I am teaching myself about the Discrete Fourier Transform. For this reason, I am going over an example, and I would like to interpret the results, for which I have a hard time.
Consider the sequence $(x_n)_{n=0}^7$ where $x_n = \sin(\frac{\pi n}{4})$ on $[0, 2\pi]$ with $\frac{\pi}{4}$ intervals. After applying DFT to this sequence $x_n \xrightarrow[]{\text{DFT}} X_k$ we get the following values:
$$X_k = \begin{cases} 4i \quad \text{when $k = 7$} \\ -4i \quad \text{when $k = 2$} \\ 0 \quad \text{otherwise} \end{cases}$$ I have plotted the sequence as a function of time and the resulting amplitudes:
Questions:
By applying the DFT, we get values that could be completely imaginary. For this reason, we compute its norm to get a real value. What does the plain value of DFT tell us? Why doesn't the definition of DFT include taking the norm? (If you notice, I did not label the $y$-axis, since I was not sure what they are called - Are they called amplitudes for both?)
DFT gives a sequence that is a function of frequency. What is this function? Is there an explicit formula?
The result of the plot is somewhat unexpected for me. There are two peaks at $\frac{\pi}{4}$ and $\frac{7\pi}{4}$, but $x_1 = x_3$ and $x_5 = x_7$. So why are $X_3 $ and $X_5$ just $0$?
Similarly, the maximum and minimum value in the sequence $x_2$ and $x_6$ are being mapped to $0$. Why does not DFT preserve maximum and minimum of the original sequence?
My goal by asking these questions is to build intuition, so if I see a sequence, I can create a reasonable picture of its DFT in my head. If you can provide explanations that can help with building intuition, please go for it.