Oversampled sigma–delta LMS adaptive FIR filters

Two filter designs for adaptive least mean squares (LMS) filtering with sigma–delta modulated input signals are described. One implementation is multibit multiplier-free and operates entirely at the oversampling frequency of the sigma–delta signals; in the other design only the FIR filter operates at the oversampled frequency while the adaptive filtering algorithm is performed at the Nyquist rate. To circumvent any aliasing problems that may be caused by the downsampling process in the architecture and ensure convergence of the adaptive FIR filter it is necessary to attenuate the high-frequency sigma-delta quantisation noise that is present. To perform this task a multiplier-free, multistage IIR filter structure is used that requires considerably fewer computations than an equivalent FIR filter. The two adaptive LMS filter designs are analysed and their performance is compared with a conventional PCM system in terms of achievable minimum MSE and adaptation speed.