The aims of this thesis are to reduce the complexity and increasethe accuracy of rotations carried out inthe fast Fourier transform (FFT) at algorithmic and arithmetic level.In FFT algorithms, rotations appear after every hardware stage, which are alsoreferred to as twiddle factor multiplications. At algorithmic level, the focus is on the development and analysisof FFT algorithms. With this goal, a new approach based on binary tree decompositionis proposed. It uses the Cooley Tukey algorithm to generate a large number ofFFT algorithms. These FFT algorithms have identical butterfly operations and data flow but differ inthe value of the rotations. Along with this, a technique for computing the indices of the twiddle factors based on the binary tree representation has been proposed. We have analyzed thealgorithms in terms of switching activity, coefficient memory size, number of non-trivial multiplicationsand round-off noise. These parameters have impact on the power consumption, area, and accuracy of the architecture.Furthermore, we have analyzed some specific cases in more detail for subsets of the generated algorithms. At arithmetic level, the focus is on the hardware implementation of the rotations.These can be implemented using a complex multiplier,the CORDIC algorithm, and constant multiplications. Architectures based on the CORDIC and constant multiplication use shift and add operations, whereas the complex multiplication generally uses four real multiplications and two adders.The sine and cosine coefficients of the rotation angles fora complex multiplier are normally stored in a memory.The implementation of the coefficient memory is analyzed and the best possible approaches are analyzed.Furthermore, a number of twiddle factor multiplication architectures based on constant multiplications is investigated and proposed. In the first approach, the number of twiddle factor coefficients is reduced by trigonometric identities. By considering the addition aware quantization method, the accuracy and adder count of the coefficients are improved. A second architecture based on scaling the rotations such that they no longer have unity gain is proposed. This results in twiddle factor multipliers with even lower complexity and/or higher accuracy compared to the first proposed architecture.

This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.

The identification of energy scales in the region of Isoscalar Giant Quadrupole Resonance (ISGQR) is motivated by their potential use in understanding how an ordered collective motion transforms into a disordered motion of intrinsic single-particle degrees-of-freedom in many-body quantum systems. High energy-resolution measurements of the ISGQR were obtained by proton inelastic scattering at Ep= 200 MeV using the K600 magnetic Spectrometer at iThemba LABS. The nuclei 58Ni, 90Zr, 120Sn and 208Pb, associated with closed shells, were investigated. Both the Fourier transform and Wavelet analysis were used to extract characteristic energy scales and were later compared with the results from the theoretical microscopic Quasi-particle Phonon Model (QPM), including contributions from collective and non-collective states. The scales found in the experimental data were in good agreement with the QPM. This provides a strong argument that the observed energy scales result from the decay of the collective modes into 2p-2h states. The different scale regions were tested directly by reconstruction of measured energy spectra using the Inverse Fourier Transform and the Continuous Wavelet Transform (CWT), together with a comparison to a previously available reconstruction using the Discrete Wavelet Transform (DWT).

L’imagerie hyperspectrale est maintenant très développée dans les applications de télédétection. Il y a principalement deux manières de construire les imageurs associés : la première méthode utilise un réseau et une fente, et l’image spectrale est acquise ligne par la ligne le long de la trajectoire du porteur. La seconde est basée sur le principe de la spectrométrie par transformée de Fourier (TF). Certains des systèmes utilisés sont construits de manière à enregistrer l’interférogramme de chaque point de la scène suivant le déplacement dans le champ. Le spectre de la lumière venant d’un point de la scène est alors calculé par la transformée de Fourier de son interférogramme. Les imageurs classiques basés sur des réseaux sont plus simples à réaliser et les données qu’ils fournissent sont souvent plus faciles à interpréter. Cependant, les spectro-imageurs par TF fournissent un meilleur rapport signal sur bruit si la source principale de bruit vient du détecteur.Dans la première partie de cette thèse, nous étudions l’influence de différents types de bruit sur les architectures classiques et TF afin d’identifier les conditions dans lesquelles ces dernières présentent un avantage. Nous étudions en particulier l’influence des bruits de détecteur, de photons, des fluctuations de gain et d’offset du détecteur et des propriétés de corrélation spatiale des fluctuations d’intensité du spectre mesuré. Dans la seconde partie, nous présentions la conception, la réalisation et les premiers résultats d’un imageur basé sur un interféromètre de Michelson à dièdres