Homework 2: Fourier Transform

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  1. I need to give you an important hint for Part 3. In Part 1, you plotted $|F_r|$, which is a real value (magnitude). In reality, however, all of your $F_r$ are complex numbers. In part 3 it is crucially important that you use the full complex number in the inverse DFT summation. If you look at your spreadsheet, you’ll most likely see that the full-period sine wave gives you $F_1 = 0 – j/2$ (check the real = cosine part and the imaginary = sine part).

    Moreover, you also have $F_{-1} = 0 + j/2$, which you plotted as $|F_{N-1}|=0.5$. Side note: The Fourier transform of a real and even function is real and even. The F.T. of a real and odd function (e.g., sine) is imaginary and odd.

    Only when you use the full complex values of $F_r$ will the summation in the inverse DFT resolve to a real-valued sine function of amplitude 1. Otherwise, the $j$ will not cancel out. Also note that you’ll make use of the odd symmetry, $\sin(-x) = -\sin(x)$ at one point.

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