I am asked to determine the Fourier transform of $(1,1,1,1)$.
In the solution I found this:
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.