Shift-variant Systems Research Articles

Inversion Algorithm of Insulator Surface Charge Density Based on Hybrid LSMR Algorithm for Shift-Variant System

Inversion Algorithm of Insulator Surface Charge Density Based on Hybrid LSMR Algorithm for Shift-Variant System

Recursive filtering for two-dimensional systems with amplify-and-forward relays: Handling degraded measurements and dynamic biases

This paper is concerned with the recursive filtering issue for a type of two-dimensional shift-varying systems communicated via imperfect networks. The communication network undergoes degraded measurements, stochastic biases, and channel noises, all reflecting real-world constraints. The degraded measurements are governed by three random variable sets with known statistical details, while the stochastic biases follow specific dynamic equations. An amplify-and-forward (AF) relay is implemented to bolster the reliability of transmission from the sensor to the distant filter. The main goal of this study is to formulate an AF relay-based recursive filtering algorithm that can predict system states with the desired performance specification. By employing the inductive technique and meticulous stochastic analysis, an upper bound for the filtering error variance is initially set based on the solutions to recursive difference equations, following which the filter gain is intricately designed to minimize this upper bound at each point in time. Furthermore, a theoretical analysis is conducted on the effect of degradation coefficients on the filtering performance. Finally, the effectiveness of the proposed recursive filtering approach is demonstrated using a numerical example.

Set-membership filtering for two-dimensional shift-varying systems with stochastic communication protocol and uniform quantization

This paper is concerned with the set-membership filtering problem for the two-dimensional shift-varying systems subject to the stochastic communication protocol and uniform quantization effects. To prevent the data transmission from collisions, only one sensor can transmit the measured information to the filter at each sampling shift instant, and the selected sensor is determined by the scheduling strategy of the stochastic communication protocol. On the contrary, the uniform quantization mechanism has been employed to mitigate the influence of quantization error on filtering performance. Incorporating the stochastic communication protocol and uniform quantization mechanism, this paper proposes a set-membership filter design framework for the two-dimensional shift-varying systems with unknown-but-bounded noises. Sufficient conditions are derived for the existence of desired set-membership filter by utilizing double mathematical induction, such that the estimation error resides within the ellipsoidal set. Moreover, the optimal filtering algorithm is given by minimizing the ellipsoidal constraints. Finally, several examples are provided to illustrate the effectiveness of the proposed filter design algorithm.

Neural-Network-Based Set-Membership Fault Estimation for 2-D Systems Under Encoding-Decoding Mechanism.

In this article, the simultaneous state and fault estimation problem is investigated for a class of nonlinear 2-D shift-varying systems, where the sensors and the estimator are connected via a communication network of limited bandwidth. With the purpose of relieving the communication burden and enhancing the transmission security, a new encoding-decoding mechanism is put forward so as to encode the transmitted data with a finite number of bits. The aim of the addressed problem is to develop a neural-network (NN)-based set-membership estimator for jointly estimating the system states and the faults, where the estimation errors are guaranteed to reside within an optimized ellipsoidal set. With the aid of the mathematical induction technique and certain convex optimization approaches, sufficient conditions are derived for the existence of the desired set-membership estimator, and the estimator gains and the NN tuning scalars are then presented in terms of the solutions to a set of optimization problems subject to ellipsoidal constraints. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed estimator design method.

From Surface Potential to Surface Charge: An Inversion Algorithm for Shift-Variant System Based on Hybrid LBD-Tikhonov Method

From Surface Potential to Surface Charge: An Inversion Algorithm for Shift-Variant System Based on Hybrid LBD-Tikhonov Method

Distributed Set-Membership Fusion Filtering for Nonlinear 2-D Systems Over Sensor Networks: An Encoding-Decoding Scheme.

In this article, the distributed set-membership fusion filtering problem is investigated for a class of nonlinear 2-D shift-varying systems subject to unknown-but-bounded noises over sensor networks. The sensors are communicated with their neighbors according to a given topology through wireless networks of limited bandwidth. With the purpose of relieving the communication burden as well as enhancing the transmission security, a logarithmic-type encoding-decoding mechanism is introduced for each sensor node so as to encode the transmitted data with a finite number of bits. A distributed set-membership filter is designed to determine the local ellipsoidal set that contains the system state by only utilizing the data from the local sensor node and its neighbors, where the proposed filter scheme is truly distributed with desirable scalability. Then, a new ellipsoid-based fusion rule is developed for the designed set-membership filters in order to form the fused ellipsoidal set that has a globally smaller volume than all local ellipsoidal sets. With the aid of the mathematical induction technique, the set theory, and the convex optimization approach, sufficient conditions are derived for the existence of the desired distributed set-membership filters and the fusion weights. Then, the filter parameters and the fusion weights are acquired by solving a set of constrained optimization problems. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed fusion filtering algorithm.

