Design procedure for high-performance stable Sigma-Delta Modulators

Sigma-Delta modulators are the standard for analog to digital conversion (ADC) nowadays. When using high oversampling ratios sigma-delta modulators (SDM) can achieve very high signal to noise ratio (SNR). They shape the noise and push it to frequencies higher than the operational band of interest. Thanks to its simplicity, single bit code shaping SDM are of greatest interest, because their performance is influenced only by the loopfilters transfer function and the modulators oversampling ratio (OSR). Despite the widespread use of Sigma-Delta modulators theoretical understanding of Sigma-Delta concept is still very limited, because these systems are nonlinear, due to the presence of a discontinuous nonlinearity - the quantizer. Up to now Prof. Mladenov spent 11 months in the MsM group with a NWO scholarship. During his stay in 2008 and 2009 the already in the MsM group developed results regarding the use of multiple-input describing functions has been extended by Prof. Mladenov in order to incorporate additional problems in the industrial application of SDMs. Prof. Mladenov developed a three-sinusoid-describing functions method for analysis and design of Sigma-Delta modulators and contributed to previously developed multiple-input describing functions method and considered the cases of application of the three-sinusoid-input describing functions in the analysis and design of Sigma-Delta modulators. Although in many cases the describing functions method is very useful for the design and analysis of Sigma-Delta modulators, it is an approximate approach that could not be applied generally. The most general approach is to use the nonlinear systems analysis to study the limit cycle behavior of the Sigma-Delta modulators. The topic of the proposed research is related to stability investigations and limit cycles verifications of high order Sigma-Delta modulators. During the stay in MsM group in 2010, 2011 and 2012 Prof. Mladenov developed an approach for characterization and validation of potential limit cycles of one bit high order Sigma-Delta modulators. This approach is generalized and the corresponding conditions for validation of limit cycles are verified. There has been much research into stability issues in Sigma-Delta modulators, but many essential questions remain unsolved. At its core, one would like to derive values of constant input such that, for certain initial conditions, the magnitude of the quantizer input will diverge toward infinity. A similar question is, given initial conditions and constant input, determine if this leads to stable behavior. To the best of our knowledge, there is no successful analytical approach to stability in high-order Sigma-Delta modulators (order greater than 2). There are several alternative approaches to stability in second-order Sigma-Delta modulators, some preliminary work on third-order designs, and only sketched approaches to stability in higher-order SDMs. In 2001, Prof. Mladenov used a transformation and has shown promising results on simple but high order Sigma-Delta modulators. This transformation in fact leads to the parallel presentation of the loop filter transfer function. The group of Prof. Mladenov continues investigating the potential of this approach. This approach has been generalized for the range of the DC input signal that guarantee stability during the stay of Prof. Mladenov in 2012. That time he also described systematically and reported all research related to limit cycles and stability issues of SDMs using the parallel presentation form of the loop filter transfer function, for all cases of poles and coefficients in this presentation. In practical design the modulators maximal DC input signal range and its SNR are determined mostly by simulations. Furthermore a lot of engineers experiment with the loopfilter coefficients in order to achieve more SNR, but up to date there is still no such thing as an optimal loopfilter transfer function for specific modulator order that provides both high performance and stable modulator behavior. All of the realistic loopflter transfer functions have their poles grouped into complex conjugated pairs and one real pole when having odd modulator order. In order to increase the modulator performance some authors move one of the complex conjugate pair of poles or the real poles somewhat outside of the unit circle, while keeping the other poles inside, resulting in increased SNR and reduced stability limit for maximal DC input signal amplitude beyond which the modulator becomes unstable. The objective of the proposed visit of Prof. Mladenov is to apply the theoretical results developed during his previous stays in MsM group and to propose a design technique for SDMs taking into account the stability and SNR performance. Such a design technique should offer high performance SDMs (with highest SNR) taking into account the stability issues. The expectaion of the proposed visit of prof. Maldenov is to motivate him to use the already developed stability analysis approach in the design of high performance sigma-delta modulators.