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.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.