FFT sampling: why is Nyquist frequency enough for inverse transform? I am trying to understand an aspect of the Fast Fourier Transform (FFT) that is not making sense to me, but I'm struggling to find the flaw in my reasoning.
Suppose I model a continuous signal at a sampling rate $f$. According to Nyquist, the maximum frequency I will be able to "recover" from this signal via a FFT is $f/2$. How come when I apply the inverse transform, I can recover the original sample in real space with a sampling rate $f$, and not a "worsened" version of the original sample that only has frequencies up to $f/2$.
In other words: the intuition I am getting is that when applying an FFT, one looses half the frequency information. However, this cannot be right, otherwise one could not invert the transform.
Could someone help me realize where is the flaw here?
 A: You start with a signal having frequencies in the interval $[0,f]$.  You take the FFT and get a spectrum with frequencies in $[-f/2,f/2]$.  Then you apply the inverse FFT and recover the signal with frequencies in the interval $[0,f]$.  Why do you think half the information vanishes in the middle step?
You might ask, how can signal components varying as fast as $f$ be represented in frequencies from $[-f/2, f/2]$?  The answer is the wagon wheel effect and aliasing: all the spectrum originally in the interval $[f/2,f]$ is aliased into the $[-f/2,0]$ components.  Then the inverse transform recovers the aliases back to their original frequencies.
This is not, however, magic.  Time-domain frequencies in the interval $[f, 2f]$ also alias into the spectral interval $[-f/2,f/2]$ and your inverse transform will reconstruct them as if they were in the interval $[0,f]$.  But it is exceptionally important to realize -- the discrete-time samples for aliased frequencies are the same.  There is nothing in the time domain signal that distinguishes aliased frequencies.  So the infinite family of aliases are all sent to one spectral bin and then reconstructed as if they had the smaller (positive) frequency (although the reconstructed sample values are the same as if the frequency was correctly recovered).
To convince yourself of some of this, actually sample signals of sinusoids having frequencies $-f/2$, $3f/2$, and $5f/2$.  Then look at other sequences of aliases to firmly convince yourself that the sampled signals are identical.  So there is no way for the inverse transform to know which member of an alias family to reconstruct -- and it doesn't have to.  In the sample, there is no difference among the aliased frequencies.
So what is this all about?  Suppose the original source was actually a pair of sinusoids at aliased frequencies.  Apply the FFT and the inverse FFT and the result is at the lowest positive alias of both -- it is no longer a sum of two sinusoids, it's just one.
