Extensive testing is performed when a new aircraft is developed. Flight testing is costly and time consuming but there are aspects of the process that can be made more efficient. A program that estimates aerodynamic parameters during flight could be used as a tool when deciding to continue or abort a flight from a safety or data collecting perspective. The algorithm of such a program must function in real time, which for this application would mean a maximum delay of a couple of seconds, and it must handle telemetric data, which might have missing samples in the data stream. Here, a conceptual program for real-time estimation of aerodynamic parameters is developed. Two estimation methods and four methods for handling of missing data are compared. The comparisons are performed using both simulated data and real flight test data. The first estimation method uses the least squares algorithm in the frequency domain and is based on the chirp z-transform. The second estimation method is created by adding boundary terms in the frequency domain differentiation and instrumental variables to the first method. The added boundary terms result in better estimates at the beginning of the excitation and the instrumental variables result in a smaller bias when the noise levels are high. The second method is therefore chosen in the algorithm of the conceptual program as it is judged to have a better performance than the first. The sequential property of the transform ensures functionality in real-time and the program has a maximum delay of just above one second. The four compared methods for handling missing data are to discard the missing data, hold the previous value, use linear interpolation or regard the missing samples as variations in the sample time. The linear interpolation method performs best on analytical data and is compared to the variable sample time method using simulated data. The results of the comparison using simulated data varies depending on the other implementation choices but neither method is found to give unbiased results. In the conceptual program, the variable sample time method is chosen as it gives a lower variance and is preferable from an implementational point of view.

The flight testing and evaluation of collected data must be efficient during intensive flight-test programs such as the ones conducted during development of new aircraft. The aim of this thesis has thus been to produce a first version of an aerodynamic derivative estimation program that is to be used during real-time flight tests. The program is to give a first estimate of the aerodynamic derivatives as well as check the quality of the data collected and thus serve as a decision support during tests. The work that has been performed includes processing of data in order to use it in computations, comparing a batch and a sequential estimation method using real-time data and programming a user interface. All computations and programming has been done in Matlab. The estimation methods that have been compared are both built on transforming data to the frequency domain using a Chirp z-transform and then estimating the aerodynamic derivatives using complex least squares with instrumental variables.The sequential frequency domain method performs estimates at a given interval while the batch method performs one estimation at the end of the maneuver. Both methods compared in this thesis produce equal results. The continuous updates of the sequential method was however found to be better suited for a real-time application than the single estimation of the batch method. The telemetric data received from the aircraft must be synchronized to a common frequency of 60 Hz. Missing samples of the data stream must be linearly interpolated and different units of measured parameters must be corrected in order to be able to perform these estimations in the real-time test environment.

Recently, there has been increasing interest within the potential user community of Global Positioning System (GPS) for high precision navigation problems such as aircraft non precision approach, river and harbor navigation, real-time or kinematic surveying. In view of more and more GPS applications, the reliability of GPS is at this issue. The Global Positioning System (GPS) is a space-based radio navigation system that provides consistent positioning, navigation, and timing services to civilian users on a continuous worldwide basis. The GPS system receiver provides exact location and time information for an unlimited number of users in all weather, day and night, anywhere in the world. The work in this thesis will mainly focuss on how to model a Mathematical expression for tracking GPS Signal using Phase Locked Loop filter receiver. Mathematical formulation of the filter are of two types: the first order and the second order loops are tested successively in order to find out a compromised on which one best provide a zero steady state error that will likely minimize noise bandwidth to tracks frequency modulated signal and returns the phase comparator characteristic to the null point. Then the Z-transform is used to build a phase-locked loop in software for digitized data. Finally, a Numerical Methods approach is developed using either MATLAB or Mathematica containing the package for Gaussian elimination to provide the exact location or the tracking of a GPS in the space for a given a coarse/acquisition (C/A) code.

