System Theory

Adaptive Filtering by L. Morales

By L. Morales

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26 Adaptive Filtering where O  ea  denotes third and higher-power terms of ea . 2, we get where  TEMSE is defined by (5). 2, we get 1 2 Eq e , e  v   TEMSE 2 Eq  e   Eq  v   (28) Substituting (27) and (28) into (11), and using Lemma 2, we can obtain (24), where parameters A, B, C are defined by (18b). 1 is satisfied, we can obtain (16) for real-valued cases. ,  EMSE . Finally, Differentiating both-hand sides of (16) with respect to  , and letting it be zero, we get   TEMSE      opt 1    Tr  Q   C  u      A u   Bu  0.

2 is satisfied, the tracking EMSE expression of (38) can be obtained while in nonstationary environments. Next, we consider the real-valued cases. 5, we get  E  u q  e    CE u  BE u   2 2 where B and C are defined by (18b). g. , 2 2 ea 2 . 8), we obtain 2 E  u q  e    CM u2  B  M  2  u2 TEMSE . 10) into the real-form equation of (10) yields  A   B  M  2   u2   TEMSE  CM u2   1Tr  Q  . 11) where A is defined by (18b). 2 is satisfied, we can obtain (37) and (38) while   2 , respectively.

16, pp. 176-179, 2009. 3 The Ultra High Speed LMS Algorithm Implemented on Parallel Architecture Suitable for Multidimensional Adaptive Filtering Marwan Jaber Université du Québec à Trois-Rivières Canada 1. Introduction Over the past decades a number of new adaptive filter algorithms have been elaborated and applied to meet demands for faster convergence and better tracking properties than earlier techniques could offer. The Filtered LMS algorithm is currently the most popular method for adapting a filter, due to its simplicity and robustness, which have made it widely adopted in many applications.

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