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

连续时间系统的输入和输出信号均为连续信号，时间连续变化，通常由微分方程或代数方程定义。
例如，在一个 RC 电路中，输入电压与输出电压之间的关系通过微分方程表示，该方程基于欧姆定律和电容器的电压-电流关系推导得出。
离散时间系统在特定时间间隔内处理输入和输出信号，其行为由在离散时间点上的差分方程描述。这类系统的一个例子是银行账户月度余额的简单模型。
另一种类型的系统是时变系统和时不变系统。
时变系统的参数随时间发生变化。连续时间时变系统由具有时变系数的微分方程描述，而离散时间时变系统则由具有时变参数的差分方程描述。
时不变系统的输出在输入信号发生时间平移时，会随之发生相同的平移，而不会改变系统的特性。
如果电阻和电容值保持不变，则RC电路是时不变的。如果这些参数发生波动，系统将变为时变系统，因为实验结果会因实验进行的时间不同而发生变化。

系统分类-ii

Classification of Systems-II

连续时间系统具有连续的输入和输出信号，并且会对时间进行连续测量。这些系统通常是由微分方程或代数方程来进行定义的。例如，在 RC 电路中，输入和输出电压之间的关系可以用欧姆定律和电容关系中所得出的微分方程来进行表示：

<img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="height: 30px;">

离散时间系统具有特定时间间隔的输入信号和输出信号，这些信号会通过差分方程定义在不同的时刻。例如，可以将银行账户月度余额的简单模型表示为：

<img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="height: 14px;">

其中，B[n] 指的是第 n 个月的余额，D[n] 是存款，W[n] 是取款。

还可以将系统分为时变系统或时不变系统。时变系统的参数会随时间发生变化。连续时间的时变系统通常是由时变微分方程来进行表示，而离散时间时变系统则是由时变差分方程来进行定义的。

当输入信号发生时移时，时不变系统的输出信号便会表现出相同的时间偏移。在数学中，如果 y(t) 是输入 x(t) 的输出，那么对于时不变系统来说，y(t−t_0) 便是输入 x(t−t_0) 的输出。例如，如果电阻 R 和电容 C 值保持不变，那么 RC 电路是时不变的。如果这些值发生波动，那么系统就会变成时变的，这意味着实验结果将会根据实验时间的不同而发生变化。

了解连续时间系统和离散时间系统以及时变系统和时不变系统中的这些区别对于分析和设计各种工程系统来说是至关重要的。

系统分类-II

连续时间系统的输入和输出信号均为连续信号，时间连续变化，通常由微分方程或代数方程定义……

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

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 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

Continuous and Discrete System Models

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