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

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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

La serie de Fourier en tiempo discreto es una contraparte de la expansión de la serie de Fourier de señales periódicas en tiempo continuo.
El cálculo de los coeficientes de expansión del DTFS implica sumas en lugar de integrales. La respuesta de un sistema lineal invariante en el tiempo a una señal periódica de tiempo discreto se puede determinar a través de un proceso paso a paso.
En primer lugar, se debe calcular el DTFS de la señal de entrada. La respuesta de salida a cada término DTFS se calcula utilizando la respuesta de frecuencia del sistema. Finalmente, estos resultados se suman para obtener la señal de salida total.
Una señal periódica en tiempo continuo tiene un período con frecuencias circulares y angulares, representadas por una serie de Fourier compleja-exponencial. De manera similar, una señal periódica de tiempo discreto tiene una frecuencia angular fundamental.
La expansión DTFS consta de términos finitos, a diferencia de la serie de Fourier para tiempo continuo, que consta de un número infinito de términos.
En el procesamiento digital de señales, DTFS ayuda a examinar muestras periódicas, identificar frecuencias específicas y filtrar eficazmente el ruido no deseado.

serie de fourier de tiempo discreto

La serie de Fourier de Tiempo Discreto (DTFS) es un concepto fundamental en el procesamiento de señales, que sirve como contraparte en tiempo discreto de la serie de Fourier de tiempo continuo. Permite la representación y el análisis de señales periódicas de tiempo discreto en términos de sus componentes de frecuencia. A diferencia de su contraparte continua, que utiliza integrales, el cálculo de los coeficientes de expansión de DTFS implica sumas debido a la naturaleza discreta de la señal.

Para una señal periódica de tiempo discreto x[n] con período N_0, los coeficientes DTFS X[k] se calculan utilizando la fórmula:

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="text-align: center; height: 48px;">

Donde k=0,1,2,…,N_0−1. Estos coeficientes X[k] representan la señal en el dominio de la frecuencia, capturando la amplitud y la fase de cada componente de frecuencia.

Para determinar la respuesta de un sistema lineal invariante en el tiempo (LTI) a una señal periódica en el tiempo discreto, se sigue un enfoque sistemático:



<ol style="padding-left: 35px; margin: 0;">
 <li>Calcular el DTFS de la señal de entrada: Calcular los coeficientes DTFS X[k] para la señal de entrada x[n].</li>
 <li>
 Calcular la respuesta de salida para cada término DTFS: Utilizar la respuesta de frecuencia del sistema H(e^(jΩ)) para determinar la salida para cada componente de frecuencia. Los coeficientes DTFS de salida Y[k] se dan por
 <img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;">
 donde Ω =2πk/N
 </li>
 <li>Sumar las respuestas: Finalmente, sumar las contribuciones de todos los términos DTFS para obtener la señal de salida total en el dominio del tiempo.</li>
</ol>

En las señales de tiempo continuo, la periodicidad se define con respecto a un período T, correspondiente a frecuencias circulares y angulares. En el caso de una señal de tiempo discreto, la periodicidad está asociada a una frecuencia angular fundamental Ω_k = 2πk/N_0, donde N_0 es el período de la señal discreta. La expansión DTFS es finita y consta de términos N, en contraste con la serie infinita de las series de Fourier de tiempo continuo.

DTFS desempeña un papel crucial en el procesamiento de señales digitales (DSP), en particular en el análisis y la manipulación de señales periódicas derivadas de datos muestreados. Es fundamental en tareas como la identificación de frecuencias específicas dentro de las señales de audio, la mejora o supresión de ciertos componentes de frecuencia y el filtrado de ruido no deseado. Al transformar las señales en el dominio de la frecuencia, DTFS facilita el análisis y procesamiento eficiente de señales, lo que permite un mejor rendimiento en diversas aplicaciones, como telecomunicaciones, ingeniería de audio y sistemas de control.

Serie de Fourier de Tiempo Discreto

La serie de Fourier en tiempo discreto es una contraparte de la expansión de la serie de Fourier de señales periódicas en tiempo continuo.
Calculando la expansión ...

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

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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