Name: fast fourier transform for signal analysis
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 10 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

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

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Fast Fourier Transform

La transformée de Fourier rapide, FFT, est un algorithme de calcul permettant de calculer la transformée de Fourier discrète en divisant les calculs en sections plus petites et gérables.
Le calcul d’une DFT à N points nécessite des multiplications complexes de N carrés, tandis que l’algorithme FFT ne nécessite que N multiplications sur deux et un logarithme de base deux N multiplications, offrant des performances nettement plus rapides.
À mesure que N augmente, la FFT devient plus rapide et plus efficace en réduisant le nombre d’opérations de l’échelle quadratique à l’échelle logarithmique.
Il utilise les propriétés de symétrie et de périodicité et minimise les calculs et les multiplications redondants.
La transformée de Fourier rapide inverse, IFFT, reconstruit le signal original à partir de sa représentation dans le domaine fréquentiel avec une efficacité de calcul améliorée.
Couramment utilisé dans le traitement du signal et de l’image, il joue également un rôle essentiel dans la communication sans fil, la recherche scientifique et l’analyse de données.

transformée de fourier rapide

La transformée de Fourier rapide (FFT) est un algorithme de calcul conçu pour calculer efficacement la transformée de Fourier discrète (DFT). En décomposant les calculs en sections plus petites et plus faciles à gérer, la FFT réduit considérablement la complexité du calcul. Le calcul direct d'une DFT à N points nécessite N_2 multiplications complexes, tandis que l'algorithme FFT n'a besoin que de (N/2)log_2N multiplications, ce qui offre une performance beaucoup plus rapide.

L'efficacité de calcul de la FFT devient particulièrement évidente à mesure que N augmente. La FFT réduit le nombre d'opérations de l'échelle quadratique à l'échelle logarithmique, améliorant ainsi à la fois la vitesse et l'efficacité. L'algorithme exploite les propriétés de symétrie et de périodicité inhérentes à la transformée de Fourier pour minimiser les calculs redondants, réduisant ainsi considérablement le nombre de multiplications requises.

La transformée de Fourier rapide inverse (IFFT) est tout aussi importante, reconstruisant le signal d'origine à partir de sa représentation dans le domaine fréquentiel. L'IFFT maintient l'efficacité de calcul de la FFT, garantissant que la transformation vers le domaine temporel soit effectuée rapidement et avec précision. Cette fonctionnalité est essentielle dans diverses applications, notamment le traitement du signal et l'analyse des données.

La FFT est largement utilisée dans le traitement du signal pour analyser les signaux audio, offrant des informations sur les composantes de fréquence du son. Dans le traitement d'images, la FFT aide dans des tâches telles que le filtrage et l'amélioration de l'image. De plus, la FFT joue un rôle essentiel dans la communication sans fil, où elle aide à la modulation et à la démodulation des signaux. Dans la recherche scientifique, la FFT est utilisée pour traiter des données expérimentales, et dans l'analyse des données, elle permet d’identifier des modèles et des tendances au sein de grands ensembles de données.

En résumé, la FFT est un outil indispensable dans divers domaines, offrant un moyen puissant pour analyser et traiter efficacement les signaux. Sa capacité à transformer les données entre les domaines temporel et fréquentiel, combinée à son efficacité de calcul, en fait un véritable fondement du traitement et de l'analyse modernes du signal.

Transformée de Fourier rapide

La transformée de Fourier rapide, FFT, est un algorithme de calcul permettant de calculer la transformée de Fourier discrète en divisant les calculs en ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

The Fourier Transform

La transformée de Fourier

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

fast fourier transform for signal analysis

The Fast Fourier Transform (FFT) is a computational algorithm that makes the Discrete Fourier Transform (DFT) faster to calculate. It works by splitting the calculation into smaller parts. This reduces the amount of work needed to move data between the time domain and the frequency domain.

A direct N-point DFT needs N² complex multiplications. The FFT lowers that cost to about (N/2)log₂N multiplications. That difference becomes more important as N gets larger, because the FFT cuts the number of operations from a quadratic scale to a logarithmic one.

The FFT uses symmetry and periodicity in the Fourier Transform to avoid repeated calculations. The Inverse Fast Fourier Transform, or IFFT, does the reverse job. It rebuilds the original signal from its frequency-domain form and does so with the same kind of computational efficiency.

Because of this speed, the FFT is useful in many kinds of signal processing and data analysis. It helps study audio signals by showing their frequency components. It also supports image processing tasks such as filtering and image enhancement, and it plays a role in wireless communication for modulation and demodulation. In scientific research and data analysis, the FFT helps process experimental data and find patterns and trends in large datasets.

