Lower bound on the mean squared error in multi-loop /spl Sigma//spl Delta/ modulation with periodic bandlimited signals
Research has been devoted to optimal signal reconstruction by /spl Sigma//spl Delta/ modulation. Using classical linear filtering, the reconstruction mean squared error (MSE) asymptotically decreases with the oversampling ratio R by R/sup -(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 /spl Oscr/(R/sup -(2n+2)/). We show for the multi-loop configuration of the /spl Sigma//spl Delta/ modulation and with the same kind of input signals that the asymptotic behavior of the reconstruction MSE cannot be less than /spl Oscr/(R/sup -(2n+2)/), including optimal reconstruction.<<ETX>>