We consider a general stochastic differential delay equation (SDDE) with multiplicative colored noise. We study the limit as the time delays and the correlation times of the noises go to zero at the same rate. First, we derive the limiting equation for the equation obtained by Taylor expanding the SDDE to first order in the time delays. The limiting equation contains a noise-induced drift term that depends on the ratios of the time delays to the correlation times of the noises. We prove that, under appropriate assumptions, the solution of the equation obtained by the Taylor expansion converges to the solution of this limiting equation in probability with respect to the sup norm over compact time intervals. Next, we derive the limiting equation for the SDDE and prove a similar convergence result regarding convergence of the solution of the SDDE to the solution of this limiting equation. We see that the limiting equation corresponding to the equation obtained by the Taylor expansion is an approximation of the limiting equation corresponding to the SDDE. Finally, we study the effects of time delay on a particular model of active Brownian motion.

The Cross Ambiguity Function (CAF) is often used for passive geolocation of an emitter based on the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) of the received signals. CAF performance has been thoroughly investigated in regards to traditional single-input single-output (SISO) signals. Little is known about how the CAF will respond to signals from multiple-input multiple-output (MIMO) systems which utilize multiple antennas. This thesis focuses on characterizing the CAF's magnitude distribution in order to determine the probability of correctly determining the correct TDOA/FDOA bin, and the resulting impact on geolocation. The received signals are studied in the presence of additive white Gaussian noise (AWGN) as well as multi-channel propagation effects such as phase ambiguities and offsets due to multi-antenna transmission. Two and four transmit antennas using either a form of spatial multiplexing or space-time block coding are the focus of this work because they are mostly commonly found in currently deployed communication systems. The effects of these transmit schemes are studied with respect to TDOA/FDOA error and the resulting position error. The analysis is performed using a detection theory framework as opposed to estimation theory in order to empha- size the impact of MIMO transmission on determining the correct TDOA/FDOA bin. A simple method using the CAF magnitude as a decision statistic is also presented so that TDOA/FDOA errors can be detected and filtered in an attempt to improve positioning estimates. / Master of Science

In this dissertation it is shown that the telemetry versions of Feher-patented QPSK (FQPSK-JR) and shaped offset QPSK (SOQPSK-TG) can be interpreted as both cross-correlated, trellis-coded quadrature modulation (XTCQM) and continuous phase modulation (CPM). Based on these representations, both modulations can be detected with near optimal bit error rate performance using a common detector that is formulated as either an XTCQM detector, a traditional CPM detector, or a pulse amplitude modulation (PAM) detector (due to the PAM decomposition of the CPM representations of these modulations). In addition it is shown that the complexity of the XTCQM detector for SOQPSK-TG can be reduced by a factor of 128 with only a 0.2 dB loss in detection efficiency relative to the optimum detector. Three decoders for STC encoded OQPSK are presented. One decoder has a bit error rate performance that matches the SISO case but with much higher complexity than that of the QPSK decoder. A second decoder matches the simplicity of the decoder for STC encoded non-offset QPSK but with a loss of 3 dB relative to the single-input, single-output (SISO) case. A third decoder matches SISO performance with lower complexity than the first one. These results for STC encoded OQPSK are extended to STC SOQPSK. It is shown that the maximum likelihood decoder is not computationally feasible. Two suboptimal decoders based on the STC OQPSK decoders are presented. These decoders have much higher complexity than their OQPSK counterparts, and they provide inferior bit error rate performance. In addition, a least squares decoder for STC encoded SOQPSK is presented which is less complex and has better performance (within 1 dB of the SISO bound) than the previous two decoders. This decoder also handles the differential delays that can occur on aeronautical telemetry channels.