Set-membership filtering for two-dimensional systems with dynamic event-triggered mechanism

In this paper, the set-membership filtering problem is investigated for a class of two-dimensional (2-D) shift-varying systems described by the general Fornasini-Marchesini second model with a dynamic event-triggered mechanism. The sensors are communicated with the remote filter through a communication network of limited bandwidth. In order to improve the efficiency of utilizing the network resources, a dynamic event-triggered mechanism is exploited for the 2-D shift-varying system, and a set-membership filter is then designed for 2-D systems subject to unknown-but-bounded noises so that the actual system state is guaranteed to reside within an optimized ellipsoidal set. By means of the mathematical induction technique and the set theory, sufficient conditions are derived to ensure the existence of the desired set-membership filter. Furthermore, the filter gains are obtained by solving a set of optimization problems subject to ellipsoidal constraints. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed filter design method.

Robust set-membership filtering for two-dimensional systems with sensor saturation under the Round-Robin protocol

In this paper, the set-membership filtering problem is investigated for a class of polytopic uncertain two-dimensional shift-varying systems with delays and sensor saturation. The measurement information is transmitted between the sensors and the filter through a shared communication network. To avoid data congestions, the Round-Robin (R-R) protocol is employed to regulate the data transmission order. This paper aims to design a robust set-membership filter such that the real system states are always contained in the state estimate ellipsoid in the presence of the state delays, the sensor saturation, the parameter uncertainties, the R-R protocol and the unknown-but-bounded external perturbations. By using the mathematical induction, sufficient condition for the existence of the desired robust set-membership filter is given in the form of recursive linear matrix inequalities, and a convex optimisation method is also proposed to determine the optimal state estimation ellipsoid in the trace sense. One numerical example is also given to illustrate effectiveness of the proposed filtering strategy.

A Recursive Algorithm for Secure Filtering for Two-Dimensional State-Saturated Systems Under Network-Based Deception Attacks

This paper addresses the recursive secure filtering problem for an array of two-dimensional shift-varying systems subject to state saturation under deception attacks. The network-based deception attacks, which occur in a probabilistic manner, are modulated by random variables obeying the Bernoulli distributions. The intention of the addressed problem is to design a recursive filter for estimating the system states with a satisfactory performance against state saturations and probabilistic attacks. By utilizing the inductive method and the matrix techniques, a feasible upper bound is attained on the filtering error variance and then minimized in the trace sense at each iteration. The coveted filter gain is recursively calculated by solving three sets of difference equations. Additionally, for the established filtering algorithm, the performance analysis is conducted to show the uniform boundedness of the minimum upper bound and the monotonicity regarding the deception attacks. An illustrative simulation is also implemented to verify the usefulness of the proposed filtering scheme.

Recursive distributed filtering for two-dimensional shift-varying systems over sensor networks under stochastic communication protocols

This paper is concerned with the distributed filtering problem for a class of two-dimensional shift-varying systems over sensor networks subject to stochastic communication protocol (SCP) over a finite horizon. The communication between the sensor nodes and the filters is implemented through shared channels of limited capacity. To avoid data collisions, the SCP is applied to determine the transmission order of the signals/packets for each sensor. The considered scheduling behavior is characterized by mutually uncorrelated random variables with known probability distributions. Recursive distributed filters are proposed to estimate the system state through available information from both individual and neighboring nodes in the sensor network according to a given topology. Attention is focused on the design of distributed filters in order to ensure the locally minimal upper bound on the error variance of the state estimation. Sufficient conditions are first established, via intensive stochastic analysis and mathematical induction, on the existence of an upper bound of the estimation error variance. Then, by means of a matrix simplification technique, the desired filter gains are designed to optimize the obtained upper bound at each shift step. Finally, a practical example is given to verify the effectiveness of the proposed filter strategy.

Minimum-Variance Recursive Filtering for Two-Dimensional Systems With Degraded Measurements: Boundedness and Monotonicity

This paper addresses the recursive minimum-variance filtering problem for a class of two-dimensional shift-varying systems with stochastic nonlinearity and degraded measurements. The stochastic nonlinearity is governed by its statistical characteristics and the degraded measurements reflect the signal degradation obeying certain prescribed probabilistic distributions. The main objective of this paper is to construct a two-step recursive filter that achieves the minimum error variance of the state estimation at each step. Utilizing an inductive approach, unbiasedness of the proposed filter is first ensured and the parameters of the filter are then designed by resorting to the completing squares method. Subsequently, the filtering performances including the boundedness and the monotonicity are investigated with respect to the measurement degradations through mathematically rigorous analysis. Moreover, a computational algorithm is presented to facilitate the online implementation of the designed filter. Finally, numerical simulation illustrates the effectiveness and applicability of the proposed filtering scheme in the state estimation problem for monitoring a long transmission line in circuit systems.

