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

이산 시간 푸리에 급수는 연속 시간 주기 신호의 푸리에 급수 확장에 대응합니다.
DTFS의 팽창 계수를 계산하려면 적분이 아닌 합계가 포함됩니다. 이산시간 주기 신호에 대한 선형 시불변 시스템의 응답은 단계별 프로세스를 통해 결정할 수 있습니다.
먼저 입력 신호의 DTFS를 계산해야 합니다. 각 DTFS 항에 대한 출력 응답은 시스템의 주파수 응답을 사용하여 계산됩니다. 마지막으로, 이러한 결과를 합산하여 총 출력 신호를 얻습니다.
연속 시간의 주기적 신호는 복소수 지수 푸리에 급수로 표시되는 원형 및 각도 주파수를 갖는 주기를 갖습니다. 마찬가지로, 이산시간 주기 신호는 기본 각주파수를 갖습니다.
DTFS 확장은 무한대의 항으로 구성되는 연속 시간에 대한 푸리에 급수와 달리 유한한 항으로 구성됩니다.
디지털 신호 처리에서 DTFS는 주기적인 샘플을 검사하고, 특정 주파수를 정확히 찾아내고, 원치 않는 노이즈를 효과적으로 필터링하는 데 도움이 됩니다.

 이산 시간 푸리에 급수

이산 시간 푸리에 급수(DTFS)는 신호 처리의 기본 개념으로, 연속 시간 푸리에 급수의 이산 시간 대응물 역할을 합니다. 이산 시간 주기 신호를 주파수 성분으로 표현하고 분석할 수 있습니다. 적분을 활용하는 연속 대응물과 달리 DTFS 확장 계수 계산에는 신호의 이산적 특성으로 인한 합산이 포함됩니다.

주기가 N_0인 이산 시간 주기 신호 x[n]의 경우 DTFS 계수 X[k]는 다음 공식을 사용하여 계산합니다.

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="height: 48px;">

여기서 k=0,1,2,…,N_0−1입니다. 이러한 계수 X[k]는 주파수 영역의 신호를 나타내며 각 주파수 성분의 진폭과 위상을 포착합니다.

선형 시불변(LTI) 시스템의 이산-시간 주기 신호에 대한 응답을 결정하기 위해 체계적인 접근 방식을 따릅니다.

<ol style="padding-left: 35px; margin: 0;">
	<li>입력 신호의 DTFS 계산: 입력 신호 x[n]에 대한 DTFS 계수 X[k]를 계산합니다.</li>
	<li>각 DTFS 항에 대한 출력 응답 계산: 시스템의 주파수 응답 H(e^ jΩ)를 사용하여 각 주파수 구성 요소에 대한 출력을 결정합니다. 출력 DTFS 계수 Y[k]는 다음과 같이 주어집니다.
	<img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="height: 36px;">

	여기서 Ω =2πk/N
	</li>
	<li>응답 합산: 마지막으로 모든 DTFS 항의 기여도를 합산하여 시간 영역에서 총 출력 신호를 얻습니다.</li>
</ol>

연속 시간 신호에서 주기는 원형 및 각 주파수에 해당하는 주기 T에 대해 정의됩니다. 이산 시간 신호의 경우 주기는 기본 각 주파수 Ω_k = 2πk/N_0와 연관되며, 여기서 N_0은 이산 신호의 주기입니다. DTFS 확장은 유한하며 N 항으로 구성되며 연속 시간 푸리에 급수의 무한 급수와 대조됩니다.

DTFS는 디지털 신호 처리(DSP)에서 중요한 역할을 하며, 특히 샘플링된 데이터에서 파생된 주기 신호를 분석하고 조작하는 데 중요합니다. 오디오 신호 내의 특정 주파수를 식별하고, 특정 주파수 구성 요소를 강화하거나 억제하고, 원치 않는 노이즈를 필터링하는 것과 같은 작업에 유용합니다. DTFS는 신호를 주파수 영역으로 변환하여 효율적인 신호 분석 및 처리를 용이하게 하여 통신, 오디오 엔지니어링, 제어 시스템과 같은 다양한 응용 분야에서 성능을 개선할 수 있습니다.

이산 시간 푸리에 급수는 연속 시간 주기 신호의 푸리에 급수 확장에 대응합니다.
확장 계산 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Fourier Series

푸리에 급수

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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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.

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DTFS for Periodic Signal Analysis

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