Fractional Fourier Transform

    The Quadratic Function in the Fractional Fourier Transform Kernel

    Demonstration 1: Counter-Rotating Asymptotes of the Hyperbolic Quadradic Form

    The Fractional Fourier Transform

    kernel comprises an exponential function with argument

    that is a quadratic form hyperbolic with respect to x and y:sw

    This can be shown to be a hyperbola whose asymptotes counter-rotate with the power a as it increases through the internal [0, 4), repeating periodically. This animation shows the counter-rotating asymptotes.

    As discussed in Chapter 5, the equations for these asymptotes is given by

    Clockwise Rotation:
    (thinner black line)

    Counterclockwise Rotation:
    (thicker grey line)

     

    as adapted from equation (5.3) of Chapter 5. These trigonometric functions, and hence the slope of asymptote lines, counter-rotate as described in Chapter 5 just after equations (5.6a) and (5.6b). These slopes are also reciprocals of one another.

    Note that at even-integer values of a, the quadratic form degenerates from a hyper bolu into lines just as the counter-rotating asymptotes of the hyperbola converge. At these points of hyperbolic degeneration and asymptotic line convergence, the degenerate lines and asymptotes are given by the equations:

    x-y = 0 for

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

    These two cases may be consolidate in general as

    x + (-1)j y = 0 for with .

    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)
    (Reflection operator)

    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 described 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