© Abdelmalek Zidouri. 2010
Received: 17 June 2010
Accepted: 26 October 2010
Published: 1 November 2010
A newly developed adaptive scheme for system identification is proposed. The proposed algorithm is a mixture of two norms, namely, the -norm and the -norm ( ), where a controlling parameter in the range is used to control the mixture of the two norms. Existing algorithms based on mixed norm can be considered as a special case of the proposed algorithm. Therefore, our algorithm can be seen as a generalization to these algorithms. The derivation of the algorithm and its convexity property are reported and detailed. Also, the first moment behaviour as well as the second moment behaviour of the weights is studied. Bounds for the step size on the convergence of the proposed algorithm are derived, and the steady-state analysis is carried out. Finally, simulation results are performed and are found to corroborate with the theory developed.
The least mean square (LMS) algorithm  is one of the most widely used adaptive schemes. Several works have been presented using the LMS or its variants [2–14], such as signed LMS , the least mean fourth (LMF) algorithm and its variants , or the mixed LMS-LMF [16–18] all of which are intuitively motivated.
The LMS algorithm is optimum only if the noise statistics are Gaussian. However, if these statistics are different from Gaussian, other criteria, such as -norm ( ), perform better than the LMS algorithm. An alternative to the LMS algorithm which performs well when the noise statistics are not Gaussian is the LMF algorithm. A further improvement is possible when using a mixture of both algorithms, that is, the LMS and the LMF algorithms .
Adaptive filter algorithms designed through the minimization of equation (1) have a disadvantage when the absolute value of the error is greater than one. This makes the algorithm go unstable unless either a small value of the step size or a large value of the controlling parameter is chosen such that this unwanted instability is eliminated. Unfortunately, a small value of the step size will make the algorithm converge very slowly, and a large value of the controlling parameter will make the LMS algorithm essentially dominant.
The rest of the paper is organized as follows. In Section 2, the description of the proposed algorithm is addressed, while Section 3 deals with the convergence analysis. Section 4 details the derivation of the excess mean-square-error. The simulation results are reported in Section 5, and finally Section 6 concludes the main findings of the paper and outlines possible further work.
2. Proposed Algorithm
For , the algorithm reduces to the -norm adaptive algorithm, and moreover if results in the familiar signed LMS algorithm .
for , the gradient decreases in a positive direction, resulting in an obviously undesirable attribute for being used as a cost function. Setting the value of within the range provides a situation where the gradient at is very much lower than that for the cases with . This means that the resulting algorithm can be less sensitive to noise.
For , gives less weight for larger error and this tends to reduce the influence of aberrant noise, while it gives relatively larger weight to smaller errors and this will improve the tracking capability of the algorithm .
2.1. Convex Property of Cost Function
2.2. Analysis of the Error Surface
It can be shown as well that the error function for the feedback section will have a global minimum since the latter one is a convex function. As in the feedforward section, the adaptive process will continuously seek the bottom of the error function of the feedback section.
2.3. The Updating Scheme
In general, the step size is chosen small enough to ensure convergence of the iterative procedure and produce less misadjustment error.
3. Convergence Analysis
The step-size is small enough for the independence assumption  to be valid. As a consequence, the weight-error vector is independent of the input .
While assumptions (A1-A2) can be justified in several practical instances, assumption (A3) can only be attained asymptotically. The independence assumption  is very common in the literature and is justified in several practical instances . The assumption of small step size is not necessarily true in practice but has been commonly used to simplify the analysis .
3.1. First Moment Behavior of the Weight Error Vector
3.2. Second Moment Behavior of the Weight Error Vector
Both of these expressions are substituted in (22) to result in its simplified form (27).
Note that will result in zero in the denominator of expression (41) and therefore will make take any value in the range of positive numbers, a contradiction with the ranges of values for the step sizes of LMS and LMF algorithms. Moreover, any value for in will make of the step size set by (41) less than zero, also this condition is discarded. This concludes that it is safer to use the more realistic bounds of (39) which will guarantee stability regardless of the value of , and therefore will be considered here.
4. Derivation of the Excess Mean-Square-Error (EMSE)
Second, the following assumption is to be used in the following ensuing analysis:
5. Simulation Results
Three different noise environments have been considered namely, Gaussian, uniform, and Laplacian. The length of the adaptive filter is the same as that of the unknown system. The learning curves are obtained by averaging 600 independent runs. Two scenarios are considered for the case of the value of , that is, and . The performance measure considered here is the excess mean-square-error (EMSE).
In the case of an SNR of 20 dB, Figures 9 and 10 depict the results. The case of is shown in Figure 9 while that of is shown in Figure 10. One can see that, even though the proposed algorithm is still performing better in the uniform noise environment, as shown in Figure 9, for , however, identical performance is obtained by the different noise environments when as reported in Figure 10. The theoretical findings confirm these results as will be seen later.
From the above results, one can conclude that when the proposed algorithm is biased towards the LMF algorithm, in contrast to the case when , the proposed algorithm is biased towards the LMS algorithm.
A new adaptive scheme for system identification has been introduced, where a controlling parameter in the range is used to control the mixture of the two norms. The derivation of the algorithm is worked out, and the convexity property is proved for this algorithm. Existing algorithms, for example [16–18] can be considered as a special case of the proposed algorithm. Also, the first moment behaviour as well as the second moment behaviour of the weights are studied. Bounds for the step size on the convergence of the proposed algorithm are derived. Finally, the steady-state analysis was carried out; simulation results performed for the purpose of validating theory are found to be in good agreement with the theory developed.
The proposed algorithm has been applied so far to a system identification scenario, for example, echo cancellation. As a future extension, recent work is going on the application of the proposed algorithm to mitigate the effects of intersymbol interference in a communication system.
The author would like to acknowledge the support of King Fahd University of Petroleum and Minerals to carry out this research.
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