Name: dtfs for periodic signal analysis
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 20 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

Discrete-Time Fourier Series

La série de Fourier à temps discret est une contrepartie de l’expansion de la série de Fourier des signaux périodiques à temps continu.
Le calcul des coefficients de dilatation du DTFS implique des sommations plutôt que des intégrales. La réponse d’un système linéaire invariant dans le temps à un signal périodique à temps discret peut être déterminée par un processus étape par étape.
Tout d’abord, le DTFS du signal d’entrée doit être calculé. La réponse de sortie à chaque terme DTFS est calculée à l'aide de la réponse en fréquence du système. Enfin, ces résultats sont additionnés pour obtenir le signal de sortie total.
Un signal périodique en temps continu a une période avec des fréquences circulaires et angulaires, représentée par une série de Fourier exponentielle complexe. De même, un signal périodique à temps discret a une fréquence angulaire fondamentale.
Le développement DTFS est constitué de termes finis, contrairement à la série de Fourier pour le temps continu, qui se compose d’un nombre infini de termes.
Dans le traitement numérique du signal, DTFS aide à examiner des échantillons périodiques, à localiser des fréquences spécifiques et à filtrer efficacement le bruit indésirable.

transformation de fourier à temps discret

La transformation de Fourier à temps discret (TFtd) est un concept fondamental du traitement du signal, servant de contrepartie à temps discret à la transformation de Fourier à temps continu. Elle permet la représentation et l'analyse de signaux périodiques à temps discret en termes de leurs composantes de fréquence. Contrairement à son homologue continu, qui utilise des intégrales, le calcul des coefficients d'expansion de la série de Fourier implique des sommations en raison de la nature discrète du signal.

Pour un signal périodique à temps discret x[n] avec une période N_0, les coefficients TFtd X[k] sont calculés à l'aide de la formule :

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="height: 48px;">

Où k=0,1,2,…,N_0−1. Ces coefficients X[k] représentent le signal dans le domaine fréquentiel, capturant l'amplitude et la phase de chaque composante de fréquence.

Pour déterminer la réponse d'un système linéaire invariant dans le temps (LTI) à un signal périodique à temps discret, une approche systématique est suivie :

<ol style="padding-left: 35px; margin: 0;">
 <li>Calculer la TFtd du signal d'entrée : calculer les coefficients TFtd X[k] pour le signal d'entrée x[n].</li>
 <li>
 Calculer la réponse de sortie pour chaque terme TFtd: utiliser la réponse en fréquence H(e^(jΩ)) du système pour déterminer la sortie pour chaque composante de fréquence. Les coefficients TFtd de sortie Y[k] sont donnés par :
 <img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="height: 36px;">
 où Ω = 2πk/N
 </li>
 <li>Additionner les réponses : enfin, additionner les contributions de tous les termes TFtd pour obtenir le signal de sortie total dans le domaine temporel.</li>
</ol>

Dans les signaux à temps continu, la périodicité est définie par rapport à une période T, correspondant aux fréquences circulaires et angulaires. Pour un signal à temps discret, la périodicité est associée à une fréquence angulaire fondamentale Ω_k = 2πk/N_0, où N_0 est la période du signal discret. Le développement TFtd est fini, composé de N termes, contrairement à la série infinie des séries de Fourier à temps continu.

La TFtd joue un rôle crucial dans le traitement numérique du signal, en particulier dans l'analyse et la manipulation de signaux périodiques dérivés de données échantillonnées. Elle joue un rôle essentiel dans des tâches telles que l'identification de fréquences spécifiques dans les signaux audio, l'amélioration ou la suppression de certaines composantes de fréquence et le filtrage des bruits indésirables. En transformant les signaux dans le domaine fréquentiel, la TFtd facilite l'analyse et le traitement efficaces des signaux, ce qui permet ainsi d'améliorer les performances dans diverses applications telles que les télécommunications, l'ingénierie audio et les systèmes de contrôle.

Transformation de Fourier à temps discret

La série de Fourier à temps discret est une contrepartie de l’expansion de la série de Fourier des signaux périodiques à temps continu.
Calcul de l’expansion ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

Fourier Series

Séries de Fourier

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

dtfs for periodic signal analysis

Discrete-Time Fourier Series (DTFS) helps analyze periodic signals in discrete time by breaking them into frequency components. It is the discrete-time counterpart of the continuous-time Fourier series. Instead of using integrals, DTFS uses summations because the signal is discrete.

