I was reading about Gabor transform and how it allows to localize frequency with time. So it stands to reason that for this to be successful any shift in the function f, should result only in shift in the output spectogram, which is also what I observe when I perform this numerically on any online website.

i.e. $ \hat{f}(w,t) = STFT(f(t)) \implies \hat{f}(w, t-a) = STFT(f(t-a))$

However, solving gabor equations I'm getting an additional factor of $e^{-iwa}$ i.e.

$\hat{f}(w, t-a) = e^{-iwa}STFT(f(t-a))$$

I'm not sure if I'm making any mistakes or what happens to this factor



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