A block floating-point treatment to the LMS algorithm: efficient realization and a roundoff error analysis

An efficient scheme is presented for implementing the LMS-based transversal adaptive filter in block floating-point (BFP) format, which permits processing of data over a wide dynamic range, at temporal and hardware complexities significantly less than that of a floating-point processor. Appropriate BFP formats for both the data and the filter coefficients are adopted, taking care so that they remain invariant to interblock transition and weight updating operation, respectively. Care is also taken to prevent overflow during filtering, as well as weight updating processes jointly, by using a dynamic scaling of the data and a slightly reduced range for the step size, with the latter having only marginal effect on convergence speed. Extensions of the proposed scheme to the sign-sign LMS and the signed regressor LMS algorithms are taken up next, in order to reduce the processing time further. Finally, a roundoff error analysis of the proposed scheme under finite precision is carried out. It is shown that in the steady state, the quantization noise component in the output mean-square error depends on the step size both linearly and inversely. An optimum step size that minimizes this error is also found out.