For a discrete-time periodic signal x[n] with period N0, the DTFS coefficients X[k] are found from the DTFS formula. The index k runs from 0 to N0 − 1. These coefficients describe the signal in the frequency domain and show the amplitude and phase of each frequency component.

DTFS is also useful for finding the response of a linear time-invariant (LTI) system to a periodic input. First, compute the DTFS coefficients of the input signal x[n]. Then use the system frequency response H(e^jΩ) to find the output for each frequency term. The output coefficients Y[k] are determined from that frequency response, and the total time-domain output is found by summing all DTFS terms.

In discrete-time signals, periodicity is tied to the fundamental angular frequency Ωk = 2πk/N0, where N0 is the signal period. The DTFS expansion has a finite number of terms, unlike the infinite series used for continuous-time Fourier series. This makes DTFS an important tool in digital signal processing for studying sampled periodic signals, identifying frequencies in audio, and helping enhance or suppress frequency components. It is also used to filter unwanted noise and support applications in telecommunications, audio engineering, and control systems.

DTFS for Periodic Signal Analysis

trigonometric fourier series

Fourier Coefficients and Harmonic Terms

Fourier series describe periodic functions by writing them as sums of sine and cosine waves. The trigonometric Fourier series uses a function with period T and represents it with simple harmonic terms. This approach helps break a complex repeating signal into parts that are easier to study. In the standard form, the series includes a0, an, and bn. The term a0 gives the average value of the function over one period. The coefficients an and bn show how much each cosine and sine term contributes...

exponential fourier series

Fourier Series in Audio Signal Processing

Fourier series are used in audio signal processing to break complex sounds into simpler sinusoidal parts. This helps explain how musical notes and other periodic signals can be analyzed and rebuilt from frequency components. The exponential Fourier series writes a periodic signal as a sum of complex exponentials at positive and negative harmonic frequencies. These harmonic terms capture the repeated pattern in the signal. Euler's identity connects the exponential form to cosine and sine, which...

properties of fourier series i

Fourier Series Properties in Signal Processing

Fourier series properties help describe periodic signals in signal processing and communications. A Fourier series breaks a signal into sine and cosine parts. These properties make it easier to study how signals behave and how they can be handled in real systems. Linearity is one of the most important properties. If two periodic signals are added together or combined in a linear way, the new signal has Fourier coefficients that are the same linear combination of the original coefficients. This...

properties of fourier series ii

Fourier Series Symmetry and Time Scaling

Fourier series change in useful ways when a signal is time-scaled or when it has symmetry. Time scaling changes the fundamental frequency, but the Fourier series coefficients stay the same. This idea helps show how a signal is represented over time in signal processing. Symmetry also makes Fourier series easier to work with. A function is even if f(t) = f(-t). For even functions, the sine terms disappear because sine is an odd function, and the integral of an odd function over a symmetric...

parseval's theorem

Parseval's Theorem in Signal Power

Parseval's theorem links signal power to Fourier series coefficients. For a periodic function, it says the average power over one period equals the sum of the squared magnitudes of its complex Fourier coefficients. This makes the theorem useful for studying how energy is distributed in a signal. The theorem also works for the trigonometric form of the Fourier series. In that form, a function is written using sine and cosine terms. The Fourier coefficients can be related to the trigonometric...

convergence of fourier series

Fourier Series Truncation and Gibbs Ripples

Fourier series can represent a periodic signal as an infinite sum of complex exponentials. For practical use, that infinite series is cut off after a finite number of terms. The result is a partial sum that gives a workable approximation of the original signal. Truncating the series creates a challenge near discontinuities. The approximation develops the Gibbs phenomenon, which is the set of persistent oscillations and overshoots around a jump in the signal. These high-frequency ripples do not...

discrete-time fourier series

Discrete-Time Fourier Series (DTFS) helps analyze periodic signals in discrete time by breaking them into frequency components. It is the discrete-time counterpart of the continuous-time Fourier series. Instead of using integrals, DTFS uses summations because the signal is discrete. For a discrete-time periodic signal x[n] with period N0, the DTFS coefficients X[k] are found from the DTFS formula. The index k runs from 0 to N0 − 1. These coefficients describe the signal in the frequency domain...

key terms and concepts

Learn how DTFS represents discrete-time periodic signals, finds frequency components, and helps analyze LTI system responses in digital signal processing.

DTFS for Periodic 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

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HS-PS4-1

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HS-PS4-3

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HS-PS4-4

DTFS for Periodic Signal Analysis

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