Fast Fourier Transform for Signal Analysis

fast-fourier-transform-for-signal-analysis

continuous -time fourier transform

From Fourier Series to Fourier Transform

The Fourier transform extends Fourier series ideas to nonperiodic functions. A Fourier series breaks a periodic signal into a sum of sines and cosines. When the period becomes infinite, that discrete sum changes into a continuous integral. A pulse-train waveform shows this change clearly. With a finite period, the repeating rectangular pulses can be represented by a Fourier series. As the period grows without bound, the waveform becomes a single isolated pulse, and the description shifts to...

basic signals of fourier transform

Fourier Transform Signals in Time and Frequency

The Fourier Transform links time-domain signals to frequency-domain signals. It is a key tool in signal processing. It helps show how a waveform can be described by frequency parts instead of only by time changes. The sinc function is one of the basic signals in this topic. It is written as sinc(x) = sin(πx)/(πx). It equals 1 when x is 0, and it has even symmetry about the y-axis. In the frequency domain, it appears as the Fourier transform of a rectangular pulse. A rectangular pulse has a...

properties of fourier transform i

Fourier Transform in Radio Broadcasting

Fourier Transform properties help radio broadcasting manage signals in the time and frequency domains. These properties are useful for sending multiple audio channels at once, changing clip speed, adding broadcast delays, adjusting audio frequencies, and demodulating signals. They make it easier to transmit and receive audio clearly. When broadcasters combine several audio signals, the Fourier Transform simplifies the process. If f(t) and g(t) have Fourier Transforms F(ω) and G(ω), then the...

properties of fourier transform ii

Fourier Transform Rules in Signal Analysis

The Fourier Transform helps analyze signals by linking the time domain and the frequency domain. Its properties show how a signal changes when it is shifted, differentiated, or combined with another signal. These rules are important in signal processing, radio broadcasting, and audio work. The frequency shifting property shows that moving a signal in frequency causes a phase change in time. If x(t) has a Fourier Transform X(f), then multiplying x(t) by e^j2πf0t gives a transform of X(f−f0).

parseval's theorem for fourier transform

Signal Energy in Time and Frequency

Parseval's theorem links signal energy in the time domain and the frequency domain. In signal processing, it shows that the same energy can be calculated in either view. This makes it an important idea for comparing time-based and frequency-based analysis. To find signal energy, the standard approach uses a resistor value of 1 ohm. In that case, power equals the square of the signal's voltage or current. This gives a simple way to relate power to energy before moving into Fourier analysis. <br...

discrete-time fourier transform

DTFT and Frequency Analysis of Digital Signals

The Discrete-Time Fourier Transform (DTFT) shows how a discrete-time signal can be analyzed in the frequency domain. It takes a signal from the time domain and reveals its frequency components and spectral features. In the DTFT, the integral used for continuous-time signals is replaced by a summation to match the discrete form of the signal. The DTFT spectrum X(Ω) is periodic with a period of 2π. This periodicity means that X(Ω) can be written as a Fourier series, which helps with analysis and...

properties of dtft i

DTFT Signal Properties and Frequency Effects

Discrete-Time Fourier Transforms (DTFTs) help describe discrete-time signals in the frequency domain. The main DTFT properties include linearity, time shifting, frequency shifting, time reversal, conjugation, and time scaling. These rules make it easier to analyze and change signals in signal processing. Linearity is the most basic property. If two discrete-time signals are multiplied by constants a and b and then added together, the DTFT of the result is the weighted sum of the two individual...

properties of dtft ii

DTFT Rules for Differentiation and Energy

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain. For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time. Convolution brings out another key DTFT rule.

discrete fourier transform

DFT for Finding Signal Frequencies

The Discrete Fourier Transform (DFT) turns a finite set of time-domain samples into frequency components. It is a key tool in signal processing because it evaluates a discrete signal at evenly spaced frequency intervals. The result shows the magnitude and phase of each complex sinusoid in the signal. The DFT also follows several important properties. It is linear, so the DFT of a sum of sequences equals the sum of their individual DFTs. When a sequence shifts in time, its DFT changes by a...

fast fourier transform

The Fast Fourier Transform (FFT) is a computational algorithm that makes the Discrete Fourier Transform (DFT) faster to calculate. It works by splitting the calculation into smaller parts. This reduces the amount of work needed to move data between the time domain and the frequency domain. A direct N-point DFT needs N² complex multiplications. The FFT lowers that cost to about (N/2)log₂N multiplications. That difference becomes more important as N gets larger, because the FFT cuts the number...

key terms and concepts

Learn how the Fast Fourier Transform speeds up DFT calculations, supports the IFFT, and helps analyze audio, images, wireless signals, and data.

