I am trying to understand how this works, specifically, what the DTFT of each step looks like in each step of the chain (for understanding). I am not looking for an answer like because the input signal has a frequency of $\pi/3$, it will be reconstructed correctly.
I plotted the DTFT of each step in MATLAB and simply don't understand the result. (DTFT of $x_d[n]$ and $x_r[n]$)
I would also like to know what the transfer functions of the downsampler and upsampler are.
The following pictures shows the DTFT of $x[n]$, $x_d[n]$ and $x_e[n]$ respectively
I don't know where the frequency components in the upsampled signal ($x_e[n]$) are coming from. I have noticed though, that it is what you would get if you convolved the original signal ($x[n]$) with an impulse train with period $2\pi/3$.
PS: Why is there no DTFT tag on this site?