電離層延遲為精密GPS定位及導航的主要誤差來源之一，為了減弱電離層延遲對GPS定位及導航的影響，可以利用雙頻GPS觀測量構建即時的區域電離層模型，以提供即時的電離層延遲誤差改正參數，修正因電離層延遲效應造成的定位及導航誤差。 本研究以台灣地區雙頻GPS觀測量，採用相位水準技術估算全電子含量（TEC）、修正的單站演算法估計各GPS衛星及接收儀之L1/L2差分延遲及以UNSW網格式演算法構建區域的電離層模型。並進而求得適合台灣南部地區網格式電離層模型之較佳網格大小及探討使用那些內政部衛星追蹤站的觀測資料，便可有效建立台灣地區的電離層模型。 / The ionospheric delay is one of the main sources of error in precise GPS positioning and navigation. The magnitude of the ionospheric delay is related to the Total Electron Content (TEC) along the radio wave path from a GPS satellite to the ground receiver. The TEC is a function of many variables, including long and short term changes in solar ionising flux, magnetic activity, season of the year, time of day, user location and viewing direction. A dual-frequency GPS receiver can eliminate (to the first order) the ionospheric delay through a linear combination of L1 and L2 observables. However, the majority of civilians use low-cost single-frequency GPS receivers that cannot use this option. Consequently, it is beneficial to estimate ionospheric delays over the region of interest, in real-time, in support of single-frequency GPS positioning and navigation applications. In order to improve real-time regional ionosphere modelling performance, a grid-based algorithm is proposed. Data from the southern Taiwan region GPS network were used to test the ionosphere modelling algorithms. From the test results described here, it is shown that the performance of real-time regional ionosphere modelling is improved significantly when the proposed algorithm is used.

Estudamos a Eletrodinâmica de Wheeler-Feynman usando um princípio variacional para um funcional de ação finito acoplado a um problema de valor na fronteira. Para trajetórias C2 por trechos, a condição de ponto crítico desse funcional fornece as equações de movimento de Wheeler-Feynman mais uma condição de continuidade dos momentos parciais e energias parciais, conhecida como condição de quina de Weierstrass-Erdmann. Estudamos em detalhe um sub-caso mais simples, onde os dados de fronteira têm um comprimento mínimo. Nesse caso, mostramos que a condição de extremo se reduz a um problema de valor na chegada para uma equação diferencial com retardo misto dependente do estado e do tipo neutro. Resolvemos numericamente esse problema usando um método de shooting e um método de Runge-Kutta de quarta ordem. Para os casos em que as fronteiras mínimas têm velocidades descontínuas, elaboramos uma técnica para resolver as condições de quina de Weierstrass-Erdmann junto com o problema de valor na chegada. As trajetórias com velocidades descontínuas previstas pelo método variacional foram verificadas por experimentos numéricos. Em um segundo desenvolvimento, para o caso mais difícil de fronteiras de comprimento arbitrário, implementamos um método de minimização com gradiente fraco para o princípio variacional e problema de fronteira acima citado. Elaboramos dois métodos numéricos, ambos implementados em MATLAB, para encontrar soluções do problema eletromagnético de dois corpos. O primeiro combina o método de elementos finitos com o método de Newton para encontrar as soluções que anulam o gradiente fraco do funcional para fronteiras genéricas. O segundo usa o método do declive máximo para encontrar as soluções que minimizam a ação. Nesses dois métodos as trajetórias são aproximadas dentro de um espaço de dimensão finita gerado por uma Galerkiana que suporta velocidades descontínuas. Foram realizados diversos testes e experimentos numéricos para verificar a convergência das trajetórias calculada numericamente; também comparamos os valores do funcional calculados numericamente com alguns resultados analíticos sobre órbitas circulares. / We study the Wheeler-Feynman electrodynamics using a variational principle for an action functional coupled to a finite boundary value problem. For piecewise C2 trajectories, the critical point condition for this functional gives the Wheeler-Feynman equations of motion in addition to a continuity condition of partial moments and partial energies, known as the Weierstrass-Erdmann corner conditions. In the simplest case, for the boundary value problem of shortest length, we show that the critical point condition reduces to a two-point boundary value problem for a state-dependent mixed-type neutral differential-delay equation. We solve this special problem numerically using a shooting method and a fourth order Runge-Kutta. For the cases where the boundary segment has discontinuous velocities we developed a technique to solve the Weierstrass-Erdmann corner conditions and the two-point boundary value problem together. The trajectories with discontinuous velocities presupposed by the variational method were verified by numerical experiments. In a second development, for the harder case with boundaries of arbitrary length, we implemented a method of minimization with weak gradient for the variational principle quoted above. Two numerical methods were implemented in MATLAB to find solutions of the two-body electromagnetic problem. The first combines the finite element method with Newtons method to find the solutions that vanish the weak gradient. The second uses the method of steepest descent to find the solutions that minimize the action. In both methods the trajectories are approximated within a finite-dimensional space generated by a Galerkian that supports discontinuous velocities. Many tests and numerical experiments were performed to verify the convergence of the numerically calculated trajectories; also were compared the values of the functional computed numerically with some known analytical results on circular orbits.

