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

P-PS4-5

Exponential and Sinusoidal Signals

La funzione esponenziale funge da strumento essenziale per caratterizzare le forme d'onda che salgono e decadono rapidamente. Questa funzione è definita utilizzando le costanti m e A. Quando entrambe le costanti sono reali, la funzione può essere rappresentata graficamente.
Un esponenziale complesso si ottiene quando la costante m è puramente immaginaria. È periodica se mantiene una grandezza di unità.
Un segnale sinusoidale a tempo continuo può essere espresso in termini di frequenza e periodo di tempo.
La relazione di Eulero può essere utilizzata per esprimere il segnale sinusoidale come esponenziali complessi periodici con la stessa frequenza fondamentale.
Allo stesso modo, la funzione esponenziale complessa può essere espressa in termini di segnali sinusoidali, tutti accomunati dalla stessa frequenza fondamentale.
Ad esempio, la somma di due esponenziali complessi può essere scritta come il prodotto di un singolo esponenziale complesso e di una singola sinusoide.
Sia i segnali sinusoidali che quelli esponenziali complessi sono ampiamente impiegati per descrivere la conservazione dell'energia in un sistema meccanico in cui una massa, collegata a un supporto stazionario tramite una molla, esibisce un semplice moto armonico.

segnali esponenziali e sinusoidali

La funzione esponenziale è fondamentale per caratterizzare le forme d'onda che aumentano e decadono rapidamente. Questa funzione esponenziale a tempo continuo è definita usando termini esponenziali con costanti a e A. Quando entrambe le costanti sono reali, la funzione è rappresentata come:

<img alt="Equation1" src="/files/ftp_upload/16042/16042_PT%20EQUATION_1.svg" style="height: 16px;">

E può essere rappresentata graficamente per mostrare la crescita o il decadimento esponenziale. Quando la costante a è puramente immaginaria, il risultato è un esponenziale complesso, espresso come:

<img alt="Equation2" src="/files/ftp_upload/16042/16042_PT%20EQUATION_2.svg" style="height: 16px;">

Dove j è l'unità immaginaria e ω_0 è la frequenza angolare. Questa funzione è periodica se mantiene una grandezza unitaria.

Un segnale sinusoidale a tempo continuo può essere descritto in termini di frequenza e periodo di tempo. La relazione di Eulero consente di esprimere il segnale sinusoidale come esponenziali complessi periodici con lo stesso periodo fondamentale. Pertanto, un segnale sinusoidale è rappresentato come:

<img alt="Equation3" src="/files/ftp_upload/16042/16042_PT%20EQUATION_3.svg" style="height: 14px;">

Può essere riscritto utilizzando gli esponenziali complessi come segue:

<img alt="Equation4" src="/files/ftp_upload/16042/16042_PT%20EQUATION_4.svg" style="height: 28px;">

Allo stesso modo, la funzione esponenziale complessa può essere espressa in termini di segnali sinusoidali, tutti con lo stesso periodo fondamentale. Ad esempio, la somma di due esponenziali complessi può essere scritta come il prodotto di un singolo esponenziale complesso e di una singola sinusoide, esemplificato da:

<img alt="Equation5" src="/files/ftp_upload/16042/16042_PT%20EQUATION_5.svg" style="height: 16px;">

Sia i segnali sinusoidali che quelli esponenziali complessi sono ampiamente usati per descrivere la conservazione dell'energia nei sistemi meccanici, come una massa collegata ad un supporto stazionario tramite una molla, che esibisce un moto armonico semplice. Questi segnali forniscono una base per analizzare il comportamento oscillatorio e i fenomeni di risonanza in tali sistemi.

Segnali esponenziali e sinusoidali

La funzione esponenziale funge da strumento essenziale per caratterizzare le forme d'onda che salgono e decadono rapidamente. Questa funzione è definita utilizzando il ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

Introduction to Signals and Systems

Introduzione ai segnali e ai sistemi

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

signal and system

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...

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-function

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...

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

Explore how exponential, complex exponential, and sinusoidal signals describe growth, decay, periodic motion, and spring-mass oscillations in high school science.

