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Resolved, 2023/2024, 1r quadrimestre

This project addresses key challenges arising from the application of randomized CUR compression methods to compress impedance matrices resulting from the application of the Method of Moments to integral equations typical of electromagnetic scattering prob- lems. Our primary focus lies in the pivot selection process, which regulates the resulting dimensions of the compressed impedance matrix. To address this concern, pivot allocation strategies are proposed. Once the overall best strategy is identified, the number of pivots required to achieve acceptable compression errors for a given interaction is determined. For simple geometries and uniform meshing, a basic weighted equation based on the 2D definition of the electric field degree of freedom is formulated to easily derive the opti- mum number of pivots. This simplified approach leads to compression times up to four times faster than those of Adaptive Cross Approximation (ACA), the competing simple compression method. However, this strategy proves inadequate for complex geometries with irregular meshing, and a priori knowledge is required to establish an upper bound on the resulting compression error. To handle complex geometries and automatically set this upper bound without prior information, a random forest model is introduced. The random forest approach aims to establish connections between the most critical geomet- ric parameters of the interaction under evaluation and the expected optimal number of pivots to achieve the predefined error. The utilization of random forest regression results in compression times faster than ACA for the same compression error.

Terahertz spectroscopy is a promising method to diagnose ocular diseases, where the cornea is typically imaged by Gaussian beams. However, the beam’s mismatch with the cornea’s spherical surface produces a 5-10 % error in analysis. We investigate cornea spectroscopy with wavefront-modified vector beams, reducing the original analysis error to less than 0.5 %. Vector beams are synthesized by our developed 3D Angular Spectrum Method expanded to vector spherical harmonic presentation, allowing wavefront modification and scattering analysis from 100-layer cornea models. We show that wavefront-modified spherical vector beams possess increased accuracy and non-sensitive focusing on cornea spectroscopy compared to the Gaussian beams. Additionally, we investigate wavefront-modified cylindrical vector beams, which show frequency-dependent scattering power arising from s- and p-polarizations. As a result, these beams are unsuitable for cornea spectroscopy, although they have potential for optical force applications. Wavefront-modified vector beams can be applied to spherical target spectroscopy and optical force applications, such as medicine, medical imaging, and optical tweezers., Assessment of the Graft 332 Rejection Using Millimeter Waves; Academy of Finland (327640), Peer Reviewed, Postprint (published version)

© 2023 Optica Publishing Group under the terms of the Open Access Publishing Agreement. Users may use, reuse, and build upon the article, or use the article for text or data mining, so long as such uses are for noncommercial purposes and appropriate attribution is maintained. All other rights are reserved., The angular spectrum method is a rigorous method to synthesize near and far-field electromagnetic beams from planar field distributions. However, this limitation of planar surfaces has restricted its applicability to beams with simple focal planes. We propose a curved boundary integral method (CBIM) to synthesize electromagnetic beams from arbitrary surfaces to address this limitation and expand the method’s scope to synthesize beams from and between shaped objects. This study presents a detailed theoretical framework behind the CBIM and validates its effectiveness and accuracy with a comprehensive set of simulations. Additionally, we present mathematical proof to support our proposal. The proposed method satisfies Maxwell’s equations and significantly benefits optical systems and inverse beam design. It allows for analyzing electromagnetic forward/backward propagation between optical elements using a single method. It is also valuable for optical force beam design and analysis., Peer Reviewed, Postprint (published version)

Lecture’s notes for the exercises class, including the solution to some exam problems., 2022/2023, 2n quadrimestre, v2023-1.3

2022/2023, 2n quadrimestre, v. 7.3.1

In this paper, we will simulate a Paul’s trap to achieve the confinement of multiple ions using a varying electric field. From multiple simulations and conditions, we will determine the range of the frequency for which trapping of one and two ions is achieved, 2022/2023

In this work, we propose a GPU parallel implementation of the randomized CUR (or Pseudo Skeleton) Approximation to compress the -matrices of linear systems that arise in the discretization of integral equations modeling electromagnetic scattering problems. This compression method is highly parallelizable, in contrast with other similar methods such as the Adaptive Cross Approximation. It involves dense linear algebra computations that can be efficiently implemented on a GPU device. Besides, a stochastic convergence criterion is introduced to minimize the communication between the host and the device. Testing the code with standard cases shows the efficiency and accuracy of the method., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019- 107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB-C21 / AEI / 10.13039/501100011033 and PID-2020-118410RB- C21 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (author's final draft)

A 16 switched beam lens based antenna with a coverage area of ±30° in the horizontal plane and from 0 to -20° in the vertical plane has been built and tested. The intended use of the antenna is as part of a smart repeater servicing the coverage area of a 5G wireless communication system operating in the n258 band of the 5G Frequency Range 2, from 24.25 to 27.5 GHz. The antenna has been built using state-of-the-art materials and components in order to assess the performance of this antenna concept. The overall antenna losses including the feeding network are of the order of 8 dB and the realized gain in the 80% of the coverage area is above 11 dB., This work was funded in part by AEI: PID2019-107885GB-C31/AEI/10.13039/501100011033 and HUAWEI TECHNOLOGIES Italia S.r.l, Peer Reviewed, Postprint (author's final draft)

Resolved, 2022/2023, 2n quadrimestre

We propose a randomized Pseudo Skeleton (or CUR) approximation method to compress the H-matrices of linear systems that arise in the discretization of integral equations in electromagnetic scattering. This method is essentially the Matrix Decomposition Algorithm (MDA) of Michielssen and Boag (1994) with equivalent basis and testing functions equal to a random subset of the original ones. It is highly parellelizable and well suited for fine-granularity architectures as GPUs. As the Adaptive Cross Approximation (ACA), CUR is purely algebraic. The method is tested with standard cases to assess its quality and efficiency., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GBC31 / AEI / 10.13039/501100011033, PID2022-136869NBC31 / AEI / 10.13039/501100011033, PID2020-113832RBC21 / AEI / 10.13039/501100011033 and PID-2020-118410RB-C21 / AEI / 10.13039/501100011033, PDC2022-133091-I00 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

