Name: system properties: linearity, causality, memory
Uploaded: 2024-09-26T13:16:54.000Z
Duration: 1 min 26 s
Channel: JoVE

Линейность — это свойство системы с прямой связью «вход-выход», сочетающее в себе однородность и аддитивность.
Однородность предполагает, что если входные данные умножаются на константу, то выходные данные умножаются на ту же константу. Аддитивность означает, что реакция на несколько входных данных является суммой их индивидуальных ответов.
Система, обладающая как однородностью, так и аддитивными свойствами, классифицируется как линейная система.
К линейным системам относятся схемы с резисторами, конденсаторами и катушками индуктивности, которые придерживаются принципа суперпозиции.
И наоборот, системы, которые не подчиняются уравнению линейности, классифицируются как нелинейные, часто включая такие устройства, как выпрямители и диоды.
Причинно-следственная система характеризуется текущей реакцией, не зависящей от будущих входных значений. В качестве примера можно привести движение автомобиля, которое не может предсказать будущие действия за рулем.
Непричинные системы, такие как идеальные фильтры, не могут быть физически реализованы и не следуют принципу причинности.
Динамические системы показывают память, где на вывод влияют как прошлые, так и будущие входные данные.
Статические системы — это системы без памяти или системы мгновенного действия, в которых выходной ток основан исключительно на текущем входе.

 Классификация систем-i

Линейность — это свойство системы, характеризующееся прямой связью вход-выход, сочетающей однородность и аддитивность.

Свойство однородности гласит, что если сигнал на входе x(t) умножается на константу c, отклик на выходе y(t) также умножается на ту же константу. Математически это выражается как:

<img alt="Equation1" src="/files/ftp_upload/16045/16045_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;">

Аддитивность означает, что отклик на сумму нескольких входов является суммой их индивидуальных откликов. Для входов x_1(t) и x_2(t), производящих выходы y_1(t) и y_2(t), соответственно:

<img alt="Equation2" src="/files/ftp_upload/16045/16045_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">

Объединение однородности и аддитивности дает уравнение линейной системы:

<img alt="Equation3" src="/files/ftp_upload/16045/16045_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">

Линейные системы включают в себя цепи с резисторами, конденсаторами и индукторами, которые придерживаются принципа суперпозиции. Наоборот, системы, которые не подчиняются уравнению линейности, классифицируются как нелинейные, такие как выпрямители и диоды.

Каузальная система — это система, в которой текущий отклик не зависит от будущих входных значений. Например, движение автомобиля не может предсказать будущие действия водителя, что делает ее каузальной системой. Напротив, некаузальные системы, такие как идеальные фильтры, не могут быть физически реализованы, поскольку они не следуют принципу каузальности.

Динамические системы демонстрируют память, где выход зависит от прошлых и настоящих входов. Примером является электрическая цепь с конденсаторами или индукторами, где на текущий выход влияют прошлые входы. Статические системы, также известные как системы без памяти или мгновенные системы, имеют выходы, основанные исключительно на текущих входах, например, простая резисторная цепь, где выходное напряжение является прямой функцией текущего входного напряжения.

Понимание этих свойств — линейности, каузальности (причинности) и памяти — необходимо для анализа и проектирования систем в различных областях техники.

 Классификация систем-I

Линейность — это свойство системы с прямой связью «вход-выход», сочетающее в себе однородность и аддитивность.
Однородность диктует, что если входные данные ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Introduction to Signals and Systems

Введение в сигналы и системы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 to a sum of inputs equals the sum of the separate responses. Together, these rules define a linear system, and circuits with resistors, capacitors, and inductors can follow the superposition principle.

Systems that do not satisfy the linearity equation are non-linear. Diodes and rectifiers are examples of non-linear systems. Causality is another key property. A causal system depends only on the present input and past inputs, not on future values.

A car’s motion is a simple example of a causal system, because it cannot depend on future driving actions. Non-causal systems, such as ideal filters, are not physically realizable because they do not follow causality. Memory is the last major property. Dynamic systems have memory, so their output depends on past and present inputs, as in circuits with capacitors or inductors. Static systems, also called memoryless or instantaneous systems, depend only on the present input. A simple resistor circuit is an example, since its output voltage is a direct function of the current input voltage.

System Properties: Linearity, Causality, Memory

system-properties-linearity-causality-memory

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 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 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...

key terms and concepts

Explore linearity, causality, and memory in system behavior. Learn how linear, non-linear, causal, dynamic, and static systems differ in engineering.

System Properties: Linearity, Causality, Memory | 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

System Properties: Linearity, Causality, Memory

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