Modelling resonances and orbital chaos in disk galaxies. Application to a Milky Way spiral model

Context: Resonances in the stellar orbital motion under perturbations from spiral arms structure play an important role in the evolution of the disks of spiral galaxies. The epicyclic approximation allows the determination of the corresponding resonant radii on the equatorial plane (for nearly circular orbits), but is not suitable in general. Aims: We expand the study of resonant orbits by analysing stellar motions perturbed by spiral arms with Gaussian-shaped profiles without any restriction on the stellar orbital configurations, and we expand the concept of Lindblad (epicyclic) resonances for orbits with large radial excursions. Methods: We define a representative plane of initial conditions, which covers the whole phase space of the system. Dynamical maps on representative planes are constructed numerically, in order to characterize the phase-space structure and identify the precise location of resonances. The study is complemented by the construction of dynamical power spectra, which provide the identification of fundamental oscillatory patterns in the stellar motion. Results: Our approach allows a precise description of the resonance chains in the whole phase space, giving a broader view of the dynamics of the system when compared to the classical epicyclic approach, even for objects in retrograde motion. The analysis of the solar neighbourhood shows that, depending on the current azimuthal phase of the Sun with respect to the spiral arms, a star with solar kinematic parameters may evolve either inside the stable co-rotation resonance or in a chaotic zone. Conclusions: Our approach contributes to quantifying the domains of resonant orbits and the degree of chaos in the whole Galactic phase-space structure. It may serve as a starting point to apply these techniques to the investigation of clumps in the distribution of stars in the Galaxy, such as kinematic moving groups.Comment: 17 pages, 15 figures. Matches accepted version in A&

Orbital Magnetism in the Ballistic Regime: Geometrical Effects

We present a general semiclassical theory of the orbital magnetic response of noninteracting electrons confined in two-dimensional potentials. We calculate the magnetic susceptibility of singly-connected and the persistent currents of multiply-connected geometries. We concentrate on the geometric effects by studying confinement by perfect (disorder free) potentials stressing the importance of the underlying classical dynamics. We demonstrate that in a constrained geometry the standard Landau diamagnetic response is always present, but is dominated by finite-size corrections of a quasi-random sign which may be orders of magnitude larger. These corrections are very sensitive to the nature of the classical dynamics. Systems which are integrable at zero magnetic field exhibit larger magnetic response than those which are chaotic. This difference arises from the large oscillations of the density of states in integrable systems due to the existence of families of periodic orbits. The connection between quantum and classical behavior naturally arises from the use of semiclassical expansions. This key tool becomes particularly simple and insightful at finite temperature, where only short classical trajectories need to be kept in the expansion. In addition to the general theory for integrable systems, we analyze in detail a few typical examples of experimental relevance: circles, rings and square billiards. In the latter, extensive numerical calculations are used as a check for the success of the semiclassical analysis. We study the weak-field regime where classical trajectories remain essentially unaffected, the intermediate field regime where we identify new oscillations characteristic for ballistic mesoscopic structures, and the high-field regime where the typical de Haas-van Alphen oscillations exhibit finite-size corrections. We address the comparison with experimental data obtained in high-mobility semiconductor microstructures discussing the differences between individual and ensemble measurements, and the applicability of the present model.Comment: 88 pages, 15 Postscript figures, 3 further figures upon request, to appear in Physics Reports 199

New Approaches in Automation and Robotics

The book New Approaches in Automation and Robotics offers in 22 chapters a collection of recent developments in automation, robotics as well as control theory. It is dedicated to researchers in science and industry, students, and practicing engineers, who wish to update and enhance their knowledge on modern methods and innovative applications. The authors and editor of this book wish to motivate people, especially under-graduate students, to get involved with the interesting field of robotics and mechatronics. We hope that the ideas and concepts presented in this book are useful for your own work and could contribute to problem solving in similar applications as well. It is clear, however, that the wide area of automation and robotics can only be highlighted at several spots but not completely covered by a single book

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims

Array Signal Processing Based on Traditional and Sparse Arrays

Array signal processing is based on using an array of sensors to receive the impinging signals. The received data is either spatially filtered to focus the signals from a desired direction or it may be used for estimating a parameter of source signal like direction of arrival (DOA), polarization and source power. Spatial filtering also known as beamforming and DOA estimation are integral parts of array signal processing and this thesis is aimed at solving some key probems related to these two areas. Wideband beamforming holds numerous applications in the bandwidth hungry data traffic of present day world. Several techniques exist to design fixed wideband beamformers based on traditional arrays like uniform linear array (ULA). Among these techniques, least squares based eigenfilter method is a key technique which has been used extensively in filter and wideband beamformer design. The first contribution of this thesis comes in the form of critically analyzing the standard eigenfilter method where a serious flaw in the design formulation is highlighted which generates inconsistent design performance, and an additional constraint is added to stabilize the achieved design. Simulation results show the validity and significance of the proposed method. Traditional arrays based on ULAs have limited applications in array signal processing due to the large number of sensors required and this problem has been addressed by the application of sparse arrays. Sparse arrays have been exploited from the perspective of their difference co-array structures which provide significantly higher number of degrees of freedoms (DOFs) compared to ULAs for the same number of sensors. These DOFs (consecutive and unique lags) are utilized in the application of DOA estimation with the help of difference co-array based DOA estimators. Several types of sparse arrays include minimum redundancy array (MRA), minimum hole array (MHA), nested array, prototype coprime array, conventional coprime array, coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) and super nested array. As a second contribution of this thesis, a new sparse array termed thinned coprime array (TCA) is proposed which holds all the properties of a conventional coprime array but with \ceil*{\frac{M}{2}} fewer sensors where $M$ is the number of sensors of a subarray in the conventional structure. TCA possesses improved level of sparsity and is robust against mutual coupling compared to other sparse arrays. In addition, TCA holds higher number of DOFs utilizable for DOA estimation using variety of methods. TCA also shows lower estimation error compared to super nested arrays and MRA with increasing array size. Although TCA holds numerous desirable features, the number of unique lags offered by TCA are close to the sparsest CADiS and nested array and significantly lower than MRA which limits the estimation error performance offered by TCA through (compressive sensing) CS-based methods. In this direction, the structure of TCA is studied to explore the possibility of an array which can provide significantly higher number of unique lags with improved sparsity for a given number of sensors. The result of this investigation is the third contribution of this thesis in the form of a new sparse array, displaced thinned coprime array with additional sensor (DiTCAAS), which is based on a displaced version of TCA. The displacement of the subarrays generates an increase in the unique lags but the minimum spacing between the sensors becomes an integer multiple of half wavelength. To avoid spatial aliasing, an additional sensor is added at half wavelength from one of the sensors of the displaced subarray. The proposed placement of the additional sensor generates significantly higher number of unique lags for DiTCAAS, even more than the DOFs provided by MRA. Due to its improved sparsity and higher number of unique lags, DiTCAAS generates the lowest estimation error and robustness against heavy mutual coupling compared to super nested arrays, MRA, TCA and sparse CADiS with CS-based DOA estimation