We propose a randomized Pseudo Skeleton (or CUR) Approximation method to compress the H-matrices of linear systems that arise in the discretization of integral equations in electromagnetic scattering. This method is highly parellelizable and well suited for fine-granularity architectures as GPUs. As the Adaptive Approximation Method (ACA), the method is purely algebraic. The method is tested with standard cases to assess its quality and efficiency, This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GBC31 / AEI / 10.13039/501100011033, PID2020-113832RBC21 / AEI / 10.13039/501100011033 and PID-2020-118410RB-C21 / AEI / 10.13039/501100011033, PDC2022-133091-I00 / AEI / 10.13039/501100011033, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

Non-destructive material measurements for structural analysis of building elements are a valuable tool in situations where invasive techniques are unfeasible. In this work, a free-space penetration loss measurement technique is applied to the study of wall structure, relying on educated guesses about the internal composition of the material under test (MUT). Using a setup with antennas and a vector network analyzer (VNA), the sample's transmission coefficients are captured in the 1-18 GHz frequency range and compared to the loss model presented in the ITU-R (International Telecommunication Union-Radiocommunication Sector) Recommendation. Tests are run in a laboratory environment on three wood samples, and in the field on two internal partitions of a standard office building in Finland. The method's limitations and operation are assessed in terms of root mean squared error (RMSE), and the results allow for detection of material anomalies and discontinuities and a general understanding of the MUT's electrical behavior., Las medidas no destructivas para el análisis estructural de elementos de construcción representan una herramienta valiosa en situaciones donde las técnicas invasivas son inviables. En este trabajo se aplica una técnica de medición de pérdidas de penetración en espacio libre al estudio estructural de paredes, basándose en suposiciones fundamentadas sobre la composición interna del material bajo estudio (MUT). Utilizando una configuración con antenas y un analizador de redes vectorial (VNA), se capturan los coeficientes de transmisión de la muestra en la banda de 1 a 18 GHz y se comparan con el modelo de pérdidas presentado en las Recomendaciones de la UIT-R (Unión Internacional de Telecomunicaciones-Sector de Radiocomunicaciones). Las pruebas se realizan en un entorno de laboratorio con tres muestras de madera, y en campo con dos tabiques internos de un edificio de oficinas estándar en Finlandia. Las limitaciones y el funcionamiento del método se evalúan en términos de error cuadrático medio de la raíz (RMSE); los resultados permiten detectar anomalías y discontinuidades del material, y comprender el comportamiento eléctrico del MUT en general., Les mesures no destructives per a l'anàlisi estructural d'elements de construcció suposen una eina valuosa en situacions on les tècniques invasives són inviables. En aquest treball s'aplica una tècnica de mesurament de pèrdues de penetració en espai lliure a l'estudi estructural de parets, basant-se en suposicions fonamentades sobre la composició interna del material sota estudi (MUT). Utilitzant una configuració amb antenes i un analitzador de xarxes vectorial (VNA), es capturen els coeficients de transmissió de la mostra en la banda de 1 a 18 GHz i es comparen amb el model de pèrdues presentat en les Recomanacions de la UIT-R (Unió Internacional de Telecomunicacions-Sector de Radiocomunications). Els tests es realitzen en un entorn de laboratori amb tres mostres de fusta, i en el camp amb dos tabics interns d'un edifici d'oficines estàndard a Finlàndia. Les limitacions i el funcionament del mètode s'avaluen en termes d'error quadràtic mitjà de l'arrel (RMSE); els resultats permeten detectar anomalies i discontinuïtats del material, i entendre el comportament elèctric del MUT en general.

Mie theory is a powerful method to model electromagnetic scattering from a multilayered sphere. Usually, the incident beam is expanded to its vector spherical harmonic representation defined by beam shape coefficients, and the multilayer sphere scattering is obtained by the T-matrix method. However, obtaining the beam shape coefficients for arbitrarily shaped incident beams has limitations on source locations and requires different methods when the incident beam is defined inside or outside the computational domain or at the scatterer surface. We propose a 3D angular spectrum method for defining beam shape coefficients from arbitrary source field distributions. This method enables the placement of the sources freely within the computational domain without singularities, allowing flexibility in beam design. We demonstrate incident field synthesis and spherical scattering by comparing morphology-dependent resonances to known values, achieving excellent matching and high accuracy. The proposed method has significant benefits for optical systems and inverse beam design. It allows for the analysis of electromagnetic forward/backward propagation between optical elements and spherical targets using a single method. It is also valuable for optical force beam design and analysis., Agencia Estatal de Investigación (PID2019-107885GB282 C31/AEI/10.13039, PRE2018-084326); Assessment of the Graft 281 Rejection Using Millimeter Waves; Academy of Finland (327640)., Peer Reviewed, Postprint (published version)

Almost hemispherical scanning capabilities are required for modern wireless communication systems to produce broad coverage without performance degradation. Phased array antennas are commonly used as a fully-electronic beam-steering solution for its rapidity in beam switching. However, the effective aperture for high scanning angles is reduced, causing a reduction of the gain with the cosine of the elevation angle. Quasi-optical systems are used to achieve high directivity for wide scanning in combination with phased arrays. An interesting solution is the dielectric dome antenna, where rotationally symmetric dielectric lenses are used to enhance the scanning performance of an antenna with limited scanning capabilities. Using a full-wave simulator to evaluate lenses combined with arrays is very time-consuming, making the lens design inefficient and laborious. In this work is presented a Ray-Tracing tool used to simulate in a fast and efficient way the far-field of two-dimensional dielectric lenses. While a full-wave simulation of a three-dimensional lens could take approximately 3 hours, the Ray-Tracing evaluation takes less than 3 minutes, making possible the full optimization and design of these lenses. A numerical calculation of the ray path is used to evaluate the phase of the electric field at the lens aperture, while the amplitude is evaluated using ray-tube power theory. The far-field radiation pattern of the lens antenna is calculated using the Kirchhoff-Fresnel diffraction formula. In this work, it is also presented a full study of the reflection and absorption losses, which is something that was not previously done in the state of the art. Matching layers are designed and evaluated to reduce the reflection losses at each interface. To demonstrate the effectiveness of this approach, we compare the radiation patterns produced by a two-dimensional dielectric dome antenna with those computed using COMSOL, showing a significant reduction in time and computationa

