サインイン

The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.

The sinc function, defined as sinc(x) = sin(πx)/(πx), is particularly notable for its symmetry and behavior at zero. It achieves a value of one when its argument is zero and exhibits even symmetry about the y-axis. This function emerges prominently in the frequency domain as the Fourier transform of a rectangular pulse. A rectangular pulse, characterized by its constant amplitude over a specific interval, transforms into a sinc function. The resulting sinc function is symmetric with a pronounced peak at the origin, and its lobes diminish in amplitude as they move away from the center. This transformation shows that a rectangular pulse in the time domain is composed of an infinite series of harmonic frequencies.

The delta function, or Dirac delta function, is another critical element in the study of Fourier transforms. It is defined to be zero everywhere except at zero, where it is infinitely large such that its integral over the entire real line is equal to one. The Fourier transform of a delta function yields a constant value across all frequencies, indicating that the delta function encompasses all frequencies with equal magnitude. This property makes the delta function an essential tool for analyzing and synthesizing signals, as it serves as the foundation for constructing other functions through convolution.

Exponential signals, represented by complex-valued functions of the form ejωt, are fundamental in describing sinusoidal oscillations at specific frequencies. When an exponential signal undergoes a Fourier transform, the result is a single impulse at the corresponding frequency in the frequency domain. This transformation highlights the pure frequency content of the exponential signal, illustrating that it consists of a single frequency component without any harmonics.

タグ

Fourier TransformSignal ProcessingTime domain SignalsFrequency domain RepresentationsSinc FunctionDelta FunctionDirac Delta FunctionRectangular PulseHarmonic FrequenciesExponential SignalsSinusoidal OscillationsImpulse ResponseConvolutionFrequency Content

章から 17:

article

Now Playing

17.2 : Basic signals of Fourier Transform

The Fourier Transform

423 閲覧数

article

17.1 : Continuous -time Fourier Transform

The Fourier Transform

218 閲覧数

article

17.3 : Properties of Fourier Transform I

The Fourier Transform

131 閲覧数

article

17.4 : Properties of Fourier Transform II

The Fourier Transform

125 閲覧数

article

17.5 : Parseval's Theorem for Fourier transform

The Fourier Transform

622 閲覧数

article

17.6 : Discrete-time Fourier transform

The Fourier Transform

198 閲覧数

article

17.7 : Properties of DTFT I

The Fourier Transform

301 閲覧数

article

17.8 : Properties of DTFT II

The Fourier Transform

133 閲覧数

article

17.9 : Discrete Fourier Transform

The Fourier Transform

168 閲覧数

article

17.10 : Fast Fourier Transform

The Fourier Transform

180 閲覧数

JoVE Logo

個人情報保護方針

利用規約

一般データ保護規則

研究

教育

JoVEについて

Copyright © 2023 MyJoVE Corporation. All rights reserved