Fast Fourier Transform for Signal Analysis | 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

HS-PS4-5

Travelling Waves

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

Wave Parameters

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

Equations of Wave Motion

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Energy and Power of a Wave

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Doppler Effect - I

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propagation-speed-of-electromagnetic-waves

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 ...

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

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The Wave Nature of Light

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms ...

Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media. [Clarification Statement: Examples of data could include electromagnetic radiation traveling in a vacuum and glass, sound waves traveling through air and water, and seismic waves traveling through the earth.] [Assessment Boundary: Assessment is limited to algebraic relationships and describing those relationships qualitatively.]

HS-PS4-1

The Electromagnetic Spectrum

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

Classification of Signals

Signals are categorized as Continuous-time or discrete-time, Periodic or aperiodic, Analog or digital, and Causal or noncausal.
A continuous-time signal ...

Discrete Fourier Transform

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal.
However, in reality, the sensor can only ...

Discrete-Time Fourier Series

The Discrete-Time Fourier Series is a counterpart to the Fourier-series expansion of continuous-time periodic signals.
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Sampling Theorem

Consider the continuous-time signal x(t), and a train of impulses where Ts,&#xA0; is the sampling interval, and fs, is the sampling frequency.
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Properties of Fourier Transform I

In radio broadcasting, Fourier Transform properties are applied in simultaneous multi-channel transmission, adjusting audio clip speeds, live broadcast ...

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Signals are sets of data or information.
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Methods of Documentation VII: EMR

Electronic Medical Records (EMRs) primarily center around electronically documenting patients&#39; health information within a single healthcare ...

The Fast Fourier Transform, FFT, is a computational algorithm for calculating the Discrete Fourier Transform by breaking the calculations into smaller, ...

reconstruction-of-signal-using-interpolation

Reconstruction of Signal using Interpolation

Consider a continuous signal and its sampled counterpart with a sampling period T.
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Downsampling

Consider a sampled sequence with zero values between sampling instants. Replace it by taking every N-th value of the sampled sequence.
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Design Example

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Consider a decimated signal with a reduced frequency range due to its lower sampling rate.
Insert zeros between each sample to upsample, that introduces ...

Properties of DTFT I

Consider two discrete-time signals, each with their respective Discrete-Time Fourier Transforms DTFTs.
The signals are first multiplied by constants a and ...

Relation of DFT to z-Transform

The Discrete Fourier Transform (DFT) analyzes the frequency content of discrete-time signals.
It maps the N-sampled discrete time-domain sequence to its ...

Feedback control systems

Feedback control systems are categorized as linear or nonlinear, time-varying or time-invariant, and classified by signal types as continuous or ...

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Certain properties provide a solid foundation for analyzing discrete-time systems using the Z-transform.
Considering two discrete-time signals, the ...

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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 ...

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Interference: Path Lengths

Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two ...

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concept-of-resonance-and-its-characteristics

Concept of Resonance and its Characteristics

If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano ...

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Sound Waves: Resonance

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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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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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Interference and Diffraction

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

Reflection and Refraction

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

Electromagnetic Fields

Electric fields generated by static charges, often referred to as electrostatic fields, are characteristically different from electric fields created by ...

Photoelectric Effect

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

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Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other. [Clarification Statement: Emphasis is on how the experimental evidence supports the claim and how a theory is generally modified in light of new evidence. Examples of a phenomenon could include resonance, interference, diffraction, and photoelectric effect.] [Assessment Boundary: Assessment does not include using quantum theory.]

HS-PS4-3

The electromagnetic spectrum consists of all the types of electromagnetic radiation arranged according to their frequency and wavelength. Each of the ...

Biological Effects of Radiation

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Intensity Of Electromagnetic Waves

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Evaluate the validity and reliability of claims in published materials of the effects that different frequencies of electromagnetic radiation have when absorbed by matter. [Clarification Statement: Emphasis is on the idea that photons associated with different frequencies of light have different energies, and the damage to living tissue from electromagnetic radiation depends on the energy of the radiation. Examples of published materials could include trade books, magazines, web resources, videos, and other passages that may reflect bias.] [Assessment Boundary: Assessment is limited to qualitative descriptions.]

HS-PS4-4

Fast Fourier Transform for Signal Analysis

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