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The fourier transform of arbitrary real data can (usually will) result in complex data. If the real data represents samples in time, then the complex FT data represent frequencies with the magnitudes representing the amplitude and the phase angles representing the phase.

If I perform some arbitrary manipulation on the complex FT data prior to applying an inverse fourier transform, the result of the IFT can also be complex. My question is, how do I interpret the imaginary part of the IFT result? My guess is that a complex IFT result is non-physical, and that this must imply that my original arbitrary manipulation was also therefore non-physical.

Is this correct? If so, are there equations which describe whether such a manipulation would be non-physical.

Richard M
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1 Answers1

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There are manipulations where it is desired that the inverse corresponds to a complex-valued signal, such as the computation of the analytic signal. If, however, the resulting signal after manipulation is desired to be real-valued then the conjugate symmetry condition of the Fourier transform of a real-valued signal must remain satisfied:

$$X_k=X^*_{N-k}$$

where $X_k$ is the DFT of a real-valued sequence $x_n$, and $N$ is the length of the DFT.

Note that due to numerical inaccuracies in the computation of the inverse DFT (FFT), there is always some residual imaginary part in the order of the machine precision.

Matt L.
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