# How can I represent this expression as a Fourier Transform?

1. Dec 22, 2015

### Xyius

Hello,

I hope I am posting this in the correct forum topic. It really is more of a "mathy" type of question, but I am posting it here because it deals with radar, and this type of math is used a lot in radar. To the mods, feel free to move it to a more suitable location if desired.

I have come across an expression in my research that I believe can be represented in terms of a Fourier transform. My research is a theoretical investigation on the effectiveness of a quantum radar to see a target in relation to a normal radar. A quantum radar is a new (theoretical only at the moment) concept to use quantum states of photons to detect targets at a distance. The expression is,

$$\sigma = \gamma \left| \sum_{n=1}^{N} e^{i k \Delta R_n} \right|^2$$

Where $\gamma$ is a constant, $k$ is the wave number and $\Delta R_n = \sqrt{(x-x_n)^2+(y-y_n)^2+(z-z_n)^2}$, which is the distance from the receiver (monostatic) to each atom. Basically what this expression is, is the summation of the photon wave function from each atom in the object. So $N$ represents the total number of atoms. This summation can absolutely be changed to be a continuous integral to obtain the desired result easier. However for simulation purposes for arbitrary objects, the summation form is what is required.

This problem seems to be identical to topics done in radar where an object is represented as a summation of isotropic point scatterers. Would anyone have a recommendation on where would be a good source to read up on this problem, or perhaps show me how to do the transform?

2. Dec 22, 2015