GAL and LSL Revisited: New Convergence Results

Two popular adaptive lattice algorithms-the gradient adaptive lattice (GAL) and the least squares lattice (LSL)-are revisited. Using the well-known ordinary differential equation (ODE) approach, two new results are presented in this paper. First, a global convergence proof of both algorithms is developed, which is, surprisingly, still missing in literature despite the popularity of the two algorithms. Second, in contrast to a general belief, it is shown that neither algorithm is completely immune to data condition or eigenvalue spread of the input correlation matrix, although significant improvement is achieved over the LMS algorithm.

[1]  Alan Weiss,et al.  Digital adaptive filters: Conditions for convergence, rates of convergence, effects of noise and errors arising from the implementation , 1979, IEEE Trans. Inf. Theory.

[2]  Lloyd J. Griffiths,et al.  A comparison of two fast linear predictors , 1981, ICASSP.

[3]  M. Morf,et al.  Recursive least squares ladder estimation algorithms , 1981 .

[4]  L. Sibul,et al.  Stochastic convergence properties of the adaptive gradient lattice , 1984 .

[5]  F. Ling,et al.  New forms of LS lattice algorithms and an analysis of their round-off error characteristics , 1985, ICASSP '85. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[6]  M. Morf,et al.  Recursive least squares ladder forms for fast parameter tracking , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[7]  Bernard Widrow,et al.  Adaptive Signal Processing , 1985 .

[8]  David G. Messerschmitt,et al.  Convergence properties of an adaptive digital lattice filter , 1981 .

[9]  Hong Fan,et al.  Application of Benveniste's convergence results in the study of adaptive IIR filtering algorithms , 1988, IEEE Trans. Inf. Theory.

[10]  S. Haykin,et al.  Learning characteristics of adaptive lattice filtering algorithms , 1980 .

[11]  Benjamin Friedlander Recursive lattice forms for spectral estimation , 1982 .

[12]  David G. Messerschmitt,et al.  Convergence properties of an adaptive digital lattice filter , 1980, ICASSP.

[13]  H. Kushner,et al.  Asymptotic Properties of Stochastic Approximations with Constant Coefficients. , 1981 .

[14]  D. Perriot-Mathonna,et al.  On the use of Ljung's results for studying the convergence properties of Hampton's adaptive filter , 1980 .

[15]  Harold J. Kushner,et al.  Weak Convergence and Properties of Adaptive Asymptotic F ilters w ith 177 Constant Gains , 1998 .

[16]  Bhaskar D. Rao,et al.  Tracking characteristics of the constrained IIR adaptive notch filter , 1987, 26th IEEE Conference on Decision and Control.

[17]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[18]  L. Griffiths A continuously-adaptive filter implemented as a lattice structure , 1977 .

[19]  Henry D'Angelo,et al.  Linear time-varying systems : analysis and synthesis , 1970 .

[20]  Walter Y. Chen,et al.  High Bit Rate Digital Subscriber Line Echo Cancellation , 1991, IEEE J. Sel. Areas Commun..

[21]  Bo Egardt,et al.  Convergence analysis of ladder algorithms for AR and ARMA models , 1986, Autom..

[22]  B. Friedlander,et al.  Lattice filters for adaptive processing , 1982, Proceedings of the IEEE.

[23]  A. Benveniste,et al.  Analysis of stochastic approximation schemes with discontinuous and dependent forcing terms with applications to data communication algorithms , 1980 .

[24]  Lloyd J. Griffiths,et al.  An adaptive lattice structure for noise-cancelling applications , 1978, ICASSP.

[25]  David G. Messerschmitt,et al.  Convergence models for adaptive gradient and least squares algorithms , 1981, ICASSP.

[26]  Lennart Ljung,et al.  Analysis of recursive stochastic algorithms , 1977 .

[27]  G. Goodwin,et al.  Improved finite word length characteristics in digital control using delta operators , 1986 .

[28]  M. Morf,et al.  Ladder forms for identification and speech processing , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.