On the contrary, it's most normal to consider the time domain function [1] as a complex valued function, even if it happens to take only real values. And the re

On the contrary, it's most normal to consider the time domain function [1] as a complex valued function, even if it happens to take only real values. And the reason for that is exactly for symmetry with its transform! The transform is certainly complex valued, thanks to the e^ix\xi factor (unless you take the sine and cosine transforms separately, but of course those are just the real and imaginary components). For example, you can see the first definition on Wikipedia [2] starts with a function f:R->C. Edit: Oh, by "symmetric" you meant the Fourier transform being symmetric about 0. I thought you meant the natural symmetry of the forms of f and \hat{f}. Still, it seems natural to me to treat the two functions similarly, including use of complex values. But I'm a (very) pure mathematician, at least in this area. [1] Or whatever function you want to take the transform of, it doesn't have to be the time domain and is in fact often spatial. This is probably why OP used the phrase "signal domain". [2] https://en.m.wikipedia.org/wiki/Fourier_transform https://en.m.wikipedia.org/wiki/Fourier_transform