I am running into a problem with the phase shifted impulse function and its fourier transform:
\begin{array}{l} X( \omega ) =\int _{-\infty }^{\infty } x( t) e^{-iwt} dt\\ \\ x( t) =\delta ( t-T_{0}) \\ \\ X( \omega ) =\int _{-\infty }^{\infty } \delta ( t-T_{0}) e^{-iwt} dt\\ \end{array}
I then use the sifting property of the impulse function to get:
\begin{array}{l} X( \omega ) =e^{-iwT_{0}}\\ \end{array}
How do I then interpret this in the frequency domain ? I read this as a cosine over the frequency spectrum. But it should be a impulse in the frequency domain?
