Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $10^{-12}$ can be obtained in the parameter region $(x,y,\mu) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).Comment: Accepted for publication in ACM Trans. Math. Soft

Algorithm 939: Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum Q.function (QƒÊ(x, y)) and the complementary function (PƒÊ(x, y)) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to 10.12 can be obtained in the parameter region (x, y, ƒÊ) ¸ [0, A] ~ [0, A] ~ [1, A], A = 200, while for larger parameters the accuracy decreases (close to 10.11 for A = 1000 and close to 5 ~ 10.11 for A = 10000)

Design and performance analysis of quadratic-form receivers for fading channels

Ph.DDOCTOR OF PHILOSOPH

A landscape assessment to identify potential elk restoration sites in Arkansas

The Arkansas Game and Fish Commission (AGFC) is considering translocation to expand its established elk population. I conducted a feasibility study that identified potential restoration sites in Arkansas to reduce the probability of reintroduction failure. I developed 2 landscape-scale predictive models using geographical information system (GIS) technology to identify potential elk restoration sites in Arkansas, one to identify suitable elk habitat and the other to assess the potential for elk-human conflict. I assessed winter habitat for elk using empirical data consisting of 239 elk-group locations collected from helicopter surveys in the Buffalo National River area. Those surveys were conducted by the AGFC in February-March, 1992-2002. A suite of 9 habitat variables were developed to characterize the habitat and landscape conditions associated with those elk-group locations. Variables were generated at multiple spatial scales, representing different orders of habitat selection, so that I could select the most appropriate scale to evaluate each variable. From those data, I then applied the Mahalanobis distance statistic to evaluate winter habitat suitability in Arkansas based on 90- x 90-m pixels. Lower Mahalanobis distance values indicated a greater similarity to the habitat conditions associated with the elk-group locations. More suitable elk habitat was associated with areas of high landscape heterogeneity, heavy forest cover, and gentle sloping ridge tops and valleys in western and northwestern Arkansas, where human population and road densities also were relatively low. Areas of intensive agriculture in the Mississippi River Delta generally were least suitable. I tested model performance by recording the frequency of occurrence of elk scat within 19 fixed-width transects surveyed in March 2002. Linear regression analysis indicated that the frequency of scat occurrence increased with decreasing mean Mahalanobis distance values (F = 9.65, P = 0.039). Those results suggest that elk presence was more likely in areas predicted by the model to be more suitable habitat. Finally, I assessed the potential for elk-human conflict in Arkansas with a GIS adaptation of the Analytical Hierarchy Process. Five elk experts in Arkansas ranked the relative importance of 8 criteria that could influence the potential for elk-human conflict in a series of pairwise comparisons. Those rankings were then applied in a weighted linear summation of 8 variables representing those criteria, resulting in a single map delineating the relative potential for elk-human conflict. Public land forage availability was determined to have the strongest influence on the potential for elk-human conflict, contributing 33% to the overall conflict potential, followed by human population growth rate (22%) and the amount of private land in row crops (18%) . Elk-human conflict potential in Arkansas ranged from 0.14 to 0.72 (x = 0.54 ± 0.009). Conflict potential was classified as low (:S 0.49), medium (0.49- 0.59), and high (\u3e 0.59), representing intervals of 6 standard deviations from the mean conflict potential value for radio-locations of nuisance elk cows. I combined contours of those conflict potential intervals with the winter habitat suitability model to identify regions where suitable elk habitat corresponded with low potential for elk-human conflict. Those regions mainly were associated with public lands in western and northwestern Arkansas. Large, contiguous patches of suitable habitat within areas of low elk-human conflict potential tended to correspond with public and private land boundaries in northern and northwestern Arkansas. The combined map provides a tool for natural resource managers to identify and rank potential elk restoration areas in Arkansas

Performance analysis of the parallel acquisition of weak GPS signals

This thesis provides a comprehensive analysis of the acquisition performance of the mobile-embedded Global Positioning System (GPS) receiver. Particular emphasis is given to the analysis of differentially coherent processing techniques and parallel acquisition strategies. New analytical expressions for the distribution of the decision variable of differentially coherent detectors are derived. In addition, new Gaussian approximations are derived and shown to be more accurate than existing approximations. Using these Gaussian approximations it is demonstrated that the traditional noncoherent combining detector is the best choice when the signal to noise ratio is large, but that differentially coherent combining is a superior choice at low signal to noise ratios. An analysis of the effects of carrier Doppler, code Doppler and data modulation on detector performance is also conducted. For the noncoherent combining detector, new expressions are obtained for the mean and worst case power attenuation due to the combined effects of carrier Doppler and data modulation. Approximate expressions are also derived for the differentially coherent combining detector. New expressions are also obtained for the mean and variance of the time to first hit using a Markov chain model and matrix methods. These models permit the use of numerical techniques to determine the optimal choice of receiver parameters for a given performance requirement. Finally the effect of unknown power levels and multi-access interference (MAI) are considered. A novel technique for detecting MAI, referred to as the power level detector, is introduced and its performance analysed. All results are verified by Monte Carlo computer simulation using a simplified signal model. The simulations were implemented on a 100 processor computer cluster

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