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

線形性は、直接的な入出力関係を持つシステムプロパティであり、均質性と加法性が組み合わされています。
同次性は、入力に定数を乗算すると、出力に同じ定数が乗算されることを意味します。加法性とは、複数の入力に対する応答が個々の応答の合計であることを意味します。
均質性と加法性の両方の特性を持つシステムは、線形システムとして分類されます。
線形システムには、重ね合わせの原理に準拠した抵抗器、コンデンサ、インダクタを備えた回路が含まれます。
逆に、線形性方程式に従わないシステムは非線形として分類され、多くの場合、整流器やダイオードなどのデバイスが含まれます。
因果システムは、将来の入力値に依存しない現在の応答によって特徴付けられます。例としては、将来の運転行動を予測することができない車の動きがあります。
理想的なフィルターなどの非因果システムは、物理的に実現できず、因果関係の原理に従っていません。
動的システムはメモリを示し、出力は過去と将来の入力の両方の影響を受けます。
静的システムは、現在の出力が現在の入力のみに基づいているメモリレスまたは瞬間的なシステムです。

システムの分類 - i

線形性は、同質性と加法性を組み合わせた、直接的な入力と出力の関係を特徴とするシステム特性です。

同質性は、入力 x(t) に定数 c を掛けた場合、出力 y(t) にも同じ定数を掛けることを意味します。数学的には、次のように表されます:

<img alt="Equation1" src="/files/ftp_upload/16045/16045_PT%20EQUATION_1.svg" style="height: 14px;">
加法性とは、複数の入力の合計に対する応答が、個々の応答の合計であることを意味します。入力 x_1(t) と x_2(t) に対して、それぞれ出力 y_1(t) と y_2(t) が生成される場合、次のように表されます:


<img alt="Equation2" src="/files/ftp_upload/16045/16045_PT%20EQUATION_2.svg" style="height: 14px;">
同質性と加法性を組み合わせると、線形システムの方程式が得られます:


<img alt="Equation3" src="/files/ftp_upload/16045/16045_PT%20EQUATION_3.svg" style="height: 14px;">
線形システムには、重ね合わせの原理に従う抵抗器、コンデンサ、インダクタを含む回路が含まれます。逆に、整流器やダイオードなど、線形方程式に従わないシステムは非線形として分類されます。

因果システムとは、現在の応答が将来の入力値に依存しないシステムのことです。たとえば、車の動きは将来の運転動作を予測できないため、因果システムになります。対照的に、理想的なフィルタなどの非因果システムは、因果の原理に従わないため、物理的に実現できません。

動的システムはメモリを持ち、出力は過去と現在の入力に依存します。一例として、コンデンサやインダクタを備えた電気回路が挙げられます。この場合、現在の出力は過去の入力の影響を受けます。静的システムはメモリレスシステムまたは瞬時システムとも呼ばれ、出力電圧が現在の入力電圧の直接関数である単純な抵抗回路など、出力は現在の入力のみに基づいています。

これらの特性 (線形性、因果関係、メモリ) を理解することは、さまざまな工学分野でシステムを分析および設計するために不可欠です。

システムの分類 - I

線形性は、直接的な入出力関係を持つシステムプロパティであり、均質性と加法性が組み合わされています。
均質性は、入力が...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Introduction to Signals and Systems

信号とシステム概論

KEY TERMS AND CONCEPTS

主要用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

System Properties: Linearity, Causality, Memory

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