Colección de problemas resueltos, 2022/2023, 1r quadrimestre

Incluye todos los problemas de examen, 2022/2023, 1r quadrimestre, v 1.19

Full classnotes, 2022/2023, 1r quadrimestre, 3.9

Automotive radars can use the wide band at extremely high frequencies (EHF) between 30 GHz and 300 GHz, which has a higher amount of free frequency bands than the lower part of the electromagnetic spectrum. The focus of this thesis is to develop a horn antenna to work in this band with the central frequency of 77 GHz, a Field of View (FOV) of 20º on Elevation and 130º on Azimuth and a reflection coefficient (S11) below -15 dB. It was possible to accomplish the requirements set by the company with a single horn antenna, which is a less complex and more compact solution. This thesis presents the results with a brief review of a two-element antenna array for possible future works., Los radares de automoción pueden utilizar la banda ancha a frecuencias extremadamente altas (EHF) entre 30 GHz y 300 GHz, que tiene una mayor cantidad de bandas de frecuencia libres que la parte inferior del espectro electromagnético. El objetivo de esta tesis es desarrollar una antena de bocina para trabajar en esta banda con la frecuencia central de 77 GHz, un campo de visión de 20º en elevación y 130º en azimut y un coeficiente de reflexión (S11) por debajo de los - 15 dB. Fue posible cumplir los requisitos establecidos por la empresa con una sola antena de bocina, que es una solución menos compleja y más compacta. Esta tesis presenta los resultados con una breve revisión de una matriz de antenas de dos elementos para posibles trabajos futuros., Els radars d'automoció poden utilitzar la banda ampla a freqüències extremadament altes (EHF) entre 30 GHz i 300 GHz, que té una major quantitat de bandes de freqüència lliures que la part inferior de l'espectre electromagnètic. L'objectiu d'aquesta tesi és desenvolupar una antena de botzina per treballar en aquesta banda amb la freqüència central de 77 GHz, un camp de visió de 20º en elevació i 130º en azimut i un coeficient de reflexió (S11) per sota dels -15 dB. Va ser possible complir els requisits establerts per l'empresa amb una sola antena de botzina, que és una solució menys complexa i més compacta. Aquesta tesi presenta els resultats amb una breu revisió d'una matriu d'antenes de dos elements per a possibles treballs futurs.

Detection of bioparticles is of great importance in electrophoresis, identification of biomass sources, food and water safety, and other areas. It requires a proper model to describe bioparticles’ electromagnetic characteristics. A numerical study of Escherichia coli bacteria during their functional activity was carried out by using two different geometrical models for the cells that considered the bacteria as layered ellipsoids and layered spheres. It was concluded that during cell duplication, the change in the dielectric permittivity of the cell is high enough to be measured at radio frequencies of the order of 50 kHz. An experimental setup based on the capacitive Wheatstone bridge was designed to measure relative changes in permittivity during cell division. In this way, the theoretical model was validated by measuring the dielectric permittivity changes in a cell culture of Escherichia coli ATTC 8739 from WDCM 00012 Vitroids. The spheroidal model was confirmed to be more accurate., This research was funded by the Agencia Estatal de Investigación PID2019-107885GB-C31/AEI/10.13039, PRE2018-084326., Peer Reviewed, Postprint (published version)

En ingeniería de antenas, uno de los aspectos más apreciados es la capacidad de apuntar el haz principal electrónicamente. Una de las técnicas más ampliamente usadas para conseguirlo es el escaneo mediante agrupaciones de antenas, aunque requiere el uso de redes de alimentación complejas que sufren considerables pérdidas a altas frecuencias. Por esta razón se está llevando a cabo investigación en el campo del escaneo frecuencial, basado en la radiación de LWA. Su desempeño a altas frecuencias, la alimentación simple y el rastreo rápido las convierten en una opción idónea para aplicaciones como radar de bajo coste, sistemas de comunicaciones 5G, sistemas de vigilancia y seguimiento y sistemas de imagen. El propósito principal del proyecto es diseñar un prototipo de LWA 1D periódica que sirva como base para fabricar un modelo más avanzado con uso potencial para un sistema de imagen radar. La antena está formada por corrugaciones de línia Goubau y parches rectangulares. Se han comparado dos diseños para la linia de transmisión, con corrugaciones con simetría longitudinal y simetría de deslizamiento, tomando especial atención a sus diagramas de dispersión. También se ha llevado a cabo un estudio paramétrico de las dimensiones de la estructura. La disposición de los parches se ha realizado en dos tiras, desplazadas una respecto de la otra, con el fin de estudiar el efecto de este desplazamiento sobre el diagrama de radiación. El escaneo por frecuencia de la antena se ilustra más adelante. Finalmente, la longitud de los parches ha sido modificada por tal de lograr una reducción del nivel de los lóbulos secundarios. En este trabajo se presentan dos prototipos. El primero opera a una frecuencia central de 9,6 GHz, con una eficiencia del 80%. Es capaz de escanear desde -50 a 50 grados en un margen frecuencial que varía desde 8,5 hasta 10,2 GHz. El segundo prototipo se cararcteriza por una reducción de los lóbulos secundarios, un haz principal más estrecho y cobertura uniform, En enginyeria d'antenes, un dels aspectes més apreciats és la capacitat d'apuntar el feix principal electrònicament. L'escaneig fent ús d'agrupacions d'antenes és una de les tècniques més àmpliament utilitzades per aquest fi, tot i que requereix l'ús de xarxes d'alimentació complexes que pateixen moltes pèrdues a altes freqüències. És per això que s'està duent a terme recerca en antenes d'escaneig per freqüència, que es basen en la radiació de LWA. El seu rendiment a altes freqüències, la simplicitat de la xarxa d'alimentació i el rastreig ràpid les converteix en una opció perfecta per aplicacions com radars de baix cost, sistemes de comunicacions 5G, sistemes de vigilància i seguiment i sistemes d'imatge. El principal propòsit del projecte és dissenyar un prototip de LWA 1D periòdica que serveixi com a base per fabricar un producte més avançat amb ús potencial per un sistema d'imatge radar. L'antena està formada per corrugacions de línia Goubau i elements radiants rectangulars. S'han comparat dos dissenys de la línia de transmissió amb corrugacions amb simetria longitudinal i simetria lliscant, prenent especial atenció als seus diagrames de dispersió. També s'ha dut a terme un estudi paramètric de les dimensions de l'estructura. La disposició dels elements radiants s'ha realitzat en dues tires, desplaçades una respecte l'altra, per estudiar l'efecte d'aquest desplaçament al diagrama de radiació. L'escaneig per freqüència de l'antena s'il·lustra més endavant. Finalment, la llargada dels elements radiants ha estat modificada per tal d'aconseguir reduir el nivell dels lòbuls secundaris. En aquest treball es presenten principalment dos prototips. El primer d'ells opera a una freqüència central de 9,6 GHz, amb una eficiència del 80%. És capaç d'escanejar, aproximadament, des de -50 a 50 graus en un rang freqüencial d'entre 8,5 fins a 10,2 GHz. El segon prototip es caracteritza per una reducció dels lòbuls secundaris un feix principal més estret i cobertura uniforme. Ope, In antenna engineering, one of the most valuable assets is the capability of steering the main beam direction electronically. Phased-array scanning is a well known technique to do so, although it requires a complex feeding network which becomes very lossy at high frequencies. This is why research is being carried out in the field of frequency scanning, which is based on radiation of leaky-wave antennas. Their performance at high frequencies, simple feeding and fast tracking features makes them the perfect choice for applications such as low cost radar, 5G communication systems, surveillance and imaging systems. The project main purpose is to design a prototype of a one dimensional periodic leaky-wave antenna (LWA) which will serve as a basis to future manufacturing of a more advanced product with potential use for an imaging radar system. The antenna consists of Goubau line corrugations with rectangular radiation patches. Longitudinal symmetric (LS) and glide symmetric (GS) corrugations of the transmission line have been compared, paying special attention to their dispersion diagrams. A parametric study of the dimensions of the structure has been conducted as well. Two lines of radiation patches have been added with different offsetings between each other, in order to study the effect of the separation between them to the radiation pattern. The frequency beam scanning characteristic of the antenna is also depicted. Finally, the length of the patches is tailored so as to obtain side lobe level (SLL) reduction. Two main prototypes are presented. The first one operates at a center frequency of 9.6 GHz, with an efficiency of 80 %. It scans over an approximate angle width from -50 to 50° in a frequency range from 8.5 to 10.2 GHz. The second one features SLL reduction, a narrow main beam and even coverage. It operates at a center frequency of 9.6 GHz with an efficiency over 78 % scanning from -35 to 20°. Both designs have been modeled using CST Microwave Studio and Matlab

