I need to the find the inverse continuous time Fourier transform for unitary angular frequency of the following signal:
$e^{a\omega^2 - b\omega + c}$ where $a$ and $b$ and $c$ are real numbers and I want to find the corresponding time-domain signal.
I was able to get the $e^{a\omega^2 + c}$ term transformed, but I was unable to find a transform for $e^{- b\omega}$ that didn't result in a distribution.
Without doing math and looking at the symmetry, I want to say that the $e^{- b\omega}$ term does a complex time shift, but now my mind hurts. Can somebody look up the identity?
--edit-- This spectrum doesn't need to be symmetric, or causal, so expecting some solutions to have imaginary time could be correct. Can somebody comment on this?