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

フーリエ変換の主要な要素の中で、sinc 関数は、引数が 0 のときに 1 に等しく、均一な対称性を示すという点で独特です。
周波数領域では、矩形パルスのフーリエ変換がこのsinc関数に変換され、原点にピークがあり、両側にローブが徐々に小さくなる対称性を示します。
指数信号は複素数値関数であり、特定の周波数での正弦波振動を表します。フーリエ変換を受けると、その周波数成分は特定の周波数に位置する単一のインパルスになります。 
これは、完全な方形波には、sinc関数のローブで表される無限の高調波周波数が組み込まれていることを示しています。
デルタ関数は、無限大であるゼロを除いて、どこでもゼロです。デルタ関数のフーリエ変換は定数であり、デルタ関数に同じ大きさのすべての周波数が含まれていることを示します。

フーリエ変換の基本信号

フーリエ変換は信号処理における極めて重要な数学的ツールであり、時間領域信号を周波数領域表現に変換できます。この領域内の多数の要素の中で、sinc関数、デルタ関数、指数信号などの特定の関数は、その固有の特性と意味により、非常に重要な意味を持ちます。

sinc関数は、sinc(x) = sin(πx)/(πx) として定義され、その対称性とゼロでの動作が特に顕著です。引数がゼロのときに値が 1 になり、y 軸を中心に対称性さえ示します。この関数は、矩形パルスのフーリエ変換として周波数領域で顕著に現れます。特定の間隔にわたって一定の振幅を特徴とする矩形パルスは、sinc関数に変換されます。結果として得られるsinc関数は対称で、原点に顕著なピークがあり、中心から離れるにつれてローブの振幅が減少します。この変換は、時間領域における矩形パルスが、無限の高調波周波数の連続で構成されていることを示しています。

デルタ関数、またはディラックのデルタ関数は、フーリエ変換の研究におけるもう 1 つの重要な要素です。デルタ関数は、ゼロ以外のすべての場所でゼロであると定義されます。ゼロでは、デルタ関数は無限に大きくなり、実数線全体にわたる積分が 1 に等しくなります。デルタ関数のフーリエ変換では、すべての周波数にわたって一定の値が生成され、デルタ関数がすべての周波数を等しい大きさで包含していることを示します。この特性により、デルタ関数は、畳み込みによって他の関数を構築するための基礎として機能するため、信号の分析と合成に不可欠なツールになります。

指数信号は、e^jωt 形式で表される複素数値関数で、特定の周波数での正弦波振動を記述する上で基本となります。指数信号をフーリエ変換すると、周波数領域内で対応する周波数で単一のインパルスが得られます。この変換により、指数信号の純粋な周波数内容が強調され、高調波のない単一の周波数成分で構成されていることが示されます。

フーリエ変換の主要な要素の中で、sinc 関数は、引数が 0 のときに 1 に等しく、均一な対称性を示すという点で独特です。
で。。。

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

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

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electrical engineering

Fourier Transform Signals in Time and Frequency

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