Given a configuration of t indistinguishable pebbles on the n vertices of a graph G, we say that a vertex v can be reached if a pebble can be placed on it in a finite number of "moves". G is said to be pebbleable if all its vertices can be thus reached. Now given the n-path Pn how large (resp. small) must t be so as to be able to pebble the path almost surely (resp. almost never)? It was known that the threshold th(Pn) for pebbling the path satisfies n2clgn≤th(Pn)≤n22lgn, where lg=log2 and c<1/2 is arbitrary. We improve the upper bound for the threshold function to th(Pn)≤n2dlgn, where d>1 is arbitrary.

Due to the growth of wireless communication many communication frequencies have grown increasingly dense. This density requires higher Q-factor to receive only the signal of interest. With the rise of smaller integrated circuits previous solutions used for filtering have become viable again. This paper explores whether the N-path filter is viable in the modern day for radio frequency receiver purposes. A non-differential N-path filter was created by utilizing Cadence Virtuoso with a working center frequency range of 750MHz to 1GHz while using TSMC technology. The desired quality factor of over 1,000 was reached while maintaining a total area of 800 by 800 micrometers. Through the analysis of the N-path filter new techniques for mixed signal analysis were used for simulation. This included parametric analysis in Cadence ADE-L and additional analysis in MATLAB, and the addition of a bootstrapping circuit to decrease simulation time. Future applications regarding analyzing mixed signals could use these methods to provide frequency response data and automated processing.

As the number of wireless devices in the world increases, so does the demand for flexible radio receiver architectures capable of operating over a wide range of frequencies and communication protocols. The resonance-based channel-select filters used in traditional radio architectures have a fixed frequency response, making them poorly suited for such a receiver. The N-path filter is based on 1960s technology that has received renewed interest in recent years for its application as a linear high Q filter at radio frequencies. N-path filters use passive mixers to apply a frequency transformation to a baseband low-pass filter in order to achieve a high-Q band-pass response at high frequencies. The clock frequency determines the center frequency of the band-pass filter, which makes the filter highly tunable over a broad frequency range. Issues with harmonic transfer and poor attenuation limit the feasibility of using N-path filters in practice. The goal of this thesis is to design an integrated active N-path filter that improves upon the passive N-path filter’s poor harmonic rejection and limited outof- band attenuation. The integrated circuit (IC) is implemented using the CMRF8SF 130nm CMOS process. The design uses a multi-phase clock generation circuit to implement a harmonic rejection mixer in order to suppress the 3rd and 5th harmonic. The completed active N-path filter has a tuning range of 200MHz to 1GHz and the out-ofband attenuation exceeds 60dB throughout this range. The frequency response exhibits a 14.7dB gain at the center frequency and a -3dB bandwidth of 6.8MHz.

Ce travail de thèse porte sur la conception de dispositifs filtrants accordables, c'est-à-dire pouvant commuter leurs caractéristiques d'un standard à un autre afin de réduire l’encombrement des chaines émission/réception d’un dispositif multistandard. Les filtres les plus utilisés actuellement sont des filtres acoustiques. En effet, ces filtres sont difficilement intégrables dans une technologie silicium et ils restent parmi les dispositifs passifs les plus encombrants du front-end RF. De plus, ils ne permettent pas d’avoir de bonnes performances en pertes d’insertion, en sélectivité et en accordabilité fréquentielle. De ce fait, des solutions alternatives sont à l’origine de ces travaux de thèse. Nous avons considéré tout d’abord un filtre actif qui a la possibilité de régler sa fréquence de résonnance, d’une part à la fréquence de résonnance du système Wifi et d’autre part, à la fréquence de résonnance du système Zigbee. Ensuite, une autre solution a été proposée dans le dernier chapitre qui consiste à réaliser une structure active filtrante bi-bande intégrée. Cette solution a pour avantage de récolter simultanément les données des systèmes opérant dans les deux bandes de fréquences visés. Les résultats obtenus des circuits réalisés sont validés par des simulations et de mesures. / This thesis concerns the design of tunable filter devices that can switch theirs characteristics from one standard to another in order to reduce the congestion of emission/reception chain of multi-standard systems. The most commonly used filters are acoustic filters. Indeed, these filters are difficult to be integrated in silicon technology and they remain one of the most bulky passive devices of the RF front-end. In addition, they don’t achieve good performance in insertion loss, frequency selectivity and tunability. Therefore, alternative solutions are at the origin of this thesis. An active filter is considered with the possibility of adjusting the resonance frequency: the resonant frequency of the Wifi system and the resonance frequency of the Zigbee system. Moreover, another solution is proposed in the last chapter, which consists to achieve a dual band structure of integrated active filter. This solution has the advantage to simultaneously collect data provided from the operating systems located in the two specified frequencies bands. Simulations and measurements validate the results of the realized circuits.