Name: modeling pulses with rectangles and triangles
Uploaded: 2025-05-22T09:06:20.000Z
Duration: 1 min 19 s
Channel: JoVE

K12 Standards

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South Carolina College and Career Ready Science Standards

High School

Physics

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Waves and Their Applications in Technologies for Information Transfer

P-PS4-5

rectangular-and-triangular-pulse-function

Rectangular and Triangular Pulse Function

单位矩形脉冲函数在数学上由以原点为中心的矩形函数表示，高度为 1 个单位。
两个参数定义了这个函数: T，指定脉冲沿时间轴的中心位置，以及 τ，确定脉冲持续时间。
一个例子可以是幅度为 5V、持续时间为 3 秒且位于时间等于 2 秒的中心的矩形脉冲。这个脉冲可以用 rectangular 函数来表示。
合成矩形脉冲涉及按顺序添加两个时移阶跃函数的图形演示。
一般来说，单位矩形函数总是可以用单位步长函数来表示。
单位三角函数通过三角函数在数学上表示。它具有单位高度，并以原点为中心。
实例是一个以等于 3s 的时间为中心的三角脉冲，其幅度为 2，宽度为 2s。要绘制三角形脉冲，请将每个 t 替换为 t-3，并将宽度设置为 2。定义的信号以图形方式演示。

矩形脉冲函数和三角形脉冲函数

单位矩形脉冲函数在数学上可以表示为一个以原点为中心、高度为一个单位的矩形函数表示形式。该函数通常会使用两个参数来进行定义：T（脉冲沿时间轴的中心位置）和 τ（脉冲的持续时间）。

例如，假设有一个幅度为 5V，持续时间为 3 秒，以 t=2 秒为中心的矩形脉冲。该脉冲可以用矩形函数来进行表示，其表示方法为：

<img alt="Equation1" src="/files/ftp_upload/16041/16041_PT%20EQUATION_1.svg" style="height: 36px;">

将两个时移阶跃函数依次相加，便能够以图形化的方式来合成矩形脉冲。一般来说，单位矩形函数始终可以使用单位阶跃函数来进行表示，其表示方法为：

<img alt="Equation2" src="/files/ftp_upload/16041/16041_PT%20EQUATION_2.svg" style="height: 36px;">

单位三角函数在数学上可以使用三角形函数来进行表示。它具有单位高度，并以原点为中心。例如，假设有一个以 t=3 秒为中心、幅度为 2、宽度为 2 秒的三角形脉冲。如果要对这个三角形脉冲进行表示，则可以用 t−3 来替换其中所有的 t，并将宽度设置为 2。可以将定义的信号表示为：

<img alt="Equation3" src="/files/ftp_upload/16041/16041_PT%20EQUATION_3.svg" style="height: 36px;">

这个三角形脉冲函数能够用图形的方式来进行表示，可以表示为：其高度如何在中心达到 2 并在边缘处逐渐减小到零时，其总宽度为 2 秒。

单位矩形函数和三角函数都是信号处理中用来表示各种波形的基本函数，通常会将其用于对信号和系统进行建模和分析的多种应用中。这些函数对于理解更为复杂的信号行为和操作来说是至关重要的。

单位矩形脉冲函数在数学上由以原点为中心的矩形函数表示，高度为 1 个单位。
二。。。

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

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 at t = 2 seconds can be written with the rectangular function. The same shape can be built graphically by adding two time-shifted step functions in sequence. In general, a unit rectangular function can be written using the unit step function.

The unit triangular function is another common pulse shape. It has unit height and is centered at the origin. For a triangular pulse centered at t = 3 seconds, with a magnitude of 2 and a width of 2 seconds, each t value is replaced with t − 3 and the width is set to 2.

A triangular pulse rises to its peak at the center and then falls to zero at the edges. In this example, the height reaches 2 at the center and spans a total width of 2 seconds. Rectangular and triangular functions are both important in signal processing because they help represent waveforms and support the modeling and analysis of signals and systems. They also provide a foundation for understanding more complex signal behavior and operations.

Modeling Pulses with Rectangles and Triangles

modeling-pulses-with-rectangles-and-triangles

signal and system

Signals and Systems in Everyday Data

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classification of signals

Signal Types by Time and Amplitude

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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 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 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

Learn how rectangular and triangular pulse functions model signals, use center and width parameters, and connect to step functions in signal processing.

Modeling Pulses with Rectangles and Triangles | 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.

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Modeling Pulses with Rectangles and Triangles

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