A numerical approach for the modelling of dipole Photoconductive Antennas (PCAs) is presented, based on the classical Drude model with extension to include the bias field screening due to both the space charge and the THz radiation fields. By employing a fixed-point iterative algorithm, good computational efficiency and convergence are obtained for typical PCA configuration parameters. Results unveiling the role played by each of the different phenomena and the radiated THz spectra for different dipole lengths are shown and discussed. © 2022 IEEE., Peer Reviewed, Postprint (published version)

Apuntes de teoría, 2022/2023, 1r quadrimestre, v 5.0.1

In this work, we propose Monte Carlo sampling as a technique to avoid communication overhead in the numerical solution of complex electromagnetic scattering problems. The method is based on statistically sampling a formal representation of the solution of the scattering problem, based on a modified Born series. The sampling process can be independently carried out in different processors with no need for communication between them. This lack of communication makes the algorithm suitable for massively parallel computational environments., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019-107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB- C21 / AEI / 10.13039/501100011033 and TEC2017- 84817-C2-2-R / AEI / 10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM-2016- 0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigaci ´on, Spain, and Catalan Research Group 2017 SGR 219, and Ag`encia de Gesti ´o d’Ajuts Uni- versitaris i de Recerca with grant 2021 FI B2 00096., Peer Reviewed, Postprint (published version)

In this work, we develop a novel Monte Carlo method for solving the electromagnetic scattering problem for clusters of photonic particles. The method is based on a formal solution of the scattering problem as a modified Born series whose coefficients are found by a conformal transformation. The terms of the Born series are approximated by sampling random elements of its matrix representation, computed by the Method of Moments. Unlike other techniques like the Fast Multiple Method, this Monte Carlo method does not require communication between processors, which makes it suitable for large parallel executions., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects PID2019- 107885GB-C31 / AEI / 10.13039/501100011033, PID2020-113832RB-C21 / AEI / 10.13039/501100011033 and TEC2017-84817-C2-2-R / AEI / 10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM- 2016-0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigación, Spain, and Catalan Research Group 2017 SGR 219 and grant 2021 FI B2 00096., Peer Reviewed, Postprint (author's final draft)

The solution of the electromagnetic scattering problem by the Method of Moments often leads to huge systems of equations that require large computational resources. In parallel computing environments, communication between computing units constitute the main bottleneck that limits the scalability of the problem. In this work, we use randomized techniques to relax the need for communication, thus achieving better scalability. The method is based on a formal solution of the scattering problem as a Neumann series modified by a set of coefficients who ensure its convergence. In the case of the Volume Integral Equation for dielectric objects, these coefficients are obtained by a conformal transformation that depends on the spectrum of the scattering operator, which is known to be confined in a certain region of the complex plane thanks to the spectral localization theorem. Then, the terms of the Neumann series are approximated by randomly sampling elements of its matrix representation computed by the method of moments. We show that this estimator is unbiased and, unlike other Monte Carlo techniques based on the Neumann series, always converges to the solution. The main advantages of this method are its ease of formulation and its potential parallelism. Since the coefficients of the modified Neumann series are known beforehand, the Monte Carlo samples can be computed independently without the need of communication between different processors or computing units, apart from a final reduction operation. Thus, the method is scalable and well suited for highly parallel architectures. Besides, the granularity of the parallel method can be adapted in a versatile way to exploit the capabilities of a particular computational environment.

The conventional scattering analysis of perfectly conducting sheets ignores the scattering contribution of the rim. The so-called thin-surface scheme arises from the discretization of the Electric-field Integral Equation by assuming the sheet thickness to be null. This gives rise to many less unknowns than the full approach, arising from modelling the whole plate. Although good accuracy is observed in many practical applications, the thin-plate scattering analysis of thick enough sheets, especially under oblique incidences, with low grazing angles, exhibits great inaccuracies. A recent approach, so-called thick-surface, has proved to show similar accuracy as the full scheme, also in those cases where the thin-surface scheme fails, with moderate computational effort. In this paper, we show RCS results computed with the thick-surface approach for several sheets with nonzero thickness where the thin-surface scheme fails. Several types of meshes are adopted, triangular, quadrangular or hybrid. The required symmetries for the generation of the meshes are pointed out in each case.

