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

Dyskretny szereg Fouriera jest odpowiednikiem rozszerzenia szeregów Fouriera sygnałów okresowych w czasie ciągłym.
Obliczanie współczynników rozszerzalności DTFS obejmuje raczej sumowanie niż całki. Odpowiedź liniowego układu niezmienniczego w czasie na dyskretny sygnał okresowy w czasie może być określona w procesie krok po kroku.
Najpierw należy obliczyć DTFS sygnału wejściowego. Odpowiedź wyjściowa na każdy składnik DTFS jest obliczana przy użyciu charakterystyki częstotliwościowej systemu. Na koniec wyniki te są sumowane w celu uzyskania całkowitego sygnału wyjściowego.
Sygnał okresowy w czasie ciągłym ma okres o częstotliwościach kołowych i kątowych, reprezentowany przez zespolony wykładniczy szereg Fouriera. Podobnie, dyskretny sygnał okresowy ma podstawową częstotliwość kątową.
Rozwinięcie DTFS składa się ze skończonych wyrazów, w przeciwieństwie do szeregu Fouriera dla czasu ciągłego, który składa się z nieskończonej liczby wyrazów.
W cyfrowym przetwarzaniu sygnałów DTFS pomaga w badaniu okresowych próbek, określaniu określonych częstotliwości i skutecznym odfiltrowywaniu niepożądanych szumów.

dyskretny szereg fouriera

Dyskretny szereg Fouriera (DTFS) to koncepcja przetwarzania sygnałów, będąca dyskretnym odpowiednikiem szeregu Fouriera. Umożliwia ona reprezentację i analizę dyskretnych sygnałów w odniesieniu do ich składowych częstotliwości. W przeciwieństwie do ciągłego odpowiednika, który wykorzystuje całki, obliczanie współczynników rozszerzenia DTFS obejmuje sumowanie ze względu na dyskretną naturę sygnału.

W przypadku dyskretnego sygnału okresowego x[n] o okresie N_0 współczynniki DTFS X[k] są obliczane przy użyciu wzoru:

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="text-align: center; height: 48px;">

Gdzie k=0,1,2,…,N_0−1. Te współczynniki X[k] reprezentują sygnał w dziedzinie częstotliwości, rejestrując amplitudę i fazę każdej składowej częstotliwości.

Aby określić odpowiedź liniowego układu niezmienniczego w czasie (LTI) na dyskretny sygnał okresowy, stosuje się podejście systematyczne:

<ol style="padding-left: 35px; margin: 0;">
 <li>Oblicz DTFS sygnału wejściowego: oblicz współczynniki DTFS X[k] dla sygnału wejściowego x[n].</li>
 <li>Oblicz odpowiedź wyjściową dla każdego członu DTFS: użyj odpowiedzi częstotliwościowej układu H(e^jΩ), aby określić wyjście dla każdej składowej częstotliwości. Współczynniki DTFS wyjściowe Y[k] są podane przez
 <img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="height: 36px;">
 gdzie Ω =2πk/N
</li>
 <li>Zsumuj odpowiedzi: Na koniec zsumuj wkłady wszystkich członków DTFS, aby uzyskać całkowity sygnał wyjściowy w dziedzinie czasu.</li>
</ol>

W sygnałach ciągłych okresowość jest definiowana względem okresu T, odpowiadającego częstotliwościom kołowym i kątowym. W przypadku sygnału dyskretnego okresowość jest związana z podstawową częstotliwością kątową Ω_k = 2πk/N_0 , gdzie N_0 jest okresem sygnału dyskretnego. Rozszerzenie DTFS jest skończone i składa się z wyrazów N, w przeciwieństwie do nieskończonego szeregu w ciągłym szeregu Fouriera.

DTFS odgrywa kluczową rolę w cyfrowym przetwarzaniu sygnałów (DSP), w szczególności w analizie i manipulowaniu sygnałami okresowymi pochodzącymi z próbkowanych danych. Jest on pomocny w takich zadaniach, jak identyfikacja określonych częstotliwości w sygnałach audio, wzmacnianie lub tłumienie określonych składowych częstotliwości i filtrowanie niepożądanego szumu. Poprzez transformację sygnałów do dziedziny częstotliwości, DTFS ułatwia analizę i przetwarzanie sygnału, umożliwiając poprawę wydajności w różnych zastosowaniach, takich jak telekomunikacja, systemy dzwiękowe i systemy sterowania.

Dyskretny szereg Fouriera

Dyskretny szereg Fouriera jest odpowiednikiem rozszerzenia szeregów Fouriera sygnałów okresowych w czasie ciągłym.
Obliczanie rozszerzenia ...

Education

JoVE Core

Electrical Engineering

Elektrotechnika

Unit 1

Jednostka 1

Fourier Series

Szeregi Fouriera

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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...

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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...

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...

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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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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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