Impulse Response Sequence Research Articles

Tensor z‐Transform

The multi‐input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z‐transform of a discrete‐time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z‐transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z‐transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary‐dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z‐transform and approximate the corresponding discrete‐time tensor sequences. Our proposed new tensor z‐transform in this work can be applied to design digital tensor filters including infinite‐impulse‐response (IIR) tensor filters (involving global feedback mechanisms) and finite‐impulse‐response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z‐transform.

Uniform rectangular distribution of far-field intensity by optical phased array

In this paper, to solve the problem of uneven distribution of far-field intensity of Flash Lidar, a method of amplitude modulation for optical phased array (OPA) is proposed, which makes the far-field intensity distribution close to the uniform rectangular distribution. We use the impulse response sequence of Finite Impulse Response (FIR) digital filter designed by the window functions to modulate the amplitude of the OPA. This kind of uniform intensity distribution can simultaneously detect a wide range with the same intensity. The modulation sequence can also be extended to two-dimensional situation. In the case of considering the radiation pattern of optical waveguide, the amplitude modulation sequence can also be modified to obtain the far-field pattern that is close to the uniform rectangular distribution.

Height Adjustable Triangular (HAT) Window Function for Impulse Response Modification of Signal Processing Systems

A widow function, in signal processing and statistics, is a mathematical function that has zero values outside its chosen interval or limit of sequence, normally symmetric around the middle of the interval. Usually the middle of the window is either maximum or near maximum and tappers smoothly as it moves away to the sides. When another function or sequence of data is mathematically multiplied by the window function it forces the product to assume its nature of zero-value outside the interval and tapering from middle to the sides. Windows are finite functions and their main function is to modify an infinite impulse response sequence so as to make it finite within its chosen interval in system design. Several windows are in existence and they include Hamming, rectangular, Han, Kaiser, Triangular, Blackman, Sine, Blackman-Harris, Gaussian, Doph-Chebyshev and Lanczos, windows. Others are Parzen, Nuttall, flat top, Turkey, windows and many more. The window to apply in the design depends on the characteristics of the signal to be processed, types of system to be implemented and quality of output desired. In this paper, a new window called Height Adjustable triangular (HAT) window function is developed and added to the list of windows for signal processing system designs. The effectiveness of the window is tested by examining its characteristics. The adjustment parameter varies the amplitude or height of the window. Result shows that it is stable and linear.

Height Adjustable Triangular (HAT) Window Function for Impulse Response Modification of Signal Processing Systems

A widow function, in signal processing and statistics, is a mathematical function that has zero values outside its chosen interval or limit of sequence, normally symmetric around the middle of the interval. Usually the middle of the window is either maximum or near maximum and tappers smoothly as it moves away to the sides. When another function or sequence of data is mathematically multiplied by the window function it forces the product to assume its nature of zero-value outside the interval and tapering from middle to the sides. Windows are finite functions and their main function is to modify an infinite impulse response sequence so as to make it finite within its chosen interval in system design. Several windows are in existence and they include Hamming, rectangular, Han, Kaiser, Triangular, Blackman, Sine, Blackman-Harris, Gaussian, Doph-Chebyshev and Lanczos, windows. Others are Parzen, Nuttall, flat top, Turkey, windows and many more. The window to apply in the design depends on the characteristics of the signal to be processed, types of system to be implemented and quality of output desired. In this paper, a new window called Height Adjustable triangular (HAT) window function is developed and added to the list of windows for signal processing system designs. The effectiveness of the window is tested by examining its characteristics. The adjustment parameter varies the amplitude or height of the window. Result shows that it is stable and linear.