Les cordes de flux magnètic heliosfèriques (MFRs) són considerades les estructures internes fonamentals de les Ejeccions de Massa Coronal Interplanetàries (ICMEs). Aquest treball es centra en determinar si donada una corda de flux magnètic amb frontera lliure, aquesta és estable o no, trobant el ritme al qual una inestabilitat pot créixer en el temps. El mètode de la velocitat de creixement va ser introduït per Linton et. al. (1996) per la inestabilitat de tipus kink, i va ser aplicat per Florido-Llinas et. al. (2020) per la predicció de rotacions en ICMEs en expansió. En la present tesi, treballem en la justificació matemàtica i la identificació de les assumpcions clau darrere aquest mètode, a la vegada que el generalitzem per tal d’analitzar altres inestabilitats del plasma que podem afectar a la secció transversal de la corda de flux magnètic. Aquest estudi ha estat aplicat a l’equilibri de configuracions de camp magnètic inspirades en el model de Gold-Hoyle, amb l’objectiu d’explorar si la torsió i forces de Lorentz estan relacionades amb l’estabilitat de les cordes de flux magnètic., Las cuerdas de flujo magnético heliosféricas (MFRs) son consideradas las estructuras internas fundamentales de les Eyecciones de Masa Coronal Interplanetarias (ICMEs). Este trabajo se centra en determinar si dada una cuerda de flujo magnético con frontera libre, esta es estable o no, encontrando el ritmo al cual la inestabilidad puede crecer en el tiempo. El método de la velocidad de crecimiento fue introducido por Linton et. al. (1996) para la inestabilidad de tipo kink, y fue aplicado por Florido-Llinas et. al. (2020) para la predicción de rotaciones en ICMEs en expansión. En la presente tesis, trabajamos en la justificación matemática y la identificación de las asunciones clave detrás de este método, a la vez que lo generalizamos con el fin de analizar otras inestabilidades del plasma que pueden afectar a la sección transversal de la cuerda de flujo magnético. Este estudio ha sido aplicado al equilibrio de configuraciones de campo magnético inspiradas en el modelo de Gold-Hoyle, con el objetivo de explorar si la torsión y fuerzas de Lorentz están relacionadas con la estabilidad de las cuerdas de flujo magnético., Heliospheric magnetic flux ropes (MFRs) have been found to be a fundamental internal structure of Interplanetary Coronal Mass Ejections (ICMEs). This work focuses on determining whether a given free-boundary flux rope structure is stable or not, by finding the rate at which an instability may evolve in time. The growth rate method was introduced by Linton et al. (1996) for the kink instability, and was applied by Florido-Llinas et al. (2020) to the forecasting of rotations in expanding ICMEs. In the present thesis, we work on the mathematical justification and the identification of the key assumptions behind this method, as well as generalizing it to analyze other plasma instabilities that affect the cross-section of the flux rope. Such a study has been applied to equilibrium magnetic field configurations inspired in the Gold-Hoyle model, in an attempt to explore whether the internal twist and Lorentz forces are related to the flux rope stability., Outgoing

Resolved, 2021/2022

Improving the longitudinal modes coupling in layered spherical structure contributes significantly to corneal terahertz sensing, which plays a crucial role in the early diagnosis of cornea dystrophies. Using a steel sphere to calibrate reflection from the cornea sample assists in enhancing the resolution of longitudinal modes. The requirement and challenges toward applying the calibration sphere are introduced and addressed. Six corneas with different properties are spotted to study the effect of perturbations in the calibration sphere in a frequency range from 100 GHz to 600 GHz. A particle-swarm optimization algorithm is employed to quantify corneal characteristics considering cases of accurately calibrated and perturbed calibrated scenarios. For the first case, the study is carried out with signal-to-noise values of 40 dB, 50 dB and 60 dB at waveguide bands WR-5.1, WR-3.4, and WR-2.2. As expected, better estimation is achieved in high-SNR cases. Furthermore, the lower waveguide band is revealed as the most proper band for the assessment of corneal features. For perturbed cases, the analysis is continued for the noise level of 60 dB in the three waveguide bands. Consequently, the error in the estimation of corneal properties rises significantly (around 30%)., This research was funded by the Agencia Estatal de Investigación PID2019-107885GB-C31/AEI/10.13039, PRE2018-084326, and Academy of Finland Project number 327640., Peer Reviewed, Postprint (author's final draft)

Coupling to longitudinal modes of thin spherical shells, under Gaussian-beam illumination, was explored with a theoretical method based on Fourier-optics analysis and vector spherical harmonics and was scrutinized with an experimental setup. For the theory part, the illumination frequency band was fixed between 100–600 GHz and the outer spherical shell radius of curvature and thickness are 7.5 mm and 0.5 mm, respectively. The shell material was either the lossless cornea or an aqueous effective media representing the cornea. Six different beam-target strategies were introduced being potential candidates for maximum coupling. Two dispersion-tuned beam ensembles with strongly frequency-dependent phase center location have been created with a fixed incident beam 1/e radius and radius of curvature called forward strategies. These computations of different alignments were continued with four beam ensembles of frequency-invariant phase center, constructed from fits to experimental data, oriented at four different axial locations with respect to the spherical shell center of curvature, they are called reverse strategies. Coupling efficiency for all strategies was calculated for different targets including perfect electrical conductor (PEC) sphere, PEC core covered by a cornea loss-free layer and cornea. All scattering strategies contrasted to scattering from equivalent planar targets as a reference with maximum coupling. The results show that, under an ideal calibration, forward strategies are a closer approximation to the plane-wave condition for the cornea. An experimental setup was assembled to explore the simulation approach in a frequency range between 220 GHz to 330 GHz. Two different quartz samples with permittivity of 4.1 were mounted on a water core, acting for a cornea. The first and second quartz radius and thickness were 7.5 mm and 0.5 mm and 8 mm and 1 mm, respectively. An adequate agreement between theory and experiment was confirmed. A particle optimisation, Funding. Agencia Estatal de Investigación (PID2019-107885GB-C31/AEI/10.13039, PRE2018-084326); Academy of Finland (327640)., Peer Reviewed, Objectius de Desenvolupament Sostenible::3 - Salut i Benestar, Postprint (author's final draft)