statique nommé DéSIIR (Démonstrateur de Spectro-Imagerie Infrarouge). Les premiers résultats montrent, qu’en mode spectromètre simultané, DéSIIR permet la restitution du spectre avec les spécifications requises dans le cadre des applications recherchées, c'est-à-dire détecter avec une résolution d environ 25 cm-1 un object de quelques degrés plus chaud que le fond de la scène et présentant une signature spectrale entre 3 et 5 juin. En mode spectromètre imageur, après recalage des images, il est possible de reconstruire le spectre de chaque point de la scène observée. / Hyperspectral imaging is now very important in remote sensing applications. There are two main ways to build such imagers : the first one uses a grating and a slit, and the spectral image is acquired line by line along the track of the carrier. The second way is to use the principle of Fourier transform (FT) spectrometry. Some of these systems are built in such a way that they record the interferogram of each point of the scene as it moves through the field of view. The spectrum of the light coming from a particular point is then calculated by the Fourier transform of its interferogram. Classical gratting-based spectral imagers are easier to build and the data they provide a better signal to noise ratio if the main source of noise comes from the detector.In the first part of this thesis, we study the influence of various types of noise on the classic and TF-based architectures to identify the conditions in which these last ones present an advantage. We study particularly the influence of detector noise, photons noise, detector gain and offset fluctuations and spatial correlation properties of the intensity fluctuations. In the second part, we present the conception, the realization and the first results of an imager bases on a Michelson interferometer with dihedrons named DéSIIR (“Démonstrateur de Spectro Imagerie Infrarouge”). The first results show that, in simultaneous spectrometer mode, DéSIIR allows the reconstruction of the spectrum with respect to the specific requirements, which are to be able to detect an objet of some degrees warmer than the background of the scene observed with a resolution of about 25 cm -1. In imager mode, this reconstruction is performed for each point of the scene.

Ce manuscrit de thèse traite de l’étude de verres ophtalmiques microstructurés et plus particulièrement pixellisés, ces derniers pouvant présenter un intérêt particulier en optique active pour la correction de la presbytie. Une étude théorique est proposée, permettant d’analyser les performances optiques d’une lentille pixellisée en termes de transport d’image et d’identifier les paramètres qui ont un impact direct sur ces performances. Après validation expérimentale des résultats obtenus, nous constatons puis expliquons l’effet sur l’observation d’une scène de l’excentrement de l’œil par rapport à la fonction de phase du verre. Nous étudions l’effet du repliement de phase inhérent aux limites des technologies de fabrication, qui vient ajouter un chromatisme axial aux défauts visuels engendrés par la pixellisation. Nous nous intéressons ensuite aux applications possibles de la pixellisation en optique passive. Nous prouvons que pour une application visée, des lentilles binaires non pixellisées, dont nous optimisons la qualité optique, conduisent à de meilleurs résultats que les lentilles pixellisées. L’impact sur l’acuité visuelle des phénomènes diffractifs parasites induits par la pixellisation est évalué au moyen d’un banc de mesure utilisant la simulation de certaines images telles qu’elles seraient vues au travers de verres ophtalmiques pixellisés. Enfin, nous menons une étude de l’aspect esthétique d’un verre pixellisé vu par un observateur externe, à partir de modèles de calcul hybrides mêlant optique géométrique et optique de Fourier. / This thesis investigates microstructured and more particularly pixelated ophthalmic glasses, the latter raising some hope for the active correction of presbyopia. A theoretical study is developed for the analysis of the optical performances of a pixelated lens in terms of image transport and leads to the identification of the parameters which have an impact on these performances. After experimental validation of the results obtained, we note and then explain the effect on the observation of a scene of eye displacement with respect to the optical function of the eyeglass. We study the effect of phase wrapping, which is inherent in the limits of the technologies implied in the manufacturing process and adds an axial chromatism effect to the visual defects generated by pixelation. We are also interested in the potential applications of pixelation in the field of passive optics. We prove that for a given application, non pixelated binary lenses, the optical quality of which we optimize, lead to better results than pixelated lenses. The impact on visual acuity of the parasitic diffractive phenomena induced by pixelation is evaluated with an optical bench using the simulation of test images seen through pixelated lenses. Finally, we study the aesthetic aspect of a pixelated component as seen by an external observer, using hybrid calculation models based on both geometrical and Fourier optics.

