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

K12 Standards

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

High School

Physics

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

P-PS4-5

rectangular-and-triangular-pulse-function

Rectangular and Triangular Pulse Function

פונקציית הדופק המלבני של היחידה מיוצגת מתמטית על ידי הפונקציה המלבנית שמרכזה במקור בגובה של יחידה אחת.
שני פרמטרים מגדירים פונקציה זו: T, המציין את מיקום מרכז הדופק לאורך ציר הזמן, ו-τ, הקובע את משך הדופק.
דוגמה יכולה להיות פולס מלבני עם משרעת 5V, משך 3s ומרכז הממוקם בזמן שווה 2s. פולס זה יכול לבוא לידי ביטוי באמצעות הפונקציה המלבנית.
סינתזה של הפולס המלבני כרוכה בהדגמה גרפית של הוספה רציפה של שתי פונקציות צעד שהוסטו בזמן.
באופן כללי, פונקציה מלבנית יחידה יכולה תמיד לבוא לידי ביטוי באמצעות פונקציית צעד היחידה.
הפונקציה המשולשת של היחידה מבוטאת מתמטית באמצעות הפונקציה המשולשת. יש לו גובה יחידה והוא ממורכז במקור.
מופע הוא פולס משולש שמרכזו בזמן השווה ל-3s, בעוצמה של 2 וברוחב של 2s. כדי לשרטט פולס משולש, החלף כל t ב- t-3 והגדר את הרוחב שווה לשניים. האות המוגדר מודגם באופן גרפי.

פונקציית פולס מלבני ופולס משולש

<div style="direction: rtl; unicode-bidi: embed;">
 פונקציית הפולס המלבני היחידתי מיוצגת מתמטית על ידי פונקציה מלבנית הממוקמת במרכז הציר בזמן, עם גובה של יחידה אחת. פונקציה זו מוגדרת על פי שני פרמטרים: T, שמציין את מיקום מרכז הפולס על ציר הזמן, ו-τ, שמגדיר את משך הפולס.
 
 לדוגמה, נניח פולס מלבני עם משרעת של 5V, משך של 3 שניות, ומרכז ב-\(t=2\) שניות. ניתן לייצג את הפולס הזה באמצעות הפונקציה המלבנית:
 
 <img alt="Equation1" src="/files/ftp_upload/16041/16041_PT%20EQUATION_1.svg" style="text-align: center; height: 36px;">
 הסינתזה של פולס מלבני יכולה להיות מוצגת גרפית על ידי הוספת שתי פונקציות מדרגה המוזזות בזמן בצורה סדרתית. באופן כללי, פונקציה מלבנית יחידתית תמיד יכולה להיות מוצגת באמצעות פונקציית מדרגת היחידה כדלקמן:
 
 <img alt="Equation2" src="/files/ftp_upload/16041/16041_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;">
 פונקציית הפולס המשולש היחידתית מיוצגת מתמטית על ידי פונקציה משולשת. יש לה גובה יחידת אחד וממוקמת במרכז הציר בזמן. לדוגמה, נניח פולס משולש הממוקם ב- t=3 שניות, עם משרעת של 2 ורוחב של 2 שניות. כדי לייצג את הפולס המשולש הזה, מחליפים את t ב- t−3 ומגדירים את הרוחב ל-2. ניתן לכתוב את האות המוגדר כך:
 
 <img alt="Equation3" src="/files/ftp_upload/16041/16041_PT%20EQUATION_3.svg" style="text-align: center; height: 36px;">
 פונקציית הפולס המשולש הזו יכולה להיות מוצגת גרפית, כאשר גובהה מגיע ל-2 במרכז ויורד לאפס בקצוות, כאשר הרוחב הכולל שלה הוא 2 שניות.
 
 שתי פונקציות הפולס המלבני והמשולש הן יסודיות בעיבוד אותות לייצוג צורות גל שונות, ומשמשות ביישומים רבים למידול וניתוח של אותות ומערכות. פונקציות אלו חיוניות להבנת התנהגויות אותות מורכבות יותר ותהליכי עיבוד אותות מתקדמים.
 
</div>

פונקציית הדופק המלבני של היחידה מיוצגת מתמטית על ידי הפונקציה המלבנית שמרכזה במקור בגובה של יחידה אחת.
שניים...

Education

JoVE Core

Electrical Engineering

הנדסה חשמלית

Unit 1

יחידה 1

Introduction to Signals and Systems

מבוא לאותים ומערכות

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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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Signal Types by Time and Amplitude

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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

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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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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

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Modeling Pulses with Rectangles and Triangles

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