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

High School

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

Ряды Фурье с дискретным временем являются аналогами ряда Фурье для непрерывных периодических сигналов.
Вычисление коэффициентов разложения DTFS включает в себя суммирование, а не интегралы. Реакция линейной инвариантной по времени системы на дискретный периодический сигнал может быть определена с помощью пошагового процесса.
Во-первых, необходимо вычислить DTFS входного сигнала. Выходной отклик на каждый член DTFS рассчитывается с использованием частотной характеристики системы. Наконец, эти результаты суммируются для получения общего выходного сигнала.
Периодический сигнал в непрерывном времени имеет период с круговыми и угловыми частотами, представленный комплексно-экспоненциальным рядом Фурье. Аналогично, дискретный периодический сигнал имеет основную угловую частоту.
Разложение DTFS состоит из конечных членов, в отличие от ряда Фурье для непрерывного времени, которое состоит из бесконечного числа членов.
При цифровой обработке сигналов DTFS помогает исследовать периодические выборки, точно определять определенные частоты и эффективно отфильтровывать нежелательные шумы.

Дискретно-временной ряд Фурье

Дискретно-временной ряд Фурье (ДВРФ) — это фундаментальное понятие в обработке сигналов, выступающее в качестве дискретного аналога непрерывного ряда Фурье. Он позволяет представлять и анализировать дискретные периодические сигналы с точки зрения их частотных компонентов. В отличие от своего непрерывного аналога, который использует интегралы, расчет коэффициентов разложения ДВРФ включает суммирование из-за дискретной природы сигнала.

Для дискретного периодического сигнала x[n] с периодом N_0 коэффициенты ДВРФ X[k] вычисляются по формуле:

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="text-align: center; height: 48px;">

Где k=0,1,2,…,N_0−1. Эти коэффициенты X[k] представляют сигнал в частотной области, фиксируя амплитуду и фазу каждого частотного компонента.

Чтобы определить реакцию линейной системы, инвариантной по времени (LTI), на дискретный периодический сигнал, применяется систематический подход:

<ol style="padding-left: 35px; margin: 0;">
	<li>Вычислите ДВРФ входного сигнала: вычислите коэффициенты ДВРФ X[k] для входного сигнала x[n].</li>
	<li>Вычислите выходной отклик для каждого члена ДВРФ: используйте частотную характеристику системы H(e^jΩ) для определения выходного сигнала для каждого частотного компонента. Выходные коэффициенты ДВРФ Y[k] задаются как
	<img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;">

	где Ω = 2πk/N
	</li>
	<li>Суммируйте отклики: наконец, суммируйте вклады всех членов ДВРФ, чтобы получить общий выходной сигнал во временной области.</li>
</ol>

В непрерывных сигналах периодичность определяется относительно периода T, соответствующего круговым и угловым частотам. Для дискретного сигнала периодичность связана с фундаментальной угловой частотой Ω_k = 2πk/N_0 , где N_0 — период дискретного сигнала. Расширение ДВРФ конечно, состоит из N членов, в отличие от бесконечного ряда в непрерывном ряду Фурье.

ДВРФ играет важную роль в цифровой обработке сигналов (DSP), особенно в анализе и манипулировании периодическими сигналами, полученными из выборочных данных. Он играет важную роль в таких задачах, как определение определенных частот в аудиосигналах, усиление или подавление определенных частотных компонентов и фильтрация нежелательных шумов. Преобразуя сигналы в частотную область, ДВРФ облегчает эффективный анализ и обработку сигналов, обеспечивая повышение производительности в различных приложениях, таких как телекоммуникации, аудиотехника и системы управления.

Ряды Фурье с дискретным временем являются аналогами ряда Фурье для непрерывных периодических сигналов.
Расчет расширения...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Fourier Series

Ряды Фурье

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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