Name: dft for finding signal frequencies
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 15 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Discrete Fourier Transform

Рассмотрим датчик вибрации, который непрерывно захватывает данные в виде непрерывного сигнала, зависящего от времени.
Однако в действительности датчик может записывать только конечное количество вибраций в виде дискретных точек данных через определенные интервалы времени.
Здесь можно использовать дискретное преобразование Фурье, DFT, для анализа частотных составляющих вибраций.
DFT разлагает любой сигнал на сумму простых синусоидальных и косинусоидальных волн, для которых можно измерить частоту, амплитуду и фазу.
В амплитудном спектре DFT сигнал, полученный от датчика, может быть представлен в виде гистограммы, соответствующей четырем синусоидальным волнам.
Высота стержня после нормализации — это амплитуда соответствующего синусоидального сигнала во временной области.
Математически ДФТ представляется как конечное суммирование произведения сигнала во временной области с комплексной экспонентой, зависящей от частоты k.
Если X как функция k велика для определенного k, это указывает на то, что вибрационный сигнал имеет сильные частотные компоненты на этих частотах.

Дискретное преобразование Фурье

Дискретное преобразование Фурье (ДПФ) является фундаментальным инструментом в обработке сигналов, расширяющим дискретное преобразование Фурье за ​​счет оценки дискретных сигналов на равномерно распределенных частотных интервалах. Это преобразование преобразует конечную последовательность выборок временного домена в частотные компоненты, каждая из которых представляет собой комплексные синусоиды, упорядоченные по частоте. ДПФ переводит эти последовательности в частотный домен, эффективно указывая величину и фазу каждого частотного компонента, присутствующего в сигнале.

Одним из ключевых свойств ДПФ является его линейность. Это свойство подразумевает, что ДПФ суммы последовательностей равно сумме их индивидуальных ДПФ. Другим важным свойством является сдвиг по времени. Когда последовательность сдвигается во временном домене, ее ДПФ претерпевает соответствующий сдвиг фазы.

Сдвиг частоты во временном домене приводит к сдвигу индексов ДПФ. Если последовательность умножается на комплексную экспоненту, ее ДПФ соответствующим образом сдвигается в частотной области. Обращение времени, которое инвертирует последовательность во временном домене, влияет на симметрию ДПФ. Если последовательность обращена, компоненты ДПФ переупорядочиваются и сопряжены.

Свойство сопряжения гласит, что если последовательность сопряжена, компоненты ДПФ также сопряжены и переупорядочены. Теорема о свёртке особенно эффективна, поскольку она упрощает процесс свертки во временном домене до простого умножения в частотном домене.

Благодаря своей периодической природе ДПФ широко используется в приложениях обработки сигналов для перехода между временным и частотным домена 
ми. Эта периодичность возникает из-за неотъемлемого процесса дискретизации в ДПФ, что делает его универсальным инструментом для анализа и обработки сигналов. Способность упрощать сложные операции и обеспечивать четкое представление о частотных компонентах сигнала подчеркивает важность ДПФ в различных задачах обработки сигналов.

Рассмотрим датчик вибрации, который непрерывно захватывает данные в виде непрерывного сигнала, зависящего от времени.
Однако в реальности датчик может только...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

The Fourier Transform

Преобразование Фурье

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 phase shift. If the sequence is shifted in frequency in the time domain, the DFT indices shift as well.

Other DFT rules describe how different changes affect the frequency domain. Multiplying a sequence by a complex exponential shifts its DFT in frequency. Time reversal flips a sequence in time and changes the symmetry of the DFT, while conjugation produces conjugated and reordered DFT components. The convolution theorem is especially useful because it turns convolution in the time domain into multiplication in the frequency domain.

Because the DFT is periodic, it is widely used to move between time and frequency views of a signal. That periodicity comes from the sampling process built into the transform. These features make the DFT useful for analyzing and manipulating signals in many signal-processing tasks.

DFT for Finding Signal Frequencies

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

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...

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

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 the Discrete Fourier Transform converts sampled signals into frequency components and how shifting, reversal, and convolution affect the result.

DFT for Finding Signal Frequencies | 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

HS-PS4-5

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HS-PS4-1

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HS-PS4-3

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HS-PS4-4

DFT for Finding Signal Frequencies

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