How do I, a prefinal year student doing EEE, get a summer research internship with IISC Bangalore in DSP, related particularly to “compressed sampling”?

You can directly apply for the summer research internship program at the official website of IISc Bangalore. Another way to secure a summer internship at IISc Bangalore is by direct email. 

The professors at IISc receive numerous internship applications. Many of them hire interns much before your mail gets to them. Professors hire interns 2-3 months before the start of the internship period. 

One of my seniors interned at IISc Bangalore. According to her-

1. SOP counts a lot. Try to sound very convincing about how much you love that research area.
2. The recommendation letter format offered by IISc contains an optional column to send the key publications of the professor who is giving you the LOR. Start the application in advance and coax your professor into giving the key publications of his research duly signed and attested.
3. CGPA holds a lot of importance. Make sure to score well.
4. A good courier service must be used to send the required papers. Track the documents. Your profile and write-up might be very good but unless they need to reach the right place.

If you are interested in research in compressed sampling and the institute is not your priority, you can also apply for international internships such as DAAD, MITACS, SN BOSE, and Charpak France. There are numerous foreign universities involved in extensive research on compressed sampling.