Uppsala universitet

Analysis of Stability and Performance of
Adaptation Algorithms with Time-invariant Gains.

IEEE Transactions on Signal Processing, vol. 52, Jan. 2004, pp. 103-116.   © IEEE

Anders Ahlén , Uppsala University
Lars Lindbom, Ericsson Infotech and
Mikael Sternad Uppsala University

Presentation slides (pdf).

A part of the paper,
Adaptation with Constant Gains: Analysis for Slow Variations.
by L. Lindbom, M. Sternad and A. Ahlén, is published at
IEEE International Conference on Acoustics, Speech and Signal Processing,
Salt Lake City, May 7-11 2001, pp. 3865-3868. © IEEE

Another part,
Adaptation with Constant Gains: Analysis for Fast Variations.
by L. Lindbom, M. Sternad and A. Ahlén is published at
IEEE International Conference on Acoustics, Speech and Signal Processing,
Orlando, Florida, May 13-17, 2002, pp. II-1097-1100 © IEEE

When tracking time-varying parameters of linear regression models, LMS is one of the simplest adaptation laws, while Kalman algorithms are the most powerful linear estimators. A third, intermediate, alternative is proposed here: The integration of the instantaneous gradient vector in used LMS is generalized to a linear time-invariant filter. Well-tuned filters provide estimates with an appropriate amount of coupling and inertia, resulting in high performance at low computational complexity.

We will here present results for the analysis of stability, performance and convergence in MSE, while a companion paper presents a novel Wiener optimization of the structure and the gains of such adaptation laws.

The difficult problem of accurately tracking time-varying radio channels in IS-136 cellular systems was an original motivating application. Here, LMS and RLS adaptation provide inadequate performance while the use of Kalman algorithms has so far been precluded, due to their computational complexity. An early version of the proposed algorithm has successfully been used on D-AMPS 1900 channels and a case study on this application can be found in a related paper.

Adaptation laws that track parameters of linear regression models are investigated. The considered class of algorithms apply linear time-invariant filtering on the instantaneous gradient vector and includes LMS as its simplest member.

The steady-state tracking performance for prediction and smoothing estimates is analyzed for parameter variations described by stochastic processes with zero mean and time-invariant statistics. The analysis is based on a novel technique that decomposes the inherent feedback of adaptation algorithms into one time-invariant loop and one time-varying loop.

The impact of the time-varying feedback on the tracking error can be neglected under certain conditions, and performance analysis then becomes straightforward.

Performance analysis in the presence of a non-negligible time-varying feedback is performed for algorithms that use scalar measurements. Convergence in MSE and the MSE tracking performance is investigated assuming independent consecutive regression vectors. Closed-form expressions for the tracking MSE are thereafter derived without this independence assumption for a subclass of algorithms applied to FIR models with white inputs. This class includes Wiener LMS adaptation.

Related publications:
Design of the general constant-gain adaptation algorithms.
The Wiener LMS adaptation algorithm, a special case with low complexity.
A Case Study on IS-136 channels.

PhD Thesis by Lars Lindbom.
Licenciate Thesis by Lars Lindbom, on averaged Kalman designs (KLMS)
and on deterministic sinusoid modelling of fading channels.


Conference version (ICASSP 2001): Postscript, 123K ; Pdf, 301K
Conference version (ICASSP 2002): Postscript, 100K ; Pdf, 425K
Slides (ICASSP 2002): Postscript, 307K ; Pdf, 208K
Slides: Large presentation 2001 (pdf).

Paper: Pdf, 419K ; Postscript, manuscript, 422 K.

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