Robust Remaindering and Signal Processin

The Chinese remainder theorem (CRT) provides a reconstruction of an integer from its remainders modulo several moduli. In this dissertation, we consider a robust remaindering problem when the remainders have errors. Its applications can be found in many areas, such as phase unwrapping in radar signal processing, frequency detection from udersampling, and computational neuroscience.;First, we present our new results on the robust remaindering problem in three aspects: 1) A robust CRT with a general set of moduli is proposed, called multi-stage robust CRT. With this method, an improvement in the robustness is obtained. 2) A trade-off between the dynamic range of the large integer and the robustness bound is obtained. It basically says that a decrease in the dynamic range may lead to an increase of the robustness bound. 3) We propose a new robust CRT that an integer can be robustly reconstructed from its erroneous remainders when a combined occurrence of multiple unrestricted errors and small errors is in the remainders.;Second, we investigate robust reconstruction and error correction for polynomials. The above results for integers are generalized to polynomials. Moreover, we obtain two different decoding algorithms, based on which we can obtain stronger residue error correction capability for polynomial remainder codes with non-pairwise co-prime moduli. With this newly obtained result, improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated.;Finally, we present some new results for multiple integer determination from their unordered residue sets. We study the determination of two integer case with a better dynamic range and simple closed-form determination algorithm. Moreover, we study the conditions on the multiple integers and moduli such that the dynamic range for the generalized CRT of multiple integers reaches the lcm of all the moduli. We propose two new such conditions, and the corresponding determination algorithms are also proposed.