Novel schemes for sigma-delta modulation: from improved exponential accuracy to low-complexity design
Percy A. Deift, C. Si̇nan Güntürk, Felix Krahmer · 2009
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 Gunturk 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.