Aliasing at $f_0 + kf_s$ In this question, it has been proved that $x(t) = \sin(2\pi f_0 t)$ and $x_k(t) = \sin(2\pi (f_0 + k f_s) t)$ have the same sample points. So sinusoids with frequencies $f_0$ and $f_0 + kf_s$ will identify as the same signals. In other words, we keep sampling frequency constant and increase frequency by $kf_s$. Apparently, only frequencies $f_0 + kf_s$ lead to aliasing. Why this frequency is special? I tried to see this result in the frequency domain but it wasn't useful. I know by sampling in time domain we get shifted replicas of the original spectrum which is two delta functions at $f_0$ and $-f_0$ but how this corresponds to $f_s$ and aliasing at $f_0 + kf_s$? What I'm trying to get here is a visual explanation of mentioned result. In the case of the sampling theorem it's really easy to see aliasing occurs when there is an overlap.
You've made an error in this statement: &quot;I know by sampling in time domain we get shifted replicas of the original spectrum which is two delta functions at $f_0$ and $−f_0$ but how this corresponds to $f_s$ and aliasing at $f_0+kf_s$?&quot;
The two delta functions at $-f_0$ and $f_0$ are the result of taking the Fourier transform of a sinusoid $x(t)$:
$$ x(t) = \sin(2{\pi}f_0t)$$
$$\mathcal{F}(x(t)) = \frac{1}{2i} \big(\delta(f - f_0) + \delta(f + f_0) \big)$$
We haven't introduced sampling yet, this is simply the result of taking the Fourier transform. When we start sampling, then we introduce replicas of the signal's spectrum.
Let's assume we sample the $x(t)$ above at a frequency of $f_s$. We will have copies of that signal's spectrum at $f_0 + kf_s$ for all integer values of $k$. Assuming we met Nyquist, there's no problem here. The problem now comes when we have the sinusoid
$$ s(t) = \sin(2{\pi}(f_0 + f_s)t)$$
$$\mathcal{F}(s(t)) = \frac{1}{2i}\big( \delta(f - [f_0 + f_s]) + \delta(f + [f_0 + f_s]) \big)$$
When sampled, we have the spectrum copies centered at $(f_0 + f_s) + kf_s$ for all integer values of $k$. If we choose $k = -1$ then we have a copy at
$$ (f_0 + f_s) + kf_s|_{k = -1} = f_0$$
So if we had a combination of $x(t)$ and $s(t)$ and sampled it, a spectrum copy of $s(t)$ would step on $x(t)$'s copy at $f_0$. This is aliasing.