Fourier transform of a time discrete signal

Question

I would like some help to better understand the Fourier transform of a discrete time signal. My doubts are:

1. The sampling of a signal can be seen as $x_s(t)=x(t) \cdot \sum_{k=-\infty}^{+\infty} \delta(t-kT_s)$ so its Fourier transform is $X_s(f)= f_s \cdot \sum_{k=-\infty}^{+\infty} X(f-k f_s)$ where $f_s$ is the sampling rate. If $x(t)$ is aperiodic, then the result is pretty straightforward; what if $x(t)$ is a periodized or a periodic signal?
2. If $x[n]$ is a periodic discrete signal, it can be expressed as a Fourier Series (DFS) $x[n]= \sum_{k=<N_0>} {X_k \cdot e^{j2 \pi \nu_0 kn}}$ where $X_k = \frac{1}{N_0} \cdot \sum_{n=<N_0>} {x[n] \cdot e^{-j2 \pi \nu_0 kn}} $ and $\nu_0= \frac{1}{N_0}$. Since the spectrum of a periodic signal is discrete, can I apply to the DFS the Fourier transform (like I would do with a continuous time signal) and write it as a sum of Dirac pulses ?

Let's take as example following exercise : The signal $x(t)= 12\text{sinc}^2(4t) + \cos(4 \pi t)$ is sampled with a rate $f_c = 3$. It then becomes $$y[n]=y(nT_c)= 12\text{sinc}^2(\frac{4}{3} n) + \cos(\frac{4 \pi}{3} n) = y_{1s}(t) + y_{2s}(t) $$. What is the Fourier transform of $y[n]$? Sampling an aperiodic signal in the time domain brings us a periodicity in the frequency domain, so $$Y_{1s}(f) = f_c \cdot \sum_{k=-\infty}^{+\infty} {Y_1(f) * \delta(f-kf_c)}= 3 \cdot \sum_{k=-\infty}^{+\infty} {tr(\frac{f-3k}{4})}$$

Now, $y_{2}[n]=4 \cos(\frac{4}{3} \pi n) $ is a periodic signal: to obtain its Fourier transform do I use the same method as before, getting $$Y_{2s}(f)= \frac{3}{2} \cdot \sum_{k=-\infty}^{+\infty} {\delta(f-2-3k) + \delta(f+2-3k)}$$ ? Is it correct?

If I then decide to send $Y(f)$ in input to a low pass filter with frequency response $H_{LP}(f)= \frac{1}{f_c} \cdot rect(\frac{f}{f_c})$, I should have $Z(f)=Y(f) \cdot H_{LP}(f)= 3 tr(\frac{f}{4}) \cdot rect(\frac{f}{3}) + 2[\delta(f-1) + \delta(f +1)] $. Now I have to get $z[n]$ from this. We know that $\mathscr{F}[x[n]]= \sum_{n=-\infty}^{+\infty}{x[n] \cdot e^{-j2 \pi fT_c n}}$ and $x[n]= T_c \cdot \int_{\frac{-f_c}{2}}^{\frac{f_c}{2}} {X(f) \cdot e^{j2 \pi fT_c n}}df$ , so $z[n]=12sinc^2(4n)*sinc(3n)+ \frac{4}{3}\cos(2 \pi n)$. Is there any mistake?

Note: is the way this post is written ok, or should the various questions each have a dedicated post?

Answer 1

The OP wants to learn about the various Fourier Transforms. Nothing will bypass reading a few chapters in a good DSP book (along with requisite chapters if those are confusing), so I don't recommend skipping that step. I can't reduce it to a short post here but will offer the following companion graphics I have that help illuminate the differences and features of each, hoping that this will be helpful to the OP's educational pursuits (and reading those chapters!):

With the FSE below, I also use this to demonstrate the property that what is periodic in one domain is discrete in the other domain. With the FSE we can restrict the time domain to be just $0$ to $T$, or we can extend the time axis to $\pm \infty$ and in either case we get the same relative values in frequency (so an implied periodicity due to mathematical equivalence). But to be clear, it is important to note that the waveform extending to $\pm infty$ in time would actually be "continuous" but zero everywhere except for the discrete frequencies shown. "Discrete" as shown in these plots means discrete non-zero values.