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

Question

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?

Answer 1

If I number your equations (1), (2) and (3) starting from the first, then your Eq (1) is correct as it represents a conceptual model of the ideal sampling process: multiplying the continuous signal with an impulse train. Consequently, Eq (3) is also correct since multiplication in time domain is convolution in frequency domain (recall that the Fourier transform of an impulse train in time domain is also an impulse train in frequency domain with a different scaling factor). Convolving the two is what causes the spectrum to repeat at multiples of the sample rate.

Your Eq (2) however is not entirely correct. Fourier transforming the signal first and sampling it later implies sampling in frequency domain. Interestingly, this should create 'aliases' in time domain, i.e., repetition of the same waveform at multiples of 'sample rate'.

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.

Update: A complex spectrum can be drawn in 3D with frequency, I and Q dimensions, but the most common practice (and that reveals the most information) is to draw two separate graphs: one for magnitude response against frequency and the other phase response against the frequency.