Fast, recursive-least-squares transversal filters for adaptive filtering

Fast transversal filter (FTF) implementations of recursive-least-squares (RLS) adaptive-filtering algorithms are presented in this paper. Substantial improvements in transient behavior in comparison to stochastic-gradient or LMS adaptive algorithms are efficiently achieved by the presented algorithms. The true, not approximate, solution of the RLS problem is always obtained by the FTF algorithms even during the critical initialization period (first N iterations) of the adaptive filter. This true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic-gradient algorithms (factor of 1.6 to 3.5, depending upon application). Additionally, the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing, fast-RLS algorithms, such as the fast Kalman algorithms of Morf, Ljung, and Falconer (1976) and the fast ladder (lattice) algorithms of Morf and Lee (1977-1981). They are further shown to attain (steady-state unnormalized), or improve upon (first N initialization steps), the very low computational requirements of the efficient RLS solutions of Carayannis, Manolakis, and Kalouptsidis (1983). Finally, several efficient procedures are presented by which to ensure the numerical Stability of the transversal-filter algorithms, including the incorporation of soft-constraints into the performance criteria, internal bounding and rescuing procedures, and dynamic-range-increasing, square-root (normalized) variations of the transversal filters.

[1]  N. Levinson.  The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[2]  Robert Bartle,et al.  The Elements of Real Analysis , 1977, The Mathematical Gazette.

[3]  Karl Johan Åström,et al.  BOOK REVIEW SYSTEM IDENTIFICATION , 1994, Econometric Theory.

[4]  G. Stewart.  Introduction to matrix computations , 1973 .

[5]  Dominique Godard,et al.  Channel equalization using a Kalman filter for fast data transmission , 1974 .

[6]  A. Gray,et al.  Roundoff noise characteristics of a class of orthogonal polynomial structures , 1975 .

[7]  Bernard Widrow,et al.  A comparison of adaptive algorithms based on the methods of steepest descent and random search , 1976 .

[8]  M. Morf,et al.  Fast algorithms for recursive identification , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

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

[10]  G. Bierman.  Factorization methods for discrete sequential estimation , 1977 .

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

[12]  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.

[13]  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.

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

[15]  Lennart Ljung,et al.  Application of Fast Kalman Estimation to Adaptive Equalization , 1978, IEEE Trans. Commun..

[16]  L. Ljung,et al.  Fast calculation of gain matrices for recursive estimation schemes , 1978 .

[17]  Lloyd J. Griffiths.  Adaptive structures for multiple-input noise cancelling applications , 1979, ICASSP.

[18]  J. Mazo.  On the independence theory of equalizer convergence , 1979, The Bell System Technical Journal.

[19]  Randy L. Haupt,et al.  Introduction to Adaptive Arrays , 1980 .

[20]  Daniel T. L. Lee.  Canonical ladder form realizations and fast estimation algorithms , 1980 .

[21]  M. S. Mueller,et al.  Least-squares algorithms for adaptive equalizers , 1981, The Bell System Technical Journal.

[22]  M. S. Mueller.  On the rapid initial convergence of least-squares equalizer adjustment algorithms , 1981, The Bell System Technical Journal.

[23]  E. Satorius,et al.  Application of Least Squares Lattice Algorithms to Adaptive Equalization , 1981, IEEE Trans. Commun..

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

[25]  K. Senne,et al.  Performance advantage of complex LMS for controlling narrow-band adaptive arrays , 1981 .

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

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

[28]  Jr. S. Marple.  Efficient least squares FIR system identification , 1981 .

[29]  R. D. Gitlin,et al.  Fractionally-spaced equalization: An improved digital transversal equalizer , 1981, The Bell System Technical Journal.

[30]  Jean-Marc Delosme,et al.  Highly concurrent computing structures for matrix arithmetic and signal processing , 1982, Computer.

[31]  Lloyd J. Griffiths,et al.  Further results of a least squares and gradient adaptive lattice algorithm comparison , 1982, ICASSP.

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

[33]  R. Gitlin,et al.  The tap-leakage algorithm: An algorithm for the stable operation of a digitally implemented, fractionally spaced adaptive equalizer , 1982 .

[34]  M. Morf,et al.  Square root covariance ladder algorithms , 1982 .

[35]  C. Samson.  A unified treatment of fast algorithms for identification , 1982 .

[36]  Ioannis Dologlou,et al.  A new generalized recursion for the fast computation of the Kalman gain to solve the covariance equations , 1982, ICASSP.

[37]  T Kailath.  Time-Variant and Time-Invariant Lattice Filters for Nonstationary Processes. , 1982 .

[38]  John G. Proakis,et al.  Digital Communications , 1983 .

[39]  S. Ljung.  Fast Algorithms for Integral Equations and Least Squares Identification Problems , 1983 .

[40]  Thomas Kailath,et al.  Application of modified least-squares algorithms to adaptive echo cancellation , 1983, ICASSP.

[41]  E. Eweda,et al.  Second-order convergence analysis of stochastic adaptive linear filtering , 1983 .

[42]  George Carayannis,et al.  Fast Kalman type algorithms for sequential signal processing , 1983, ICASSP.

[43]  M. Honig.  Convergence models for lattice joint process estimators and least squares algorithms , 1983 .

[44]  T. Kailath,et al.  Normalized lattice algorithms for least-squares FIR system identification , 1983 .

[45]  Thomas Kailath,et al.  Fast, fixed-order, least-squares algorithms for adaptive filtering , 1983, ICASSP.

[46]  George Carayannis,et al.  A fast sequential algorithm for least-squares filtering and prediction , 1983 .

[47]  S. Tewksbury,et al.  Multiprocessor Implementation of Adaptive Digital Filters , 1983, IEEE Trans. Commun..

[48]  E. Eweda,et al.  Convergence of an adaptive linear estimation algorithm , 1984 .

[49]  Thomas Kailath,et al.  Lattice filter parameterization and modeling of nonstationary processes , 1984, IEEE Trans. Inf. Theory.

[50]  Thomas Kailath,et al.  Least-squares adaptive lattice and transversal filters: A unified geometric theory , 1984, IEEE Trans. Inf. Theory.

[51]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[52]  Bernard Widrow,et al.  On the statistical efficiency of the LMS algorithm with nonstationary inputs , 1984, IEEE Trans. Inf. Theory.

[53]  Thomas Kailath,et al.  An Efficient Exact-Least-Squares Fractionally Spaced Equalizer Using Intersymbol Interpolation , 1984, IEEE Journal on Selected Areas in Communications.

[54]  D. Lin.  On digital implementation of the fast kalman algorithms , 1984 .

[55]  S. Qureshi,et al.  Adaptive equalization , 1982, Proceedings of the IEEE.

[56]  M. Morf,et al.  A unified derivation for fast estimation algorithms by the conjugate direction method , 1985 .

[57]  J. J. Werner,et al.  Effects of biases on digitally implemented data-driven echo cancelers , 1985, AT&T Technical Journal.