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

I sistemi a tempo continuo hanno segnali di ingresso e uscita continui, con il tempo misurato continuamente, e sono generalmente definiti da equazioni differenziali o algebriche.
Ad esempio, in un circuito RC, la relazione tra la tensione di ingresso e quella di uscita è espressa attraverso un'equazione differenziale derivata utilizzando la legge di Ohm e la relazione tensione-corrente del condensatore.
I sistemi a tempo discreto elaborano segnali di input e output a intervalli specifici e il loro comportamento è descritto da equazioni alle differenze in istanze temporali distinte. Un esempio di tale sistema è un semplice modello per il saldo di un conto bancario da un mese all'altro.
Un altro tipo di sistema è variabile nel tempo e invariante nel tempo.
Un sistema variabile nel tempo ha parametri che cambiano nel tempo. I sistemi a tempo continuo e variabile nel tempo sono descritti da equazioni differenziali con coefficienti dipendenti dal tempo, mentre i sistemi a tempo discreto e variabili nel tempo sono descritti da equazioni alle differenze con parametri dipendenti dal tempo.
I sistemi tempo-invarianti producono un'uscita che, quando il segnale di ingresso è spostato nel tempo, si sposta in modo identico senza alterare le caratteristiche del sistema.
Un circuito RC è invariante nel tempo se i valori di resistenza e capacità rimangono costanti. Se questi fluttuano, il sistema diventa variabile nel tempo poiché i risultati dell'esperimento variano a seconda di quando l'esperimento viene condotto.

classificazione dei sistemi-ii

I sistemi a tempo continuo hanno segnali d’ingresso e d’uscita continui, con il tempo misurato in modo continuo. Questi sistemi sono generalmente definiti da equazioni differenziali o algebriche. Ad esempio, in un circuito RC, la relazione tra la tensione d’ingresso e quella d’uscita è espressa attraverso un'equazione differenziale derivata dalla legge di Ohm e dalla relazione del condensatore:





<img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="height: 30px;">





I sistemi a tempo discreto hanno segnali d’ingresso e d’uscita a intervalli specifici, definiti in istanti distinti da equazioni differenziali. Un esempio è un semplice modello per il saldo di un conto bancario da un mese all'altro, rappresentato come:





<img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="height: 14px;">





dove B[n] è il saldo al mese n, D[n] è il deposito e W[n] è il prelievo.







I sistemi possono anche essere classificati come variabili o invarianti nel tempo. Un sistema variabile ha dei parametri che cambiano nel tempo. I sistemi tempo-varianti a tempo continuo sono descritti da equazioni differenziali variabili nel tempo, mentre i sistemi tempo-varianti a tempo discreto sono definiti da equazioni differenziali variabili nel tempo.





Un sistema invariante nel tempo mostra identici spostamenti temporali nei segnali di output quando il segnale di input è spostato nel tempo. Matematicamente, se y(t) è l'output per l'input x(t), allora per un sistema invariante nel tempo, y(t−t_0) è l'output per l'input x(t−t_0). Ad esempio, un circuito RC è invariante nel tempo se i valori di resistenza R e capacità C rimangono costanti. Se questi valori fluttuano, il sistema diventa tempo-variante, questo significa che i risultati varieranno a seconda di quando viene condotto l'esperimento.





Comprendere queste distinzioni nei sistemi a tempo continuo e a tempo discreto, così come nei sistemi variabili nel tempo e invarianti nel tempo, è fondamentale per analizzare e progettare un'ampia gamma di sistemi ingegneristici.

Classificazione dei sistemi-II

I sistemi a tempo continuo hanno segnali di ingresso e uscita continui, con il tempo misurato continuamente, e sono generalmente definiti da segnali differenziali o algebrici ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

Introduction to Signals and Systems

Introduzione ai segnali e ai sistemi

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

Signals and Systems in Everyday Data

Signals and systems help describe how data moves through the world. A signal x(t) is a set of data or a time function that represents a variable of interest. It can carry information about atmospheric temperature, humidity, human voice, television images, a dog’s bark, or birdsongs. A signal can also depend on more than one independent variable. An image, for example, depends on horizontal and vertical position, so it can be treated as a two-dimensional signal. In this course, the focus is on...

classification of signals

Signal Types by Time and Amplitude

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...

energy and power signals

Power and Energy in Electrical Signals

Power and energy in electrical signals are measured from voltage and current across a resistor. These signals help describe how much energy a signal carries and how much power it delivers over time. For a continuous-time signal, total energy is found by integrating the square of the signal's magnitude over a chosen time interval. Time-averaged power is then the total energy divided by the length of that interval. For a discrete-time signal, total energy is found by summing the squares of the...

even and odd signals

Signal Symmetry and Decomposition

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...

basic continuous time signals

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

Rectangular and triangular pulse functions are basic signal shapes used to model waveforms in signal processing. The rectangular pulse has a flat top and is centered at the origin with a height of one unit. It is described by two parameters: T, which gives the center location on the time axis, and τ, which sets the pulse duration. A rectangular pulse can also represent a real signal with a chosen amplitude and length. For example, a pulse with a 5 V amplitude, a 3-second duration, and a center...

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...

basic operations on signals

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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