Theory on the Speed of Convergence in Adaptive Equalizers for Digital Communication

Ungerboeck, G.

doi:10.1147/rd.166.0546

Cited by 162 publications

(51 citation statements)

References 6 publications

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“…13) corresponds to the model shown in Figure 3. The total output power in Figure 3 Ungerboeck [21] and Widrow [22) showed that for the LMS algorithm, the misadjustment is linearly dependent on the order N. The exponential dependence on N for the lattice suggests that, in general, a higher misadjustment can be expected for the lattice filter.…”

Section: (38)mentioning

confidence: 98%

Stochastic convergence properties of the adaptive gradient lattice

Sohie

Sibul²

1984

IEEE Trans. Acoust., Speech, Signal Process.

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Section: (38)mentioning

confidence: 98%

Stochastic convergence properties of the adaptive gradient lattice

Sohie

Sibul²

1984

IEEE Trans. Acoust., Speech, Signal Process.

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“…Although, in most cases, it is straightforward to extend the results to complex-valued processes if difficulties arise, results for the complex-valued case will be pointed out. While deterministic approaches have proven l 2 stability for any kind of driving signal u k [4][5][6][7], results from stochastic approaches are restricted to specific classes of random processes (unfiltered independent identically distributed (IID) [8], Gaussian [9,10], and spherically invariant random processes (SIRP) [11]). A recent historical overview is provided in [12].…”

Section: M×1mentioning

confidence: 99%

“…Nevertheless, such stochastic analysis is useful since it provides information about how the speed of convergence and the steady-state error depend on the step-size μ. The resulting stability bounds [8][9][10][11]13] are typically conservative:…”

Section: M×1mentioning

confidence: 99%

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Asymptotic equivalent analysis of the LMS algorithm under linearly filtered processes

Rupp

2016

EURASIP J. Adv. Signal Process.

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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.

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“…In [1], many arguments were demonstrated to question the usefulness of the mean squared error (MSE) in image and audio processing due to our complex human perception and these arguments were nicely supported by many practical examples and observations. Such a quadratic error measure has also been employed in adaptive-filter theory as a practical means to derive convergence in the mean-square sense, starting with Ungerböck in 1972 [2] who applied the technique onto Widrow and Hoff 's famous least-mean-square (LMS) algorithm [3] 1 . He also introduced the so-called independence assumption that is not well argued for [4] but a necessity once MSE techniques are being applied.…”

Section: Introduction: Some Historical Background On Adaptive-filter mentioning

confidence: 99%

Adaptive filters: stable but divergent

Rupp

2015

EURASIP J. Adv. Signal Process.

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The pros and cons of a quadratic error measure in the context of various applications have often been discussed. In this tutorial, we argue that it is not only a suboptimal but definitely the wrong choice when describing the stability behavior of adaptive filters. We take a walk through the past and recent history of adaptive filters and present 14 canonical forms of adaptive algorithms and even more variants thereof contrasting their mean-square with their l 2 −stability conditions. In particular, in safety critical applications, the convergence in the mean-square sense turns out to provide wrong results, often not leading to stability at all. Only the robustness concept with its l 2 −stability conditions ensures the absence of divergence.

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Theory on the Speed of Convergence in Adaptive Equalizers for Digital Communication

Cited by 162 publications

References 6 publications

Stochastic convergence properties of the adaptive gradient lattice

Stochastic convergence properties of the adaptive gradient lattice

Asymptotic equivalent analysis of the LMS algorithm under linearly filtered processes

Adaptive filters: stable but divergent

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