Tangential interpolation-based eigensystem realization algorithm for MIMO systems

ABSTRACTThe eigensystem realization algorithm (ERA) is a commonly used data-driven method for system identification and reduced-order modelling of dynamical systems. The main computational difficulty in ERA arises when the system under consideration has a large number of inputs and outputs, requiring to compute a singular value decomposition (SVD) of a large-scale dense Hankel matrix. In this work, we present an algorithm that aims to resolve this computational bottleneck via tangential interpolation. This involves projecting the original impulse response sequence onto suitably chosen directions. The resulting data-driven reduced model preserves stability and is endowed with an a priori error bound. Numerical examples demonstrate that the modified ERA algorithm with tangentially interpolated data produces accurate reduced models while, at the same time, reducing the computational cost and memory requirements significantly compared to the standard ERA. We also give an example to demonstrate the limitations of the proposed method.

Digital and analog integer delayed modeling and control for multivariable systems with multiple time delays in states, inputs and outputs

This paper presents a new approximated integer delayed modeling and control technique for Continuous-time Fractional Delay Differential Equations (CFDDEs). Using the impulse response sequences of the CFDDE and the balanced model-reduction method, a delay-free discrete-time state-space model is constructed. Then, an equivalent Discrete-time Integer Delay Difference Equation (DIDDE) is obtained by transforming the obtained discrete-time state-space model into a controller-type block companion form. Furthermore, based on the obtained DIDDE, an equivalent Continuous-time Integer Delay Differential Equation (CIDDE) is determined by means of the newly developed Chebyshev's bilinear approximation method. For digital control of the CFDDE, an optimal Discrete-time Integer Delayed Control Law (DIDCL) is designed using the conventional discrete-time LQR approach together with the obtained delay-free discrete-time state-space model. On the other hand, for continuous-time control of the CFDDE, a Continuous-time Integer Delayed Control Law (CIDCL) is determined from the designed DIDCL by means of the inverse Chebyshev's bilinear approximation method. Finally, digital and analog integer delayed observers are constructed for the implementations of the developed DIDCL and CIDCL, respectively. An illustrative example is given to demonstrate the effectiveness of the proposed method.

A Constructive Positive Realization with Sparse Matrices for a Continuous-Time Positive Linear System

This paper discusses a computational method to construct positive realizations with sparse matrices for continuous-time positive linear systems with multiple complex poles. To construct a positive realization of a continuous-time system, we use a Markov sequence similar to the impulse response sequence that is used in the discrete-time case. The existence of the proposed positive realization can be analyzed with the concept of a polyhedral convex cone. We provide a constructive algorithm to compute positive realizations with sparse matrices of some positive systems under certain conditions. A sufficient condition for the existence of a positive realization, under which the proposed constructive algorithm works well, is analyzed.

Investigating Feedback Loops in Synthetic Neural Networks by a Nonparametric Identification Method

Abstract Feedback circuits are crucial dynamic motifs which occur in many intra-cellular and inter-cellular regulatory networks. In this paper, an effective nonparametric identification method, Non-causal Impulse Response Component Method (NIRCM) is developed to identify feedback loops embedded in biological neural networks, which uses only time-series experimental data. The NIRCM, based on correlation identification and spectral factor analysis, provides a non-causal component criterion for the identification of feedback loops. Significant non-causal components of the impulse response sequences observed in the negative time axis imply an existence of feedback loop. The proposed identification method was applied to several 2-node SRM (Spike Response Model) networks. For these synthetic models, NIRCM correctly implies the existence of feedback loops and shows their effectiveness of feedback loop identifications.

Identification of feedback loops embedded in cellular circuits by investigating non-causal impulse response components

