Avec le besoin croissant en bande-passante, les technologies dites de radio-cognitive sont de plus en plus étudiées par la communauté scientifique. L’enjeu est d’utiliser au mieux le spectre disponible. L'une de ces technologies, Transform Domain Communication System (TDCS), dont les performances en termes d’efficacité énergétique et spectrale étaient jusqu'à présent méconnues, constitue le sujet d'étude de cette thèse. Après une présentation du contexte scientifique et industriel de la thèse, le système TDCS est introduit, ainsi que ses similarités et différences avec OFDM et MC-CDMA. Le système est ensuite décrit sous le formalisme algébrique des modulations linaires. Cela a permis d’établir une expression de l’efficacité spectrale du système. Plusieurs techniques sont alors proposées pour améliorer celle-ci tout en améliorant, dans certains cas, le taux d’erreur binaire. Étant composé d’un de plusieurs composantes sinusoïdales, le signal TDCS souffre d’un fort Peak-to-Average Power Ratio (PAPR). La théorie ensembliste est alors présentée puis mise à profit en troisième partie de cette thèse pour proposer les algorithmes Douglas-Rachford et ROCS de réduction du PAPR des signaux TDCS. Ces algorithmes convergent plus rapidement et vers des valeurs plus basses que l’algorithme POCS précédemment utilisé dans la littérature / For about ten years, spectrum scarcity and the growing need of bandwidth have pushed the studies on cognitive-radio technologies to counter this waste. Among them: the Transform Domain Communication System (TDCS), on which this thesis focuses. Until now, TDCS’ performance in terms of spectral and power efficiency was largely unknown. After introducing the thesis’ industrial and scientific context, the TDCS is introduced and compared with popular technologies such as OFDM and MC-CDMA. The system is then studied by means of the linear modulations’ algebraic framework. This has led to the TDCS’ spectral efficiency determination and to new design rules to jointly achieve a better spectral efficiency and a lower BER. Several methods are then proposed to further increase the spectral efficiency by means of a dense multidimensional modulation. Since a TDCS signal is made of several sines, it suffers from a strong Peak-to-Average Power Ratio (PAPR). Set theoretic estimation is then introduced in a third part and new PAPR-reduction algorithms such as Douglas-Rachford and Reflection Onto Convex Sets are brought to light and achieve better performance than the usual POCS algorithm regarding to the convergence rate, as well as the achieved PAPR

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This master’s thesis deals with the restoration of audio signals damaged by quantization. The theoretical part starts with a description of quantization and dequantization in general, few existing methods of dequantization of audio signals and theory of sparse representations of signals are also presented. The next part introduces algorithms suitable for dequantization, specifically Douglas–Rachford, Chambolle–Pock, SPADEQ and implementation of these algorithms in MATLAB application in the next chapter. In the last part of this thesis, testing of reconstructed signals using the algorithms takes place and results are evaluated by objective measures SDR, PEMO-Q, PEAQ and subjective listening test MUSHRA.

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The JPEG codec is currently the most widely used image format. This work deals with the design and implementation of an alternative JPEG codec using proximal algorithms in combination with the fixation of points from the original image to suppression of artifacts created in common JPEG coding. To solve the problem, the prox_TV and then the Douglas-Rachford algorithm were used, for which special functions using l_1-norm for image reconstruction were derived. The results of the proposed solution are very good because they can effectively suppress the artefacts created and the result corresponds to the image with a higher set qualitative factor. The proposed method achieves very good results for both simple images and photos, but in the case of large images (1024 × 1024 px) and larger, a large amount of computing time is required, so the method is more suitable for smaller images.

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