Fixed point error analysis of the normalized ladder algorithm

An attempt is made to analyze the fixed point error performance of the normalized ladder algorithm, for autoregressive system identification, assuming rounding arithmetic. A preliminary simulation study of this algorithm has shown that the bias in the estimated reflection coefficients is much more predominant than the variance of the error in the estimate. The study, therefore, is directed to find a model for predicting the bias in the estimated reflection coefficients. The analysis shows that the roundoff errors associated with the square root operations in one of the algorithm equations are mainly responsible for the bias in the estimated reflection coefficients. These errors arise because of the normalization procedure that makes the quantities under the square root operations very close to one. Two main results are presented in the paper. 1) A simplified theoretical expression for predicting the average bias in the estimated reflection coefficients at any stage is derived. 2) A recursive relation for the average error, arising from the finite precision arithmetic in the squared residuals, is derived. This relation illustrates how the errors made in one stage affect the errors in the succeeding stages. Simulations are performed to check the theoretical models. The experimental results agree very closely with the theoretical predictions.