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

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Rectangular and Triangular Pulse Function

单位矩形脉冲函数在数学上由一个高度为一个单位、中心位于原点的矩形函数表示。
该函数由两个参数定义：T，表示脉冲在时间轴上的中心位置；τ，决定脉冲的持续时间。
一个示例是幅度为 5V、持续时间为 3 秒、中心位于 2 秒时刻的矩形脉冲。该脉冲可使用矩形函数表示。
合成矩形脉冲涉及通过图形方式依次叠加两个时间平移的阶跃函数.
一般来说，单位矩形函数始终可以用单位阶跃函数来表示。
单位三角函数通过三角函数进行数学表达，其高度为1，且以原点为中心。
一个实例是一个以时间为 3s 为中心的三角脉冲，其幅值为 2，宽度为 2s。要绘制该三角脉冲，需将每个t 替换为t-3，并将宽度设为 2。所定义的信号以图形方式展示。

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表示为一个以原点为中心、高度为一个单位的矩形函数。该……

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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

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signal and system

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classification of signals

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energy and power signals

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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

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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

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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

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Modeling Pulses with Rectangles and Triangles

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