What is an example of using aliasing to your advantage when recovering an input signal? Suppose you have an arbitrary analog input signal $x_a(t)$ guaranteed to have frequencies within a bandwidth $[f_1,f_2]$ Hz. 

Suppose your sampling frequency $F_s$Hz, and sample $x_a(t)$ to produce $x(n)=x_a(\frac{n}{F_s})$. 

Then by the sampling theorem, you can successfully recover (or reconstruct) the signal so long as $f_2 &lt; \frac{F_s}{2}$. Otherwise you will experience aliasing and cannot recover the original signal $x_a(t)$. 

It appears (from this article) that there are "tricks" that can be employed that take advantage of aliasing scenario and can still recover the signal $x_a(t)$ completely.  

The specific part of the article of concern is pasted below: 


  Using Nyquist aliasing as benefit
  
  The trick is to use the aliasing (or frequency folding) to your
  advantage. By undersampling the data converter, higher-frequency
  content will be aliased into all of the lower Nyquist zones (see
  Figure 2). You will need to make absolutely sure that nothing ends up
  in the lower bands – any noise or frequency components in the lower
  zones will also be aliased into the first Nyquist. The good news is
  that the data rate from the data converter is only a fraction of the
  required RF input sample rate if this were a first Nyquist system.
  Under sampling greatly reduces the data rate of the samples supplied
  to the digital signal processor (DSP) or FPGA.


Could somebody explain this to me? Why is the Texas Instruments link not violating Nyquist's Theorem?
you can successfully recover (or reconstruct) the signal so long as $f_2&lt;F_s/2$


Nope. The sampling theorem says that you need 2 samples per Hz of bandwidth, so in this case you'd need 

$$Fs &gt; 2*(f_2-f_1)$$

For more info on how this works google "bandpass sampling" or just read https://en.wikipedia.org/wiki/Undersampling.

The basic idea is that sampling creates a periodic repetition of the original spectrum with a period of $F_s$. As long as none of the repeated spectra falls on the same frequencies as the original spectrum, you don't get aliasing.