Complex Conjugate of the comb function

[sahand_n9]

Homework Statement

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:

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

Homework 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.

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:
$I(x) = comb^2(2x) \ast e^{i\Phi(x)} \ast e^{-i\Phi(x)}$

Now, isn't $e^{i\Phi(x)} \ast e^{-i\Phi(x)}$ just the auttcorrelation of $e^{i\Phi(x)}$?

 

[strangerep]

Homework Statement

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:

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

Homework 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.

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

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:
$I(x) = comb^2(2x) \ast e^{i\Phi(x)} \ast e^{-i\Phi(x)}$

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...