Novel Schemes for Sigma-Delta Modulation: From Improved Exponential Accuracy to Low-Complexity Design

The central theme of this thesis is one-bit quantization of bandlimited signals via Sigma-Delta modulation. In this commonly used analog-to-digital conversion method, the signal of interest is represented by a ±1-valued sequence that is computed recursively from regular samples of the signal via a difference equation. The key feature of the method is that the low frequency content of the quantized representation approximates the signal: The larger the oversampling rate λ with respect to the Nyquist frequency, the higher the accuracy of the reconstruction that is achievable. It is known that exponential accuracy with an error decay rate O(2−rλ) for some rate constant r > 0 is achievable via Sigma-Delta modulation with modulators of increasing order. In this thesis, we first construct a family of schemes which gives a better rate constant r than is known for oversampled one-bit quantization. The construction builds on an idea by Güntürk and proceeds by solving an optimization problem posed in his work. En route, the solution establishes a connection between Sigma-Delta modulation and the theory of orthogonal polynomials. Second, we prove stability results for Sigma-Delta modulators involving recursion filters with rational transfer functions; stability is crucial to achieve satisfactory approximation. Such modulators are commonly used in practice because the associated analog circuits are of low complexity. Nevertheless, prior to this thesis, a rigorous stability analysis for such modulators was not available. We construct the first family of provably stable modulators of this type for all orders. Also, we introduce a novel, generalized stability criterion for Sigma-Delta modulation.