Can someone explain negative frequencies when doing the Fourier transform?

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.

• how you get confused ? – Cardinal Oct 29 '15 at 20:27
• I listed two questions at the bottom of my post. What is the meaning of negative frequencies? And how do you plot an amplitude spectrum with only positive frequencies while maintaining energy conservation? – Darcy Oct 29 '15 at 23:24
• 1) Negative frequency suggest conter clock wise spin or whatever ... 2) You need only a multiplication of $exp(-j2\pi f_0t)$ to shift the frequency while maintaining same energy. Do you deal with PSD or this sort of things ? – Cardinal Oct 30 '15 at 6:37
• See What is the physical significance of negative frequencies?... you haven't found this question because you were looking on a wrong site. – user147263 Oct 31 '15 at 4:28