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Say I take a real valued function in the time domain, take the Fourier transform to get its frequency domain representation, do some work on it in the frequency domain, which leads to imaginary values, and then take the inverse Fourier transform to get a new time domain signal. What does it mean if the values of the time domain signal are imaginary? I understand that imaginary values in the frequency domain allow for the representation of phase of each frequency (https://stackoverflow.com/questions/10304532/why-does-fft-produce-complex-numbers-instead-of-real-numbers). But what do imaginary numbers mean in the time domain?

I can tell you what my particular application is, but I believe that this is a general question that others might come across in the future.

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Imaginary values in the time domain simply mean that you actually have two signals, the real-part and the imaginary part. If you start out with a real-valued signal, take the Fourier transform, do some processing in the frequency domain, then transform back, and you end up with a complex-valued signal, then 9 out of 10 times you've done something wrong (or, if the imaginary part is very small, it is numerical noise due to round-off errors). In other words, in most cases the frequency domain processing should not make your originally real-valued time domain signal complex-valued, i.e. it should retain the conjugate symmetry of the signal's Fourier transform.

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