Name: parseval's theorem in signal power
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 18 s
Channel: JoVE

パーセバルの定理では、関数が周期的である場合、1 つの周期にわたる信号の平均パワーは、すべての複素フーリエ係数の二乗振幅の合計に等しくなると述べています。
パーセバルの定理を検証するには、関数が標準形式の複素フーリエ級数を持つと仮定します。これを代入してさらに解くと、定理の証明が得られます。
興味深いことに、パーセバルの定理は、三角関数フーリエ級数のフーリエ係数で表すこともできます。
オーディオ処理では、Parsevalの定理を使用して、元の音波のエネルギーとその圧縮バージョンを比較します。
この定理の工学的解釈は、実用的な洞察を提供します。関数が電流や電圧などの電気信号を表す場合、関数の 2 乗は 1 オームの抵抗の瞬時電力を表します。
この定理は、1つの周期に抵抗器で消費されるエネルギーをフーリエ級数に関連付け、三角関数のフーリエ級数と振幅位相のフーリエ級数の2つの異なる式を提供します。

パーセバルの定理

パーセバルの定理は、信号処理と高調波解析における基本的な概念です。周期関数の場合、1 周期の信号の平均電力は、そのすべての複素フーリエ係数の 2 乗の和に等しいと主張します。マルク＝アントワーヌ・パーセバルにちなんで名付けられたこの定理は、信号のエネルギー分布を分析するための強力なツールを提供します。

興味深いことに、パーセバルの定理は、関数を正弦関数と余弦関数で表現するフーリエ級数の三角形式にも当てはまります。ここで、フーリエ係数は三角級数の係数と関連付けることができるため、この定理を別の形式で適用できます。

パーセバルの定理を検証するには、まず、複素フーリエ級数表現を持つ関数 x(t) を考えます。

<img alt="Equation1" src="/files/ftp_upload/16063/16063_PT%20EQUATION_1.svg" style="height: 30px;">

ここで、c_n は複素フーリエ係数、ω_0 は基本角周波数です。定理は次のように述べます。


<img alt="Equation2" src="/files/ftp_upload/16063/16063_PT%20EQUATION_2.svg" style="height: 36px;">

ここで、T は関数の周期です。フーリエ級数を左辺に代入して解くと、等式が確認され、定理が証明されます。

パーセバルの定理は、特にオーディオ処理における実用的なアプリケーシで重要です。これにより、元の音波に含まれるエネルギーと圧縮バージョンのエネルギーを比較できます。この比較は、圧縮プロセスでエネルギーが失われすぎてオーディオ信号の品質が大幅に低下しないようにするために不可欠です。


工学的観点から、パーセバルの定理は貴重な洞察を提供します。たとえば、問題の関数が電流や電圧などの電気信号を表す場合、この関数の 2 乗は 1 オームの抵抗器で消費される瞬間電力を表します。したがって、この定理は、1 周期にわたって抵抗器で消費されるエネルギーを信号のフーリエ級数表現に結び付けます。この関係は、三角関数のフーリエ級数を使用する形式と、フーリエ級数の振幅位相形式を使用する形式の 2 つの異なる形式で表現されます。したがって、パーセバルの定理は強力な分析ツールとして機能するだけでなく、理論的な概念と実際の工学的アプリケーションを結び付ける役割も果たします。

パーセバルの定理は、関数が周期的である場合、1 つの期間にわたる信号の平均電力は二乗の合計に等しいと述べています ...

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

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 series coefficients, so the same idea can be applied in this alternate form.

To verify Parseval's theorem, start with a function x(t) written as a complex Fourier series. In this form, Cn are the complex Fourier coefficients, and ω0 is the fundamental angular frequency. The theorem states the equality between average power over a period T and the sum of the squared coefficient magnitudes. Substituting the Fourier series into the left-hand side and simplifying confirms the result.

Parseval's theorem is important in audio processing. It can compare the energy in an original sound wave with the energy in a compressed version. This helps show whether compression has kept most of the signal energy and avoided major loss in sound quality.

The theorem is also useful in engineering with current or voltage signals. In that case, the square of the function represents the instantaneous power dissipated in a 1-ohm resistor. Parseval's theorem then connects the energy dissipated over one period to the Fourier series form of the signal. It can be written using either the trigonometric Fourier series or the amplitude-phase form, which makes it a practical bridge between signal theory and real engineering problems.

Parseval's Theorem in Signal Power

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 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 Parseval's theorem connects a periodic signal’s average power to Fourier coefficients, with uses in audio compression and electrical signals.

Parseval's Theorem in Signal Power | 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

Parseval's Theorem in Signal Power

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