Optimal MSE signal reconstruction in oversampled A/D conversion using convexity

Signal reconstruction in oversampled analog-to-digital conversion is classically performed by a lowpass filtering of the quantized signal. This leads to a mean squared error (MSE) inversely proportional to R ²ⁿ⁺¹, where R is the oversampling rate and n is the order of the converter. It is shown that this reconstruction does not necessarily lead to a signal which gives the same digital sequence as that of the original input signal. It is shown that, if this were the case, one would obtain an MSE inversely proportional to R²ⁿ⁺² instead. A way to achieve such an estimate with the same bandwidth and same digital conversion is proposed. This made it possible to perform numerical experiments which confirm the R²ⁿ⁺² behavior of the MSE