Name: fourier coefficients and harmonic terms
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 17 s
Channel: JoVE

Una serie de Fourier es una técnica matemática que descompone las funciones periódicas en una serie infinita de armónicos sinusoidales.
La serie trigonométrica de Fourier representa una función periódica con un período específico utilizando funciones de seno y coseno. La frecuencia de estas funciones es inversamente proporcional al período de la función original.
Los términos constantes y la contribución individual de cada función seno y coseno a la función original se entienden utilizando coeficientes de Fourier. El cálculo de estos coeficientes implica la integración a lo largo de un período.
Para que una serie de Fourier represente una función periódica, deben cumplirse las condiciones de Dirichlet.
La primera condición exige que la función tenga una integral finita en un período.
La segunda condición establece que la variación de la función debe limitarse dentro de un rango dado.
Por último, la función debe tener un número finito de discontinuidades dentro de cualquier rango específico, ninguna de las cuales puede ser infinita.
La serie de Fourier se puede utilizar para aproximar funciones: no puede recuperar completamente las funciones que no cumplen con las condiciones de Dirichlet.

serie trigonométrica de fourier

La serie de Fourier es una técnica matemática fundamental que descompone las funciones periódicas en una serie infinita de armónicos sinusoidales. Este método permite la representación de señales periódicas complejas como sumas de funciones seno y coseno simples, lo que facilita su análisis e interpretación en varios campos, incluidos el procesamiento de señales, la acústica y la ingeniería eléctrica.
La serie trigonométrica de Fourier expresa específicamente una función periódica con un período definido T utilizando las funciones seno y coseno. La forma general de la serie trigonométrica de Fourier para una función x(t) es:
<img alt="Equation1" src="/files/ftp_upload/16059/16059_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;">
Aquí, a_0 representa el valor promedio de la función durante un período, mientras que a_n y b_n son los coeficientes de Fourier que cuantifican la contribución de cada función seno y coseno, respectivamente. Estos coeficientes se determinan mediante la integración durante un período T:
<img alt="Equation2" src="/files/ftp_upload/16059/16059_PT%20EQUATION_2.svg" style="text-align: center; height: 46px;">
<img alt="Equation3" src="/files/ftp_upload/16059/16059_PT%20EQUATION_3.svg" style="text-align: center; height: 48px;">
<img alt="Equation4" src="/files/ftp_upload/16059/16059_PT%20EQUATION_4.svg" style="text-align: center; height: 48px;">
Estas integrales son esenciales para calcular los coeficientes exactos que reconstruyen la función original a partir de sus componentes sinusoidales.
Para representar con precisión una función periódica mediante una serie de Fourier, se deben cumplir las condiciones de Dirichlet. La primera condición estipula que la función debe tener una integral finita durante un período, lo que garantiza que la función general esté acotada. La segunda condición requiere que la función tenga un número limitado de máximos y mínimos dentro de un rango determinado, lo que garantiza que la función no presente oscilaciones excesivas. La tercera condición exige que la función posea un número finito de discontinuidades, ninguna de las cuales sea infinita. Estas condiciones garantizan que la serie de Fourier converja adecuadamente a la función original.
En aplicaciones prácticas, incluso si no se cumplen estrictamente estas condiciones, a menudo es posible construir representaciones de series de Fourier. Estas representaciones, aunque potencialmente menos precisas, pueden proporcionar aproximaciones útiles para analizar y sintetizar funciones periódicas. Esta flexibilidad subraya la solidez y utilidad de las series de Fourier en diversas aplicaciones matemáticas y de ingeniería.

Serie trigonométrica de Fourier

Una serie de Fourier es una técnica matemática que descompone las funciones periódicas en una serie infinita de armónicos sinusoidales.
El trigonométrico ...

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Electrical Engineering

Ingeniería Eléctrica

Unit 1

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Fourier Series

Series de Fourier

KEY TERMS AND CONCEPTS

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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 to the final result.

These coefficients are found by integration over one full period T. The integrals give the exact values needed to rebuild the original function from its sinusoidal pieces. This is why Fourier coefficients are central to the method.

To match a periodic function well, the Dirichlet conditions should be satisfied. The function must have a finite integral over one period, a limited number of maxima and minima in any range, and a finite number of discontinuities that are not infinite. These conditions help the Fourier series converge to the original function.

Even when the Dirichlet conditions are not perfectly met, a Fourier series can often still be written. The result may be less accurate, but it can still give a useful approximation of the periodic function. This makes Fourier series a practical tool for analyzing and synthesizing repeating signals in mathematics and engineering.

Fourier Coefficients and Harmonic Terms

trigonometric fourier series

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...

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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...

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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...

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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...

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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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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...

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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 Fourier series use sine and cosine terms, Fourier coefficients, and Dirichlet conditions to model periodic functions and signals.

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Fourier Coefficients and Harmonic Terms

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