Today I came across Fourier series. The lecturer told us that apart from solving PDEs (as we are using them), they constitute the foundation of signal processing. Hence I went online and had a look in the Wikipedia page and I saw a quote saying that signal processing can be done with 17th century mathematics.
So my question is:
Are there unexplored fields of mathematical signal processing (I wouldn't know how else to call it)? What I want to know is, apart from the engineering problems or technological problems, is there some advancement in the mathematical methods for signal processing? I want to do two dissertations in my final year (it is possible at my university) and so I wanted to know if there would anything new to explore or talk about in my dissertation, regarding signal processing.
For example, of course the technology of it keeps advancing, but is the maths behind it staying the same? Or is the advancement in signal processing just a matter of engineeristic improvement?