Name: signal energy in time and frequency
Uploaded: 2024-09-26T13:17:03.000Z
Duration: 1 min 15 s
Channel: JoVE

パーセバルの定理は、信号のエネルギーを計算するために信号処理で使用される原理です。これにより、時間データまたは周波数データを使用して同じエネルギー値を計算でき、これら 2 つのドメイン間のエネルギー保存が実証されます。
信号の電力を考えると、そのエネルギーを計算できます。通常、この計算のベースとして1Ωの抵抗が使用され、電力は電圧または電流の2乗に相当します。
この定理は、エネルギーが周波数領域で決定できることを示しており、フーリエ変換がエネルギーを保存することを証明しています。
この定理は、1 Ω の抵抗に供給される合計エネルギーが、信号の 2 乗の下の総面積に等しいか、フーリエ変換の 2 乗の大きさの下の総面積の 2 倍以上の 1 に等しいことを示唆しています。
この定理は時間領域と周波数領域のエネルギーを結び付け、フーリエ変換の二乗の大きさが信号のエネルギー密度を反映していることを示唆しています。
信号のエネルギーは、時間領域で直接計算することも、周波数領域のフーリエ変換から間接的に計算することもできます。

フーリエ変換におけるパーセバルの定理

Parseval's Theorem for Fourier transform

パーセバルの定理は、信号処理の基本原理であり、時間領域または周波数領域のいずれかで信号のエネルギーを計算できます。この定理は、これら 2 つの領域間のエネルギー保存を証明する上で極めて重要であり、計算されたエネルギー値が分析領域に関係なく一定であることを保証します。

パーセバルの定理を理解するには、まず信号エネルギーが通常どのように計算されるかを理解することが重要です。信号の電力を考慮する場合、そのエネルギーは、通常 1 オームに設定された標準抵抗値に基づいて計算できます。この文脈での電力は、信号の電圧または電流の 2 乗に相当します。このアプローチにより、エネルギー計算が簡素化され、信号の電力とエネルギーの関係がわかりやすくなります。パーセバルの定理は、エネルギー計算の概念を周波数領域に拡張し、信号分析のための強力なツールを提供します。この定理は、信号の総エネルギーは、時間領域で信号の二乗を積分するか、周波数領域でフーリエ変換の二乗を​​積分することによって決定できると述べています。

この定理は、時間領域と周波数領域を橋渡しし、信号に存在するエネルギーがどちらの領域でも正確に反映されることを示すという点で重要です。フーリエ変換の二乗の大きさは、信号のエネルギー密度と呼ばれることが多く、信号の周波数成分から間接的に信号のエネルギーを計算する代替方法を提供します。

実際のアプリケーションでは、パーセバルの定理は、フィルタリング、変調、スペクトル分析などの信号処理タスクでエネルギーの保存を保証します。この定理は、信号の時間領域表現と周波数領域特性間の固有の関係を強調し、信号の分析と操作に不可欠なものとなっています。パーセバルの定理を活用することで、エンジニアや科学者は時間領域と周波数領域の分析を自信を持って切り替えることができ、異なる領域にわたるエネルギー計算の精度と一貫性を確保できます。

パーセバルの定理は、信号のエネルギーを計算するために信号処理で使用される原理です。それは同じエネルギー値の計算を可能にします...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

The Fourier Transform

フーリエ変換

KEY TERMS AND CONCEPTS

主要用語と概念

signal energy in time and frequency

Parseval's theorem links signal energy in the time domain and the frequency domain. In signal processing, it shows that the same energy can be calculated in either view. This makes it an important idea for comparing time-based and frequency-based analysis.

To find signal energy, the standard approach uses a resistor value of 1 ohm. In that case, power equals the square of the signal's voltage or current. This gives a simple way to relate power to energy before moving into Fourier analysis.

Parseval's theorem extends that idea to the frequency domain. It states that a signal's total energy can be found by integrating the square of the signal in time or by integrating the square of its Fourier transform in frequency. The squared magnitude of the Fourier transform is often called the energy density, since it shows how the signal's energy is distributed across frequency components.

This relationship is useful in many signal processing tasks. Filtering, modulation, and spectral analysis all depend on moving between time and frequency representations. Parseval's theorem helps confirm that energy is conserved during that shift, so calculations stay consistent in either domain.

