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

선형성은 동질성과 가산성을 결합하는 직접 입력-출력 관계가 있는 시스템 속성입니다.
균질성은 입력에 상수를 곱하면 출력에 동일한 상수를 곱하도록 지시합니다. 가산성은 여러 입력에 대한 응답이 개별 응답의 합임을 의미합니다.
동질성 속성과 가산성 속성을 모두 가진 시스템은 선형 시스템으로 분류됩니다.
선형 시스템에는 중첩 원리를 준수하는 저항기, 커패시터 및 인덕터가 있는 회로가 포함됩니다.
반대로, 선형성 방정식을 따르지 않는 시스템은 비선형으로 분류되며, 종종 정류기 및 다이오드와 같은 장치를 포함합니다.
인과 관계는 미래의 입력 값에 독립적인 현재 응답을 특징으로 합니다. 예를 들어 미래의 주행 동작을 예측할 수 없는 자동차의 움직임을 들 수 있습니다.
이상적인 필터와 같은 비인과 시스템은 물리적으로 실현할 수 없으며 인과 관계의 원리를 따르지 않습니다.
동적 시스템은 출력이 과거와 미래의 입력에 의해 영향을 받는 메모리를 표시합니다.
정적 시스템은 현재 출력이 현재 입력만을 기반으로 하는 메모리가 없거나 순간적인 시스템입니다.

시스템 분류-i

선형성은 동질성과 가법성을 결합한 직접적인 입력-출력 관계를 특징으로 하는 시스템 속성입니다.

동질성은 입력 x(t)에 상수 c를 곱하면 출력 y(t)에 같은 상수를 곱한다는 것을 나타냅니다. 수학적으로 이는 다음과 같이 표현됩니다.

<img alt="Equation1" src="/files/ftp_upload/16045/16045_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;">

가법성은 여러 입력의 합에 대한 응답이 개별 응답의 합이라는 것을 의미합니다. 입력 x_1(t) 및 x_2(t)가 각각 출력 y_1(t) 및 y_2(t)를 생성하는 경우:

<img alt="Equation2" src="/files/ftp_upload/16045/16045_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">

동질성과 가법성을 결합하면 선형 시스템의 방정식이 생성됩니다.

<img alt="Equation3" src="/files/ftp_upload/16045/16045_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">

선형 시스템에는 중첩 원리를 따르는 저항기, 커패시터 및 인덕터가 있는 회로가 포함됩니다. 반대로 선형성 방정식을 따르지 않는 시스템은 정류기 및 다이오드와 같이 비선형으로 분류됩니다.

인과 시스템은 현재 응답이 미래 입력 값과 독립적인 시스템입니다. 예를 들어 자동차의 움직임은 미래의 운전 행동을 예측할 수 없으므로 인과 시스템입니다. 반면, 이상 필터와 같은 비인과적 시스템은 인과성의 원리를 따르지 않기 때문에 물리적으로 실현할 수 없습니다.

동적 시스템은 메모리를 나타내며, 여기서 출력은 과거 및 현재 입력에 따라 달라집니다. 예를 들어, 커패시터나 인덕터가 있는 전기 회로에서 현재 출력은 과거 입력의 영향을 받습니다. 무기억성 또는 순시적 시스템이라고도 하는 정적 시스템은 출력 전압이 현재 입력 전압의 직접적인 함수인 간단한 저항 회로와 같이 현재 입력에만 기반한 출력을 갖습니다.

선형성, 인과성 및 메모리와 같은 이러한 속성을 이해하는 것은 다양한 엔지니어링 분야에서 시스템을 분석하고 설계하는 데 필수적입니다.

시스템 분류-I

선형성은 동질성과 가산성을 결합하는 직접 입력-출력 관계가 있는 시스템 속성입니다.
균질성은 입력이 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Introduction to Signals and Systems

신호 및 시스템 입문

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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