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Concept# Linear time-invariant system

Summary

In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining h(t)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.
Linear time-invariant system theory is also used in , where the systems have

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EE-205: Signals and systems (for EL&IC)

This class teaches the theory of linear time-invariant (LTI) systems. These systems serve both as models of physical reality (such as the wireless channel) and as engineered systems (such as electrical circuits, filters and control strategies).

MICRO-310(b): Signals and systems I (for SV)

Présentation des concepts et des outils de base pour l'analyse et la caractérisation des signaux, la conception de systèmes de traitement et la modélisation linéaire de systèmes pour les étudiants en sciences de la vie. Application de ces techniques au traitement et à la transmission de signaux.

MICRO-310(a): Signals and systems I (for MT)

Présentation des concepts et des outils de base pour la caractérisation des signaux ainsi que pour l'analyse et la synthèse des systèmes linéaires (filtres ou canaux de transmission). Application de ces techniques au traitement du signal et aux télécommunications.

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This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes.

In this paper, we describe linear turbo equalizers (TEQ) and investigate their practical application to underwater acoustic communications. Owing to the ability to achieve a good performance-complexity trade-off, linear TEQ is a good candidate for long reverberant channels, which usually demand high computational complexity. First, we reveal a relationship between two different TEQ structures; channel estimate (CE)-based minimum mean square error (MMSE) TEQ versus direct-adaptive linear TEQ. We show that without inclusion of the second-order a priori statistics, the coefﬁcients of direct-adaptive TEQ converge to linear time-invariant form, though an optimal MMSE solution derived from a priori information is time-variant. Nevertheless, the direct-adaptive TEQ yields performance comparable to the CE-based MMSE TEQ while maintaining lower complexity. This was conﬁrmed through real experiments conducted off the coast of Martha’s Vinyard, MA (“SPACE 08”). We also discuss a practical design of a multi-channel least mean square (LMS) TEQ and experiments show that the LMS-TEQ successfully decodes data achieving up to 19.53 kbit/s for 1000 meter distance.

Giancarlo Ferrari Trecate, Mustafa Sahin Turan, Liang Xu

Consensusability of multi-agent systems (MASs) certifies the existence of a distributed controller capable of driving the states of each subsystem to a consensus value. We study the consensusability of linear interconnected MASs (LIMASs) where, as in several real-world applications, subsystems are physically coupled. We show that consensusability is related to the simultaneous stabilizability of multiple LTI systems, and present a novel sufficient condition in form of a linear program for verifying this property. We also derive several necessary and sufficient consensusability conditions for LIMASs in terms of parameters of the subsystem matrices and the eigenvalues of the physical and communication graph Laplacians. The results show that weak physical couplings among subsystems and densely connected physical and communication graphs are favorable for consensusability. Finally, we validate our results through simulations of networks of supercapacitors and DC microgrids.

2021