Analysing energy detector diversity receivers for spectrum sensing

The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived. v Acknowledgements The work in this thesis, the work that led to the work in this thesis and the work that didn’t lead anywhere at all has occupied me for five years of my life. Five years is a long time. Enough for nine countries, four bedrooms and three (and counting) grey hairs; enough for all afternoon card games, pointless arguments and far too much tea; enough for rainy days, leaky shoes and wet socks; enough for four o’clock pints and afternoon starts; enough to make new friends and neglect old ones; enough to make up for it; enough for free time and none at all; enough for frustration and satisfaction; enough to write a thesis. Things often don’t work out quite the way you expect them to. Over the past few years, I’ve run into a few of what Flann O’Brien called insoluble pancakes, conundrums of inscrutable potentialities or, put more simply, snorters. Sometimes, problems can be quite difficult. Fortunately, I had an excellent support group: my supervisor, Colin, my fellow postgraduate students, Ralph, my friends outside of university (at home and abroad) and, of course, my family. Over the years, I’ve been reassured, consoled, hugged, teased and given a healthy dose of cynicism either in person, over the phone or by email. It made the worst days tolerable and the best unforgettable. Thanks for putting up with me. Unfortunately, there’s only room for one name on the cover. I’ve also had a lot of support from others in the last few years: the academic and administrative staff in my department, who made it a nice place to be; The Irish Research Council for Science, Education and Technology, who sponsored the work that led to this thesis; the Boole Centre for Research in Informatics, who let me use their compute cluster, and Brian Clayton, who didn’t get mad when my code kept crashing it; the editors and reviewers from the various conferences and journals I’ve submitted papers to, who (mostly) criticised constructively; and Kevin McCarthy and Marco Chiani, who had the unenviable task of reading this thesis and examining me on it. Thanks for the suggestions (and also for finding what I hope was the last typo). I think it reads a lot better now. Finally, I’d like to thank Monty, who never minded when I was in a grumpy mood and doesn’t have the capacity to understand what an acknowledgement is. He’s been curled in a ball on a rug by my feet the whole time I’ve been writing this.