Fourier-Transform-and-its-applications-in-mathematical-modeling
Discrete Fourier Transform
Let
$$
k, n \in \{0, 1, \ldots, N-1\}, \quad N \in \mathbb{N}, \quad j = \sqrt{-1}
$$
DFT is defined as
$$
X_{}^{f}(k) = \sum_{n=0}^{N-1} x(n)W_{N}^{kn}
$$
where N is amount of data samples as well as DFT coefficients
$$
\left(a_n\right)_{n \in \{0, 1, \ldots, N-1\}}
$$
is a uniformly sampled sequence as to exactly the appropriate interval T called later samplingInterval, N is called later samplingRate
$$
\begin{align*}
W_{N}^{} &= \exp\left(\frac{-j2\pi}{N}\right) \\
\end{align*}
$$
is N-th root of unity
$$
\begin{align*}
W_{N}^{kn} &= \exp\left(\frac{-j2\pi}{N}kn\right) \\
\end{align*}
$$
and
$$
X_{}^{f}(k)
$$
is the k-th DFT coefficient