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Complex Conjugate of the comb function

  1. Dec 24, 2013 #1
    1. The problem statement, all variables and given/known data
    This is not exactly a HW problem but related to my thesis work where I am deriving an expression for the intensity of light after a particular spatial filtering. I have:

    [itex]I(x) = \left[ comb(2x) \ast e^{i\Phi(x)} \right] \left[ comb^*(2x) \ast e^{-i\Phi(x)} \right] [/itex]
    Where [itex]comb(x) = \sum_{N=-\infty}^{\infty} \delta(x-N)[/itex], the symbol [itex]\ast[/itex] is the convolution operator, and [itex]\Phi(x)[/itex] is some arbitrary function of x.


    2. Relevant equations
    Is the complex conjugate of the comb function the same as itself? I have not been able to find anything on the complex conjugate of the Dirac delta function or the comb function. I cannot see why it would be different but I am not sure.


    3. The attempt at a solution
    My attempt at re-arranging the terms using commutative property of the convolution with the assumption that the complex conjugate of the comb function is itself yields:
    [itex]I(x) = comb^2(2x) \ast e^{i\Phi(x)} \ast e^{-i\Phi(x)} [/itex]

    Now, isn't [itex]e^{i\Phi(x)} \ast e^{-i\Phi(x)} [/itex] just the auttcorrelation of [itex]e^{i\Phi(x)} [/itex]?
     
  2. jcsd
  3. Dec 24, 2013 #2

    strangerep

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    Write out the general defining equation of the delta distribution. How does it act on an arbitrary complex-valued function ##f(z)## ?

    Careful! Your comb##^2## function would involve squares of the delta distribution, which is mathematically ill-defined.

    Consider
    $$\Big(f(x) \ast g(x)\Big)\Big(a(x) \ast b(x)\Big).$$Write out both products separately as integrals. Then try to take the product. Also think carefully about what a product is in momentum space, and vice versa...
     
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