Discrete Fourier Transform - Some Discrete Fourier Transform Pairs

Some Discrete Fourier Transform Pairs

Some DFT pairs
Note
Shift theorem
Real DFT
\left\{ \begin{matrix} N & \mbox{if } a = e^{i 2 \pi k/N} \\ \frac{1-a^N}{1-a \, e^{-i 2 \pi k/N} } & \mbox{otherwise} \end{matrix} \right. from the geometric progression formula
from the binomial theorem
\left\{ \begin{matrix} \frac{1}{W} & \mbox{if } 2n < W \mbox{ or } 2(N-n) < W \\ 0 & \mbox{otherwise} \end{matrix} \right. \left\{ \begin{matrix} 1 & \mbox{if } k = 0 \\ \frac{\sin\left(\frac{\pi W k}{N}\right)} {W \sin\left(\frac{\pi k}{N}\right)} & \mbox{otherwise} \end{matrix} \right. is a rectangular window function of W points centered on n=0, where W is an odd integer, and is a sinc-like function (specifically, is a Dirichlet kernel)
Discretization and periodic summation of the scaled Gaussian functions for . Since either or is larger than one and thus warrants fast convergence of one of the two series, for large you may choose to compute the frequency spectrum and convert to the time domain using the discrete Fourier transform.

Read more about this topic:  Discrete Fourier Transform

Famous quotes containing the words discrete and/or transform:

    “One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
    —Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)

    “He had said that everything possessed
    The power to transform itself, or else,
    And what meant more, to be transformed.”
    —Wallace Stevens (1879–1955)