IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. zyxwvutsrqpo 41. NO. 5, MAY 1993 I947 filter will act as a set of notch filters at dc, zyxwvutsrqp U,,, and 200. This is an intuitively reasonable result given the nonlinear quadratic opera- tions applied to cos zyxwvutsrqp (uonT). Comparing these results with the cor- responding transfer function for case A, we note that the position of the zeros is not changed by the inclusion of the dc term. Also, the position of the complex conjugate poles does not change. How- ever, the pole located on the real axis is pushed closer to the unit circle. So the overall effect of the dc term in the deterministic input case is similar to the random input case-it reduces the stability limit. Explicit stability conditions for the discrete system (1 zyxwvuts 1) can be obtained by constructing a Jury table zyxwvutsrqp [ 151. The dominant con- dition for the stability of the filter from the Jury table is given by (14) A corresponding analysis for case A (no dc term) gives a very sim- ilar result: IV. CONCLUSION Overall we may summarize the effect of the bias term as follows: it always degrades performance both in terms of convergence and steady state performance although the degree of degradation may not be large. On the other hand, if we do not include a dc term then the filter produces a biased output. REFERENCES [I] I. Pitas and A. N. VenetsanoDoulos. zyxwvutsrqpo Nonlinear Dinital Filters: Prin- ciples and Applications. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Sys- tems. Wiley, 1980. T. Koh and E. J. Powers, “Second-order Volterra filtering and its application to nonlinear system identification,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 1445-1455, 1985. D. Mansour and A. H. Gray, “Frequency domain nonlinear adaptive filter,” in Proc. IEEE zyxwvutsrqponm In?. zyxwvutsrqp Conf. Acoust., Speech, Signal Processing, V. J. Mathews and J. Lee, “A fast recursive least squares second- order Volterra filter,” in Proc. IEEE Int. Con5 Acousr., Speech, Sig- nal Processing, 1988, pp. 1383-1386. P. Duvaut, “A unifying and general approach to adaptive linear- quadratic discrete time Volterra filtering,” in Proc. IEEE Int. Con5 Acoust., Speech, Signal Processing, 1989, pp. 1166-1171. S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice- Hall, 1986. M. J. Coker and D. N. Simkins, “A nonlinear adaptive noise can- celler,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process- ing, 1980, pp. 470-473. 0. Agazzi, D. G. Messerschmitt, and D. A. Hodges, “Nonlinear echo cancellation of data signals,” IEEE Trans. Commun., vol. COM- B. Widrow and S. D. Steams, Adaptive Signal Processing. Engle- wood Cliffs, NJ: Prentice-Hall, 1985. P. M. Clarkson and M. V. Dokic, “Stability and convergence be- havior of second-order LMS Volterra filter,” Proc. Inst. Elec. Eng., R. A. Hom and C. R. Johnson, Matrix Analysis. Cambridge Uni- versity Press, 1985. P. M. Clarkson and P. R. White, “Simplified analysis of the LMS adaptive filter using a transfer function approximation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 987-993, 1987. Kkwer Academic, 1996. 1981, pp. 550-553. 30, pp. 2421-2433, 1982. vol. 27, pp. 441-443, 1991. r141 1151 P. M. Clarkson, “Analysis of the LMS adaptive filter applied to pe- riodic signals,” in Mathematics in Signal Processing. Oxford Uni- versity Press, 1987, pp. 593-605. E. I. Juxy, Theory and Application of the z-transform Method. New York:Wiley, 1964. On Statistical Efficiency of the LMS Algorithm in System Modeling B. Farhang-Boroujeny Abstract-The conventional least mean square (LMS) algorithm is compared with the ideal LMS/Newton (ILMSN) algorithm. It is shown that, although, under certain conditions, for similar misadjustment, the output mean square error (MSE) of the ILMSN algorithm may converge much faster than the MSE of the LMS algorithm, the dif- ference between the two algorithms may not be that great if misalign- ments of the adaptive filter tap-gains are compared. Analytical results are presented, with computer simulations that support their validity. I. INTRODUCTION Since its advent [l], the least mean square (LMS) algorithm has been attractive to most researchers in the field of adaptive signal processing. Many researchers have pointed out the deficiency of the LMS algorithm in dealing with colored inputs and suggested many ways to combat it [2], [3]. Widrow and Walach [4] have compared the performance of the exact least squares with an ideal algorithm called orthogonalized LMS and have shown that (within the context discussed in zyxwv [4]) the efficiency of this algorithm in tak- ing advantage of the available data is almost as high as the exact least squares. The latter algorithm is also presented in a slightly different form and called ideal LMS/Newton (ILMSN) algorithm in [5]. In addition, the simulation results presented in [6] shows that the LMS type algorithms are more robust to sudden variation of the environment parameters than the least squares ones. This correspondence aims to reveal another interesting feature of the LMS algorithm when used in a modeling application such as Fig. 1. We compare the conventional LMS with the ILMSN al- gorithm under the condition that their input samples are highly cor- related, i.e., the eigenvalues of the input correlation matrix are widely spread. Although the LMS and ILMSN algorithms are both designed to minimize the output mean square error (MSE), we con- sider the model error, as performance index to compare the two algorithms. The model error gives a measure of the difference between the plant P(z) and its model W(z) with constant weighting function over all frequen- cies. On the other hand, the output MSE (and equivalently mis- adjustment) gives the latter difference with the power spectral den- sity of input process as the weighting function over various frequencies. In other words, while the output MSE is input de- pendent, the model error is input independent. Therefore, in ap- plications such as system modeling in which one may be interested Manuscript received October 27, 1991; revised September 18, 1992. The associate editor coordinating the review of this correspondence and ap- proving it for publication was Dr. James Zeidler. The author is with the Department of Electrical Engineering, National University of Singapore, Singapore 05 I 1. IEEE Log Number 9207559. 1053-587X/93$03.00 zyxwvut 0 1993 IEEE