Complex Exponentials in Signal Behavior | 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.

electrical engineering

The Wave Nature of Light

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Speed of a Transverse Wave

The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The ...

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Sound Waves and Doppler Shift

Source: Arianna Brown,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, Irvine, CA
...

Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a ...

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Propagation Speed of Electromagnetic Waves

Electromagnetic waves are consistent with Ampere&#39;s law. Assuming there is no conduction current Ampere&#39;s law is given as:


Consider a plane ...

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Dual Nature of Electromagnetic (EM) Radiation

Electromagnetic (EM) radiation consists of electric and magnetic field components oscillating in planes perpendicular to each other and mutually ...

Propagation of Waves

When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second ...

Standing Waves

Source: Arianna Brown,&#160;Asantha Cooray, PhD,&#160;Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, ...

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be ...

Velocity and Acceleration of a Wave

A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which ...

Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.

P-PS4-1

Electromagnetic Waves

James Clerk Maxwell formulated a single theory combining all the electric and magnetic effects scientists knew during that time, calling the phenomena his ...

Generating Electromagnetic Radiations

The German physicist Heinrich Hertz (1857&#8211;1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. ...

The Electromagnetic Spectrum

Electromagnetic waves are categorized according to their wavelengths and frequencies, giving the electromagnetic spectrum. These waves are classified as ...

Electromagnetic Wave Equation

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Problem-Solving: Tuning of a Guitar String

In the case of stringed instruments like the guitar, the elastic property that determines the speed of the sound produced is its linear mass density or ...

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Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a ...

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Beats

The study of music provides many examples of the superposition of waves and the constructive and destructive interference that occurs. Very few examples ...

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Shock Waves

While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the ...

Echo

The human ear cannot distinguish between two sources of sound if they happen to reach within a specific time interval, typically 0.1 seconds apart. More ...

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Standing Electromagnetic Waves

Electromagnetic waves can be reflected; the surface of a conductor or a dielectric can act as a reflector. As electric and magnetic fields obey the ...

Standing Waves in a Cavity

A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the ...

Design, evaluate, and refine a solution for improving how digital devices store and transmit information.

P-PS4-2

Travelling Waves

A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. ...

Graphing the Wave Function

Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the ...

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Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect ...

Reflection of Waves

When a wave travels from one medium to another, it gets reflected at the boundary of the second medium. A common example of this is when a person yells at ...

Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. ...

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Modes of Standing Waves: II

The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions ...

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Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of ...

Plane Electromagnetic Waves II

Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between ...

Electromagnetic Waves in Matter

Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

Intensity Of Electromagnetic Waves

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Evaluate the claims, evidence, and reasoning about how electromagnetic radiation can be described either by a wave model or a particle model, and in some situations one model is more useful than the other.

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Absorption of Radiation

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Interaction of EM Radiation with Matter: Spectroscopy

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Energy Transfer in Chemical Reactions

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Radiation Pressure: Problem Solving

The radiation pressure applied by an electromagnetic wave on a perfectly absorbing surface equals the energy density of the wave. The wave&#39;s momentum ...

Evaluate the validity and reliability of claims in published materials of the effects that different frequencies of electromagnetic radiation have when absorbed by matter.

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Photoelectric Effect

Source: Yong P. Chen, PhD, Department of Physics &#38;&#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
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Emission Spectra

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Photoluminescence: Fluorescence and Phosphorescence

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For a ...

Energy and Work

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Basic Continuous Time Signals

Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function. These are collectively referred to as ...

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Magnetic resonance imaging (MRI) is a noninvasive medical imaging technique based on a phenomenon of nuclear physics discovered in the 1930s, in which ...

Rectangular and Triangular Pulse Function

The unit rectangular pulse function is mathematically represented by a rectangular function centered at the origin with a height of one unit. This ...

Imaging Studies I: CT and MRI

Computed Tomography or CT scans use X-ray technology to generate detailed images of bones, organs, and tissues.
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Signal and System

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One-dimensional signals include single-variable functions, such as the variations in air pressure over time in ...

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Complex Exponentials in Signal Behavior

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