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

Теорема Парсеваля — это принцип, используемый при обработке сигналов для вычисления энергии сигнала. Он позволяет вычислять одно и то же значение энергии с использованием временных или частотных данных, демонстрируя сохранение энергии между этими двумя областями.
Если мы рассмотрим мощность сигнала, то его энергию можно рассчитать. Как правило, в качестве основы для этого расчета используется резистор на 1 Ом, где мощность эквивалентна квадрату напряжения или тока.
Теорема показывает, что энергия может быть определена в частотной области, доказывая, что преобразование Фурье сохраняет энергию.
Теорема предполагает, что полная энергия, подаваемая на резистор сопротивлением 1 Ом, равна общей площади под квадратом сигнала или единице более двух пи, умноженной на общую площадь под величиной квадрата преобразования Фурье.
Теорема связывает энергии во временной и частотной областях, подразумевая, что квадрат величины преобразования Фурье отражает плотность энергии сигнала.
Энергия сигнала может быть вычислена непосредственно во временной области или косвенно из преобразования Фурье в частотной области.

Теорема Парсеваля для преобразования Фурье

Parseval's Theorem for Fourier transform

Теорема Парсеваля — это фундаментальный принцип обработки сигналов, который позволяет вычислять энергию сигнала как во временном, так и в частотном домене. Эта теорема имеет решающее значение для демонстрации сохранения энергии между этими двумя доменами, гарантируя, что вычисленное значение энергии остается постоянным независимо от домена анализа.

Чтобы понять теорему Парсеваля, необходимо сначала понять, как обычно вычисляется энергия сигнала. При рассмотрении мощности сигнала его энергия может быть вычислена на основе стандартного значения резистора, обычно установленного на 1 Ом. Мощность в этом контексте эквивалентна квадрату напряжения или тока сигнала. Такой подход упрощает вычисление энергии, позволяя легко связать мощность сигнала с его энергией. Теорема Парсеваля расширяет концепцию вычисления энергии на частотный домен, предоставляя мощный инструмент для анализа сигнала. Теорема утверждает, что полная энергия сигнала может быть определена либо путем интегрирования квадрата сигнала во временном домене, либо путем интегрирования квадрата его преобразования Фурье в частотном домене.

Значение теоремы важно, поскольку она связывает временной и частотный домены, показывая, что энергия, присутствующая в сигнале, может быть точно отражена в любом из доменов. Квадратная величина преобразования Фурье, часто называемая плотностью энергии сигнала, предоставляет альтернативный метод вычисления энергии сигнала косвенно из его частотных компонентов.

В практических приложениях теорема Парсеваля обеспечивает сохранение энергии в задачах обработки сигналов, таких как фильтрация, модуляция и спектральный анализ. Она подчеркивает неотъемлемую связь между представлением сигнала во временном домене и его характеристиками в частотном домене, что делает ее незаменимой при анализе и обработке сигналов. Используя теорему Парсеваля, инженеры и ученые могут уверенно переходить между анализами во временном и частотном доменах, обеспечивая точность и согласованность расчетов энергии в различных доменах.

Теорема Парсеваля — это принцип, используемый при обработке сигналов для вычисления энергии сигнала. Это позволяет рассчитать одну и ту же энергетическую величину ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

The Fourier Transform

Преобразование Фурье

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 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.

electrical engineering

Signal Energy in Time and Frequency

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