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:

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “One definition of man is “an intelligence served by organs.””
    —Ralph Waldo Emerson (1803–1882)

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)