How to obtain the CTFT of a shifted unit impulse in terms of f in hertz (from the formula in terms of $ \omega $)

$ x(t)=\delta (t-t_0) $

$ \mathcal{X}(\omega )=e^{-iwt_0} $


To obtain X(f), use the substitution

$ \omega= 2 \pi f $.

More specifically

$ \begin{align} X(f)&=\mathcal{X}(2\pi f)=e^{-i2\pi ft_0} \end{align} $


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