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

Question

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

You can find the formula here, Eq. 1. For vector 1 just plug in $x_0=1,x_1=1,x_2=0,x_3 = 0$ and $N=4$ and you got yourself a DFT! — Winther, Nov 05 '14 at 15:31

score 1 · Accepted Answer · answered Nov 05 '14 at 16:42

$\textbf{Definition:}$ 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 $$\sum_{n=0}^{4}x_ne^\frac{-2\pi kni}{5}=e^0+e^\frac{-2\pi ki}{5}+e^\frac{-4\pi ki}{5}=1++e^\frac{-2\pi ki}{5}+e^\frac{-4\pi ki}{5}$$

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

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