Fixed-point error analysis of stochastic gradient adaptive lattice filters

Abstract-This paper presents a theoretical analysis of the stochastic gradient adaptive lattice filter used as a linear, one-step predictor, when the effects of finite precision arithmetic are taken into account. Only the fixed-point implementation is considered here. Both the unnormalized and normalized adaptation algorithms are analyzed. Expressions for the steady-state mean-squared values of the accumulated numerical errors in the computation of the reflection coefficients and the prediction errors of different orders have been developed. The results show that the dominant term in the expressions for the mean-squared values of the numerical errors is inversely proportional to the convergence parameter. Furthermore, they indicate that the quantization errors associated with the reflection coefficients are more critical than those associated with representing the prediction error sequences. Another interesting result is that signals with high correlation among samples produce larger numerical errors in the adaptive lattice filter than signals with low correlation among samples. We present several simulation examples that show close agreement with the theoretical results. We also present some comparisons between the numerical behavior of the lattice and transversal stochastic gradient adaptive filters. The numerical results support the general belief that the gradient adaptive lattice filters have better numerical properties than their transversal counterparts, even though it is conceivable that the lattice filters can produce larger numerical errors than the transversal filters under some circumstances.