Name: fourier transform signals in time and frequency
Uploaded: 2024-09-26T13:17:03.000Z
Duration: 1 min 7 s
Channel: JoVE

Среди ключевых элементов преобразования Фурье функция sinc уникальна тем, что она равна 1, когда ее аргумент равен нулю и демонстрирует четную симметрию.
В частотной области преобразование Фурье прямоугольного импульса преобразуется в эту функцию sinc, которая проявляет симметрию с пиком в начале координат и постепенно уменьшающимися лепестками по обе стороны.
Экспоненциальный сигнал, являющийся комплексной функцией, представляет собой синусоидальные колебания на определенной частоте. При воздействии преобразования Фурье его частотный состав становится одиночным импульсом, расположенным на определенной частоте. 
Это указывает на то, что идеальная прямоугольная волна включает в себя бесконечное количество гармонических частот, представленных лепестками функции sinc.
Дельта-функция равна нулю везде, кроме нуля, где она бесконечна. Преобразование Фурье дельта-функции является константой, означающей, что дельта-функция содержит все частоты с одинаковой величиной.

Основные сигналы преобразования Фурье

Преобразование Фурье является основным математическим инструментом в обработке сигналов, позволяющим преобразовывать сигналы временного домена в их представления в частотном домене. Среди многочисленных элементов в этом домене определенные функции, такие как функция sinc, дельта-функция и экспоненциальные сигналы, имеют важное значение из-за их уникальных свойств и последствий их применений.

Функция sinc, определяемая как sinc(x) = sin(πx)/(πx), особенно примечательна своей симметрией и поведением в нуле. Она достигает значения единицы, когда ее аргумент равен нулю, и демонстрирует чётную симметрию относительно оси y. Эта функция примечательна в частотном домене как преобразование Фурье прямоугольного импульса. Прямоугольный импульс, характеризующийся постоянной амплитудой в течение определенного интервала, преобразуется в функцию sinc. Результирующая функция sinc симметрична с выраженным пиком в начале координат, а ее лепестки уменьшаются по амплитуде по мере удаления от центра. Это преобразование показывает, что прямоугольный импульс во временном домене состоит из бесконечного ряда гармонических частот.

Дельта-функция, или дельта-функция Дирака, является еще одним критическим элементом в изучении преобразований Фурье. Она определяется как равная нулю везде, кроме нуля, где она бесконечно велика, так что ее интеграл по всей действительной оси равен единице. Преобразование Фурье дельта-функции дает постоянное значение на всех частотах, указывая на то, что дельта-функция охватывает все частоты с одинаковой величиной. Это свойство делает дельта-функцию важным инструментом для анализа и синтеза сигналов, поскольку она служит основой для построения других функций посредством свёртки.

Экспоненциальные сигналы, представленные комплекснозначными функциями вида e^jωt, являются основополагающими для описания синусоидальных колебаний на определенных частотах. Когда экспоненциальный сигнал подвергается преобразованию Фурье, результатом является один импульс на соответствующей частоте в частотном домене. Это преобразование подчеркивает чистое частотное содержимое экспоненциального сигнала, иллюстрируя, что он состоит из одного частотного компонента без каких-либо гармоник.

Среди ключевых элементов преобразования Фурье функция sinc уникальна тем, что она равна 1, когда ее аргумент равен нулю и демонстрирует четную симметрию.
В...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

The Fourier Transform

Преобразование Фурье

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 constant amplitude over a limited interval. When it is transformed, the result is a sinc function with a strong peak at the center. Its side lobes get smaller as they move away from the origin. This shows that a rectangular pulse in time is made of an infinite set of harmonic frequencies.

The delta function, also called the Dirac delta function, is another important signal. It is zero everywhere except at zero, where it is infinitely large in a way that makes its total area equal to one. Its Fourier transform is a constant across all frequencies. That means the delta function contains equal magnitude at every frequency and is useful for signal analysis and synthesis through convolution.

Exponential signals are also fundamental in Fourier analysis. They are complex-valued functions written in the form e^jωt and describe sinusoidal oscillations at a specific frequency. When an exponential signal is transformed, it becomes a single impulse at that same frequency in the frequency domain. This shows that the signal has one pure frequency component and no harmonics.

Fourier Transform Signals in Time and Frequency

fourier-transform-signals-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

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

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 the Fourier Transform connects time and frequency through sinc, delta, and exponential signals. Built for grades 9–12 science students.

Fourier Transform Signals 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

Fourier Transform Signals in Time and Frequency

Description

Transcript

Tags