Optimized Reconfigurable Fast Convolution-Based Transmultiplexers for Flexible Radio Access

Multirate fast-convolution (FC) processing can be used for realizing low-complexity filter banks (FBs) and transmultiplexers (TMUXs). The main advantage of the FC-based realizations when compared with the polyphase FBs is the increased configurability, that is, the number of subchannels, their bandwidths, and the center frequencies can be adjusted independently. In general, FC-based FBs are linear periodically shift-variant systems. In this brief, novel matrix representations for the FC synthesis and analysis FBs are first derived. These representations give all the shift-variant impulse responses of the FC-based FBs. Then the TMUX optimization criteria are expressed using these representations. Two examples are included to demonstrate the performance of the optimized designs as well as to illustrate the flexibility of the resulting FC-based TMUXs.

Inversion Algorithm to Calculate Charge Density on Solid Dielectric Surface Based on Surface Potential Measurement

Charge accumulation on a solid dielectric surface is one of the critical concerns for the design and optimization of the insulation system in a high-voltage power equipment, since it will lead to the overstress of electrical insulation. Therefore, it is important to obtain the charge density distribution on a solid dielectric surface with high accuracy. The acquisition of surface charge for insulators requires multipoint potential measurements to establish the inverse calculation for the determination of an unknown charge distribution. Up to now, extensive studies have been conducted on this subject; nevertheless, the methods are either too complicated and time consuming, or only applicable for specific arrangements, or with poor accuracy. In this paper, the problem is divided into two categories, i.e., shift-variant system and shift-invariant system, and the basic principle of an improved inversion algorithm is interpreted to solve the problem. The 2-D Fourier transform and Wiener filter techniques are employed in the algorithm for shift-invariant system thus the relationship between potential and charge density can be processed in spatial frequency domain, which tremendously simplifies the conventional procedure. The accuracy and resolution of the algorithm are discussed in detail with the aid of numerical examples. In the end, experiments are conducted and the effectiveness of the algorithm is verified.

Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies $M$ -channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix $\boldsymbol {A}$ . It is shown that graph filter banks represent, (linear and) periodically, shift-variant systems only when $\boldsymbol {A}$ satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when $\boldsymbol {A}$ satisfies the eigenvector structure satisfied by $M$ -block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If $\boldsymbol {A}$ is actually $M$ -block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case, the PR condition is identical to PR in classical filter banks—any classical PR example can be converted to a graph PR filter bank on an $M$ -block cyclic graph. It is shown that for $M$ -block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Shift-Variance and Nonstationarity of Linear Periodically Shift-Variant Systems and Applications to Generalized Sampling-Reconstruction Processes

We present a systematic analysis on shift-variance of linear $T$ -periodically shift-variant ( $T$ -LPSV) systems with continuous-time input and output. We first determine how far a $T$ -LPSV system $H$ is away from being shift-invariant in terms of two measures, namely, the distance and the angle between $H$ and the subspace of linear shift-invariant systems. We then consider the level and the index of shift-variance (SVI) of $H$ via commutator of $H$ and the shift-operator, and examine the amount of shift-variance of $H$ under particular input. All these quantities are characterized in terms of the so-called shift-variant part and the shift-variance kernel of $H$ . Moreover, we study nonstationarity of wide-sense cyclostationary random processes, possibly induced by $T$ -LPSV systems. We define and calculate a measure of nonstationarity (NSt) of the output as the SVI of the autocorrelation operator. The Nst can be used to characterize performance loss when the output is treated with a WSS approach. The expected shift-variance of $H$ subject to particular WSS random process is also calculated. We apply the above results to generalized sampling-reconstruction processes (GSRPs). In addition, we consider the approximation error of the GSRP and relate it to the shift-variance measures. Three sampling schemes (namely, the orthogonal, the consistent, and the minimax regret sampling) are examined in detail. We show that for both deterministic and WSS random inputs, the minimax regret GSRP always introduces less shift-variance than the consistent counterparts. Numerical results for the GSRPs of B-splines of various orders are provided.

