An extended Hubbard model with large short and long-ranged Coulomb repulsions and distant transfer is numerically investigated by use of the Lanczos exact diagonalization (ED) method to study the charge order and unconditional dimerization of a chain at density rho (ρ)= 0.4. From the analysis of the spin and charge correlation functions, a picture consistent with the formation of a dimer insulating state, which is of Wigner lattice-type (WL) charge order (CO), is obtained. The next-nearest neighbour (NNN) hopping t2 enhances the intradimer correlations and weakens the interdimer correlations. Implications for the CuO2 chains in Sr14Cu24O41 are discussed.We have also introduced a Heisenberg model which parametrically depends on hole positions. If the electrostatic hole-hole repulsion is included such a model allows to evaluate all energy eigenvalues and eigenstates (for small system size) and thus enables us to evaluate thermodynamic properties as function of temperature,magnetic field, and doping. Assuming certain exchange constants we can investigate the influence of the electrostatic hole-hole repulsion on ground state properties as well as on thermal averages like the magnetization which include contributions of low-lying spin-hole configurations.

In dieser Arbeit wurden zwei verschiedene Wärmebadankopplungen an klassische Spin-Systeme realisiert. Zum einen wurde ein stochastischer Ansatz mittels Landau-Lifshitz-Dämpfung und Fluktuationen numerisch realisiert und zum anderen wurde ein vollkommen deterministischer Ansatz entworfen und optimiert. Mit Hilfe dieser Ankopplungsmethoden ist es möglich, sowohl statische magnetische Eigenschaften klassischer Spin-Systeme als auch deren dynamische magnetische Eigenschaften zu simulieren. Als Anwendung wurden Spin-Gitter-Relaxationszzeiten und Neutronenstreuquerschnitte für molekulare Magneten wie z.B. dem "ferric wheel" berechnet und mit aktuellen experimentellen Ergebnissen verglichen. Als zweite Anwendung wird die Magnetisierungsumkehr in einem sphärischen Teilchen diskutiert.

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In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.:List of Figures List of Tables List of Abbreviations 1. Introduction 2. Magnetism of cuprates 3. Experimental methods 4. DFT-based microscopic modeling 5. Simulations of a magnetic model 6. Model spin systems: challenging the computational approach 7. Kagome lattice compounds 8. Summary and outlook Appendix Bibliography List of publications Acknowledgments

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Um formalismo da funÃÃo de Green à usado para calcular o espectro de excitaÃÃes associadas com uma impureza magnÃtica localizada intersticialmente em diferentes estruturas ferromagnÃticas descritas pelo modelo de Ising e de Heisenberg. No capÃtulo 3, descrevemos um ferromagneto de rede cÃbica simples semi-infinita atravÃs do modelo de Ising. Neste caso, as excitaÃÃes nÃo-ressonantes (isto Ã, os modos de defeito fora da regiÃo das ondas de spin de volume e de superfÃcie) e as excitaÃÃes ressonantes (os modos de defeito dentro da regiÃo das ondas de spin de volume) sÃo calculadas numericamente para a fase de alta-temperatura. Duas situaÃÃes sÃo analisadas, dependendo da posiÃÃo da impureza em relaÃÃo a seus vizinhos: a impureza està na superfÃcie; a impureza està na regiÃo de volume. Nos demais capÃtulos, usamos o modelo de Heisenberg/Ising (onde passamos do modelo de Heisenberg para o de Ising atravÃs do controle de um parÃmetro) para descrever os seguintes sistemas: ferromagneto de rede quadrada infinita (capÃtulo 4), ferromagneto de rede quadrada centrada infinita (capÃtulo 5), ferromagneto de rede cÃbica de corpo centrado infinita (capÃtulo 6) e rede favo de mel infinita (capÃtulo 7), todos contendo uma impureza magnÃtica localizada intersticialmente. Nos trÃs primeiros casos, sÃo calculados apenas os modos de defeito acima da banda de volume do material puro (modos Ãpticos). No capÃtulo 7, sÃo analisados apenas os modos de defeito abaixo da banda de volume do material puro (modos acÃsticos).

Première partie : Le retournement de mot est une opération de réécriture liée à une présentation (de semigroupe dans ce travail). Dans les bons cas, le retournement donne une solution au problème de mot. Sinon, il existe un moyen d'ajouter des relations à une présentation pour la rendre complète. D'un autre côté, les bases de Gröbner fournissent un moyen de compléter une présentation qui résout le problème de mot. On montre que les deux méthodes sont différentes ; une classification des divergences est proposée. On introduit ensuite une extension du retournement pour contourner le défaut de complétude de certaines présentations et on montre son efficacité sur la présentation d'Heisenberg — qui est incomplète. Deuxième partie : On se restreint aux présentations d'Artin-Tits des monoïdes de tresses. On montre que la distance combinatoire maximale entre deux mots de tresse équivalents est au moins quartique en leur largeur. On montre des critères simples pour qu'un diagramme de van Kampen (ou un diagramme de retournement) réalise la distance combinatoire entre deux mots équivalents. On calcule ensuite des bornes pour deux nombres liés au retournement de mot, et plus particulièrement pour les mots de tresse de largeurs arbitrairement grandes : le premier est, partant d'un mot, la longueur maximale d'une suite de retournements et le second la longueur du mot terminal (qui existe et est unique) d'une telle suite. Pour le premier, on montre une minoration quartique en la longueur du mot de départ ; pour le second, on établit une majoration cubique en la longueur.

