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.

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