An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems

AbstractThis work proposes a model reduction method, the adaptive-order rational Arnoldi (AORA) method, to be applied to large-scale linear systems. It is based on an extension of the classical multi-point Padé approximation (or the so-called multi-point moment matching), using the rational Arnoldi iteration approach. Given a set of predetermined expansion points, an exact expression for the error between the output moment of the original system and that of the reduced-order system, related to each expansion point, is derived first. In each iteration of the proposed adaptive-order rational Arnoldi algorithm, the expansion frequency corresponding to the maximum output moment error will be chosen. Hence, the corresponding reduced-order model yields the greatest improvement in output moments among all reduced-order models of the same order. A detailed theoretical study is described. The proposed method is very appropriate for large-scale electronic systems, including VLSI interconnect models and digital filter designs. Several examples are considered to demonstrate the effectiveness and efficiency of the proposed method

Incremental construction of LSTM recurrent neural network

Long Short--Term Memory (LSTM) is a recurrent neural network that uses structures called memory blocks to allow the net remember significant events distant in the past input sequence in order to solve long time lag tasks, where other RNN approaches fail. Throughout this work we have performed experiments using LSTM networks extended with growing abilities, which we call GLSTM. Four methods of training growing LSTM has been compared. These methods include cascade and fully connected hidden layers as well as two different levels of freezing previous weights in the cascade case. GLSTM has been applied to a forecasting problem in a biomedical domain, where the input/output behavior of five controllers of the Central Nervous System control has to be modelled. We have compared growing LSTM results against other neural networks approaches, and our work applying conventional LSTM to the task at hand.Postprint (published version

Hankel-norm approximation of FIR filters: a descriptor-systems based approach

We propose a new method for approximating a matrix finite impulse response (FIR) filter by an infinite impulse response (IIR) filter of lower McMillan degree. This is based on a technique for approximating discrete-time descriptor systems and requires only standard linear algebraic routines, while avoiding altogether the solution of two matrix Lyapunov equations which is computationally expensive. Both the optimal and the suboptimal cases are addressed using a unified treatment. A detailed solution is developed in state-space or polynomial form, using only the Markov parameters of the FIR filter which is approximated. The method is finally applied to the design of scalar IIR filters with specified magnitude frequency-response tolerances and approximately linear-phase characteristics. A priori bounds on the magnitude and phase errors are obtained which may be used to select the reduced-order IIR filter order which satisfies the specified design tolerances. The effectiveness of the method is illustrated with a numerical example. Additional applications of the method are also briefly discussed

Robust filtering for bilinear uncertain stochastic discrete-time systems

Copyright [2002] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper deals with the robust filtering problem for uncertain bilinear stochastic discrete-time systems with estimation error variance constraints. The uncertainties are allowed to be norm-bounded and enter into both the state and measurement matrices. We focus on the design of linear filters, such that for all admissible parameter uncertainties, the error state of the bilinear stochastic system is mean square bounded, and the steady-state variance of the estimation error of each state is not more than the individual prespecified value. It is shown that the design of the robust filters can be carried out by solving some algebraic quadratic matrix inequalities. In particular, we establish both the existence conditions and the explicit expression of desired robust filters. A numerical example is included to show the applicability of the present method

Single-ensemble-based eigen-processing methods for color flow imaging-Part I. the Hankel-SVD filter

Because of their adaptability to the slow-time signal contents, eigen-based filters have shown potential in improving the flow detection performance of color flow images. This paper proposes a new eigen-based filter called the Hankel-SVD filter that is intended to process each slow- time ensemble individually. The new filter is derived using the notion of principal Hankel component analysis, and it achieves clutter suppression by retaining only the principal components whose order is greater than the clutter eigen- space dimension estimated from a frequency-based analysis algorithm. To assess its efficacy, the Hankel-SVD filter was first applied to synthetic slow-time data (ensemble size: 10) simulated from two different sets of flow parameters that model: (1) arterial imaging (blood velocity: 0 to 38.5 cm/s, tissue motion: up to 2 mm/s, transmit frequency: 5 MHz, pulse repetition period: 0.4 ms) and 2) deep vessel imaging (blood velocity: 0 to 19.2 cm/s, tissue motion: up to 2 cm/s, transmit frequency: 2 MHz, pulse repetition period: 2.0 ms). In the simulation analysis, the post-filter clutter- to-blood signal ratio (CBR) was computed as a function of blood velocity. Results show that for the same effective stopband size (50 Hz), the Hankel-SVD filter has a narrower transition region in the post-filter CBR curve than that of another type of adaptive filter called the clutter- downmixing filter. The practical efficacy of the proposed filter was tested by application to in vivo color flow data obtained from the human carotid arteries (transmit frequency: 4 MHz, pulse repetition period: 0.333 ms, ensemble size: 10). The resulting power images show that the Hankel-SVD filter can better distinguish between blood and moving- tissue regions (about 9 dB separation in power) than the clutter-downmixing filter and a fixed-rank multi-ensemble- based eigen-filter (which showed a 2 to 3 dB separation). © 2006 IEEE.published_or_final_versio

Computation of the para-pseudoinverse for oversampled filter banks: Forward and backward Greville formulas

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2008 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience. In frame expansions, the least square signal reconstruction operator is called the dual frame, which can be obtained by choosing the synthesis filter bank as the para-pseudoinverse of the analysis bank. In this paper, we study the computation of the dual frame by exploiting the Greville formula, which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended. Here, we first develop the backward Greville formula to handle the case of row deletion. Based on the forward Greville formula, we then study the computation of para-pseudoinverse for extended filter banks and Laplacian pyramids. Through the backward Greville formula, we investigate the frame-based error resilient transmission over erasure channels. The necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived. A postfiltering structure is also presented to implement the para-pseudoinverse when the transform coefficients in one subband are completely lost

Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio

In this paper we consider the design of spectrally efficient time-limited pulses for ultrawideband (UWB) systems using an overlapping pulse position modulation scheme. For this we investigate an orthogonalization method, which was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally efficient for UWB systems, from a set of N equidistant translates of a time-limited optimal spectral designed UWB pulse. We derive an approximate L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram matrix to obtain a practical filter implementation. We show that the centered ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends to infinity. The set of translates of the Nyquist pulse forms an orthonormal basis or the shift-invariant space generated by the initial spectral optimal pulse. The ALO transform provides a closed-form approximation of the L\"owdin transform, which can be implemented in an analog fashion without the need of analog to digital conversions. Furthermore, we investigate the interplay between the optimization and the orthogonalization procedure by using methods from the theory of shift-invariant spaces. Finally we develop a connection between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201

Signal Reconstruction via H-infinity Sampled-Data Control Theory: Beyond the Shannon Paradigm

This paper presents a new method for signal reconstruction by leveraging sampled-data control theory. We formulate the signal reconstruction problem in terms of an analog performance optimization problem using a stable discrete-time filter. The proposed H-infinity performance criterion naturally takes intersample behavior into account, reflecting the energy distributions of the signal. We present methods for computing optimal solutions which are guaranteed to be stable and causal. Detailed comparisons to alternative methods are provided. We discuss some applications in sound and image reconstruction

Fractional biorthogonal partners in channel equalization and signal interpolation

The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners