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How does the shape of a frequency domain function (Fourier Transformed time domain function) change when a time shift is applied in the time domain?

Note that what you plotted is not the frequency spectrum $S_c(j\omega) =|S_c(j\omega)|e^{j\phi (\omega)}$, but the magnitude spectrum $|S_c(j\omega)|$ The shape of the magnitude spectrum is unchanged: ...
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