Resolved, 2021/2022

This article addresses the efficient numerical analysis of sparsely distributed small modifications in a large structure, that is, the sparsely distributed partial modification problem (SDPMP). A hierarchical matrix (H-matrix)-based fast direct solver is adopted to solve the SDPMP in a very short time for each new set of modifications. The main idea of this method is to reuse the parts of the compressed inverse matrix operator that do not change and only recompute the modified parts. In contrast with already existing algorithms designed to tackle a single localized modification, this new approach is still very fast for sparsely distributed small modifications, so it becomes a very attractive option to solve optimization problems efficiently, due to the very cheap analysis of the modified structure at each iteration of the optimization procedure. The efficiency of the solution of SDPMP is demonstrated for several cases involving passive and active metasurfaces and a reconfigurable antenna. As a practical example, a programmable metasurface reflector is optimized with particle swarm optimization to obtain a prescribed reflection pattern., This work was supported in part by the National Nature Science Foundation of China under Grant 61871219; in part by the China Scholarship Council under Grant 201906830042; in part by the Startup Foundation for Introducing Talent of the Nanjing University of Information Science and Technology (NUIST) under Grant 2022r079; in part by the Ministerio de Ciencia e Innovacion (MICINN) under Project PID2019- 107885GB-C31/AEI/10.13039/501100011033, Project PID2020-113832RBC21/AEI/10.13039/501100011033, and Project TEC2017-84817-C2-2-R/AEI/ 10.13039/501100011033; and in part by the Unidad de Excelencia Maria de Maeztu, which is financed by the Agencia Estatal de Investigación, Spain, under Grant MDM-2016-0600/AEI/10.13039/501100011033; and in part by the Catalan Agency for Management of University and Research Grants (AGAUR) under Research Group under Grant 2017 SGR 219., Peer Reviewed, Postprint (author's final draft)

Resolved, 2021/2022

Submillimeter-wave and THz sensing of cornea leverage the layered tissue structure for assessments of corneal water content and thickness. The cornea is bounded by air on the anterior and an optically thick body of water on the posterior and thus presents as a lossy thin film lying atop a lossy termination. Resolution of the cornea’s lossy longitudinal modes via frequency domain reflectometry in a band sufficiently low (e.g. 220 GHz – 330 GHz) for significant penetration allows simultaneous estimates of corneal tissue water content (CTWC) and central corneal thickness (CCT). However, since the cornea is spherical, efficient coupling to longitudinal modes requires normal incidence across the interrogated area and thus a converging spherical phase front whose curvature matches the corneal surface curvature [1, 2]., Peer Reviewed, Postprint (published version)

The conventional scattering analysis of perfectly conducting plates neglects the scattering contribution of the rim in the discretization of the Electric-Field Integral Equation. This so-called thin-plate scheme manages many less unknowns than the full approach, arising from modelling the whole plate, with acceptable accuracy in many practical applications. A recent approach, so-called thick-plate, has proved to show similar accuracy as the full scheme, also in those cases where the thin-surface fails; namely, the scattering analysis of thick enough plates, especially under oblique incidences, with low grazing angles. In this paper, we reveal how the thick-plate scheme shows improved computational times, especially in large scale computations, as compared to the full approach. Also, we discuss how the thick-plate analysis is amenable to parallelization, thereby leading to computational times comparable with the thin-surface approach., This work was supported by FEDER and the "Spanish Plan Estatal de Investigación Científica y Técnica y de Innovación", under projects: TEC2017-84817-C2-2-R/ AEI/10.13039/501100011033, TEC2016-78028-C3-1-P/AEI/10.1 3039/501100011033, PID2019-107885GB-C31/AEI/10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM-2016-0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigación, Spain, and Catalan Research Group 2017 SGR 219., Peer Reviewed, Postprint (author's final draft)

Resolved

The scattering analysis of perfectly conducting plates with the method-of-moment discretization of the Electric-Field Integral Equation has been traditionally carried out according to the thin-sheet approximation, which neglects the current contributions over the plate rims. The plate is then modelled as an open surface and handles a restricted number of unknowns as compared with the canonical modelling as a closed object. The thin-sheet approximation, although used in many practical applications, becomes inaccurate for thick enough plates or plates under plane wave incidence at grazing angles. Recently, a thick-plate approach has been presented to address the time-efficient and accurate scattering analysis of thick plates. This scheme exploits the symmetry properties of fields and currents thereby giving rise to two uncoupled problems with roughly half the number of unknowns of the original problem. In this paper, we extend this thick-plate scheme to the efficient scattering analysis of thick curved sheets., This work was supported by FEDER and the "Spanish Plan Estatal de Investigación Científica y Técnica y de Innovación", under projects: TEC2017-84817-C2-2-R/ AEI/10.13039/501100011033, TEC2016-78028-C3-1-P/AEI/10.13039/501100011033, PID2019-107885GB-C31/AEI/10.13039/501100011033 and the Unidad de Excelencia Maria de Maeztu MDM-2016-0600/AEI/10.13039/501100011033, which is financed by the Agencia Estatal de Investigación, Spain, and Catalan Research Group 2017 SGR 219., Peer Reviewed, Postprint (author's final draft)

In this work, we propose a method to enhance the convergence of the Born series. The Born series is widely used in scattering theory, but its convergence is only guaranteed under certain restrictive conditions which limit the cases where this formulation can be applied. The proposed method, based on modifying the singularities of the resolvent operator by a change of variables, accelerates the convergence of the series or even achieves convergence in otherwise divergent cases. Due to its low computational cost and ease of implementation, the method preserves the advantages of the Born series while it extends its range of application. It can also be applied to a variety of methods relying on the same mathematical principles as the Born series. Numerical examples are provided to show that a significant improvement can be achieved with simple techniques and with a limited knowledge about the operator spectrum., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects TEC2016-78028-C3-1-P, MDM2016-0600, TEC2017-83343-C4-2-R, TEC2017-84817-C2-2-R, PID 2019-107885GB-C31 and Catalan Research Group 2017 SGR 219; and by Secretary of Universities and Research of the Ministry of Business and Knowledge of the Generalitat of Catalonia and European Social Fund under scholarship 2019 FI B00788., Peer Reviewed, Postprint (author's final draft)

