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I understand what is meaning of Fourier transform over function that returns only real values — it can be thought of function taking time and returning real amplitude value of signal at that time, being transformed to function taking frequency and returning complex amplitude of given frequency, that denotes weighted sum of cosine and sine waves or phase-shifted cosine wave.

But what if original function returns not only real values, but also complex ones? I have troubles understanding the meaning of imaginary part when thinking about that function as signal representation. Is this somehow connected with polarization or other 2D waves? In case Fourier transform is applied to such "complex signal", what will be the meaning of transform result?

Can you please help me understand that?

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