Estudamos a Eletrodinâmica de Wheeler-Feynman usando um princípio variacional para um funcional de ação finito acoplado a um problema de valor na fronteira. Para trajetórias C2 por trechos, a condição de ponto crítico desse funcional fornece as equações de movimento de Wheeler-Feynman mais uma condição de continuidade dos momentos parciais e energias parciais, conhecida como condição de quina de Weierstrass-Erdmann. Estudamos em detalhe um sub-caso mais simples, onde os dados de fronteira têm um comprimento mínimo. Nesse caso, mostramos que a condição de extremo se reduz a um problema de valor na chegada para uma equação diferencial com retardo misto dependente do estado e do tipo neutro. Resolvemos numericamente esse problema usando um método de shooting e um método de Runge-Kutta de quarta ordem. Para os casos em que as fronteiras mínimas têm velocidades descontínuas, elaboramos uma técnica para resolver as condições de quina de Weierstrass-Erdmann junto com o problema de valor na chegada. As trajetórias com velocidades descontínuas previstas pelo método variacional foram verificadas por experimentos numéricos. Em um segundo desenvolvimento, para o caso mais difícil de fronteiras de comprimento arbitrário, implementamos um método de minimização com gradiente fraco para o princípio variacional e problema de fronteira acima citado. Elaboramos dois métodos numéricos, ambos implementados em MATLAB, para encontrar soluções do problema eletromagnético de dois corpos. O primeiro combina o método de elementos finitos com o método de Newton para encontrar as soluções que anulam o gradiente fraco do funcional para fronteiras genéricas. O segundo usa o método do declive máximo para encontrar as soluções que minimizam a ação. Nesses dois métodos as trajetórias são aproximadas dentro de um espaço de dimensão finita gerado por uma Galerkiana que suporta velocidades descontínuas. Foram realizados diversos testes e experimentos numéricos para verificar a convergência das trajetórias calculada numericamente; também comparamos os valores do funcional calculados numericamente com alguns resultados analíticos sobre órbitas circulares. / We study the Wheeler-Feynman electrodynamics using a variational principle for an action functional coupled to a finite boundary value problem. For piecewise C2 trajectories, the critical point condition for this functional gives the Wheeler-Feynman equations of motion in addition to a continuity condition of partial moments and partial energies, known as the Weierstrass-Erdmann corner conditions. In the simplest case, for the boundary value problem of shortest length, we show that the critical point condition reduces to a two-point boundary value problem for a state-dependent mixed-type neutral differential-delay equation. We solve this special problem numerically using a shooting method and a fourth order Runge-Kutta. For the cases where the boundary segment has discontinuous velocities we developed a technique to solve the Weierstrass-Erdmann corner conditions and the two-point boundary value problem together. The trajectories with discontinuous velocities presupposed by the variational method were verified by numerical experiments. In a second development, for the harder case with boundaries of arbitrary length, we implemented a method of minimization with weak gradient for the variational principle quoted above. Two numerical methods were implemented in MATLAB to find solutions of the two-body electromagnetic problem. The first combines the finite element method with Newtons method to find the solutions that vanish the weak gradient. The second uses the method of steepest descent to find the solutions that minimize the action. In both methods the trajectories are approximated within a finite-dimensional space generated by a Galerkian that supports discontinuous velocities. Many tests and numerical experiments were performed to verify the convergence of the numerically calculated trajectories; also were compared the values of the functional computed numerically with some known analytical results on circular orbits.