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

Parseval&#39;s theorem states that if a function is periodic, then the average power of the signal over one period equals the sum of the squared magnitudes of all the complex Fourier coefficients.
To validate Parseval&#39;s theorem, assume that the function has a complex Fourier series of a standard form. Substituting this and solving further gives the proof of the theorem.
Interestingly, Parseval&#39;s theorem can also be expressed in terms of the Fourier coefficients of the trigonometric Fourier series.
In audio processing, Parseval&#39;s theorem is used to compare the energy of an original sound wave with its compressed version.
The engineering interpretation of this theorem provides practical insights. If the function represents an electrical signal, such as current or voltage, then the square of the function represents the instantaneous power in a 1-ohm resistor.
This theorem also relates the energy dissipated in the resistor during one period to the Fourier series, providing two different expressions - one in terms of the trigonometric Fourier series and another in terms of the amplitude phase Fourier series.

parseval's theorem

Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a function in terms of sine and cosine functions. Here, the Fourier coefficients can be related to the trigonometric series coefficients, allowing the theorem to be applied in this alternate form.
To validate Parseval's theorem, we start by considering a function x(t) with a complex Fourier series representation:
<img alt="Fourier series equation, signal processing method, mathematical formula for periodic functions." src="/files/ftp_upload/16063/16063_PT%20EQUATION_1.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Where Cn are the complex Fourier coefficients, and &#969;0 is the fundamental angular frequency. The theorem states:
<img alt="Power calculation equation; integral representation; Fourier series; mathematical analysis." src="/files/ftp_upload/16063/16063_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
Where T is the period of the function. Substituting the Fourier series into the left-hand side and solving confirms the equality, thus proving the theorem.
Parseval's theorem is crucial in practical applications, particularly in audio processing. It allows for comparing the energy contained in an original sound wave to that in its compressed version. This comparison is essential in ensuring that the compression process does not significantly degrade the quality of the audio signal by losing too much energy.
From an engineering perspective, Parseval's theorem offers valuable insights. For instance, if the function in question represents an electrical signal such as current or voltage, then the square of this function represents the instantaneous power dissipated in a 1-ohm resistor. Consequently, the theorem links the energy dissipated in the resistor over one period to the Fourier series representation of the signal. This relationship is expressed in two different forms: one using the trigonometric Fourier series and the other using the amplitude-phase form of the Fourier series. Thus, Parseval's theorem not only serves as a powerful analytical tool but also bridges theoretical concepts with practical engineering applications.

Parseval's Theorem

Parseval&#39;s theorem states that if a function is periodic, then the average power of the signal over one period equals the sum of the squared ...

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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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Learn how Parseval's theorem connects a periodic signal’s average power to Fourier coefficients, with uses in audio compression and electrical signals.

Parseval's Theorem in Signal Power | JoVE K12

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Parseval's Theorem in Signal Power

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