Pour répondre au besoin de miniaturisation des spectromètres de terrain travaillant dans l’infrarouge, l’ONERA a développé un nouveau concept baptisé MICROSPOC. Ce dispositif est un détecteur infrarouge auquel a été intégré un interféromètre à deux ondes, constituant un spectromètre statique par transformée de Fourier. Ce plan focal infrarouge modifié, qui fusionne la fonction interférométrique et la fonction de détection, associé à une optique de tête simplifiée, permet d’envisager la réalisation d’instruments très compacts. L’objectif de cette thèse est de concevoir un spectromètre infrarouge miniature basé sur le dispositif MICROSPOC. Dans un premier temps, un travail théorique a été mené, dans l’objectif de dimensionner un système optique très compact. Notre choix s’étant orienté vers un système optique de collection (le détecteur voit une source étendue à distance finie), l’étude de l’acceptance angulaire de MICROSPOC dans ces conditions d’éclairement est indispensable afin de prévoir le contraste et la forme des franges d’interférence. Les résultats montreront la grande acceptance angulaire de MICROSPOC.Dans un second temps, un démonstrateur basé sur un composant MICROSPOC et sur le système optique simplifié a été réalisé. Ce démonstrateur a été caractérisé en laboratoire puis utilisé sur le terrain lors d’une campagne de mesures. Ces différentes exploitations ont montré la robustesse de l’instrument malgré l’obtention d’interférogrammes présentant divers défauts.Dans un troisième temps, une chaîne de traitement a été développée afin d’estimer un spectre à partir d’un interférogramme obtenu à l’aide du démonstrateur. Du fait des caractéristiques intrinsèques de MICROSPOC, la transformée de Fourier n’est pas la meilleure solution pour estimer un spectre. Nous l’avons montré en nous intéressant aux effets des disparités de longueur d’onde de coupure du détecteur sur l’estimation d’un spectre. Nous nous sommes alors tournés vers une approche consistant à utiliser la caractérisation spectrale de l’instrument pour inverser la mesure. Cette approche donne des résultats satisfaisants.Enfin, le but principal de cette thèse a été élargi par la conception et la réalisation de différents démonstrateurs combinant une fonction d’imagerie à une fonction de spectrométrie. Les premières pistes pour la conception d’un spectromètre qui tient dans la main ont été données. / In order to satisfy the need for handheld infrared spectrometers, the ONERA developed a new concept called MICROSPOC. This device is an infrared focal plane array with a built-in two-wave wedge-like interferometer and forms a static Fourier-transform spectrometer. This modified focal plane array, which merges the detection function and the interferometric function, in association with a simplified optical system, allows to consider the realisation of a much compact instrument. The goal of this thesis is to design and to realize a miniaturized infrared spectrometer based on the MICROSPOC concept.Firstly, a theoritical work has been led in order to design a compact optical system. Since we have chosen a collection optical system (the focal plane array sees an extended source placed at a finite distance), the study of MICROSPOC angular acceptance in these lightening conditions is needed in order to predict the contrast and the shape of interference fringes. The huge angular acceptance of MICROSPOC will be established with the results of this study.Secondly, a demonstrator based on MICROSPOC device and on the simplified optical system has been realized. This demonstrator has been caracterized in the laboratory and used in real conditions of a measurement campaign. These different exploitations have shown the robustness of the instrument despite some defaults on acquired interferograms.Then, a processing chain has been developed in order to estimate a spectrum from an interferogram acquired with our demonstrator. Considering the MICROSPOC’s own characteristics, the Fourier-transform is not the best way to estimate a spectrum. We have come to this conclusion by studying the effects of cut-off wavelenghts disparities of the detector on the spectrum estimation. At this point we have considered an approach that consists of using the spectral characterization of the instrument in order to inverse the measure. This approach gives satisfying results.Finaly, the main goal has been widened with the design and the realisation of other instruments that combine a spectrometric function and a imaging function. The first elements for the design of a handheld spectrometer have been given.

