Fourier transform of a real signal

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SUMMARY

The Fourier transform of a real signal produces a symmetric spectrum, which is crucial for signal processing. When applying the Fast Fourier Transform (FFT) to a real signal and discarding half of the spectrum, the result is a complex signal, denoted as rc(t). This complex signal retains all the necessary information to reconstruct the original real signal. The reconstruction process is detailed in the referenced paper, which provides a comprehensive explanation of the underlying principles.

PREREQUISITES
  • Understanding of Fourier Transform principles
  • Familiarity with Fast Fourier Transform (FFT) algorithms
  • Knowledge of complex signal representation
  • Basic signal processing concepts
NEXT STEPS
  • Read the paper on Analytic Signals for in-depth understanding
  • Explore the mathematical foundations of the Fourier Transform
  • Investigate the properties of complex signals in signal processing
  • Learn about signal reconstruction techniques from frequency domain data
USEFUL FOR

Signal processing engineers, researchers in communications, and students studying Fourier analysis will benefit from this discussion.

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