Non-Linear Reconstruction of Delta-Sigma Modulated Signals: Randomized Surrogate Constraint Decoding Algorithm

Delta-sigma (ΔΣ) A/D conversion is a popular technique used to achieve high resolution data conversion for low to moderate bandwidth applications. It is commonly accepted that the digital circuitry of the ΔΣ A/D converters should perform a linear time-invariant (LTI) filtering on the output of the ΔΣ modulator in order to reconstruct the input signal. However, it has been shown that, when higher order ΔΣ A/D converters are operated at higher oversampling ratios (OSRs), non-linear reconstruction algorithms extract more information about the input signal than LTI methods. Still, use of non-linear algorithms has been limited due to their complexity and stability issues. Two practical non-linear methods presented in the literature are fast projection onto convex sets (POCS) and direct projection (DP). In this work, we show that the non-linear reconstruction process for ΔΣ modulated sequences can be treated as a linear feasibility problem (LFP). We then show that if the reconstruction process is defined as an LFP, its implementation is less complex than in the case of the DP method. We further describe how the LFP can be well-conditioned and demonstrate the reconstruction process by using the surrogate constraint algorithm with random constraint selection. The described algorithm considers a small number of constraints at a time, which makes it suitable for serial implementation.