Name: parseval's theorem in signal power
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 18 s
Channel: JoVE

O teorema de Parseval afirma que, se uma função é periódica, então a potência média do sinal durante um período é igual à soma das magnitudes quadradas de todos os coeficientes complexos de Fourier.
Para validar o teorema de Parseval, suponha que a função tenha uma série complexa de Fourier de uma forma padrão. Substituir isso e resolver mais dá a prova do teorema.
Curiosamente, o teorema de Parseval também pode ser expresso em termos dos coeficientes de Fourier da série trigonométrica de Fourier.
No processamento de áudio, o teorema de Parseval é usado para comparar a energia de uma onda sonora original com sua versão comprimida.
A interpretação de engenharia deste teorema fornece insights práticos. Se a função representa um sinal elétrico, como corrente ou tensão, o quadrado da função representa a potência instantânea em um resistor de 1 ohm.
Este teorema também relaciona a energia dissipada no resistor durante um período com a série de Fourier, fornecendo duas expressões diferentes - uma em termos da série trigonométrica de Fourier e outra em termos da série de Fourier da fase de amplitude.

teorema de parseval

O teorema de Parseval é um conceito fundamental no processamento de sinais e análise harmônica. Ele afirma que, para uma função periódica, a potência média do sinal em um período é igual à soma das magnitudes quadradas de todos os seus coeficientes complexos de Fourier. Este teorema, nomeado em homenagem a Marc-Antoine Parseval, fornece uma ferramenta poderosa para analisar a distribuição de energia em sinais.

Curiosamente, o teorema de Parseval também é válido para a forma trigonométrica da série de Fourier, que expressa uma função em termos de funções seno e cosseno. Aqui, os coeficientes de Fourier podem ser relacionados aos coeficientes da série trigonométrica, permitindo que o teorema seja aplicado nesta forma alternativa.

Para validar o teorema de Parseval, começamos considerando uma função x(t) com uma representação complexa de série de Fourier:

<img alt="Equation1" src="/files/ftp_upload/16063/16063_PT%20EQUATION_1.svg" style="height: 30px;">

Onde c_n são os coeficientes complexos de Fourier e ω_0 é a frequência angular fundamental. O teorema afirma:

<img alt="Equation2" src="/files/ftp_upload/16063/16063_PT%20EQUATION_2.svg" style="height: 36px;">

Onde T é o período da função. Substituir a série de Fourier no lado esquerdo e resolver confirma a igualdade, provando assim o teorema.

O teorema de Parseval é crucial em aplicações práticas, particularmente no processamento de áudio. Ele permite comparar a energia contida em uma onda sonora original com aquela em sua versão comprimida. Essa comparação é essencial para garantir que o processo de compressão não degrade significativamente a qualidade do sinal de áudio ao perder muita energia.

De uma perspectiva de engenharia, o teorema de Parseval oferece insights valiosos. Por exemplo, se a função em questão representa um sinal elétrico, como corrente ou voltagem, então o quadrado dessa função representa a potência instantânea dissipada em um resistor de 1 ohm. Consequentemente, o teorema vincula a energia dissipada no resistor ao longo de um período à representação da série de Fourier do sinal. Essa relação é expressa em duas formas diferentes: uma usando a série trigonométrica de Fourier e a outra usando a forma amplitude-fase da série de Fourier. Assim, o teorema de Parseval não serve apenas como uma ferramenta analítica poderosa, mas também conecta conceitos teóricos com aplicações práticas de engenharia.

Teorema de Parseval

O teorema de Parseval afirma que, se uma função é periódica, então a potência média do sinal ao longo de um período é igual à soma do quadrado ...

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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 series coefficients, so the same idea can be applied in this alternate form.

To verify Parseval's theorem, start with a function x(t) written as a complex Fourier series. In this form, Cn are the complex Fourier coefficients, and ω0 is the fundamental angular frequency. The theorem states the equality between average power over a period T and the sum of the squared coefficient magnitudes. Substituting the Fourier series into the left-hand side and simplifying confirms the result.

Parseval's theorem is important in audio processing. It can compare the energy in an original sound wave with the energy in a compressed version. This helps show whether compression has kept most of the signal energy and avoided major loss in sound quality.

The theorem is also useful in engineering with current or voltage signals. In that case, the square of the function represents the instantaneous power dissipated in a 1-ohm resistor. Parseval's theorem then connects the energy dissipated over one period to the Fourier series form of the signal. It can be written using either the trigonometric Fourier series or the amplitude-phase form, which makes it a practical bridge between signal theory and real engineering problems.

Parseval's Theorem in Signal Power

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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...

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Parseval's Theorem in Signal Power

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