Objetivou-se avaliar a utilização da metodologia de infravermelho com transformada de Fourier (IVTF) na identificação de adulteração em leite cru. O objetivo no primeiro estudo foio de desenvolver calibração para identificação de adulteração através da comparação do espectro de leite adulterado com um espectro de referência para leite cru. Para construção do espectro referência foram coletadas 800 amostras de diferentes fazendas localizadas na região de estudo no período de agosto de 2009 a março de 2010. Através da análise de componentes principais, foram desenvolvidas oito calibrações com diferentes números de fatores (componentes principais). Para validação das diferentes calibrações, foram coletadas outras 100 amostras da mesma região de estudo. As amostras foram adulteradas com três diferentes adulterantes geralmente utilizados na adulteração do leite cru sendo eles: bicarbonato de sódio (BS), citrato de sódio (CS) e soro de queijo (SO). Para cada adulterante foram utilizadas três diferentes concentrações (BS: 0,05, 0,10 e 0,25 %; CS: 0,025, 0,050 e 0,075% e SO: 5, 10 e 20%). Para avaliação do desempenho das calibrações foram calculadas a especificidade e a sensibilidade para cada adulterante e respectivas concentrações. A especificidade foi de 92,9% na calibração com 12 fatores. Já a sensibilidade variou em função do adulterante, concentração e número de fatores. Todas as amostras adulteradas com bicarbonato foram identificadas corretamente com modelos de 10 fatores e na concentração mais baixa (0,05%). Já para o citrato, somente amostras com concentração de 0,075% foram identificadas em 93,9% dos casos para o modelo de 18 fatores. Amostras adulteradas com soro, mesmo na concentração mais alta (20%) puderam ser identificadas em menos de 47,1% dos casos, talvez pelo tratamento espectral realizado pelo software ASM em que parte do espectro não é considerado na construção das calibrações. A comparação espectral mostrou-se uma técnica interessante para identificação de leite adulterado e que também poderia antecipar a descoberta de novos adulterantes. Já no segundo estudo o objetivo foi desenvolver calibrações para determinação da concentração dos adulterantes bicarbonato de sódio, citrato de sódio e soro de queijo. Para desenvolvimento das calibrações foram utilizadas as amostras adulteradas do primeiro estudo e, para validação, foram coletadas outras 60 amostras que foram novamente adulteradas. O desempenho de cada calibração foi avaliado através da acurácia (Se), limite de detecção (LD) e coeficiente de determinação (R2). Todas as calibrações apresentaram R2 superior a 0,91 com LD de 0,015%, 0,017% e 3,9% para BS, CS e SO, respectivamente. Já a acurácia foi de 0,005%, 0,009 % e 2,26% para BS, CS e SO, respectivamente. Através dos resultados obtidos neste estudo, conclui-se que a metodologia de IVTF pode ser utilizada para determinação da concentração de bicarbonato de sódio, citrato de sódio e soro de queijo em leite cru. Associada à equipamentos automatizados, é uma opção viável no monitoramento destes adulterantes, tendo o baixo custo operacional e alto desempenho analítico como características adicionais. / The objective of this research was to evaluate the use of the methodology of Fourier Transform Infrared (FTIR) in identifying adulteration in raw milk. The objective of the first study was to develop scales to identify contamination by comparing the spectrum of adulterated milk with a reference spectrum for raw milk. To build the reference spectrum, 800 samples were collected from different farms located in the study area from August 2009 to March 2010. Through the analysis of principal components, eight scales with different numbers of factors (principal components) were developed. To validate the different calibrations, other 100 samples were collected from the same region of study. The samples were adulterated with three different adulterants commonly used in the adulteration of raw milk: sodium bicarbonate (SB), sodium citrate (SC) and whey (W). Each adulterant was used at three different concentrations (SB: 0.05, 0.10 and 0.25%; SC: 0.025, 0.050 and 0.075% and W: 5, 10 and 20%). To evaluate the performance of the calibrations, the specificity and sensitivity for each adulterant and its concentration were calculated. The specificity was 92.9% in the calibration with 12 factors. Sensitivity varied according to the adulterant, concentration and number of factors. All samples adulterated with bicarbonate were accurately identified in the 10-factor model and at the lowest concentration (0.05%). As for citrate, only samples at a concentration of 0.075% were identified in 93.9% of cases for the 18-factor model. Samples adulterated with whey, even at the highest concentration (20%) were identified in less than 47.1% of cases, which maybe attributed to the fact that part of the spectrum is not consideredin the spectral processing performed by ASM software. The spectral comparison proved to be an interesting technique for the identification of adulterated milk and that could also be used to discover new adulterants. In the second study, the aim was to develop calibrations for determining the concentration of sodium bicarbonate, sodium citrate and whey used as raw milk adulterants. To develop the calibrations, adulterated samples of the first study were used and, for validation, other 60 samples were collected and adulterated. The performance of each calibration was evaluated regarding accuracy (Se), detection limit (DL) and determination coefficient (R2). All calibrations presented R2 higher than 0.91 with DL of 0.015%, 0.017% and 3.9% for SB, SC and W, respectively. Accuracy was 0.005%, 0.009% and 2.26% for SB, SC and W, respectively. Results show that the FTIR methodology can be used for determining the concentration of sodium bicarbonate, sodium citrate and whey in raw milk. Associated with automated equipment, it is a feasible option for monitoring these adulterants, presenting low operational costs and high analytical performance as additional features.

