Name: continuous and discrete system models
Uploaded: 2024-09-26T13:16:54.000Z
Duration: 1 min 31 s
Channel: JoVE

Les systèmes à temps continu ont des signaux d’entrée et de sortie continus, avec un temps mesuré en continu, et sont généralement définis par des équations différentielles ou algébriques.
Par exemple, dans un circuit RC, la relation entre la tension d'entrée et de sortie est exprimée par une équation différentielle dérivée de la loi d'Ohm et de la relation tension-courant du condensateur.
Les systèmes à temps discret traitent les signaux d’entrée et de sortie à des intervalles spécifiques et leur comportement est décrit par des équations de différence à des instances de temps distinctes. Un exemple d’un tel système est un modèle simple pour le solde d’un compte bancaire d’un mois à l’autre.
Un autre type de système est variable dans le temps et invariant dans le temps.
Un système variant dans le temps a des paramètres qui changent au fil du temps. Les systèmes en temps continu et variant dans le temps sont décrits par des équations différentielles avec des coefficients dépendants du temps, tandis que les systèmes à temps discret et variant dans le temps sont décrits par des équations différentielles avec des paramètres dépendants du temps.
Les systèmes invariants dans le temps produisent une sortie qui, lorsque le signal d'entrée est décalé dans le temps, se décale à l'identique sans modifier les caractéristiques du système.
Un circuit RC est invariant dans le temps si les valeurs de résistance et de capacité restent constantes. Si ceux-ci fluctuent, le système devient variable dans le temps, car les résultats de l’expérience varient en fonction du moment où l’expérience est menée.

classification des systèmes - ii

Les systèmes à temps continu ont des signaux d'entrée et de sortie continus, le temps étant mesuré en continu. Ces systèmes sont généralement définis par des équations différentielles ou algébriques. Par exemple, dans un circuit RC, la relation entre la tension d'entrée et la tension de sortie est exprimée par une équation différentielle dérivée de la loi d'Ohm et de la relation du condensateur,


<img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="height: 30px;">

Les systèmes à temps discret ont des signaux d'entrée et de sortie à des intervalles spécifiques, définis à des instants distincts par des équations différentielles. Un exemple est un modèle simple pour le solde d'un compte bancaire d'un mois à l'autre, représenté par :


<img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="height: 14px;">

où B[n] est le solde au mois n, D[n] est le dépôt et W[n] est le retrait.


Les systèmes peuvent également être classés comme variables dans le temps ou invariants dans le temps. Un système variable dans le temps a des paramètres qui changent au fil du temps. Les systèmes variables dans le temps à temps continu sont décrits par des équations différentielles variables dans le temps, tandis que les systèmes variables dans le temps à temps discret sont définis par des équations aux différences variables dans le temps.


Un système invariant dans le temps présente des décalages temporels identiques dans les signaux de sortie lorsque le signal d'entrée est décalé dans le temps. Mathématiquement, si y(t) est la sortie pour l'entrée x(t), alors pour un système invariant dans le temps, y(t−t_0) est la sortie pour l'entrée x(t−t_0). Par exemple, un circuit RC est invariant dans le temps si les valeurs de résistance R et de capacité C restent constantes. Si ces valeurs fluctuent, le système devient variable dans le temps, ce qui signifie que les résultats varieront en fonction du moment où l'expérience est réalisée.


La compréhension de ces distinctions dans les systèmes à temps continu et à temps discret, ainsi que dans les systèmes variables et invariants dans le temps, est essentielle pour analyser et concevoir un large éventail de systèmes d’ingénierie.

Classification des systèmes - II

Les systèmes à temps continu ont des signaux d’entrée et de sortie continus, avec un temps mesuré en continu, et sont généralement définis par des signaux différentiels ou algébriques ...

Education

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Electrical Engineering

Génie électrique

Unit 1

Unité 1

Introduction to Signals and Systems

Introduction aux signaux et systèmes

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Classification of Systems-II

continuous and discrete system models

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations.

An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their inputs and outputs are defined at specific time steps, and they are described by difference equations.

A bank account balance from month to month is a simple discrete-time model. In that example, B[n] is the balance at month n, D[n] is the deposit, and W[n] is the withdrawal. Systems are also grouped as time-varying or time-invariant. A time-varying system has parameters that change with time, while a time-invariant system keeps the same shift in output when the input is shifted.

For a time-invariant system, if y(t) is the output for input x(t), then y(t − t0) is the output for x(t − t0). An RC circuit is time-invariant when the resistance R and capacitance C stay constant. If those values change, the system becomes time-variant. Then the result depends on when the experiment is done. Knowing these categories helps in analyzing and designing many engineering systems.

Continuous and Discrete System Models

signal and system

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Signals in signal processing are grouped by time behavior, amplitude, and whether they start before or after time zero. These signal classes help students describe and compare signals before they are analyzed or processed. A continuous-time signal has a value at every instant in time. A discrete-time signal has values only at specific moments, often written as x(n), where n is an integer. Discrete-time signals often come from systems with naturally discrete values, such as digital audio...

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Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time. An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis. Any continuous-time signal can be split into even and odd...

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Step, Impulse, and Ramp Signals

Step, impulse, and ramp signals are the main basic continuous-time signals. They are also called singularity functions. These signals are known for discontinuities or for derivatives that are discontinuous. The unit step function, written as u(t), is zero for negative time values and one for positive time values. It has a jump at t = 0. In many cases, it can model an abrupt change, such as the step voltage that appears when a car ignition key is turned. The derivative of the unit step...

rectangular and triangular pulse function

rectangular-and-triangular-pulse-function

Modeling Pulses with Rectangles and Triangles

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exponential and sinusoidal signals

Complex Exponentials in Signal Behavior

Exponential and sinusoidal signals are used to describe how waveforms change over time. The continuous-time exponential function uses constants α and A. When both are real, it shows exponential growth or decay. Graphs of these signals make the rise or drop easy to see. When α is purely imaginary, the signal becomes a complex exponential. In this form, j is the imaginary unit and ω0 is the angular frequency. A complex exponential is periodic when its magnitude is unity. That periodic behavior...

basic discrete time signals

Discrete-Time Signal Basics and Shapes

Discrete-time signals include several basic sequences used in signal processing. The unit step sequence is 1 for zero and positive values of the integer n. A graph of eight sample points shows it as a step that begins at n = 0 and stays constant after that. The unit impulse, also called the sample sequence, is 0 for every n except at n = 0, where it equals 1. This sequence is written as δ(n). It is the first difference of the unit step sequence, and the unit step sequence u(n) is the...

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Signal Transformations in Time and Amplitude

Signal transformations in time and amplitude are key tools in signal processing. They include time reversal, time scaling, time shifting, and amplitude changes. These operations help students describe how a continuous-time signal changes when it is mirrored, stretched, moved, or resized. Time reversal flips a continuous-time signal about the vertical axis at t = 0. It is done by replacing t with -t, so x(t) becomes x(-t). A graph of the signal shows this mirrored shape clearly. Time scaling...

classification of systems-i

System Properties: Linearity, Causality, Memory

System properties such as linearity, causality, and memory help describe how a system responds to input. These ideas are useful for studying and designing systems in engineering. A system can be checked by looking at how its output changes with different inputs. Linearity means the input-output relationship follows homogeneity and additivity. Homogeneity says that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Additivity means the response...

classification of systems-ii

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations. An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their...

key terms and concepts

Learn how continuous-time and discrete-time systems differ, plus how time-varying and time-invariant behavior is identified using RC circuits and bank account examples.

Continuous and Discrete System Models | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Continuous and Discrete System Models

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