I apologize if this question has been asked before. I have looked and have not found a clear explanation.
When doing the discrete Fourier transform (e.g. fft in MATLAB) for a vector of discrete time signals $f(t)$ with length $N$ and sample rate $\Delta t$, the result is a vector of complex Fourier coefficients, $F(\omega )$ with length $N$. However, I get confused when trying to plot these Fourier coefficients with respect to frequency.
My understanding is that each frequency is defined as $$f_n=\frac{n}{N\Delta t},\;\;\;\;\;\;n=-N/2\;\;...\;\; -2,-1,0,1,2\;\;...\;\;N/2.$$
I have two problems:
1) This results in a frequency set with $N+1$ elements, but $F(\omega )$ only has $N$ elements. On which side of zero do we lose an element? Wouldn't this wreck the symmetry and thereby violate conservation of energy?
2) Negative frequencies seem non-physical to me. So, if I chop the Fourier transform such that $f\geq0$, then I lose half the spectrum. Where does this extra energy go? What does it mean to have a "negative" frequency?
Any help or explanations are greatly appreciated.