Lower bound on the mean squared error in multi-loop ΣΔ modulation with periodic bandlimited signals

Research has been devoted to optimal signal reconstruction by ΣΔ modulation. Using classical linear filtering, the reconstruction mean squared error (MSE) asymptotically decreases with the oversampling ratio R by R^-(2n+1), where n is the order of the modulator. Using a nonlinear reconstruction scheme, an improvement of the MSE asymptotic behavior by 3 dB per octave of R has been observed for time-varying, periodic and bandlimited input signals. This implies an MSE of 𝒪(R^-(2n+2)). We show for the multi-loop configuration of the ΣΔ modulation and with the same kind of input signals that the asymptotic behavior of the reconstruction MSE cannot be less than 𝒪(R^-(2n+2)), including optimal reconstruction.