Stability in a general Sigma Delta modulator

A heuristic solution to the optimisation of flutter control in compression systems (and to some more binary quadratic programming problems) via ΔΣ modulation circuits

“…In fact, simplified and approximated models of ∆Σ modulators, such as the one based on the assumption shown in Figure 2(b), even if suitable from an analysis/design point of view, cannot account for the complex behaviour that is intrinsic with the highly non-linear nature of the modulators themselves. These are the reasons why designers typically need to rely on empirical criteria [10], [13]. At this point it is necessary to introduce two further considerations.…”

Higher order 1-bit Sigma-Delta modulation on a circle

“…The following stability criterion provides a sufficient condition for the stability of a Σ∆ scheme (1). This criterion is well-known to the engineering community (e.g., [21]), but had not been used in a rigorous mathematical framework until the work of Güntürk [8].…”

“…Proposition 1 (Stability Criterion). [8], [21] Consider a Σ∆ quantizer given by the recurrence relation (1). If…”

“…From (21) we see that in the second order case there is only one summand larger than of O(N −2 ) and it involves the remainder |∆u N −1 |. Therefore, we choose a constant update similar to that in Section IV.A, namely,…”

“…Essentially, Proposition 2 tells us that whilst the first term of the error estimate (21) gives the error of O(N −m ) for the m-th order Σ∆ quantizer, the second term always gives a sum of larger errors of order up to O(N −1 ), hence increasing the order of the Σ∆ scheme without making any changes to the existing scheme becomes meaningless.…”