Time average MSE analysis for the first order sigma-delta modulator with the inputs of bandlimited signals

Based on experiments and numerical simulation, it has been widely believed that the time average mean square error in the first order sigma-delta modulator with input of bandlimited signals decays like O(λ^-3) as the sampling ratio λ goes to infinity. This conjecture remains as an open problem for many years. Combining tools from number theory, harmonic analysis, real analysis and complex analysis, this paper shows that the conjecture holds in some reasonable sense.