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This dissertation presents the solution of difference equations and focuses on a method of difference equations solution with the aid of eigenvectors. The first part reminds the basic terms from area of difference equations such as dynamic of difference equations and linear difference equations of first order and higher order. Then the second section recalls also the system of difference equations including the fundamental matrix and general solution description. Afterthat, the method of solving the difference equations with a variation of constants and transform of scalar equations to the system are shown. The second part of the dissertation analyses some known algorithms and methods for the solution of linear difference equations. The Z-transform, its importance and usage for finding the solution of difference equation is recalled. Then the discrete analogue of Putzer's algorithm is mentioned because this algorithm was often used to check the results obtained by the newly described algorithm in further parts of this thesis. Also some ways of the system matrix power are stated. The next section then describes the principle of Weyr's method which is the basic point for further development of the theory including the presentation of the research results gained by Jiří Čermák in this area. The third part describes own solution of the difference equations system via eigenvectors based on the principle of Weyr's method for differential equations. The solution of system of linear homogeneous difference equtions with constant coefficients including the proof is presented and this solution is then extended to nonhomogeneous systems. Consequently to the theory, the influence of a nulity and the multiplicity of roots on the form of the solution is discussed. The last section of this part shows the implementation of the algorithm in Matlab program (for basic simpler cases) and its application to some cases of difference equations and systems with these equations. The final part of the thesis is more practical and it presents the usage of the designed algorithm and theory. Firstly, the algorithm is compared with Z-transform and the method of variation of constants and it is illustrated how to obtain the same results by using these three approaches. Then an example of current response solution in RLC circuit is demonstrated. The continuous case is solved and then the problem is transferred to discrete case and solved with the Z-transform and the method of eigenvectors. The obtained results are compared with the result of the continuous case.

Time domain analysis in vector network analyzers (VNAs) is a method to represent the frequency response, stated by the S-parameters, in time domain with apparent high resolution. Among other utilities time domain option from Agilent allows to measure microwave devices into a specific frequency range and down till DC as well with the two time domain mode: band-pass and low-pass mode. A special feature named gating is of important as it allows representing a portion of the time domain representation in frequency domain.   This thesis studies the time domain option 010 from Agilent; its uncertainties and sensitivity. The task is to find the best method to measure the antenna element matching taking care to reduce the influence of measurement errors on the results.   The Agilent 8753ES is the instrument used in the thesis. A specific matching problem in the antenna electric down-tilt (AEDT) previously designed by Powerwave Technologies is the task to be solved. This is because it can not be measured directly with 2-port VNAs. It requires adapters, extra coaxial cables and N-connectors, all of which influences the accuracy. The AEDT connects to the array antenna through cable-board-connectors (CBCs). The AEDT and the CBCs were designed before being put into the antenna-system. Their S-parameters do not coincide with the ones measured after these devices were put in the antenna block.   Time domain gating and de-embedding algorithms are two methods proposed in this thesis to measure the S-parameters of the desired antenna element while reducing the influence of measurement errors due to cables CBCs and other connectors. The aim is to find a method which causes less error and gives high confidence measurements.   For the time domain analysis, reverse engineering of the time domain option used in the Agilent VNA 8753ES is implemented in a PC for full control of the process. The results using time-domain are not sufficiently reliable to be used due to the multiple approximations done in the design. The methodology that Agilent uses to compensate the gating effects is not reliable when the gate is not centered on the analyzed response. Big errors are considered due to truncation and masking effects in the frequency response.   The de-embedding method using LRL is implemented in the AEDT measurements, taking away the influences of the CBCs, coaxial cables and N-connector. It is found to have sufficient performance, comparable to the mathematical model. Error analysis of both methods has been done to explaine the different in measurements and design.

