Development and analysis of mathematical methods for estimating statistical parameters in sensor array-based systems - Kunzler, Jonas Augusto (2020)
The direction of arrival estimation and the estimation of energy pulses in x-ray spectroscopy are based on the same parameter, the phase shift of measured signals relative to a reference. This work demonstrates the possibility of applying well-established direction-of-arrival estimation techniques to the problem of spectroscopy. The two themes are correlated in an introductory part and, then, they are discussed separately giving emphasis on the intrinsic characteristics of each topic. In the problem of direction of arrival estimation, the signal model for the sensor array is presented as well as some maximum likelihood-based methods. It consists of a uniform linear array of half-wavelength spaced sensors, which produce phase-shifted copies of a reference signal in each circuit. The time lag is closely related to the direction of arrival of a planar wave. The spatial domain shows complex exponentials whose frequencies should be estimated. The main estimation methods are MODE, MODEX, modified MODEX, and SEAD. They are discussed along this work, but special attention lies on the SEAD method, which is based on the eigenvalue decomposition of the modified spatial correlation matrix. The di˙erence between the two largest eigenvalues constitute an interesting measure, it provides an initial guess for the estimates by presenting prominent peaks near the true direction of arrival positions. By sweeping the entire angular interval with a test angle and calculating the eigenvalues the di˙erential spectrum can be traced. A mathematical analysis of the di˙erential spectrum has shown that it is basically a matrix norm calculation, leading to the definition of the total di˙erential spectrum. Therefore, a matrix-based approach is proposed to estimate the arriving angles. This procedure allows to express the angular spectrum in terms of the true angles, a test angle, the correlation between the signals, the number of sources, and the number of sensors. This mathematical formulation is general and demonstrates the principles of operation of the SEAD method, it is one of the main contributions of the present work. The use of the norm-based approach avoided the need of performing the eigenvalue-decomposition, consequently reducing the overall runtime while enhancing the performance for widely-spaced sources. SEAD method based on matrix norms outperformed the MODE method and its derivatives regarding the root mean square error. However, for a number of sources less than 4, the execution time of SEAD-based method is more time consuming than MODEX and modified MODEX. The estimation of energy pulses in x-ray spectroscopy is not a new topic, however, the microwave SQUID multiplexer is a relatively new approach and it is currently a thriving research subject. The objective of the system is to simultaneously read out the energy of energetic particles that fall into detectors for a large amount of resonators. The detectors may be a transition edge sensor or a metallic magnetic calorimeter. Metallic calorimeters are paramagnetic sensors that are situated in a weak magnetic field. They translate temperature variation into magnetic flux variation. To readout the magnetic flux variations due to charged particles, a superconducting quantum interference device is used. The SQUID is an extremely sensitive interferometer and behaves as a variable inductor. The SQUID is coupled to a termination of a superconducting transmission line and produces changes in the resonant frequency of the circuit. Therefore, particle energy can be readout electronically by demodulating a radio frequency that travels through the resonator. Modulation occurs at the amplitude and phase of the complex carrier. After the discard of intermediate frequencies, mixing down the carrier, a complex low frequency carrier is obtained. The signal can be recovered by demodulating the carrier phase. Due to the similarity with the direction of arrival estimation problem, it is proposed a sensor array-based approach, which profits from the spatial correlation matrix decomposition into its eigenvalues. In this way, an analogy is established between the two problems. For obtaining the correlation matrix, a virtual reference vector is considered, whose phase does not depend on the incidence of particles, i.e. when the system is idle. This reference vector is compared by means of the correlation function with the received data sample by sample. A convenient data vector length is defined, so that a snapshot matrix with two sensors is obtained and, consequently, a 2 × 2 dimension correlation matrix is set. The eigenvalue decomposition can be explicitly solved as a function of the matrix entries, such that a general formulation for the method is presented. In addition, simplifications were imposed that allows its implementation on dedicated hardware. The estimation error is analyzed for the proposed method and the maximum likelihood method. The eigenvalue-based method presented greater robustness against the e˙ects of noise. In some specific cases, in which the maximum likelihood method was not able to properly estimate the energy pulse, the proposed method proved to be quite eÿcient.