSITION OF COMPLEX VECTOR SPACES AND THE JORDAN CANONICAL FORM OF COMPLEX LINEAR TRANSFORMATIONS

New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.

From Translation to Linear and Linear Canonical Transformations

In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂x) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.

Classical and Quantum Behavior of Generalized Oscillators in Terms of Linear Canonical Transformations

The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.