# 2. Mathematical Definitions¶

## 2.1. Coordinate system¶

Fig. 2.1 shows the coordinate system that is used in the following chapters. A vector \(\x\) can be described by its position \((x,y,z)\) in space or by its length, azimuth angle \(\phi \in [0,2\pi[\), and elevation \(\theta \in \left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). The azimuth is measured counterclockwise and elevation is positive for positive \(z\)-values.

## 2.2. Fourier transformation¶

Let \(s\) be an absolute integrable function, \(t,\w\) real numbers, then the temporal Fourier transform is defined after [Bra00] as

(2.1)¶\[S(\w) = \mathcal{F}\left\{s(t)\right\} =
\int^{\infty}_{-\infty} s(t) \e{-\i\w t} \d t.\]

In the same way the inverse temporal Fourier transform is defined as

(2.2)¶\[s(t) = \mathcal{F}^{-1}\left\{S(\w)\right\} =
\frac{1}{2\pi} \int^{\infty}_{-\infty} S(\w)
\e{\i\w t} \d\w.\]