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

线性度是一种具有直接输入-输出关系的系统属性，结合了同质性和可加性。
同质性规定，如果输入乘以一个常数，则输出乘以相同的常数。加性意味着对多个输入的响应是它们各自响应的总和。
同时具有均匀性和可加性特性的系统被归类为线性系统。
线性系统包括带有电阻器、电容器和电感器的电路，它们遵循叠加原理。
相反，不遵守线性度方程的系统被归类为非线性系统，通常包括整流器和二极管等器件。
因果系统的特征是当前响应独立于未来输入值。一个例子是汽车的运动，它无法预测未来的驾驶动作。
非因果系统（例如理想滤波器）无法物理实现，也不遵循因果原则。
动态系统显示内存，其中输出受过去和未来输入的影响。
静态系统是无内存或瞬时系统，其中电流输出完全基于当前输入。

系统分类-i

线性是一种系统属性，其中的特征是直接的输入-输出关系，并且还结合了同质性和可加性。

同质性规定，如果输入 x(t) 乘以常数 c，则会输出 y(t) 乘以相同的常数。在数学中，可以将其表示为：

<img alt="Equation1" src="/files/ftp_upload/16045/16045_PT%20EQUATION_1.svg" style="height: 14px;">

可加性指的是由多个输入所组成的响应是它们各自的响应之和。对于输入 x_1(t) 和 x_2(t)，则会分别产生输出 y_1(t) 和 y_2(t)：

<img alt="Equation2" src="/files/ftp_upload/16045/16045_PT%20EQUATION_2.svg" style="height: 14px;">

将同质性和可加性进行结合，便能够得到线性系统方程：

<img alt="Equation3" src="/files/ftp_upload/16045/16045_PT%20EQUATION_3.svg" style="height: 14px;">

线性系统包括遵循叠加原理的电阻器、电容器和电感器电路。相反，不遵循线性方程的系统则会被归类为非线性系统，例如整流器和二极管。

因果系统指的当前响应与未来输入值无关的系统。例如，汽车的运动无法预测未来的驾驶行为，因此它是一个因果系统。相反，像理想滤波器这样的非因果系统则无法在物理上实现，因为它们不遵循因果关系原理。

动态系统具有记忆性，其输出取决于过去和现在的输入。例如，假设有一个带有电容器或电感器的电路，其当前输出会受到过去输入所产生的影响。静态系统（也称为无记忆或瞬时系统）的输出会完全基于当前的输入，例如简单的电阻电路，其输出电压是当前输入电压的直接函数。

了解这些属性（线性、因果关系和记忆性）对于分析和设计各个工程领域的系统来说是至关重要的。

系统分类-I

线性度是一种具有直接输入-输出关系的系统属性，结合了同质性和可加性。
同质性表明，如果输入是 ...

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Introduction to Signals and Systems

信号与系统导论

KEY TERMS AND CONCEPTS

关键术语与概念

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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