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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?

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    Here is a recent IEEE article: http://ieeexplore.ieee.org/abstract/document/6387646/ – 2017-02-14
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    Algorithms to solve large scale optimization problems. There are several applications for which the current compressed sensing algorithms cannot be used due to the size of the problem. – 2017-02-14
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    @AnonSubmitter85 what kind of improvements could this lead to? – 2017-02-14

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There is something called compressed sensing which is quite heavily studied nowadays, see e.g. https://en.wikipedia.org/wiki/Compressed_sensing .

The gist of it is, how many samples do we need in order to be able to reconstruct a signal? For the most part, a result known as the Nyqvist-Shannon sampling theorem (https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) has been used to answer this, as it gives us a sufficient condition for perfectly reconstructing signals from samples, and many have applied this theorem in practice without thinking too much about it.

However, in recent years people have realized that one can do better, and capture just as much information about a signal with less samples, which of course is really nice. The main task is then figuring out when and how one can do so. There is a lot of math involved in this, as well as mathematicians researching the topic (including, for example, Terry Tao), so you can try looking into it.

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Array processing is very important to mobile technologies. Advances in array processing is part of why in the last 10 years the resolution of mobile phone cameras has exploded. There are a number of other applications outlined in the Wikipedia article.