What does it mean that one frequency aliases another?

So I was watching a video on sampling and the professor asked which frequencies will alias to 0 and which to $B$ if we are sampling at frequency $f_{s1}$. What does mean that two frequencies alias each other ?

The answer is that $0, f_{s1},2f_{s1},3f_{s1},\ldots, nf_{s1}$ will alias to 0 and $f_{s1}+B, 2f_{s1}+B, \ldots, nf_{s1} + B$ will alias to $B$.

Could anyone explain this please? The image shows the frequency spectrum in question.

2 Answers

I assume that since you're watching lectures on sampling theory, then you have already grasped the core ideas of basic signal operations and Fourier transforms and signal spectrum plots.

Then, consider a signal $x(t)$ whose C.T. Fourier transform is $X(\omega)$. When this signal is multiplied by the (ideal) periodic impulse train $$\sum_{k=-\infty}^{\infty} \delta(t-kT_s)$$ where $T_s$ is the sampling period (and therefore $F_s = 1/T_s$ is the sampling frequency in Hz.) the resulting signal is called as the sampled signal $$s(t)=x(t) \sum_{k=-\infty}^{\infty} \delta(t-k T_s)$$

Then it can be shown (by applying properties of CTFT) that the spectrum $S(\omega)$ of the sampled signal $s(t)$ is the periodical extension of the spectrum $X(\omega)$ of the signal $x(t)$, i.e., $S(\omega)$ is obtained by shifting $X(\omega)$ to left and right in frequency by the amount $k \times \omega_{s} = k \times 2 \pi F_s$. ( Here I'm using radian frequency $\omega$ which is related to Hertz frequency $f$ by $\omega=2\pi f$ ) : $$S(\omega) = \frac{1}{T_s} \sum \limits_{k=-\infty}^{\infty} X(\omega -k 2\pi F_s)$$

For example consider the the shift for $k=-1$, then $X(\omega+2\pi F_s)$ is added on top of original spectrum $X(\omega)$. Now by setting $\omega=0$ in $S(\omega)$, it can be seen that $X(0+2\pi F_s)=X(2\pi Fs)$ has come to the origin in addition to $X(0)$. Setting $k=-2$ would show also that $X(4\pi F_s)$ has come to the origin. This way it can be deduced that those frequencies given by $\omega_k = k 2\pi F_s$ are all appearing at the origin in $S(\omega)$. This appearance of those frequencies $2 \pi k F_s$ at the origin (or more generally any frequency $\omega_k = 2 \pi k F_s + w_0$ at the frequency $\omega_0$) is called as aliasing in the spectrum of $S(\omega)$

It helps to think of not just "frequencies" but sinusoidal waveforms of certain specific frequencies.

Take two pure sinusoidal waveforms of any two different frequencies. If you plot them you will notice that they cross, thus have the same value, at a set of points at a sample rate in between the two sinewave frequencies. Thus, from just this set of sample points, you can't tell whether you were sampling the higher frequency sinewave, or the lower frequency sinewave, or a mix of both. Thus we call the two frequencies aliases. Because we can't tell the difference (from just those samples). One could disguise itself as the other.