[Venkatesh]

There is an excellent book, "Music, Physics, and Engineering", in which fourier analysis and basic transforms are explored. You don't need much beyond Precalc to understand their explanations.

However the book only has Fourier analysis as a tiny element (roughly 15-20 pages in the whole thing) and does not talk about Fast fourier transforms or the like, so its not worth buying if you are only interested in Fourier. Its an old book, and should be in many libraries.