Some people say elegance is the only beauty that never fades, and when we invoke Eulerโs formula to perform a simple substitution on the classical notation of the Discrete Fourier Transform (DFT) it is considerably less esoteric, and imminently more understandable given a familiarity with sine and cosine. DFT then becomes a simple - and simply elegant - exercise to perform dot products on an input signal with sine and cosine waves which vary in frequency.
\[\mathrm{DFT}[j] = \sum_{k=0}^{n-1} \mathrm{w}[k] \cdot e^{-\mathrm{i}x}\]
\[e^{-\mathrm{i}x } = cos(x) - i\, sin(x)\] \[\mathrm{DFT}[j] = \sum_{k=0}^{n-1} \mathrm{w}[k] \cdot (cos(x) - i\,sin(x)) \\\]