Name: fourier coefficients and harmonic terms
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 17 s
Channel: JoVE

フーリエ級数は、周期関数を無限級数の正弦波調和に分解する数学的手法です。
三角フーリエ級数は、正弦関数と余弦関数を使用して、特定の周期を持つ周期関数を表します。これらの関数の周波数は、元の関数の周期に反比例します。
定数項と、各サイン関数とコサイン関数の元の関数への個々の寄与は、フーリエ係数を使用して理解されます。これらの係数の計算には、1 つの期間にわたる積分が含まれます。
フーリエ級数が周期関数を表すためには、ディリクレ条件が満たされる必要があります。
最初の条件は、関数が 1 つの期間にわたって有限の積分を持つことを要求します。
2 番目の条件は、関数の変動が任意の範囲内に制限される必要があることを示します。
最後に、関数は特定の範囲内に有限個の不連続性を持つ必要があり、そのどれも無限にすることはできません。
フーリエ級数は関数の近似に使用できますが、ディリクレの条件を満たさない関数を完全に回復することはできません。

三角フーリエ級数

フーリエ級数は、周期関数を無限の正弦波や余弦波の調和成分に分解する基礎的な数学的手法です。この方法により、複雑な周期信号を単純な正弦関数と余弦関数の和として表現できるようになり、信号処理、音響、電気工学など、さまざまな分野での分析と解釈が容易になります。

三角フーリエ級数は、正弦関数と余弦関数を使用して、定義された周期 T を持つ周期関数を具体的に表現します。関数 x(t) の三角フーリエ級数の一般的な形式は次のとおりです。

<img alt="Equation1" src="/files/ftp_upload/16059/16059_PT%20EQUATION_1.svg" style="height: 40px;">

ここで、a_0 は 1 周期における関数の平均値を表し、a_n と b_n は、それぞれ各余弦関数と正弦関数の寄与を定量化するフーリエ係数です。これらの係数は、1 周期 T にわたる積分によって決定されます:

<img alt="Equation2" src="/files/ftp_upload/16059/16059_PT%20EQUATION_2.svg" style="height: 46px;">
<img alt="Equation3" src="/files/ftp_upload/16059/16059_PT%20EQUATION_3.svg" style="height: 48px;">
<img alt="Equation4" src="/files/ftp_upload/16059/16059_PT%20EQUATION_4.svg" style="height: 48px;">

これらの積分は、正弦波成分から元の関数を再構築する正確な係数を計算するために不可欠です。

フーリエ級数を用いて周期関数を正確に表すには、ディリクレ条件を満たす必要があります。最初の条件は、関数が 1 周期にわたって有限積分を持つ必要があることを規定し、関数全体が制限されることを保証します。2つ目の条件は、関数が任意の範囲内で限られた数の最大値と最小値を持つことを要求し、関数が過度の振動を示さないことを保証します。3つ目の条件は、関数が有限数の不連続性を持つべきであることで、そのいずれも無限ではありません。これらの条件により、フーリエ級数が元の関数に適切に収束することが保証されます。

実際のアプリケーションでは、これらの条件が厳密に満たされていなくても、フーリエ級数表現を構築できる場合がよくあります。このような表現は、精度は劣るかもしれませんが、周期関数の分析と合成に役立つ近似値を提供できます。この柔軟性は、さまざまな数学および工学アプリケーションにおけるフーリエ級数が強力かつ有用であることを強調しています。

フーリエ級数は、周期関数を無限級数の正弦波調和に分解する数学的手法です。
三角関数...

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

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 to the final result.

These coefficients are found by integration over one full period T. The integrals give the exact values needed to rebuild the original function from its sinusoidal pieces. This is why Fourier coefficients are central to the method.

To match a periodic function well, the Dirichlet conditions should be satisfied. The function must have a finite integral over one period, a limited number of maxima and minima in any range, and a finite number of discontinuities that are not infinite. These conditions help the Fourier series converge to the original function.

Even when the Dirichlet conditions are not perfectly met, a Fourier series can often still be written. The result may be less accurate, but it can still give a useful approximation of the periodic function. This makes Fourier series a practical tool for analyzing and synthesizing repeating signals in mathematics and engineering.

Fourier Coefficients and Harmonic Terms

trigonometric fourier series

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

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...

key terms and concepts

Learn how Fourier series use sine and cosine terms, Fourier coefficients, and Dirichlet conditions to model periodic functions and signals.

Fourier Coefficients and Harmonic Terms | 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

Fourier Coefficients and Harmonic Terms

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