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:

    Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?
    Richard Eberhart (b. 1904)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)