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

Jednostka prostokątnej funkcji impulsowej jest matematycznie reprezentowana przez funkcję prostokątną wyśrodkowaną w punkcie początkowym o wysokości jednej jednostki.
Funkcję tę definiują dwa parametry: T, określający środkowe położenie impulsu wzdłuż osi czasu, oraz τ, określający czas trwania impulsu.
Przykładem może być impuls prostokątny o amplitudzie 5 V, czasie trwania 3 s i środku znajdującym się w czasie równym 2s. Impuls ten można wyrazić za pomocą funkcji prostokątnej.
Synteza impulsu prostokątnego polega na graficznym zademonstrowaniu sekwencyjnego dodawania dwóch funkcji krokowych przesuniętych w czasie.
Ogólnie rzecz biorąc, funkcja prostokątna jednostki może być zawsze wyrażona za pomocą funkcji krokowej jednostki.
Jednostka funkcji trójkątnej jest matematycznie wyrażona za pomocą funkcji trójkątnej. Ma wysokość jednostki i jest wyśrodkowany w punkcie początkowym.
Przykładem jest trójkątny impuls wyśrodkowany w czasie równym 3s, o wielkości 2 i szerokości 2s. Aby naszkicować impuls trójkątny, zamień każde t na t-3 i ustaw szerokość równą dwóm. Zdefiniowany sygnał jest demonstrowany graficznie.

prostokątna i trójkątna funkcja impulsu

Prostokątna funkcja impulsu jednostkowego jest matematycznie reprezentowana przez prostokątną funkcję wyśrodkowaną w początku układu współrzędnych o wysokości jednej jednostki. Funkcja ta jest definiowana przez dwa parametry: T, który określa położenie środka impulsu wzdłuż osi czasu i τ, który określa czas trwania impulsu.

Na przykład rozważmy prostokątny impuls o amplitudzie 5 V, trwający 3 sekundy i wyśrodkowany w t=2 sekundy. Ten impuls można wyrazić za pomocą funkcji prostokątnej, zapisanej jako

<img alt="Equation1" src="/files/ftp_upload/16041/16041_PT%20EQUATION_1.svg" style="height: 36px;">

Syntezę prostokątnego impulsu można zademonstrować graficznie, dodając kolejno dwie przesunięte w czasie funkcje skokowe. Ogólnie rzecz biorąc, prostokątną funkcję jednostkową można zawsze wyrazić za pomocą funkcji skokowej jednostkowej w następujący sposób:

<img alt="Equation2" src="/files/ftp_upload/16041/16041_PT%20EQUATION_2.svg" style="height: 36px;">

Jednostkowa funkcja trójkątna jest matematycznie wyrażona za pomocą funkcji trójkątnej. Ma wysokość jednostkową i jest wyśrodkowana w początku układu współrzędnych. Na przykład rozważmy impuls trójkątny wyśrodkowany w t=3 sekundy, o wielkości 2 i szerokości 2 sekund. Aby wyrazić ten impuls trójkątny, zamień każdy t na t−3 i ustaw szerokość równą 2. Zdefiniowany sygnał można zapisać jako,

<img alt="Equation3" src="/files/ftp_upload/16041/16041_PT%20EQUATION_3.svg" style="height: 36px;">

Tę funkcję impulsu trójkątnego można zilustrować graficznie, pokazując, jak jego wysokość osiąga 2 w środku i zwęża się do zera na krawędziach, obejmując całkowitą szerokość 2 sekund.

Zarówno funkcje prostokątne, jak i trójkątne są podstawowe w przetwarzaniu sygnałów do reprezentowania różnych kształtów przebiegów i są używane w wielu aplikacjach do modelowania i analizowania sygnałów i układów. Funkcje te są niezbędne do zrozumienia bardziej złożonych zachowań sygnałów i działań na nich.

Prostokątna i trójkątna funkcja impulsu

Jednostka prostokątnej funkcji impulsowej jest matematycznie reprezentowana przez funkcję prostokątną wyśrodkowaną w punkcie początkowym o wysokości jednej jednostki.
Dwa...

Education

JoVE Core

Electrical Engineering

Elektrotechnika

Unit 1

Jednostka 1

Introduction to Signals and Systems

Wstęp do sygnałów i systemów

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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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

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basic operations on signals

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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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