So I am taking the Fast Fourier Transform of the following function:
$$ x[n] = \displaystyle\sum\limits_{i=0}^{5} A_{i} \cos\left(\frac{\omega_{i}}{\omega_{s}} n + \phi_{i}\right) $$
Where the sampling rate $\omega_{s} = 2 \pi 8000$, the frequencies $\omega$, the phases $\phi$ and the amplitudes $A$ are: \begin{align} \omega &= 2 \pi \{500, 522, 610, 675, 746, 825 \}\\ \phi &= \{0,0,0,0,0,0\}\\ A &= \{1, 1, -1, -1, 1, -1\} \end{align}
The length of the signal is $N = 2048$ samples. No window is applied before taking the FFT.
The resulting FFT spectrum should have only a real component since the imaginary component should be zero. However, when taking the FFT and plotting the spectrum I get the following:
Why is the imaginary part not zero? It seems to be a discontinuity. However, I am not sure why it exists. Why does the imaginary part of the spectrum look like this?