Abstract
Least-squares algorithms aim at the minimization of the sum of the squares of the difference between the desired signal and the model filter output [1, 2]. When new samples of the incoming signals are received at every iteration, the solution for the least-squares problem can be computed in recursive form resulting in the recursive least-squares (RLS) algorithms. The conventional version of these algorithms will be the topic of this chapter.
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Notes
- 1.
The a posteriori error is computed after the coefficient vector is updated, and taking into consideration the most recent input data vector \({\mathbf {{x}}}(k)\).
- 2.
The expression for \(\xi _\mathrm{min, p}\) can be negative; however, \(\xi (k)\) is always nonnegative.
- 3.
Again the reader should recall that when computing the gradient with respect to \({\mathbf {{w}}}^{*}(k)\), \({\mathbf {{w}}}(k)\) is treated as a constant.
References
G.C. Goodwin, R.L. Payne, Dynamic System Identification: Experiment Design and Data Analysis (Academic, New York, 1977)
S. Haykin, Adaptive Filter Theory, 4th edn. (Prentice Hall, Englewood Cliffs, 2002)
S.H. Ardalan, Floating-point analysis of recursive least-squares and least-mean squares adaptive filters. IEEE Trans. Circ. Syst. (CAS) 33, 1192–1208 (1986)
J.M. Cioffi, Limited precision effects in adaptive filtering. IEEE Trans. Circ. Syst. (CAS) 34, 821–833 (1987)
E. Eweda, Comparison of RLS, LMS, and sign algorithms for tracking randomly time-varying channels. IEEE Trans. Signal Process. 42, 2937–2944 (1994)
S. Haykin, A.H. Sayed, J.R. Zeidler, P. Yee, P.C. Wei, Adaptive tracking of linear time-varying systems by extended RLS algortihms. IEEE Trans. Signal Process. 45, 1118–1128 (1997)
E. Eweda, Maximum and minimum tracking performances of adaptive filtering algorithms over target weight cross correlations. IEEE Trans. Circ. Syst. II: Analog Digital Signal Proc. 45, 123–132 (1998)
E. Eleftheriou, D.D. Falconer, Tracking properties and steady-state performance of RLS adaptive filter algorithms. IEEE Trans. Acoust. Speech Signal Process. (ASSP) 34, 1097–1110 (1986)
F. Ling, J.G. Proakis, Nonstationary learning characteristics of least squares adaptive estimation algorithms, in Proceedings of the IEEE International Conference on Acoustics, Speech, Signal Processing, San Diego, CA, Mar 1984, pp. 30.3.1–30.3.4
S. Ardalan, On the sensitivity of transversal RLS algorithms to random perturbations in the filter coefficients. IEEE Trans. Acoust. Speech Signal Process. 36, 1781–1783 (1988)
C.R. Johnson Jr., Lectures on Adaptive Parameter Estimation (Prentice Hall, Englewood Cliffs, 1988)
O.M. Macchi, N.J. Bershad, Adaptive recovery of a chirped sinusoid in noise, Part 1: performance of the RLS algorithm. IEEE Trans. Signal Process. 39, 583–594 (1991)
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Diniz, P.S.R. (2020). Conventional RLS Adaptive Filter. In: Adaptive Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-29057-3_5
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DOI: https://doi.org/10.1007/978-3-030-29057-3_5
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