Signal Energy in Time and Frequency

continuous -time fourier transform

From Fourier Series to Fourier Transform

The Fourier transform extends Fourier series ideas to nonperiodic functions. A Fourier series breaks a periodic signal into a sum of sines and cosines. When the period becomes infinite, that discrete sum changes into a continuous integral. A pulse-train waveform shows this change clearly. With a finite period, the repeating rectangular pulses can be represented by a Fourier series. As the period grows without bound, the waveform becomes a single isolated pulse, and the description shifts to...

basic signals of fourier transform

Fourier Transform Signals in Time and Frequency

The Fourier Transform links time-domain signals to frequency-domain signals. It is a key tool in signal processing. It helps show how a waveform can be described by frequency parts instead of only by time changes. The sinc function is one of the basic signals in this topic. It is written as sinc(x) = sin(πx)/(πx). It equals 1 when x is 0, and it has even symmetry about the y-axis. In the frequency domain, it appears as the Fourier transform of a rectangular pulse. A rectangular pulse has a...

properties of fourier transform i

Fourier Transform in Radio Broadcasting

Fourier Transform properties help radio broadcasting manage signals in the time and frequency domains. These properties are useful for sending multiple audio channels at once, changing clip speed, adding broadcast delays, adjusting audio frequencies, and demodulating signals. They make it easier to transmit and receive audio clearly. When broadcasters combine several audio signals, the Fourier Transform simplifies the process. If f(t) and g(t) have Fourier Transforms F(ω) and G(ω), then the...

properties of fourier transform ii

Fourier Transform Rules in Signal Analysis

The Fourier Transform helps analyze signals by linking the time domain and the frequency domain. Its properties show how a signal changes when it is shifted, differentiated, or combined with another signal. These rules are important in signal processing, radio broadcasting, and audio work. The frequency shifting property shows that moving a signal in frequency causes a phase change in time. If x(t) has a Fourier Transform X(f), then multiplying x(t) by e^j2πf0t gives a transform of X(f−f0).

parseval's theorem for fourier transform

Parseval's theorem links signal energy in the time domain and the frequency domain. In signal processing, it shows that the same energy can be calculated in either view. This makes it an important idea for comparing time-based and frequency-based analysis. To find signal energy, the standard approach uses a resistor value of 1 ohm. In that case, power equals the square of the signal's voltage or current. This gives a simple way to relate power to energy before moving into Fourier analysis. <br...

discrete-time fourier transform

DTFT and Frequency Analysis of Digital Signals

The Discrete-Time Fourier Transform (DTFT) shows how a discrete-time signal can be analyzed in the frequency domain. It takes a signal from the time domain and reveals its frequency components and spectral features. In the DTFT, the integral used for continuous-time signals is replaced by a summation to match the discrete form of the signal. The DTFT spectrum X(Ω) is periodic with a period of 2π. This periodicity means that X(Ω) can be written as a Fourier series, which helps with analysis and...

properties of dtft i

DTFT Signal Properties and Frequency Effects

Discrete-Time Fourier Transforms (DTFTs) help describe discrete-time signals in the frequency domain. The main DTFT properties include linearity, time shifting, frequency shifting, time reversal, conjugation, and time scaling. These rules make it easier to analyze and change signals in signal processing. Linearity is the most basic property. If two discrete-time signals are multiplied by constants a and b and then added together, the DTFT of the result is the weighted sum of the two individual...

properties of dtft ii

DTFT Rules for Differentiation and Energy

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain. For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time. Convolution brings out another key DTFT rule.

discrete fourier transform

DFT for Finding Signal Frequencies

The Discrete Fourier Transform (DFT) turns a finite set of time-domain samples into frequency components. It is a key tool in signal processing because it evaluates a discrete signal at evenly spaced frequency intervals. The result shows the magnitude and phase of each complex sinusoid in the signal. The DFT also follows several important properties. It is linear, so the DFT of a sum of sequences equals the sum of their individual DFTs. When a sequence shifts in time, its DFT changes by a...

fast fourier transform

Fast Fourier Transform for Signal Analysis

The Fast Fourier Transform (FFT) is a computational algorithm that makes the Discrete Fourier Transform (DFT) faster to calculate. It works by splitting the calculation into smaller parts. This reduces the amount of work needed to move data between the time domain and the frequency domain. A direct N-point DFT needs N² complex multiplications. The FFT lowers that cost to about (N/2)log₂N multiplications. That difference becomes more important as N gets larger, because the FFT cuts the number...

key terms and concepts

Learn how Parseval's theorem connects signal energy in the time and frequency domains, using power, Fourier transforms, and energy conservation.

Signal Energy in Time and Frequency | 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

Signal Energy in Time and Frequency

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