Abordaremos o estudo de condições para que certas equações diferenciais parciais tenham solução. Consideraremos equações do tipo Lu = f; onde tomamos L em algumas classes de campos vetoriais em toros de dimensão maior que dois. Tais campos vetoriais são operadores que agem no espaço das funções definidas no toro e que são infinitamente diferenciáveis. A principal questão é determinar quando tais operadores têm imagem fechada. Temos também interesse em saber quando que a imagem de tais operadores e um subespaço de codimensão finita, bem como estudar a regularidade de tais operadores. As respostas de tais questões envolvem certas propriedades dos coeficientes desses operadores, onde citamos: a conexidade de sub-níveis de primitivas da parte imaginária dos coeficientes; condições Diofantinas; a ordem de anulamento dos coeficientes e relações entre as ordens de anulamento das partes real e imaginária dos coeficientes; além disso, o número de vezes que a parte imaginária de um coeficiente c muda de sinal entre dois zeros consecutivos de c também desempenha um papel. Conseguimos caracterizar a resolubilidade e a hipoelíticidade global de campos vetoriais do tipo tubo em toros de dimensão maior do que dois, estendendo os resultados em dimensão dois. Depois, em dimensões, fornecemos condições que respondem sobre a imagem ser ou não fechada, para uma outra classe de campos vetoriais que não são do tipo tubo. Uma de tais condições esta relacionada com a famosa condição (P) de Nirenberg-Treves. Em particular, obtemos o mesmo para uma classe de campos vetoriais em dimensão são dois, para os quais a codimensão da imagem foi exaustivamente estudada. / We are concerned with the study of properties so that we can solve certain partial differential equations. We will consider equations of the form Lu = f; where we take L in some classes of vector fields on tori of dimension greater than two. This vector fields are viewed as operators acting on the space of smooth functions deffned on the torus. The main questions to study the closedness of the range of L. It is also of interest to know whe ther the range has finite codimension, as well as to study the regularity of L. The answers of these questions are connected with certain properties of the coeffcients of L; such as: Diophantine conditions; the connectedness of some sublevel sets involving primitive so fthe imaginary part of the coeffcients; the order of vanishing of each coeffcient and relations between the order of vanishing of the real and imaginary parts of each coeffcient; in addition, the number of times that the imaginary part of a coeffcient c changes sign between two consecutive zeros of c also plays a role. We characterize both global solvability and hypoellipticity for vector fields of tube type on tori of dimension greater than two, extending the results in dimension two. More over, in dimension three, we find conditions for the closedness of the range for a class of vector fields which are not of tube type. One of theese conditions is related to the well known Nirenberg-Treves condition (P). In particular,we obtain the same for a class of vector fields on the two- torus,for which the codimension of the range was largely studied.

