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DFT/DTFT problem

I had this problem on a previous midterm but I can't seem to get the correct answer. The answers are shown in red provided by the professor. I've attempted it two different ways: one by having my limit from from 0 to (17-1) then tried by setting limit to 0 to (13-1) by truncating values before zero. I don't believe I'm approaching this problem correctly and need some guidance.

Thank you

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In my opinion, the problem is not clearly specified because it remains a bit unclear which DFT length $N$ you're supposed to use. From the figure and from the given solution one can guess that $N=16$. So the time domain vector is

x = [0,1,2,3,0,1,2,3,0,1,2,3,0,1,2,3]

which results in the DFT

X = [24,0,0,0,-8+8i,0,0,0,-8,0,0,0,-8-8i,0,0,0]

corresponding to the given solution. Since you're asked to specify magnitude and phase, you should compute the magnitude and phase of $X[12]=-8-8i$.

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  • $\begingroup$ Why didn't you include the last 0 at n = 12? $\endgroup$ Dec 19, 2017 at 8:39
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    $\begingroup$ @ZairMahmood: That's the problem with this exercise that you don't know what the actual signal is. It turns out that if you do not include the last zero (i.e., if you see it just as a periodic repetition of the first zero), then you obtain the desired result. Again, the problem formulation is a bit unclear. $\endgroup$
    – Matt L.
    Dec 19, 2017 at 9:42

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