Mixed analog-digital realization of the LMS algorithm using sigma-delta modulators

A multiplier-less mixed analog-digital (mixed-mode) realization of the LMS algorithm is proposed for use in linearly adaptive - recursive and non-recursive - switched-capacitor state-space filters. Analog and digital Σ-Δ modulators are used to perform the required multiplications. Digital circuitry is used to prevent the offset problems that occur in conventional analog LMS realizations. Having analog input signals, our realization provides a digital output for further processing. The dependence of the performance on the accuracy of the mixed-mode multipliers is investigated. It is shown theoretically, and by means of simulations, that the multipliers used to weight the state signals are the performance limiting elements. Nevertheless, we show that low-pass filtering of the state signals can increase the adaptation accuracy by 15 dB per doubling of the oversampling ratio - if second-order modulators are used. Using an oversampled clock rate for the modulators, which is eight times larger than the clock rate of the SC filter, our realization of the LMS algorithm can estimate the filter weights with an accuracy equivalent to 11-bits.