Fractional Fourier Transform - Definition

Definition

If the continuous Fourier transform of a function is denoted by, then, and in general ; similarly, denotes the n-th power of the inverse transform of . The FRFT further extends this definition to handle non-integer powers for any real, denoted by and having the properties:

when is an integer, and:

More specifically, is given by the equation:

\mathcal{F}_\alpha(f)(\omega) = \sqrt{\frac{1-i\cot(\alpha)}{2\pi}} e^{i \cot(\alpha) \omega^2/2} \int_{-\infty}^\infty e^{-i\csc(\alpha) \omega t + i \cot(\alpha) t^2/2} f(t)\, dt.

Note that, for, this becomes precisely the definition of the continuous Fourier transform, and for it is the definition of the inverse continuous Fourier transform.

If is an integer multiple of, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit, and leads to a Dirac delta function in the integrand. More easily, since, must be simply or for an even or odd multiple of, respectively.

Read more about this topic:  Fractional Fourier Transform

Famous quotes containing the word definition:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)