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

A série de Fourier em tempo discreto é uma contraparte da expansão da série de Fourier de sinais periódicos em tempo contínuo.
O cálculo dos coeficientes de expansão da DTFS envolve somas em vez de integrais. A resposta de um sistema invariante de tempo linear a um sinal periódico de tempo discreto pode ser determinada por meio de um processo passo a passo.
Primeiro, o DTFS do sinal de entrada deve ser calculado. A resposta de saída para cada termo DTFS é calculada usando a resposta de frequência do sistema. Finalmente, esses resultados são somados para obter o sinal de saída total.
Um sinal periódico em tempo contínuo tem um período com frequências circulares e angulares, representado por uma série de Fourier exponencial complexa. Da mesma forma, um sinal periódico de tempo discreto tem uma frequência angular fundamental.
A expansão DTFS consiste em termos finitos, ao contrário da série de Fourier para tempo contínuo, que consiste em um número infinito de termos.
No processamento de sinal digital, o DTFS auxilia no exame de amostras periódicas, identificando frequências específicas e filtrando efetivamente ruídos indesejáveis.

série de fourier de tempo discreto

A Série de Fourier em Tempo Discreto (DTFS) é um conceito fundamental no processamento de sinais, servindo como a contraparte em tempo discreto para a série de Fourier em tempo contínuo. Ela permite a representação e análise de sinais periódicos em tempo discreto em termos de seus componentes de frequência. Ao contrário de sua contraparte contínua, que utiliza integrais, o cálculo dos coeficientes de expansão DTFS envolve somas devido à natureza discreta do sinal.

Para um sinal periódico em tempo discreto x[n] com período N_0, os coeficientes DTFS X[k] são calculados usando a fórmula:

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="height: 48px;">

Onde k=0,1,2,…,N_0−1. Esses coeficientes X[k] representam o sinal no domínio da frequência, capturando a amplitude e a fase de cada componente da frequência.

Para determinar a resposta de um sistema Linear Invariante no Tempo (LTI) a um sinal periódico de tempo discreto, uma abordagem sistemática é seguida:

<ol style="padding-left: 35px; margin: 0;">
	<li style="margin: 8px 0;">Calcule o DTFS do Sinal de Entrada: Calcule os coeficientes DTFS X[k] para o sinal de entrada x[n]</li>
	<li style="margin: 8px 0;">Calcule a Resposta de Saída para Cada Termo DTFS: Use a resposta de frequência do sistema H(e^jΩ) para determinar a saída para cada componente de frequência. Os coeficientes DTFS de saída Y[k] são dados por 
	<img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;"> 
	onde Ω =2πk/N</li>
	<li style="margin: 8px 0;">Some as Respostas: Finalmente, some as contribuições de todos os termos DTFS para obter o sinal de saída total no domínio do tempo.</li>
</ol>

Em sinais de tempo contínuo, a periodicidade é definida em relação a um período T, correspondendo a frequências circulares e angulares. Para um sinal de tempo discreto, a periodicidade está associada a uma frequência angular fundamental Ω_k = 2πk/N_0, onde N_0 é o período do sinal discreto. A expansão DTFS é finita, consistindo de N termos, contrastando com a série infinita em séries de Fourier de tempo contínuo.

O DTFS desempenha um papel crucial no processamento de sinais digitais (DSP), particularmente na análise e manipulação de sinais periódicos derivados de dados amostrados. É fundamental em tarefas como identificar frequências específicas em sinais de áudio, aprimorar ou suprimir certos componentes de frequência e filtrar ruídos indesejados. Ao transformar sinais no domínio de frequência, o DTFS facilita a análise e o processamento eficientes de sinais, permitindo melhor desempenho em várias aplicações, como telecomunicações, engenharia de áudio e sistemas de controle.

Série de Fourier de Tempo Discreto

A série de Fourier em tempo discreto é uma contraparte da expansão da série de Fourier de sinais periódicos em tempo contínuo.
Calculando a expansão ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

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Fourier Series

Séries de Fourier

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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-PS4-4

DTFS for Periodic Signal Analysis

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