I'm trying to simulate the behavior of my FFT spectrum analyzer, I am using Mathematica, but my questions are more conceptual than technical implementation of code...I hope.
If we consider a simple cosine wave transient signal $y(t) = A \cos(2 \pi \nu_0 t)$, where $A = 1$ and $\nu_{0} = 24$ $\rm{Hz}$. This signal is sampled at some time interval $dt$ and this gives me 2048 points (2048 because of my device), I want to resolve this data to some frequency resolution, $\Delta f$, and from the Fourier limit: $t_{sample} = 1/\Delta{f}$ so my sample rate, $S_{r} = 1/dt$ is adjusted accordingly.
I then take the absolute value of the FFT of these 2048 points and reconstruct the frequency component using $\Delta f$ (in this example $\Delta f = 0.0625$ $\rm{Hz}$, this value is because of my device and the features I want to resolve) and the index of the resultant FFT, and get this:
I understand that the symmetric appearance of an FFT comes from the real and imaginary components having the same response to signals -- the absolute value of their responses will be the same. But the phase of the real and imaginary parts is shifted by $\pi / 4$. This gives us our mirrored appearance.
FIRST QUESTION: Usually I would just throw the right hand frequencies away and multiply the amplitude of what is left by $2$. Is this the correct approach or should I reverse the order of one half of these 2048 points and then add them together (intuitively I would say no, because both halves should be identical so multiplying by $2$ should suffice)? What is the best method of reconstructing the frequency component/axis?
SECOND QUESTION: This is something I just don't understand. If I increase the frequency of my transient signal $\nu_{0}$ then the two mirrored peaks move closer to each other, converging in the center. This makes sense. If I keep increasing $\nu_{0}$ of the transient arbitrarily (past $124$ $\rm{Hz}$ which corresponds to the limit of frequencies) however I still see a peak, in fact I see this oscillating behavior of the peaks just bouncing back and forth as I further increase the frequency -- Why?! Is this natural or because of the way I define my frequencies.