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

El teorema de Parseval establece que si una función es periódica, entonces la potencia promedio de la señal durante un período es igual a la suma de las magnitudes al cuadrado de todos los coeficientes complejos de Fourier.
Para validar el teorema de Parseval, supongamos que la función tiene una serie compleja de Fourier de una forma estándar. Sustituyendo esto y resolviendo aún más se obtiene la demostración del teorema.
Curiosamente, el teorema de Parseval también se puede expresar en términos de los coeficientes de Fourier de la serie trigonométrica de Fourier.
En el procesamiento de audio, el teorema de Parseval se utiliza para comparar la energía de una onda sonora original con su versión comprimida.
La interpretación de ingeniería de este teorema proporciona información práctica. Si la función representa una señal eléctrica, como corriente o voltaje, entonces el cuadrado de la función representa la potencia instantánea en una resistencia de 1 ohmio.
Este teorema también relaciona la energía disipada en la resistencia durante un período con la serie de Fourier, proporcionando dos expresiones diferentes: una en términos de la serie trigonométrica de Fourier y otra en términos de la serie de Fourier de fase de amplitud.

teorema de parseval

El teorema de Parseval es un concepto fundamental en el procesamiento de señales y el análisis armónico. Afirma que, para una función periódica, la potencia promedio de la señal durante un período es igual a la suma de las magnitudes al cuadrado de todos sus coeficientes complejos de Fourier. Este teorema, llamado así por Marc-Antoine Parseval, proporciona una herramienta poderosa para analizar la distribución de energía en señales.

Curiosamente, el teorema de Parseval también es válido para la forma trigonométrica de la serie de Fourier, que expresa una función en términos de funciones seno y coseno. Aquí, los coeficientes de Fourier se pueden relacionar con los coeficientes de la serie trigonométrica, lo que permite aplicar el teorema en esta forma alternativa.

Para validar el teorema de Parseval, comenzamos considerando una función x(t) con una representación compleja de la serie de Fourier:

<img alt="Equation1" src="/files/ftp_upload/16063/16063_PT%20EQUATION_1.svg" style="text-align: center; height: 30px;">

Donde c_n son los coeficientes complejos de Fourier y ω_0 es la frecuencia angular fundamental. El teorema establece:

<img alt="Equation2" src="/files/ftp_upload/16063/16063_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;">

Donde T es el período de la función. Sustituir la serie de Fourier en el lado izquierdo y resolver confirma la igualdad, lo que demuestra el teorema.

El teorema de Parseval es crucial en aplicaciones prácticas, particularmente en el procesamiento de audio. Permite comparar la energía contenida en una onda de sonido original con la de su versión comprimida. Esta comparación es esencial para garantizar que el proceso de compresión no degrade significativamente la calidad de la señal de audio al perder demasiada energía.

Desde una perspectiva de ingeniería, el teorema de Parseval ofrece información valiosa. Por ejemplo, si la función en cuestión representa una señal eléctrica como la corriente o la tensión, entonces el cuadrado de esta función representa la potencia instantánea disipada en una resistencia de 1 ohmio. En consecuencia, el teorema vincula la energía disipada en el resistor durante un período con la representación de la señal en forma de serie de Fourier. Esta relación se expresa de dos formas diferentes: una utilizando la serie trigonométrica de Fourier y la otra utilizando la forma amplitud-fase de la serie de Fourier. De este modo, el teorema de Parseval no sólo sirve como una poderosa herramienta analítica, sino que también conecta conceptos teóricos con aplicaciones prácticas de ingeniería.

Teorema de Parseval

El teorema de Parseval establece que si una función es periódica, entonces la potencia media de la señal durante un período es igual a la suma de la ...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Fourier Series

Series de Fourier

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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 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

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...

key terms and concepts

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

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

Parseval's Theorem in Signal Power

Description

Transcript

Tags