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Zeitschriftenartikel:

M. Rupp:
"Asymptotic equivalent analysis of the LMS algorithm under linearly filtered processes";
EURASIP Journal on Advances in Signal Processing, 2016:18 (2016), 18; S. 1 - 16.



Kurzfassung deutsch:
While the least mean square (LMS) algorithm has been widely explored for some specific statistics of the driving
process, an understanding of its behavior under general statistics has not been fully achieved. In this paper, the mean
square convergence of the LMS algorithm is investigated for the large class of linearly filtered random driving
processes. In particular, the paper contains the following contributions: (i) The parameter error vector covariance
matrix can be decomposed into two parts, a first part that exists in the modal space of the driving process of the LMS
filter and a second part, existing in its orthogonal complement space, which does not contribute to the performance
measures (misadjustment, mismatch) of the algorithm. (ii) The impact of additive noise is shown to contribute only to
the modal space of the driving process independently from the noise statistic and thus defines the steady state of the
filter. (iii) While the previous results have been derived with some approximation, an exact solution for very long filters
is presented based on a matrix equivalence property, resulting in a new conservative stability bound that is more
relaxed than previous ones. (iv) In particular, it will be shown that the joint fourth-order moment of the decorrelated
driving process is a more relevant parameter for the step-size bound rather than, as is often believed, the second-order
moment. (v) We furthermore introduce a new correction factor accounting for the influence of the filter length as well
as the driving process statistic, making our approach quite suitable even for short filters. (vi) All statements are
validated by Monte Carlo simulations, demonstrating the strength of this novel approach to independently assess the
influence of filter length, as well as correlation and probability density function of the driving process

Kurzfassung englisch:
While the least mean square (LMS) algorithm has been widely explored for some specific statistics of the driving
process, an understanding of its behavior under general statistics has not been fully achieved. In this paper, the mean
square convergence of the LMS algorithm is investigated for the large class of linearly filtered random driving
processes. In particular, the paper contains the following contributions: (i) The parameter error vector covariance
matrix can be decomposed into two parts, a first part that exists in the modal space of the driving process of the LMS
filter and a second part, existing in its orthogonal complement space, which does not contribute to the performance
measures (misadjustment, mismatch) of the algorithm. (ii) The impact of additive noise is shown to contribute only to
the modal space of the driving process independently from the noise statistic and thus defines the steady state of the
filter. (iii) While the previous results have been derived with some approximation, an exact solution for very long filters
is presented based on a matrix equivalence property, resulting in a new conservative stability bound that is more
relaxed than previous ones. (iv) In particular, it will be shown that the joint fourth-order moment of the decorrelated
driving process is a more relevant parameter for the step-size bound rather than, as is often believed, the second-order
moment. (v) We furthermore introduce a new correction factor accounting for the influence of the filter length as well
as the driving process statistic, making our approach quite suitable even for short filters. (vi) All statements are
validated by Monte Carlo simulations, demonstrating the strength of this novel approach to independently assess the
influence of filter length, as well as correlation and probability density function of the driving process

Schlagworte:
Adaptive gradient-type filters, Mismatch, Misadjustment


"Offizielle" elektronische Version der Publikation (entsprechend ihrem Digital Object Identifier - DOI)
http://dx.doi.org/10.1186/s13634-015-0291-1

Elektronische Version der Publikation:
http://publik.tuwien.ac.at/files/PubDat_248333.pdf


Erstellt aus der Publikationsdatenbank der Technischen Universität Wien.