Fractional Fourier Transform

    The Quadratic Function in the Fractional Fourier Transform Kernel

    Demonstration 3: Combined Contour Plot and Counter-Rotating Asymptotes

    The Fractional Fourier Transform

     

     

    kernel comprises an exponential function with argument that is hyperbolic with respect to x and y:

     

     

    This can be shown to be a hyperbola whose asymptotes counter-rotate with the power α as it increases through the internal [0, 4) and repeating periodically. This demonstration shows an animation of the counter-rotating asymptotes superimposed upon the contour plot of hyperbolic quadratic form as the power α cycles though its periodic range.

    The colors depict relative values of the kernel exponential's quadratic-form argument in each frame as a function of x and y. A color wheel relative color scheme is used to indicate the relative values within each frame of the animation:

    • Orange and yellow hues depict the positive-most values within the given frame
    • Violet and blue hues depict the negative-most values within the given frame
    • Red hues depict (positive or negative) near-infinite to infinite values

    Note that at even-integer values of α , the hyperbola itself degenerates into a line:

    x-y = 0 for ,

     

    (that is, where α even-integer multiples of 2) and

    x+y = 0 for

     

    (that is, where α odd-integer multiples of 2).

    At either of these even-integer values of α, the fractional Fourier transform kernel becomes one of two types of delta function (in the sense of distributions within Schwartz Space extension of )

     
    (Identity operator)
    		 α = 4k
    (Reflection operator)
    		 α = 4k+2.

     

    Note that the counter-rotating asymptotes converge at angles 45-degrees from the x and y axes. This reflects the fact that the natural cannonical variables for the fractional Fourier transform are

     

    as discussed in Chapter 1, and a fact that is also reflected by the arguments of the delta functions (6.5a) and (6.5b) above.

    As discussed in Chapter 5, their are a number of special values worth noting (with caveats of variable changes and limit processes). These are tabulated in Table 1, below

    Example Value of

    General Value of

    Effect of the operator

    Structure of the Kernal

    Graphical Behavior

    0

    4k

    Identity Operator

    Delta function , cross-product term
    vanishes

    1/2

    4k + 1/2

    Similar to Bargmann Transform

    Symmetric in x and y

    1

    4k+1

    Fourier Transform

    Quadratic terms vanish, cross-product term is negative

    2

    4k+2

    Reflection Operator

    Delta function, cross-product term vanishes

    3

    4k+3

    Inverse Fourier

    Quadratic terms vanish, cross-product term is positive

    7/2

    4k+ 7/2

    Similar to Inverse Bargmann Transform

    Symmetric in x and (-y)

      REFERENCES

      [1] ... roots and zeros of hyperbolic behavior