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By L. Morales

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1 with   0 8) in [11]. 1 in [11] are all second-order approximate. Example 2: Consider the real-valued data in Gaussian noise environments. 1 in [11].  v k:odd    2  (46) where  k  1  !!  v   p: odd . (47) Then, substituting (47) into Theorem 1 and Theorem 2 or substituting (46) into (45a) - (45c), yield the steady-state performance results for real LMP algorithm in Gaussian noise environments.  v u   .  v u    (48) The above expression is also applicable for LMS algorithm by means of  1  !!

10] Y. Wang, C. Zhang, and Z. Wang, “A new variable step-size LMS Algorithm with application to active noise control”, Proc. IEEE ICASSP, pp. 573-575, 2003. [11] T. Arnantapunpong, T. Shimamura, and S. A. Jimaa,” A New Variable Step Size for Normalized LMS Algorithm”, NCSP'10 - 2010 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing, Honolulu, Hawaii, USA March 3-5, 2010. [12] Simon Haykin, Adaptive Filter Theory, Prentice- hall, 3rd Edition, 1996 [13] S. A.

143, pp137-142, 1996. [8] B. Lin, R. He, X. Wang and B. Wang, “Excess MSE analysis of the concurrent constant modulus algorithm and soft decision-directed scheme for blind equalization,” IET Signal Processing, vol. 2, pp. , 2008. [9] D. L. Duttweiler, “Adaptive filter performance with nonlinearities in the correlation multiplier,” IEEE Trans. , Speech, Signal Processing, vol. , 1982. 46 Adaptive Filtering [10] S. C. Douglas and T. -Y. Meng, “Stochastic gradient adaptation under general error criteria,” IEEE Trans.

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