Mean-Square Performance of Adaptive Filter Algorithms in Nonstationary Environments

Employing a recently introduced unified adaptive filter theory, we show how the performance of a large number of important adaptive filter algorithms can be predicted within a general framework in nonstationary environment. This approach is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. This general performance analysis can be used to evaluate the mean square performance of the Least Mean Square (LMS) algorithm, its normalized version (NLMS), the family of Affine Projection Algorithms (APA), the Recursive Least Squares (RLS), the Data-Reusing LMS (DR-LMS), its normalized version (NDR-LMS), the Block Least Mean Squares (BLMS), the Block Normalized LMS (BNLMS), the Transform Domain Adaptive Filters (TDAF) and the Subband Adaptive Filters (SAF) in nonstationary environment. Also, we establish the general expressions for the steady-state excess mean square in this environment for all these adaptive algorithms. Finally, we demonstrate through simulations that these results are useful in predicting the adaptive filter performance.

Authors:

• Mohammad Shams Esfand Abadi
• John Hakon Husøy

Keywords:

• Adaptive filter
• general framework
• energy conservation
• mean-square performance
• nonstationary environment.

References:

[1] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood
Cliffs, NJ: Prentice-Hall, 1985.
[2] S. Haykin, Adaptive Filter Theory. NJ: Prentice-Hall, 4th edition, 2002.
[3] A. H. Sayed, Fundamentals of Adaptive Filtering. Wiley, 2003.
[4] B. Widrow, J. M. McCool, M. Larimore, and C. R. Johnson, "Stationary
and nonstationry learning characteristics of the LMS adaptive filter," in
Proc. IEEE, 1976, pp. 1151-1162.
[5] N. J. Bershad, P. Feintuch, A. Reed, and B. Fisher, "Tracking charcteristics
of the LMS adaptive line enhancer: Response to a linear chrip signal
in noise," IEEE Trans. Acoust., Speech, Signal Processing, vol. 28, pp.
504-516, 1980.
[6] S. Marcos and O. Macchi, "Tracking capability of the least mean square
algorithm: Application to an asynchronous echo canceller," IEEE Trans.
Acoust., Speech, Signal Processing, vol. 35, pp. 1570-1578, 1987.
[7] E. Eweda, "Analysis and design of a signed regressor LMS algorithm for
stationary and nonstationary adaptive filtering with correlated Gaussian
data," IEEE Trans. Circuits, Syst., vol. 37, pp. 1367-1374, Nov. 1990.
[8] ÔÇöÔÇö, "Optimum step size of the sign algorithm for nonstationary
adaptive filtering," IEEE Trans. Acoust., Speech, Signal Processing,
vol. 38, pp. 1897-1901, 1990.
[9] ÔÇöÔÇö, "Comparsion of RLS and LMS, and sign algorithms for tracking
randomly time-varying channels," IEEE Trans. Signal Processing,
vol. 42, pp. 2937-2944, 1994.
[10] N. R. Yousef and A. H. Sayed, "Steady-state and tracking analyses of the
sign algorithm without the explicit use of the independence assumption,"
IEEE Signal Processing Letters, vol. 7, pp. 307-309, 2000.
[11] ÔÇöÔÇö, "A unified approach to the steady-state and tracking analyses of
adaptive filters," IEEE Trans. Signal Processing, vol. 49, pp. 314-324,
2001.
[12] A. H. Sayed and M. Rupp, "A time-domain feedback analysis of adaptive
algorithms via the small gain theorem," in Proc. SPIE, vol. 2563, 1995,
pp. 458-469.
[13] M. Rupp and A. H. Sayed, "A time-domain feedback analysis of filtered-
error adaptive gradient algorithms," IEEE Trans. Signal Processing,
vol. 44, pp. 1428-1439, 1996.
[14] M. S. E. Abadi and A. M. Far, "A unified approach to steady-state
performance analysis of adaptive filters without using the independence
assumptions," Signal Processing, vol. 87, pp. 1642-1654, 2007.
[15] H.-C. Shin and A. H. Sayed, "Mean-square performance of a family of
affine projection algorithms," IEEE Trans. Signal Processing, vol. 52,
pp. 90-102, Jan. 2004.
[16] H.-C. Shin, W. J. Song, and A. H. Sayed, "Mean-square performance
of data-reusing adaptive algorithms," IEEE Signal Processing Letters,
vol. 12, pp. 851-854, Dec. 2005.
[17] J. H. Hus├©y and M. S. E. Abadi, "A common framework for transient
analysis of adaptive filters," in Proc. 12th IEEE Mediterranean Electrotechnical
Conference, Dubrovnik, Croatia, May 2004, pp. 265-268.
[18] ÔÇöÔÇö, "Transient analysis of adaptive filters using a general framework,"
Automatika, Journal for Control, Measurement, Electronics, Computing
and Communications, vol. 45, pp. 121-127, 2004.
[19] J. H. Hus├©y, "A streamlined approach to adaptive filters," in Proc.
EUSIPCO, Firenze, Italy, Sept. 2006, published online by EURASIP
at http://www.arehna.di.uoa.gr/Eusipco2006/papers/1568981236.pdf.
[20] P. S. R. Diniz, Adaptive Filtering: Algorithms and practical implementation,
2nd ed. Kluwer, 2002.
[21] S. S. Pradhan and V. E. Reddy, "A new approach to subband adaptive
filtering," IEEE Trans. Signal Processing, vol. 47, pp. 655-664, 1999.
[22] M. de Courville and P. Duhamel, "Adaptive filtering in subbands using a
weighted criterion," IEEE Trans. Signal Processing, vol. 46, pp. 2359-
2371, 1998.
[23] K. A. Lee and W. S. Gan, "Improving convergence of the NLMS
algorithm using constrained subband updates," IEEE Signal Processing
Letters, vol. 11, pp. 736-739, 2004.
[24] J. Apolinario, M. L. Campos, and P. S. R. Diniz, "Convergence analysis
of the binormalized data-reusing LMS algorithm," IEEE Trans. Signal
Processing, vol. 48, pp. 3235-3242, Nov. 2000.
[25] S. G. Sankaran and A. A. L. Beex, "Normalized LMS algorithm with
orthogonal correction factors," in Proc. Asilomar Conf. on Signals,
Systems, and Computers, 1997, pp. 1670-1673.
[26] B. A. Schnaufer and W. K. Jenkins, "New data-reusing LMS algorithms
for improved convergence," in Proc. Asimolar Conf.,, Pacific Groves,
CA, May 1993, pp. 1584-1588.
[27] R. A. Soni, W. K. Jenkins, and K. A. Gallivan, "Acceleration of
normalized adaptive filtering data-reusing methods using the Tchebyshev
and conjugate gradient methods," in Proc. Int. Symp. Circuits Systems,
1998, pp. 309-312.
[28] S. L. Gay and J. Benesty, Acoustic Signal Processing for Telecommunication.
Boston, MA: Kluwer, 2000.
[29] T. K. Moon and W. C. Sterling, Mathematical Methods and Algorithms
for Signal Processing. Upper Saddle River: Prentice Hall, 2000.
[30] H. Malvar, Signal Processing with Lapped Transforms. Artech House,
1992.