Made available in DSpace on 2014-12-17T14:54:59Z (GMT). No. of bitstreams: 1 FabianaTS_TESE.pdf: 1364206 bytes, checksum: 5e147adc9ca5829c7a40ed214ab434d2 (MD5) Previous issue date: 2011-12-02 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given / Neste trabalho utiliza-se a matem?tica intervalar para estabelecer os conceitos intervalares das principais ferramentas utilizadas em processamento digital de sinais. Mais especificamente, foram desenvolvidos aqui as abordagens intervalares para sinais, sistemas, amostragem, quantiza??o, codifica??o, transformada Z e transformada de Fourier. ? feito um estudo de algumas aritm?ticas que lidam com n?meros complexos sujeitos ? imprecis?es, tais como: aritm?tica complexa intervalar (ou retangular), aritm?tica complexa circular, aritm?tica setorial e aritm?tica intervalar polar. A partir da?, investiga-se algumas propriedades que ser?o relevantes para o desenvolvimento e aplica??o no processamento de sinais discretos intervalares. Mostra-se que nos conjuntos IR e R(C), seja qual for a aritm?tica correta adotada, n?o se tem um corpo, isto ?, os elementos desses conjuntos n?o se comportam como os n?meros reais ou complexos com suas aritm?ticas cl?ssicas e que isso ir? requerer uma avalia??o matem?tica dos conceitos necess?rios ? teoria de sinais e a rela??o desses com as aritm?ticas intervalares. Tamb?m tanto ? introduzido o conceito de amplitude intervalar complexa, como alternativa ? defini??o cl?ssica quanto utiliza-se a ordem de Kulisch-Miranker para n?meros complexos afim de que se escreva n?meros complexos intervalares na forma de intervalos, o que torna poss?vel as opera??es atrav?s dos extremos. Essa rela??o ? utilizada em propriedades de somas de intervalos de n?meros complexos. O uso de sinais e sistemas intervalares foi motivado pela representa??o intervalar num sistema de ponto flutuante abstrato. Isto ?, se um n?mero x 2 R n?o ? represent?vel em um sistema de ponto flutuante F, ele ? mapeado para um intervalo [x;x], tal que x ? o maior dos n?meros menores que x represent?vel em F e x ? o menor dos n?meros maiores que x represent?vel em F. A representa??o intervalar ? importante em processamento digital de sinais, pois a imprecis?o em dados ocorre tanto no momento da medi??o de determinado sinal, quanto no momento de process?-los computacionalmente. A partir da?, define-se sinais e sistemas intervalares que assumem tanto valores reais quanto complexos. Para isso, utiliza-se o estudo feito a respeito das aritm?ticas complexas intervalares e mostram-se algumas propriedades dos sistemas intervalares, tais como: causalidade, estabilidade, invari?ncia no tempo, homogeneidade, aditividade e linearidade. Al?m disso, foi definida a representa??o intervalar de fun??es complexas. Tal fun??o estende sistemas cl?ssicos a sistemas intervalares preservando as principais propriedades. Um conceito muito importante no processamento digital de sinais ? a quantiza??o, uma vez que a maioria dos sinais ? de natureza cont?nua e para process?-los ? necess?rio convert?-los em sinais discretos. Aqui, este processo ? descrito detalhadamente com o uso da matem?tica intervalar, onde se prop?em, inicialmente, uma amostragem intervalar utilizando as id?ias de representa??o no sistema de ponto flutuante. Posteriormente, s?o definidos n?veis de quantiza??o intervalares e, a partir da?, ? descrito o processo para se obter o sinal quantizado intervalar e s?o definidos o erro de quantiza??o intervalar e o sinal codificado intervalar. ? mostrado que os n?veis de quantiza??o intervalares representam os n?veis de quantiza??o cl?ssicos e o erro de quantiza??o intervalar representa o e erro de quantiza??o cl?ssico. Uma estimativa para o erro de quantiza??o intervalar ? apresentada. Utilizando a aritm?tica retangular e as defini??es e propriedades de sinais e sistemas intervalares, ? introduzida a transformada Z intervalar e s?o analisadas as condi??es de converg?ncia e as principais propriedades. Em particular, quando a vari?vel complexa z ? unit?ria, define-se a transformada de Fourier intervalar para sinais discretos no tempo, al?m de suas propriedades. Por fim, foram apresentadas as implementa??es dos resultados que foram feitas no software Matlab