This thesis is a study of different plasma antennas through a parametric analysis. The goal is to understand the particularities of plasma antennas. Such devices use an energized gas as a conductor to transmit and receive electromagnetic waves. By altering the properties of the gas, the radiation and reception properties of the antenna can be modified. Plasma antennas were first conceived in the early twentieth century. However, they still present some implementation challenges. In this thesis, models ranging between different degrees of complexity are simulated and studied. Finally, some conclusions regarding the implementation, applications and widespread use are drawn. The results obtained along with publications about the state-of-the-art are a glimpse into the many possibilities of plasma antennas., Este TFG es un estudio de diferentes antenas de plasma mediante el análisis paramétrico. El objetivo era comprender las particularidades de las antenas de plasma. Este tipo de dispositivos utilizan un gas energizado como conductor para transmitir y recibir ondas electromagnéticas. Alterando las propiedades del gas, se pueden modificar los parámetros de radiación y recepción de la antena. Las antenas de plasma fueron concebidas a principios del siglo veinte. Sin embargo, todavía presentan retos a la hora de ser implementadas. En esta tesis, se simulan y estudian modelos con diferentes grados de complejidad. Finalmente, se obtienen algunas conclusiones sobre la implementación, aplicaciones y uso generalizado. Los resultados obtenidos junto con artículos sobre el estado del arte, conforman un buen resumen sobre las muchas posibilidades de las antenas de plasma., Aquest TFG és un estudi de diferents antenes de plasma mitjançant l?anàlisi paramètric. L?objectiu era entendre les particularitats de les antenes de plasma. Aquest tipus de dispositius utilitzen un gas energitzat com a conductor per a transmetre i rebre ones electromagnètiques. Alterant les propietats del gas, es poden modificar els paràmetres de transmissió i recepció de l?antena. Les antenes de plasma es van concebre a començaments del segle vint. Tot i això, encara presenten obstacles a l?hora de ser implementades. En aquest TFG se simulen i estudien models amb diferents nivells de complexitat Finalment, s?obtenen algunes conclusions sobre la implementació, aplicacions i ús generalitzat. Els resultats obtinguts juntament amb publicacions sobre l?estat de l?art són resum sobre les àmplies possibilitats de les antenes de plasma.

The present article consists in the modelling an design of a Paul ion trap using the Method of Moments for numerical integration. Firstly, we have developed some preliminary work with simple geometries in two and three dimensions. Then we have designed the geometry of the trap, consisting in two hiperboloid plates which have been discretized, triangulated, and solved by the Method of Moments. The potential in the plates has been set to vary harmonically, since sinusoidal signals have proved to be more e ective than triangular and square signals. We have then simulated the con nement of one, two and multiple ions, and we have studied how the chosen frequency and potential a ect the e ectiveness of the trap, 2020/2021

In this work, we develop a novel Monte Carlo method for solving the electromagnetic scattering problem. The method is based on a formal representation of the solution as a modified Born series whose coefficients are found by a conformal transformation. The terms of the Born series are approximated by sampling random elements of its matrix representation, computed by the Method of Moments. This method requires almost no communications between cores, which makes it suitable for large scale parallelization., This work was partly funded by Unidad de Excelencia Maria de Maeztu MDM-2016- 0600/AEI/10.13039/501100011033, ID2019-107885GBC31/ AEI/10.13039/501100011033, TEC2017-83343-C4- 2-R/AEI/10.13039/501100011033, TEC2017-84817-C2-2- R/AEI/10.13039/501100011033, Catalan Research Group 2017 SGR 219 and Secretary of Universities and Research of the Ministry of Business and Knowledge of the Generalitat of Catalonia and European Social Fund under scholarship 2019 FI B00788., Peer Reviewed, Postprint (published version)

Electrical and Telecommunication Engineering is nowadays playing a fundamental role in biomedical research and its applications. Microwave and Antenna Group at EPFL are working resolutely to help in this vital field for society, as it has been unfortunately proven once again during the sanitary crisis caused by the COVID-19 outbreak in the last months. With this final degree thesis, I have tried to make my humble contribution to this broad topic, focusing on the field of Radio Frequency Identification (RFID). To do so, the main objective of this work was to study the frequency splitting phenomenon and verify whether it is plausible to identify a 2mm electrically small passive loop antenna using another active antenna of the same characteristics. It has been found that for close distances with no dielectric boundaries (up to 2.5mm for this configuration), analysing the frequency splitting phenomenon is an effective way to correctly confirm the presence of the passive antenna (or transponder), and it is even possible to determine its distance relative to the active antenna (or reader). When a dielectric environment is added, frequency splitting can be detected as well. However, in this case, the detectable distance range decreases due to the high relative permittivity conditions. These findings lead to a feasible way to implement a low-complexity 1-bit transponder for short-range applications., La Ingeniería Eléctrica y de Telecomunicaciones tiene actualmente un papel fundamental en la investigación biomédica y sus aplicaciones. El Grupo de Microondas y Antenas de la EPFL trabaja decididamente para ayudar en este campo vital para la sociedad, como ha sido lamentablemente demostrado una vez más durante la crisis sanitaria provocada por la pandemia de COVID-19 en los últimos meses. Con esta tesis de fin de grado, he tratado de hacer mi humilde aportación a este amplio tema, centrándome en el campo de la Identificación por Radiofrecuencia (RFID). Para ello, el objetivo principal de este trabajo ha sido estudiar el fenómeno de división frecuencial (conocido en inglés como frequency splitting) y verificar si es posible identificar una antena pasiva eléctricamente pequeña del tipo espira elemental de 2 mm utilizando otra antena activa de las mismas características. Se ha encontrado que para distancias cortas sin condiciones de dieléctrico (hasta 2.5 mm para esta configuración), analizar el fenómeno de división frecuencial es una forma efectiva de confirmar correctamente la presencia de la antena pasiva (o transpondedor), e incluso es posible determinar su distancia relativa a la antena activa (o lector). Cuando se añade un entorno dieléctrico, también se puede detectar la división frecuencial, pero en distancias más limitadas para condiciones de permitividad relativa alta. Estos hallazgos conducen a una forma viable de implementar un transpondedor de 1 bit de baja complejidad para aplicaciones de corto alcance., L'Enginyeria Elèctrica i de Telecomunicacions té actualment un paper fonamental en la recerca biomèdica i les seves aplicacions. El Grup de Microones i Antenes (MAG) de l'EPFL treballa decididament per tal d'ajudar en aquesta àrea vital per al conjunt de la societat, com ha estat malauradament demostrat una vegada més durant la crisi sanitària derivada de la pandèmia causada pel COVID-19 els darrers mesos. Amb aquest treball de fi de grau, he intentat fer la meva modesta aportació a aquest ampli tema, centrant-me en el camp de la Identificació per Ràdio Freqüència (RFID). A tal efecte, el principal objectiu d'aquest treball ha estat estudiar el fenomen de la divisió freqüencial (conegut en anglès com frequency splitting) i verificar si és possible identificar una antena elèctricament petita passiva del tipus espira elemental de 2mm de diàmetre mitjançant una altra antena activa de les mateixes característiques. S'ha comprovat que per a distàncies properes sense condicions de dielèctric (fins a 2,5mm en aquesta configuració), analitzar el fenomen de divisió freqüencial és una manera eficaç de confirmar correctament la presència de l'antena passiva (o transponedor), i és fins i tot possible determinar-ne la distància respecte a l'antena activa (o lector). Quan s'hi afegeix un entorn dielèctric, el fenomen de la divisió freqüencial pot ser detectat també, encara que en distàncies més limitades en condicions de permitivitats relatives elevades. Aquests resultats porten a una implementació viable d'un detector d'un bit de baixa complexitat per a aplicacions de curt abast.

