Name: complex exponentials in signal behavior
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 18 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

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Exponential and Sinusoidal Signals

La fonction exponentielle sert d’outil essentiel pour caractériser les formes d’onde qui montent et décroît rapidement. Cette fonction est définie à l’aide des constantes m et A. Lorsque les deux constantes sont réelles, la fonction peut être représentée graphiquement.
Une exponentielle complexe est obtenue lorsque la constante m est purement imaginaire. Elle est périodique si elle maintient une grandeur d’unité.
Un signal sinusoïdal en temps continu peut être exprimé en termes de fréquence et de période.
La relation d'Euler peut être utilisée pour exprimer le signal sinusoïdal sous forme d'exponentielles complexes périodiques avec la même fréquence fondamentale.
De même, la fonction exponentielle complexe peut être exprimée en termes de signaux sinusoïdaux, partageant tous la même fréquence fondamentale.
Par exemple, la somme de deux exponentielles complexes peut être écrite comme le produit d’une seule exponentielle complexe et d’une seule sinusoïde.
Les signaux exponentiels sinusoïdaux et complexes sont largement utilisés pour décrire la conservation de l’énergie dans un système mécanique où une masse, reliée à un support fixe par un ressort, présente un mouvement harmonique simple.

signaux exponentiels et sinusoïdaux

La fonction exponentielle est essentielle pour caractériser les formes d'onde qui augmentent et diminuent rapidement. Cette fonction exponentielle à temps continu est définie à l'aide de termes exponentiels avec des constantes a et A. Lorsque les deux constantes sont réelles, la fonction est représentée par,

<img alt="Equation1" src="/files/ftp_upload/16042/16042_PT%20EQUATION_1.svg" style="height: 16px;">

et peut être représentée graphiquement pour montrer une croissance ou une décroissance exponentielle. Lorsque la constante a est purement imaginaire, le résultat est une exponentielle complexe, exprimée par,

<img alt="Equation2" src="/files/ftp_upload/16042/16042_PT%20EQUATION_2.svg" style="height: 16px;">

où j est l'unité imaginaire et ω_0 est la fréquence angulaire. Cette fonction est périodique si elle conserve une amplitude égale à l'unité.

Un signal sinusoïdal à temps continu peut être décrit en termes de fréquence et de période de temps. La relation d'Euler permet d'exprimer le signal sinusoïdal sous forme d'exponentielles complexes périodiques ayant la même période fondamentale. Ainsi, un signal sinusoïdal est représenté comme suit,

<img alt="Equation3" src="/files/ftp_upload/16042/16042_PT%20EQUATION_3.svg" style="height: 14px;">

peut être réécrit en utilisant des exponentielles complexes comme suit,

<img alt="Equation4" src="/files/ftp_upload/16042/16042_PT%20EQUATION_4.svg" style="height: 28px;">

De même, la fonction exponentielle complexe peut être exprimée en termes de signaux sinusoïdaux, tous partageant la même période fondamentale. Par exemple, la somme de deux exponentielles complexes peut être écrite comme le produit d'une seule exponentielle complexe et d'une seule sinusoïde, comme le montre l’exemple suivant,

<img alt="Equation5" src="/files/ftp_upload/16042/16042_PT%20EQUATION_5.svg" style="height: 16px;">

Les signaux sinusoïdaux et exponentiels complexes sont largement utilisés pour décrire la conservation de l'énergie dans les systèmes mécaniques, tels qu'une masse reliée à un support stationnaire par un ressort, présentant un mouvement harmonique simple. Ces signaux constituent une base pour l'analyse du comportement oscillatoire et des phénomènes de résonance dans de tels systèmes.

Signaux exponentiels et sinusoïdaux

La fonction exponentielle sert d’outil essentiel pour caractériser les formes d’onde qui montent et décroît rapidement. Cette fonction est définie à l’aide de la commande ...

Education

Enseignement

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Electrical Engineering

Génie électrique

Unit 1

Unité 1

Introduction to Signals and Systems

Introduction aux signaux et systèmes

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 links it closely to sinusoidal motion.

A continuous-time sinusoidal signal can be described by its frequency and time period. Using Euler’s relation, a sinusoid can be written as a set of periodic complex exponentials with the same fundamental period. The same idea also works in reverse, so a complex exponential can be expressed in terms of sinusoidal signals.

The sum of two complex exponentials can also be rewritten in a simpler form. It becomes the product of one complex exponential and one sinusoid. These signal forms are widely used to model energy conservation in mechanical systems, such as a mass attached to a fixed support by a spring. In that setting, they help describe simple harmonic motion, oscillation, and resonance.

Complex Exponentials in Signal Behavior

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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...

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Power and Energy in Electrical 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

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

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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

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...

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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

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Complex Exponentials in Signal Behavior | JoVE K12

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Complex Exponentials in Signal Behavior

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