Discrete Fourier transform of $(1,1,1,1)$ I am asked to determine the Fourier transform of $(1,1,1,1)$. In the solution I found this: ![solution]( I don't get how is he transitioning from the $\omega$'s to $-i, i, -...--prophetes.ai

Discrete Fourier transform of $(1,1,1,1)$ I am asked to determine the Fourier transform of $(1,1,1,1)$. In the solution I found this: ![solution]( I don't get how is he transitioning from the $\omega$'s to $-i, i, -1, 1$ etc... How to break it down, so that it's more understandable? Which middle-step (which could explain the path to solution better) is the professor not writing? Any hints?

Since we do a 4 sample DFT, we will be looking at powers of the complex 4th root of unity.

$w$ is the complex fourth root of unity $w^4 = 1$, we can pick $w = i$ or $w=-i$

It seems this aufgabe chooses the negative one. Now substitute $-i$ everywhere and you will get the matrix.

$$\cases{(-i)^0 = 1\\\\(-i)^1 = -i\\\\(-i)^2 = -1\\\ (-i)^3=i}$$

And then it will be periodic, $w^5=w$ et.c.