Feedback circuits are crucial dynamic motifs which occur in many biomolecular regulatory networks. They play a pivotal role in the regulation and control of many important cellular processes such as gene transcription, signal transduction, and metabolism. In this study, we develop a novel computationally efficient method to identify feedback loops embedded in intracellular networks, which uses only time-series experimental data and requires no knowledge of the network structure. In the proposed approach, a non-parametric system identification technique, as well as a spectral factor analysis, is applied to derive a graphical criterion based on non-causal components of the system's impulse response. The appearance of non-causal components in the impulse response sequences arising from stochastic output perturbations is shown to imply the presence of underlying feedback connections within a linear network. In order to extend the approach to nonlinear networks, we linearize the intracellular networks about an equilibrium point, and then choose the magnitude of the output perturbations sufficiently small so that the resulting time-series responses remain close to the chosen equilibrium point. In this way, the impulse response sequences of the linearized system can be used to determine the presence or absence of feedback loops in the corresponding nonlinear network. The proposed method utilizes the time profile data from intracellular perturbation experiments and only requires the perturbability of output nodes. Most importantly, the method does not require any a priori knowledge of the system structure. For these reasons, the proposed approach is very well suited to identifying feedback loops in large-scale biomolecular networks. The effectiveness of the proposed method is illustrated via two examples: a synthetic network model with a negative feedback loop and a nonlinear caspase function model of apoptosis with a positive feedback loop.

Multiple Multiplicative Fault Diagnosis for Dynamic Processes via Parameter Similarity Measures

In this paper, a systematic approach that employs novel parameter similarities is proposed to detect, isolate, and identify multiplicative faults in a multi-input multi-output (MIMO) dynamic system. These multiplicative faults are usually difficult to deal with using conventional statistics-based methods. Similarity measures based on impulse response sequences of dynamic elements are defined. By using the proposed similarity measures, the overall and local faults, including dead time, gain, and other dynamic parameters in a multivariate process, can be detected and isolated. The method has the potential to be used for on-line fault diagnosis. Simulated numerical and industrial examples are used to demonstrate the methodology.

A New Variable Tap-Length LMS Algorithm to Model an Exponential Decay Impulse Response

This letter proposes a new variable tap-length least-mean-square (LMS) algorithm for applications in which the unknown filter impulse response sequence has an exponential decay envelope. The algorithm is designed to minimize the mean-square deviation (MSD) between the optimal and adaptive filter weight vectors at each iteration. Simulation results show the proposed algorithm has a faster convergence rate as compared with the fixed tap-length LMS algorithm and is robust to the initial tap-length choice

Pseudo-fractional ARMA modelling using a double Levinson recursion

The modelling of fractional linear systems through ARMA models is addressed. To perform this study, a new recursive algorithm for impulse response ARMA modelling is presented. This is a general algorithm that allows the recursive construction of ARMA models from the impulse response sequence. This algorithm does not need an exact order specification, as it gives some insights into the correct orders. It is applied to modelling fractional linear systems described by fractional powers of the backward difference and the bilinear transformations. The analysis of the results leads to propose suitable models for those systems.

The Blind Identification of Multi-Inputs and Multi-Outputs Shallow-Water Acoustic Channel

Blind channel identification/estimation is very important for object detection, trace, localization in the ocean acoustics. Time domain blind identification algorithm requiring exact length of the channel being identification. Due to the characteristics of the shallow-water channel, the length of channel impulse response sequence is uncertain, Hence a frequency domain method for the blind MIMO (Multiple-Input Multiple-Output) underwater identification based on higher order statistics (HOS) is used to estimate the original acoustic channel from received signals on hydrophones only, with the low signal to noise ratio (SNR). The simulation results in the acoustic environment proved this work is effective and efficient for blind identification of the shallow-water acoustic channel.

Algorithm for positive realization of transfer functions

The aim of this brief is to present a finite-step algorithm for the positive realization of a rational transfer function H(z). In comparison with previously described algorithms we emphasize that we do not make an a priori assumption on (but, instead, include a finite step procedure for checking) the nonnegativity of the impulse-response sequence of H(z). For primitive transfer functions a new method for reducing the pole order of the dominant pole is also proposed.

