Signal processing research idea
This is very simple. You need an inter-disciplinary domain where all (and more) of your interests are satisfied, while being very publishable, employable and fun. Easy!! Jump into the bleeding edge of Speech Recognition.

There is TONS of current research is Automatic Speech Recognition (ASR). The domain is very multi-disciplinary (applied math, linguistics, computer science, optimization, electrical engineering) with applications to just about every part of life where a speech interface can work (healthcare, mobile assistants, in-car, entertainment/gaming, education, etc)

For example, state of the art systems like Baidu's Deep Speech 2 use convolutional neural networks (CNNs) to process speech spectrograms (DSP!). CNNs have revolutionized image processing since 2012. The CTC objective that is optimized during training is a very interesting loss function for supervised training of sequences. 

Finally, have a look at these 2016 Stanford University student projects, notably those on speech recognition. This should give you some ideas of what scope or size of project makes sense.


Conditioning WaveNet on Learned Formant Characterizations for Speech
Audio Enhancement
Music-Speech Discrimination
Semantic Analysis of Political Speeches 
Language Recognition and using Speech Recordings
Reinforcement Learning for Feature Selection in Affective Speech Classification 


Good luck!!