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Determine the 8-point DFT of the sequence x(n)={1,1,1,1,1,1,0,0}
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Solution:

X(k)=N1n=0x(n)ej2πkn/Nk=0,1,N1

For N=8

X(k)=7n=0x(n)ejπkn/4k=0,1,2..N1

For k=0

$ \begin{aligned} X(0) & =\sum_{n=0}^7 x(n) \\\\ \mathrm{X}(0) & =\mathrm{x}(0)+\mathrm{x}(1)+\mathrm{x}(2)+\mathrm{x}(3)+\mathrm{x}(4)+\mathrm{x}(5)+\mathrm{x}(6)+\mathrm{x}(7) \\\\ & =1+1+1+1+1+1+0+0 \\\\ & =6 …

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