Filter-generating systems

Combining the methods of generating functions and discrete-time linear systems, we can develop multidimensional (M-D) digital systems for systematic generation of entire families of interrelated digital filters. These M-D systems, dubbed filter-generating systems, are mostly of infinite-impulse response (IIR) type and can produce the impulse response sequence of any member of the family. Temporal and spatio-temporal implementation of filter-generating systems are studied. A temporal implementation scheme is devised for causal filter-generating systems in order to implement an arbitrary filter member using (M-D) signal processing techniques. It is shown that this may result in low order recursive filtering structures that can implement arbitrarily high order members of the family. A spatio-temporal implementation scheme is developed for generation of regular signal flow-graphs for the generated family of finite-impulse response (FIR) filters. It is shown that the signal flow-graphs give rise to cellular array structures. Concrete examples are provided for maximally flat FIR filters.

Nonparametric estimation of transfer functions: rates of convergence and adaptation

The paper deals with estimating the transfer functions of stable linear time-invariant systems under stochastic assumptions. We adopt a nonparametric minimax approach for measuring the estimation accuracy. The quality of an estimator is measured by its worst case error over a family of transfer functions. The families with polynomially and exponentially decaying impulse response sequences are considered. We establish nonasymptotic upper bounds on the accuracy of the least squares estimator for finite impulse response approximation. It is shown that the attainable estimation accuracy is determined essentially by the rate at which the true impulse response tends to zero. Lower bounds on the estimation accuracy are presented. An adaptive estimator which does not exploit any a priori information about the true system, is developed.

Non-parametric identification of generalized Hammerstein models

The Hammerstein model is considered in a generalized form, where its nonlinear element can have multi-inputs and a finite memory. The identification of the multi-input finite memory nonimearity and the impulse response sequence of the model is treated using a non-parametric approach. A numerical example is given. The identification results of the example illustrate the effectiveness of the developed technique.

Mixed Objective Control Synthesis: Optimal $\ell_ 1/{\cal H}_2$ Control

In this paper we consider the problem of minimizing the $\ell_1$ norm of the transfer function from the exogenous input to the regulated output over all internally stabilizing controllers while keeping its $\cal{H}$$_2$ norm under a specified level. The problem is analyzed for the discrete-time, single-input single-output (SISO), linear-time invariant case. It is shown that an optimal solution always exists. Duality theory is employed to show that any optimal solution is a finite impulse response sequence, and an a priori bound is given on its length. Thus, the problem can be reduced to a finite-dimensional convex optimization problem with an a priori determined dimension. Finally, it is shown that, in the region of interest of the $\cal{H}$$_2$ constraint level, the optimal is unique and continuous with respect to changes in the constraint level.

Parameterization of stable systems from partial impulse response sequences

The authors address the problem of identifying the class of all stable system transfer functions that interpolate the given partial impulse response sequence. In this context, classical Pade approximations that are also stable, are shown to be a special case of this general formulation. The theory developed in this connection is utilized to obtain a new criterion for determining the model order and system parameters for rational systems, and, further, to generate nonminimum phase optimal stable rational approximations of nonrational systems from its impulse response sequence.

A multivariable Steiglitz-McBride method: Stationary points and a priori error bound

We present two dual versions of a multi-input/multi-output (MIMO) SteiglitzMcBride identification method, and give an analytic description of the set of the possible stationary points. As in the scalar case (Regalia and Mboup 1992, Regalia 1995), the description is given in terms of first- and second-order interpolation constraints on the model impulse response and covariance sequences, respectively. The constraints are related to the theory of M -Markov covariance equivalent realizations and generalize the works of Inouye (1983) and King et al. (1988). It is shown that the description is intimately connected to a class of first- and secondorder matrix-valued interpolation problems of tangential Nevanlinna-Pick type. Such problems are studied by Alpay et al. (1996). We also examine the quality of the model furnished in reduced order cases, and show in particular that the mismodelling error at any stationary point of the method can be bounded in terms of the Hankel singular values of the unknown system. The bound so obtained compares favourably with known bounds from L - and Hankel-norm model 2 reduction.