Evaluate the DFT of the vectors $(1,1,0,0)$ and $(1,1,1,0,0)$

Question

Evaluate the DFT of the vectors $(1,1,0,0)$ and $(1,1,1,0,0)$

I toke Fourier Analysis last semester but I do not remember how to approach the problem. Can someone give me a re-fresher?

Answer 1

$\textbf{Definition:}$ The DFT of a vector is $X_k \equiv \sum_{n=0}^{N-1}x_ne^\frac{-2\pi kni}{N}$ where $k \in \mathbb{Z}$

The DFT for vector $(1,1,0,0)$ is $$\sum_{n=0}^{3}x_ne^\frac{-2\pi kni}{4}=e^0+e^\frac{-2\pi ki}{4}=1+e^\frac{-\pi ki}{2}$$

The DFT for vector $(1,1,1,0,0)$ is $$\sum_{n=0}^{4}x_ne^\frac{-2\pi kni}{5}=e^0+e^\frac{-2\pi ki}{5}+e^\frac{-4\pi ki}{5}=1++e^\frac{-2\pi ki}{5}+e^\frac{-4\pi ki}{5}$$