# Is sampling a Fourier transformed signal and fourier transforming a sampled signal the same?

I'm having a hard time understanding an assignment that states:

Draw the complex spectrum of the sampled signal $f(t)$ (periodic and continuous). Do this, by first calculating the Fourier transformation and sample it afterwards by multiplying with the impulse train.

The way I understand it, I need to calculate $$\begin{equation*} \mathcal{F}\left[f(t) \cdot \sum_{k=-\infty}^{\infty}\delta(t-kT)\right] \end{equation*}$$ however the second sentence suggests this is the same as $$\mathcal{F}[f(t)] \cdot \sum_{k=-\infty}^{\infty}\delta(t-kT)$$ However, Wikipedia says this is not the case and instead suggests $$\begin{equation*}\mathcal{F}\left[f(t) \cdot \sum_{k=-\infty}^{\infty}\delta(t-kT)\right] = \mathcal{F}[f(t)] *\left[\frac{1}{T} \cdot \sum_{k=-\infty}^{\infty}\delta\left(t-\frac{k}{T}\right)\right] \end{equation*}$$ are maybe both formulas correct? and how do I draw a complex spectrum? is it 3D?

Why it is not completely incorrect here is that your signal is given to be continuous and periodic. A periodic signal has a sampled spectrum due to the reason described above; technically, it is said to have a Fourier series. So when you compute $F(f)$, either you should have in time domain those complex sinusoids with Fourier series coefficients, or you should have in frequency domain a frequency domain impulses with those coefficients. This might have caused confusion to whoever made the assignment.