Resolved

Wave scattering from a human cornea illuminated with a submillimeter-wave Gaussian beam is explored with Fourier analysis. This new approach enabled us to investigate the cornea as a coated sphere rather than a homogenized one. The cornea was modeled as an aqueous spherical shell using effective medium theory, with 60 percent water, enclosing a sphere of pure water. The corneal model was illuminated at 220 GHz – 330 GHz. The interaction of the incident and back-reflected beam, back-scattered field, and back-scattering from one usual beam-cornea alignment scheme were evaluated; beam waist collocated with the surface apex. The result indicates the amount of difference between the reflection from planar stratified and back-scattering from the cornea in the case of focusing the beam waist at the corneal apex., Peer Reviewed, Postprint (author's final draft)

The accuracy of the Adaptive Cross Approximation (ACA) algorithm, a popular method for the compression of low rank matrix blocks in Method of Moment computations, is sometimes seriously compromised by unpredictable errors in the convergence criterion. This paper proposes an alternative criterion, based on global sampling of the error in the elements of the ACA compressed matrix. The sampling error depends on the size of the sample but also on the population distribution of the error, which makes it difficult to control the error independently of the underlying problem. However, as argued and demonstrated in the paper, the distribution of the error converges to the same unique probability distribution function for all low rank matrices. Complementing the sampling criterion with a simple mechanism to detect this convergence, we arrive at a criterion that controls the error irrespective of the underlying problem. As a practical example the RCS of a moderate size metallic ogive is computed to illustrate the merits of the proposed criterion. The proposed algorithm may also be useful in other methods that approximate low-rank matrices by interpolation of a reduced set of its elements., This work was partly funded by the Ministerio de Ciencia e Innovacion (MICINN) under projects TEC2016-78028-C3-1-P, TEC2017-84817-C2-2-R, TEC2017-83343-C4-2-R, PID 2019-107885GBC31, MDM2016-0600, and Catalan Research Group 2017 SGR 219., Peer Reviewed, Postprint (author's final draft)

An antenna array is considered. The objective is to find suitable excitation of elements of the array, which approximates the required radiation pattern to the one achieved by the array. The quality of the approximation will be evaluated as a quadratic error with respect to the required radiation pattern. In addition, it is considered that excitation can only have quantized states and, in the limit case, it would have only two states. It is also intended to find different methods which avoids the usage of brutal force when deciding which quantized state provides the lowest quadratic error. Furthermore, the formalisation of the different procedures is required in order to provide reliable and consistent synthesis methods. Finally, it will also provide an optimized Matlab software capable of performing the necessary calculations for quantized synthesis methodologies, Considerem una agrupació d’antenes. L’objectiu és trobar quines són les excitacions adequades que s’han d’aplicar als elements de l’agrupació, les quals aproximen el diagrama de radiació requerit amb el que l’agrupació pot aconseguir. La qualitat de l’aproximació serà avaluada per l’error quadràtic respecte el diagrama de radiació requerit. A més, es considera que les excitacions només poden tindre estats quantificats i, en el cas límit, estarà formada només per dos estats. Es buscarà també trobar diferents mètodes els quals evitin l’ús de força bruta en el moment de decisió entre quin estat quantificat proporciona el menor error quadràtic. Tanmateix, la formalització dels diferents mètodes s’haurà d’assolir per tal de proporcionar mètodes de síntesi que siguin fiables i consistents. Finalment, es realitzarà un programa de Matlab optimitzat que serà capaç de realitzar els càlculs pertinents per dur a terme les metodologies de la síntesi quantificada, Consideramos una agrupación de antenas. El objetivo es encontrar cuáles son las excitaciones adecuadas que se deben aplicar a los elementos de la agrupación, las cuales aproximan el diagrama de radiación requerido con el que la agrupación puede conseguir. La calidad de la aproximación será evaluada por el error cuadrático respecto al diagrama de radiación requerido. Además, se considera que las excitaciones solo pueden tener estados cuantificados y, en el caso límite, estará formada por solo dos estados. Se buscará también encontrar diferentes métodos los cuales eviten el uso de fuerza bruta en el momento de decisión entre qué estado cuantificado proporciona el menor error cuadrático. Asimismo, la formalización de los diferentes métodos se tendrá que conseguir con el fin de proporcionar métodos de síntesis que sean fiables y consistentes. Finalmente, se realizará un programa de Matlab optimizado que será capaz de realizar los cálculos pertinentes para llevar a cabo las metodologías de la síntesis cuantificada.