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Doesn't the simple one sided representation give enough information about the signal?

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First, Fourier transform involves complex numbers. Complex functions is the spce of choice for this. Hence, a Fourier transformation of a real or a complex signal is often complex. Hence generally, the magnitude of it does not "give enough information" about a signal. Only amplitude misses information on the sign or the phase of the complex spectrum.

Second, there is no a priori reason why a Fourier transformation should be symmetric. However, when the signal is real, there is some symmetry in the spectrum (depending on what flavor of Fourier transform one is using).

So, knowing that the original signal is real, displaying half of an amplitude spectrum (say one-sided for simplicity) may convey as much information as the two-sided version. But it is really not enough by itself.

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