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

離散時間フーリエ級数は、連続時間周期信号のフーリエ級数展開に対応するものです。
DTFSの膨張係数の計算には、積分ではなく合計が含まれます。離散時間周期信号に対する線形時不変システムの応答は、段階的なプロセスを通じて決定できます。
まず、入力信号のDTFSを計算する必要があります。各 DTFS 項に対する出力応答は、システムの周波数応答を使用して計算されます。最後に、これらの結果を合計して、合計出力信号を求めます。
連続時間の周期信号には、複素指数フーリエ級数で表される円周周波と角周波数の周期があります。同様に、離散時間周期信号には基本的な角周波数があります。
DTFS 展開は有限項で構成されますが、連続時間のフーリエ級数は無限の項で構成されます。
デジタル信号処理では、DTFSは周期的なサンプルの調査、特定の周波数の特定、望ましくないノイズの効果的なフィルタリングに役立ちます。

離散時間フーリエ級数

離散時間フーリエ級数 (DTFS) は信号処理の基本概念であり、連続時間フーリエ級数の離散時間版として機能します。これにより、離散時間周期信号は周波数成分で表現および分析できます。連続時間フーリエ級数が積分を利用するのに対し、DTFS では信号が離散的であるため、係数の計算には和を用います。

周期 N_0 の離散時間周期信号 x[n] の場合、DTFS 係数 X[k] は次の式で計算されます。

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="height: 48px;">
ここで、k=0,1,2,…,N_0−1 です。これらの係数 X[k] は周波数領域で信号を表し、各周波数成分の振幅と位相を示します。

離散時間周期信号に対する線形時不変 (LTI) システムの応答を決定するには、体系的なアプローチに従います。

入力信号の DTFS を計算：入力信号 x[n] の DTFS 係数 X[k] を計算します。

各 DTFS 項の出力応答を計算：システムの周波数応答 H(e^jΩ) を使用して、各周波数成分の出力を決定します。出力 DTFS 係数 Y[k] は、


<img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="height: 36px;">

ここで、Ω =2πk/N

応答を合計：最後に、すべての DTFS 項の寄与を合計して、時間領域での総出力信号を取得します。

連続時間信号では、周期性は周期 T に基づいて定義され、これは円周周波数と角周波数に対応します。離散時間信号の場合、周期性は基本角周波数 Ω_k = 2πk/N_0 に関連付けられます。ここで、N_0 は離散信号の周期です。DTFS の展開は有限で、N 項で構成され、連続時間フーリエ級数の無限級数とは対照的です。

DTFS は、デジタル信号処理 (DSP)、特にサンプルデータから得られた周期信号の分析と操作において重要な役割を果たします。例えば、オーディオ信号内の特定の周波数の識別、特定の周波数成分の強調または抑制、不要なノイズの除去などのタスクに役立ちます。信号を周波数領域に変換することにより、DTFS は効率的な信号分析と処理を容易にし、通信、音響工学、制御システムなどのさまざまな分野で性能の向上を可能にします。

離散時間フーリエ級数は、連続時間周期信号のフーリエ級数展開に対応するものです。
拡張を計算する...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Fourier Series

フーリエ級数

KEY TERMS AND CONCEPTS

主要用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

HS-PS4-5

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HS-PS4-4

DTFS for Periodic Signal Analysis

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