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The growth of technology continually increases demand for electromagnetic spectrum in both civilian and military applications. As a result this resource is increasingly scarce, usage is overlaid, and the electromagnetic environment (EME) is increasingly complex, congested and contested. 

The EME is essential to the UK across all domains; space, air, land, sea and cyber. To solve problems encountered when operating in the EME there are currently a range of approaches and traditional engineering techniques. However, applying new or alternative mathematics presents an opportunity to create new, innovative approaches.

The Defence, Science and Technology Laboratory (Dstl), PA Consulting and the Isaac Newton Institute are collaborating to engage the mathematical community to work on challenges in the EME.

A workshop was held in January 2020 to scope possible solutions to 6 challenges (selected from 70 different challenges that scientists and engineers in Dstl were trying to solve):

Autonomous Collaborative Spectrum Management in a Competed and Congested RF environment 

This involved the formulation of a new simplified optimisation model of how a variety (high and low priority) of users can best be accommodated in a multi-channel setting. The next steps are to implement solution techniques and perhaps elicit some typical real-world parameters. The harder problem of spatial re-use (mesh networks etc.) remains for future work.

Optimum search route to discover mesh sensor network

In this scenario we consider a mobile platform which is trying to locate a state mesh sensor network by receiving the RF communications between nodes. We want to determine the optimum search route to provide the most efficient discovery of the sensor network. The nodes are positioned in Star configuration with a central node communicating to a set of mesh nodes.
The mathematical approach iteratively refines the optimum route based on the information available and gathered as the mobile sensors traverse the environment.  The mathematical approach starts by calculating the optimum route based on any information available at the start of the operation
This information is sufficient to provide an initial starting point and route across the environment. The algorithm then moves the sensor along the route, takes measurements and updates the route based on this new information.
A critical element of the route finding is that the algorithm determines the route which, based on the prior information will provide the maximum information gain, in other words most likely to gather information which helps collapse the many possible locations of the nodes to a smaller set of probable locations.

Sensing and transmitting using large swarms of drones 

The challenge was broken down into three constituent parts: Beamforming, imaging and autonomous drone control. Each element was explored to the degree that mathematical approaches were identified. It was recommended that the next steps would be to use synthetic data to provide a low risk opportunity to build this problem solver and test its viability. This would provide a complete ground truth for algorithm testing and simulate a future with enhanced sensor technology. A sensitivity analysis can be performed where the effect of variations in the target on the ability to DTRI can be assessed and inform any experimental work that may follow. Uncertainty quantification and propagation can also be assessed. 

Evaluation and classification of known and unknown signals in a congested environment 

The team working on this mathematised the challenge statement. They then identified a number of approaches to the problem using the following methods: Classification approaches, Iterative predict and compare, Matrix decomposition image processing approaches and Machine learning. The team found the challenge difficult and the next step suggested was to have access to real or simulated data. 

Autonomous generation of communications protocols

In considering the problem, it was assumed that there are a finite number of protocols available. This reduced a problem with apparently infinite when looking across the full OSI stack 4, to something more tractable. The team therefore abstracted the problem to the theory of graphs over a finite set of protocols. Although a solution was not identified, a number of mathematical approaches are relevant to the reformed problem: Optimisation theory, Consensus forming, Complex network theory (random graphs), Decision making under uncertainty, Multi-layered networks / adaptive networks, Bayesian exploration, Dynamical systems and Bifurcation theory.
The recommendation was to go back to the original challenge and formulate a more specific question/scenario to test out possible algorithms/analyses.

Understanding the boundaries of resilience of a signal coding scheme – OFDM (Orthogonal Frequency Division Multiplexing)

Challenge 6 was the most technical of the set and required the team to be supported by domain experts. This was a conscious decision to understand whether technical topics could be addressed. 
Professor Chandler-Wilde of Reading University mathematised the principle of OFDM to enable the team to approach the problem. They then explored its vulnerabilities and recommended following up by expanding on their discussions and simulation.


This research scoping workshop proved successful in engaging applied mathematicians from multiple disciplines to bring a fresh perspective. The longer term plan is that this activity will build closer links and collaborations between applied mathematicians and the owners of complex challenges, establishing a joined up multi-disciplinary UK community for this area. More information on this workshop can be found here. 

As a direct result of this workshop, many of the academics have since started small funded projects (as detailed above), to further develop the work and a follow-on workshop has been scheduled for 16th-18th September 2020. More details on this workshop can be found here.

We are very keen to build a community that involves those that wouldn’t normally engage with these challenges, particularly women and those from ethnic minorities and also particularly welcome applications from mathematicians with expertise in areas such as real/complex analysis, algebra, topology, probability theory, geometry, or combinatorics.