Fourier transform of a real signal

[FrogPad]

Taking a Fourier-transform of a real signal, gives me a spectrum that has symmetry.

If I take the FFT of a real signal, then throw away half of the spectrum, and then do an inverse transform I get a complex-signal.

I go from r(t) to rc(t) where rc(t) is a complex-signal.

Now this complex-signal supposedly contains all the information to reconstruct the original real-signal. My question is, how?

 

[FrogPad]

If anyone is interested, this question is answered in the following paper:
http://classes.engr.oregonstate.edu/eecs/winter2009/ece464/AnalyticSignal_Sept1999_SPTrans.pdf