Name: dtft rules for differentiation and energy
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 24 s
Channel: JoVE

Рассмотрим пару преобразований Фурье с дискретным временем (DTFT), дифференцируем обе стороны относительно Ω, а затем умножим на j.
Правая часть становится преобразованием Фурье nx[n], иллюстрирующим свойство частотного дифференцирования.
 Примените дискретную свертку времени к парам DTFT, измените порядок суммирования справа и примените свойство time-shifting.
Подстановка уравнений дает свойство свертки времени.
Умножьте сигналы в двувременной области и поменяйте местами порядок интегрирования и суммирования с правой стороны.
Это соответствует периодической свертке в частотной области, масштабированной обратно пропорционально периоду, демонстрируя частотную свертку.
Свойство накопления фокусируется на суммировании сигнала дискретного времени.
DTFT накопленного сигнала имеет исходный DTFT, масштабированный на экспоненциальный множитель плюс член дельта-функции, обозначающий периодические компоненты с кратностью 2π.
Энергия сигнала Ex равна сумме квадратов величин во временной области. Это равно интегралу от квадратов величин его DTFT, демонстрирующего соотношение Парсеваля.

Свойства ДВПФ ii

Properties of DTFT II

При изучении обработки дискретных сигналов понимание свойств дискретного временного преобразования Фурье (ДВПФ) имеет решающее значение для анализа и обработки сигналов в частотной области. Несколько свойств, включая частотную дифференциацию, свертку, накопление и соотношение Парсеваля, предлагают мощные инструменты для анализа сигналов.

Свойство частотного дифференцирования иллюстрируется рассмотрением пары ДВПФ и дифференцированием обеих сторон по ω. Умножая на j (мнимая единица), правая часть преобразуется в преобразование Фурье nx[n]. Математически, если X(e^jω) — это ДВПФ x[n], то j(d/dω(X(e^jω)) — это ДВПФ nx[n]. Это свойство полезно для нахождения частотных характеристик, связанных с наклоном спектра сигнала.

Применяя дискретную временную свёртку к парам ДВПФ, мы наблюдаем еще одно важное свойство. Изменяя порядок суммирования в правой части и применяя свойство сдвига во времени, мы приходим к свойству свёртки во времени. Оно гласит, что свёртка двух сигналов во временном домене соответствует умножению их ДВПФ в частотном домене. И наоборот, умножение двух сигналов во временном домене приводит к периодической свёртке их ДВПФ в частотном домене, масштабированной обратной периода.

Свойство накопления фокусируется на суммировании дискретного по времени сигнала с течением времени. Дискретное по времени преобразование Фурье (ДВПФ) этого накопленного сигнала связано с ДВПФ исходного сигнала, но изменено экспоненциальным масштабным коэффициентом. Кроме того, есть термин, который включает дельта-функцию, которая вводит периодические компоненты с интервалом 2π в частотной области. Это свойство подчеркивает, как накопление во временном домене влияет на частотное представление, приводя к периодическим особенностям.

Соотношение Парсеваля является ключевым результатом, который связывает энергию сигнала во временном домене с его представлением в частотном домене. В частности, полная энергия сигнала x[n], которая является суммой квадратов величин во временном домене, равна интегралу квадратов величин его ДВПФ. Это соотношение является основополагающим при анализе мощности и энергии сигнала в обеих доменах.

Эти свойства в совокупности повышают способность анализировать, проектировать и понимать дискретные по времени системы, делая их незаменимыми в цифровой обработке сигналов.

Свойства ДВПФ II

Рассмотрим пару преобразований Фурье с дискретным временем (DTFT), дифференцируем обе стороны относительно Ω, а затем умножим на j.
Правая сторона ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

The Fourier Transform

Преобразование Фурье

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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. When two signals are convolved in time, their DTFTs multiply in frequency. The reverse is also true in a periodic form: multiplying two signals in time leads to a periodic convolution of their DTFTs, scaled by the inverse of the period.

The accumulation property looks at the DTFT of a signal after its values are summed over time. The transformed result includes an exponential scaling factor, along with a delta-function term. That delta term adds periodic components at intervals of 2π in the frequency domain.

Parseval's relation connects signal energy in both domains. The total energy of x[n], found by summing the squared magnitudes in time, equals the integral of the squared magnitude of its DTFT. Together, these DTFT properties make it easier to analyze, design, and understand discrete-time systems in digital signal processing.

DTFT Rules for Differentiation and Energy

dtft-rules-for-differentiation-and-energy

continuous -time fourier transform

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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...

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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).

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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. <br...

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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

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.

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key terms and concepts

Learn DTFT properties for frequency differentiation, convolution, accumulation, and Parseval's relation. See how time-domain changes appear in frequency.

DTFT Rules for Differentiation and Energy | 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

DTFT Rules for Differentiation and Energy

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