O método das matrizes operacionais é expandido para o corpo complexo a ordens arbitrárias pela abordagem de Riemann-Liouville e Caputo com ênfase nas séries de Fourier complexas. Elabora-se uma adaptação do formalismo bra-ket de Dirac à linguagem tensorial no espaço de Hilbert de funções com expansões finitas para uso específico na teoria de equações diferenciais e matrizes operacionais, denominado \\Formalismo Coadjunto\". Estende-se o tratamento aos operadores fracionais de Weyl para períodos genéricos a fim de determinar as matrizes operacionais de derivação e integração de ordem arbitrária na base complexa de Fourier. Introduz-se um novo método de resolução de equações diferenciais ordinárias lineares e fracionais não-homogêneas, denominado \\Modelagem Operacional\", que permite a obtenção de soluções de equações de alta ordem com grande precisão sem a necessidade de imposição de condições iniciais ou de contorno. O método apresentado é aperfeiçoado por meio de um novo tipo de expansão, que denominamos \"Séries Associadas de Fourier\", a qual apresenta convergência mais rápida que a série de Fourier original numa restrição de domínio, possibilitando soluções de EDOs e EDFs de alta ordem com maior precis~ao e ampliando a esfera de casos passíveis de resolução. / Operational matrices method is expanded to complex field and arbitrary orders by using the Riemann-Liouville and Caputo approach with emphasis on complex Fourier series. Dirac\'s bra-ket notation is associated to tensor procedures in Hilbert spaces for finite function expansions to be applied specifically to dfferential equations and operational matrices, being called \\Coadjoint Formalism\". This treatment is extended to Weyl fractional operators for generic periods in order to establish the integral and derivative operational matrices of fractional order to complex Fourier basis. A new method to solve linear non-homogeneous ODEs and FDEs, called \\Operational Modelling\"is introduced. It yields high precision solutions on high order dfferential equations without assumption of boundary or initial conditions. The presented method is improved by a new kind of function expansion, called \\Fourier Associated Series\", which yields a faster convergence than original Fourier in a restrict domain, enabling to obtain solutions of high order ODEs and FDEs with excellent precision and broadening the set of solvable equations.

Abordaremos o estudo de condições para que certas equações diferenciais parciais tenham solução. Consideraremos equações do tipo Lu = f; onde tomamos L em algumas classes de campos vetoriais em toros de dimensão maior que dois. Tais campos vetoriais são operadores que agem no espaço das funções definidas no toro e que são infinitamente diferenciáveis. A principal questão é determinar quando tais operadores têm imagem fechada. Temos também interesse em saber quando que a imagem de tais operadores e um subespaço de codimensão finita, bem como estudar a regularidade de tais operadores. As respostas de tais questões envolvem certas propriedades dos coeficientes desses operadores, onde citamos: a conexidade de sub-níveis de primitivas da parte imaginária dos coeficientes; condições Diofantinas; a ordem de anulamento dos coeficientes e relações entre as ordens de anulamento das partes real e imaginária dos coeficientes; além disso, o número de vezes que a parte imaginária de um coeficiente c muda de sinal entre dois zeros consecutivos de c também desempenha um papel. Conseguimos caracterizar a resolubilidade e a hipoelíticidade global de campos vetoriais do tipo tubo em toros de dimensão maior do que dois, estendendo os resultados em dimensão dois. Depois, em dimensões, fornecemos condições que respondem sobre a imagem ser ou não fechada, para uma outra classe de campos vetoriais que não são do tipo tubo. Uma de tais condições esta relacionada com a famosa condição (P) de Nirenberg-Treves. Em particular, obtemos o mesmo para uma classe de campos vetoriais em dimensão são dois, para os quais a codimensão da imagem foi exaustivamente estudada. / We are concerned with the study of properties so that we can solve certain partial differential equations. We will consider equations of the form Lu = f; where we take L in some classes of vector fields on tori of dimension greater than two. This vector fields are viewed as operators acting on the space of smooth functions deffned on the torus. The main questions to study the closedness of the range of L. It is also of interest to know whe ther the range has finite codimension, as well as to study the regularity of L. The answers of these questions are connected with certain properties of the coeffcients of L; such as: Diophantine conditions; the connectedness of some sublevel sets involving primitive so fthe imaginary part of the coeffcients; the order of vanishing of each coeffcient and relations between the order of vanishing of the real and imaginary parts of each coeffcient; in addition, the number of times that the imaginary part of a coeffcient c changes sign between two consecutive zeros of c also plays a role. We characterize both global solvability and hypoellipticity for vector fields of tube type on tori of dimension greater than two, extending the results in dimension two. More over, in dimension three, we find conditions for the closedness of the range for a class of vector fields which are not of tube type. One of theese conditions is related to the well known Nirenberg-Treves condition (P). In particular,we obtain the same for a class of vector fields on the two- torus,for which the codimension of the range was largely studied.