Now that we have a cursory understanding of the number i, we can return to that particularly tricky part of the Discrete Fourier Transform which involves both the imaginary number i and the transcendental number e,

$$e^{-\varphi \mathrm{i}}$$

In the same way that we can choose to represent points on a two-dimensional plane using either polar or Cartesian coordinates, we can also choose to represent points on the complex plane using polar or Cartesian coordinates. When we use e, we’re using the polar form of representation. The polar form is somehow more beautiful, but the Cartesian form is easier for us to work with. The relationship between the Cartesian and polar form is given by Euler’s Formula, which states that,

$$ e^{\varphi \mathrm{i}} = cos(\varphi) + sin(\varphi) i$$

Euler’s formula establishes the relationship between e and the unit-circle on the complex plane. It tells us that e raised to any imaginary number will produce a point on the unit circle. As we already know, points on the unit circle can always be defined in terms of sine and cosine. So, writing eφi is a shorthand for writing out the Cartesian form which uses sine and cosine. The visualization below shows the value of eφi for various values of φ. Note how similar this looks to a phasor. In fact, this construction can be referred to as a complex phasor.

Let’s recall the definition of the Discrete Fourier Transform and see how we might simplify it given our awareness of Euler’s Formula. Recall that the classical notation is written as,
$$ \mathrm{DFT}[k] = \sum_{n=0}^{N-1} \mathrm{x}[n] \cdot e^{-\varphi\mathrm{i}} $$
When we invoke Euler's formula to perform a simple substitution,
$$ e^{-\varphi\mathrm{i} } = cos(\varphi) - sin(\varphi)i$$
We arrive at a formula which is considerably less esoteric, and imminently more understandable given our familiarity with sine and cosine. We now see that the DFT is somehow all about performing dot products on an input signal with sine and cosine waves which vary in frequency.
$$ \mathrm{DFT}[k] = \sum_{n=0}^{N-1} \mathrm{x}[n] \cdot (cos(\varphi) - sin(\varphi)i) \\ $$
The remaining piece of the puzzle is understanding how exactly the dot product helps us construct phasors which represent our signal. In the next three sections we’ll explore this topic in detail.