Cette thèse concerne la mise en place de méthodes numériques pour déterminer les configurations magnétiques d'équilibre, et leur utilisation pour des systèmes possédant des hétérogénéités structurales et des singularités magnétiques.<br /><br />Nous décrivons tout d'abord un algorithme fondé sur une méthode multipolaire rapide et qui permet de calculer efficacement le champ dipolaire dans une assemblée de spins dans le cadre du modèle de Heisenberg classique. <br /><br />En utilisant ce modèle, nous étudions l'interaction de parois magnétiques avec des défauts structuraux dans des couches minces de FePt. Nous traitons le cas des parois d'antiphase et les micromacles. Nous analysons les valeurs des champs de décrochage des parois magnétiques, notamment en les comparant avec des données expérimentales.<br /><br />Nous détaillons ensuite une méthode multiéchelle que nous développée. Cette méthode permet, dans un formalisme unifié, de décrire un système avec le modèle de Heisenberg et le modèle micromagnétique.<br /><br />La dernière partie de cette thèse est consacrée à l'étude de systèmes de grande taille possédant des variations spatiales rapides d'aimantation, en utilisant la méthode multiéchelle : vortex dans un élément magnétique, configurations avec un point de Bloch dans un cube, bulle magnétique dans une couche mince de FePd. Dans ce dernier cas, les résultats sont comparés à des observations récentes par microscopie de Lorentz.

Cette thèse présente des développements de méthodes applicables au traitement théorique de réseaux de spins périodiques. Une méthode (Self-Consistent Perturbation), est inspirée par une expansion perturbative de la fonction d'onde à partir d'une fonction de référence très localisée. Cette variante d'un formalisme Coupled Cluster conduit à des équations polynomiales couplées, aisément résolues. Les autres méthodes sont basées sur des changements d'échelle, dans l'esprit du Groupe de Renormalisation dans l'Espace Réel, le réseau étant vu comme des blocs en interaction. La théorie des Hamiltoniens effectifs, utilisant le spectre exact de dimères ou trimères de blocs, permet de définir des interactions effectives. On a considéré soit des blocs à nombre impair de sites, qu'on peut voir comme des quasi-spins, ce qui est susceptible de produire des réseaux isomorphes et permet, d'itérer le processus et de garder l'élégance et les concepts du formalisme du Groupe de Renormalisation, soit des blocs à nombre pair de sites, qui conduisent à une description excitonique renormalisée des états excités. Les méthodes ont été testées sur des réseaux simples, puis appliquées à la recherche de transitions de phase sur une série de réseaux bidimensionnels (carré anisotrope, 1/5-depleted, plaquette, Shastry-Sutherland) et à des rubans graphitiques. Les localisations des transitions de phase (et les valeurs des gaps) sont prédites de façon très cohérentes par les diverses méthodes utilisées et en bon accord avec les meilleurs évaluations disponibles. L'hypothèse de l'existence d'une phase intermédiaire dans le réseau Shastry-Sutherland est confortée par nos calculs.

Les modulations multiporteuses ont prouvé leur intérêt pour les transmissions à haut débit, avec fil (ADSL) ou sans fil (DAB, DVB-T, HIPERLAN,...). Les applications citées mettent toutes en oeuvre un cas particulier de modulation multiporteuse, l'OFDM avec un filtre de mise en forme rectangulaire. Le but de cette thèse est de déterminer de nouvelles formes de modulations multiporteuses plus générales et utilisant des filtres de mise en forme mieux adaptés à certains types de canaux de transmission. Les techniques BFDM/QAM suréchantillonnées et BFDM/OQAM qui utilisent des filtres de modulation (dits prototypes) orthogonaux ou non sont ainsi étudiées avec une approche à temps discret qui prend en compte leur réalisation matérielle. Cette étude se base d'une part sur l'analogie existant entre les systèmes de Weyl-Heisenberg et les modulations multiporteuses, et d'autre part sur l'analogie existant entre les bancs de filtres et les modulations multiporteuses. Une écriture à l'aide de familles de Weyl-Heisenberg et une réalisation sous la forme d'un transmultiplexeur sont fournies pour chacune de ces modulations. Dans chacun de ces deux cas, on établit des conditions de biorthogonalité (c'est-à-dire des conditions de démodulation parfaite sur canal parfait) équivalentes portant sur les composantes polyphases des filtres prototypes. On en déduit alors des relations de dualité entre les bancs de filtres MDFT et les modulations BFDM/OQAM. Ceci est aussi équivalent au fait que, pour un type particulier de système de Weyl-Heisenberg dans $\lZ$ considéré comme un espace de Hilbert sur $\R$, deux familles de Weyl-Heisenberg sont biorthogonales, si et seulement si elles constituent deux frames duales. Par ailleurs, l'étude des transmultiplexeurs associés aux modulations BFDM/QAM et BFDM/OQAM conduit à plusieurs schémas de réalisation équivalents mettant en oeuvre des algorithmes de FFT et IFFT et la dualité établie entre les modulations BFDM/OQAM et les bancs de filtres MDFT aboutit à un nouveau codeur en sous-bandes MDFT généralisé et à délai de reconstruction réduit. Deux critères d'optimisation des prototypes sont utilisés : la maximisation de la localisation temps-fréquence et la minimisation de l'énergie hors-bande. Des techniques d'optimisation permettant d'obtenir des filtres orthogonaux ou biorthogonaux quasi-optimaux vis-à-vis de l'un de ces deux critères sont décrites. Enfin, les intérêts respectifs des prototypes orthogonaux et biorthogonaux, optimisés vis-à-vis de l'un ou l'autre de ces critères sont illustrés par des simulations sur différents types de canaux de transmission.