The objective of the proposed research is to develop a framework for the signal reconstruction algorithm with sub-Nyquist sampling rate and the low-cost hardware design in system level. A further objective of the proposed research is to monitor the device-under-test (DUT) and to adapt its behaviors. The key contribution of this research is that the high-speed signal acquisition is done by direct subsampling. As the signal is directly sampled without any front-end radio-frequency (RF) components such as mixers or filters, the cost of hardware is reduced. Furthermore, the distortion and the nonlinearity from the RF components can be avoided. The first proposed work is wideband signal reconstruction by dual-rate time-interleaved subsampling hardware and Multi-coset signal reconstruction. Using the combination of the dual-rate hardware and the multi-coset algorithm, the number of sampling channel is significantly reduced compared to the conventional multi-coset works. The second proposed work is jitter tracking by accurate period estimation with incoherent subsampling. In this work, the long-term jitter in PRBS is tracked without hardware synchronization and clock-data-recovery (CDR) circuits. The third proposed work is eye-monitoring and time-domain-reflectometry (TDR) by monobit receiver signal reconstruction. Using a monobit receiver based on incoherent subsampling and time-variant threshold signal, high resolution of reconstructed signal in both amplitude and time is achieved. Compared to a multibit-receiver, the scalability of the test-system is significantly increased.

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.

Avec la multiplication des terminaux mobiles et le foisonnement des objets dits « connectés », on assiste à la montée d'un besoin de moyens de communication à tout endroit et en toute situation, accompagné d'un encombrement spectral toujours plus important. Dans ce contexte, si la capacité d'adaptation au canal des modulations multiporteuses permet de bien s'accommoder du besoin de communication en tout endroit, les techniques actuelles, en particulier l'OFDM, souffrent d'une mauvaise localisation fréquentielle et d'un mauvais facteur de crête, ce qui limite leur utilisation dans un contexte embarqué et/ou en présence de fortes contraintes spectrales. Dans cette thèse, nous étudions les modulations multiporteuses au-delà de la cadence de Nyquist. En augmentant la densité de signalisation, ces dernières permettent d'augmenter l'efficacité spectrale. Cela est cependant contrebalancé par l'apparition d'auto-interférence, ce qui rend la réception plus délicate.Sur canal à bruit additif blanc gaussien, on montre comment choisir des impulsions de mise en forme maximisant le rapport signal à interférence plus bruit. On montre que ces dernières permettent d'obtenir une turbo-égalisation linéaire de l'auto-interférence minimisant l'erreur quadratique moyenne. Nos travaux mettent en évidence que ces mêmes impulsions permettent également de réduire le facteur de crête à mesure que la densité augmente. Enfin, sur canal sélectif en fréquence, on vérifie que l'approximation du canal par un coefficient par sous-porteuse est toujours possible. Ces résultats montrent que ce nouveau type de modulation permet d'augmenter l'efficacité spectrale tout en conservant la capacité d'adaptation au canal intrinsèque aux modulations multiporteuses. / With an increasing number of mobile terminals coupled with a large spreading of so-called "smart devices", we can see a growing demand for effective communication means in any place and in any situation.This goes with a more and more overcrowded spectrum.In this context, multicarrier modulations are good candidates to allow effective communication in any place.However current techniques, OFDM in particular, suffer from a bad time--frequency localization and peak-to-average power ratio, limiting their relevancy in an embedded context, or in scenarios with severe spectral constraints.In this thesis, we study faster-than-Nyquist multicarrier modulations.This kind of modulation allow for an increase in spectral efficiency by means of an increase in signaling density.This, in compensation, comes at the price of unavoidable self-interference, which makes demodulation harder.On an additive white Gaussian noise channel, we show how to carefully chose pulse-shapes that maximize signal-to-interference-plus-noise ratio.We show that these particular pulse-shapes yields a linear turbo-equalization of self-interference minimizing the mean squared error.Next, our work highlights the capability of these optimal pulse-shapes to reduce peak-to-average power ratio as density rises.Lastly, on frequency selective channels, we confirm that low complexity equalization using one tap by subcarrier is still possible.These results show how this new modulation technique helps increasing spectral efficiency while keeping what made multicarrier modulations popular: good adaptation to transmission channels.