DFT can be thought as a simple change of basis. In the time domain, every sample of a signal represents its amplitude at that very instant.

\[ y\left[n\right] = \sin{\left( \frac{2\pi k }{N} n\right)} \]

\[ k = 2, N = 64\]

Nevertheless, when applying a transformation such as the DFT, every sample means something different.

The DFT is computed as the scalar product between the signal and a orthogonal base. This base is as follows:

\[ \{\boldsymbol{w}^{(k)}\}_{\{k=0,1,…,N-1\}} \]

\[ w_n = e^{j \frac{2\pi}{N}nk}\]

If some of these base vectors are plotted with the imaginary part in the Y axe and the real part in the X axe:

Real and imaginary part of the DFT basis with N=64

DFT calculation

The DFT of a signal can be computed then, applying a change of basis over the desired signal.

\[ X\left[k\right] = \sum_{n=0}^{N-1} x\left[n\right] e^{-j\frac{2\pi}{N}nk}, \text{      } k = 0,1,…,N-1 \]

DFT of a \(\delta\left[n\right]\)

\[ x\left[n\right] = \delta[n] = \left\{\begin{matrix}
1 & n = 0\\
0 & otherwise
\end{matrix}\right. \] \[X\left[k\right] = \sum_{n=0}^{N-1}{\delta\left[n\right] e^{-j\frac{2\pi}{N}nk}}\]

\[ X\left[k\right] =  1 \cdot e^{-j\frac{2\pi}{N}\cdot 0 \cdot k} = 1 \]

DFT of a cosine

\[ x\left[n\right] = 4 \cos{\left(\frac{2\pi}{4}n\right)} =  4 \cos{\left(\frac{2\pi 4}{16}n\right)} = \frac{4}{2} \left[e^{ \frac{2\pi}{16} 4n} + e^{-j \frac{2\pi}{16} 4n} \right]\]

[To be finished]