Efficient Determination of the Uncertainty for the Optimization of SPECT System Design: A Subsampled Fisher Information Matrix

System designs in single photon emission tomography (SPECT) can be evaluated based on the fundamental trade-off between bias and variance that can be achieved in the reconstruction of emission tomograms. This trade off can be derived analytically using the Cramer-Rao type bounds, which imply the calculation and the inversion of the Fisher information matrix (FIM). The inverse of the FIM expresses the uncertainty associated to the tomogram, enabling the comparison of system designs. However, computing, storing and inverting the FIM is not practical with 3-D imaging systems. In order to tackle the problem of the computational load in calculating the inverse of the FIM, a method based on the calculation of the local impulse response and the variance, in a single point, from a single row of the FIM, has been previously proposed for system design. However this approximation (circulant approximation) does not capture the global interdependence between the variables in shift-variant systems such as SPECT, and cannot account e.g., for data truncation or missing data. Our new formulation relies on subsampling the FIM. The FIM is calculated over a subset of voxels arranged in a grid that covers the whole volume. Every element of the FIM at the grid points is calculated exactly, accounting for the acquisition geometry and for the object. This new formulation reduces the computational complexity in estimating the uncertainty, but nevertheless accounts for the global interdependence between the variables, enabling the exploration of design spaces hindered by the circulant approximation. The graphics processing unit accelerated implementation of the algorithm reduces further the computation times, making the algorithm a good candidate for real-time optimization of adaptive imaging systems. This paper describes the subsampled FIM formulation and implementation details. The advantages and limitations of the new approximation are explored, in comparison with the circulant approximation, in the context of design optimization of a parallel-hole collimator SPECT system and of an adaptive imaging system (similar to the commercially available D-SPECT).

Simulation of modulation transfer function using a rendering method

We propose a novel simulation method based on rendering to evaluate the modulation transfer function (MTF) of optical imaging systems. The new simulation method corresponds to an experimental measurement of the MTF using imaging resolution test charts, and therefore allows an analysis of the resolving power of shift-variant optical systems that are difficult to evaluate with conventional methods based on the point spread function (PSF). Furthermore, the effects of stray light, such as from reflection or scattering, on the imaging performance can be analyzed. In contrast to methods based on illumination optics using Monte Carlo methods, the proposed method calculates the intensities on an image surface with rendering techniques used in three-dimensional computer graphics (3D CG), which results in calculations that are faster and have a higher precision. The proposed method is highly effective in analyzing the MTF of optical imaging systems through simulations.

Stability of Shift-Varying 2-D State-Space Digital Filters

Stability conditions for 2-D shift-varying systems are presented. The shift-varying nature of such systems emerges in applications such as adaptive filtering or adaptive image processing, where the coefficients are neither static nor periodic. The forms considered in this paper are the Givone-Roesser and the Fornasini-Marchesini models, both of which are discrete 2-D state-space filters. The sufficient conditions for BIBO stability that are proven herein are an outgrowth of the 1-D time-varying state-space conditions that have been previously established. The nature of feedback in the 2-D space is explored and found to be much more complex than for the 1-D case. However, it is also shown that when every feedback path is guaranteed to satisfy a variation on exponential stability, then BIBO stability of these two models can be assured. Further conditions are also established which engage the Lyapunov equation and guarantee the exponential stability requirement.

Slant Plane CSAR Processing Using Householder Transform

Fourier analysis-based focusing of synthetic aperture radar (SAR) data collected during circular flight path is a recent advancement in SAR signal processing. This paper uses the Householder transform to obtain a ground plane circular SAR (CSAR) signal phase history from the slant plane CSAR phase history by inverting the linear shift-varying system model, thereby circumventing the need for explicitly computing a pseudo-inverse. The Householder transform has recently been shown to have improved error bounds and stability as an underdetermined and ill-conditioned system solver, and the Householder transform is computationally efficient.

TU-FF-A4-06: Motion Blurring Correction for a Cone Beam CT System with Image Decomposition and Deconvolution Method

Purpose: To improve the image resolution of a cone beam breast CT system by correcting the image blurring caused by gantry motion for a system works under continuous fluoroscopy acquisition mode. Method and Materials: Computer simulation models were built to simulate the effects on CT system resolution from different subcomponents, including the focal spot distribution and gantry motion. The system MTF results showed that image resolution degraded from the center towards the edge of the FOV, along the azimuthal direction. The major cause of this degradation was the gantry motion during continuous fluoroscopy acquisition. Azimuthal MTF results from computer simulation were used to generate a 2-D MTF response function to model the system resolution property by fitting discrete data points into a set of 3rd order polynomials. To preserve the image resolution of CT images on the radial direction, the original CT images were converted from the Cartesian coordinates into a new orthogonal coordinate system by the following transformation: For each radial position, the image data were de-convolved with corresponding system MTF in frequency domain along the azimuthal direction (along T axis). CT Images were converted back to Cartesian coordinates after deconvolution by: Results: A full view 2D MTF function was generated from computer simulations. With the presented method, the azimuthal resolution of the CT image was improved while the radial resolution was preserved. This method was applied to simulated phantom images and clinical breast CT images, qualitative observations showed improved spatial resolution along azimuthal direction. Conclusion: The full view 2D MTF function proved to be sufficient to model the spatial resolution of a cone beam CT scanner, which is a non-isotropic, shift-variant system. The presented novel image decomposition and deconvolution method demonstrated the promising potential to correct for the spatial resolution degradation.