Calculate $($by hand$)$ the $8-$point FFT of the following sampled sinusoids:

$($i$)$ Two period sine; samples $=[0,1,0,-1,0,1,0,-1]$;

$($ii$)$ Three$-$period cosine; samples $=[1,12,0,12,1,12,0,12]$;

$($iii$)$ Two period sine with $/4$ phase offset; samples $=[12,12,12,12,12,12,12,12]$;

$($a$)$ For each case, you should calculate the FFT of the sampled data to find the complex coefficients.

$($b$)$ Explain why the coefficients had the structure you found $($are they pure real, pure imaginary, a combi$-$

nation? Why would such a result occur?$)$

$($c$)$ Bonus: For the following sampled data, calculate the $8-$point FFT$,$ and then express the samples as a

decomposition with the coefficients multiplying the appropriate columns of the FFT matrix. Sum up the

results to show that all of the correct cancellations occur and you get back the original sample values.

samples $=[1,0,1,2,1,0,1,2$