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

連続時間システムは、連続的な入力信号と出力信号を持ち、時間は連続的に測定され、一般に微分方程式または代数方程式によって定義されます。
たとえば、RC回路では、入力電圧と出力電圧の関係は、オームの法則とコンデンサの電圧-電流関係を使用して導出される微分方程式で表されます。
離散時間システムは、入力信号と出力信号を特定の間隔で処理し、それらの動作は、異なる時間インスタンスでの差分方程式によって記述されます。このようなシステムの例としては、月ごとの銀行口座の残高の単純なモデルがあります。
別のタイプのシステムは、時間変動と時間不変です。
時間とともに変化するシステムには、時間とともに変化するパラメーターがあります。連続時間時変システムは、時間依存の係数を持つ微分方程式で記述され、離散時間の時変システムは、時間依存のパラメーターを持つ差分方程式で記述されます。
時間不変システムは、入力信号がタイムシフトされると、システムの特性を変更せずに同じようにシフトする出力を生成します。
RC回路は、抵抗と容量の値が一定のままである場合、時間不変です。これらが変動すると、実験を行うタイミングによって実験結果が変動するため、システムは時間変動になります。

システムの分類 - ii

連続時間システムには連続的な入力信号と出力信号があり、時間は連続的に測定されます。これらのシステムは、通常、微分方程式や代数方程式で定義されます。たとえば、RC 回路では、入力電圧と出力電圧の関係は、オームの法則とコンデンサの関係から導かれる微分方程式によって表されます。

<img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="height: 30px;">
離散時間システムには、特定の間隔で定義され、特定の瞬間に入力および出力される信号があります。例として、銀行口座の残高を月ごとにモデル化する場合、次のように表されます。


<img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="height: 14px;">
ここで、B[n] は月 n の残高、D[n] は預金、W[n] は引き出しです。

システムは、時間変動型または時間不変型に分類することもできます。時間変動システムには、時間の経過とともに変化するパラメータがあります。連続時間の時間変動システムは時間変動の微分方程式で記述され、離散時間の時間変動システムは時間変動の差分方程式で定義されます。

時間不変システムは、入力信号が時間シフトされると、出力信号も同一の時間シフトを示します。数学的には、y(t) が入力 x(t) に対する出力である場合、時間不変システムでは、y(t−t_0) が入力 x(t−t_0) に対する出力です。たとえば、抵抗 R と静電容量 C の値が一定である場合、RC 回路は時間不変です。これらの値が変動する場合、時間変動システムとなり、実験が行われるタイミングに応じて結果が変わります。

連続時間システムと離散時間システム、および時間変動システムと時間不変システムにおけるこれらの違いを理解することは、幅広い工学システムを分析および設計する上で非常に重要です。

システムの分類 - II

連続時間システムは、連続的な入力信号と出力信号を持ち、時間は連続的に測定され、一般に微分または代数によって定義されます...

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-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 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

Continuous and Discrete System Models

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