**9. Conclusion**

In this chapter, we present the proposed MAS-MTNOGA and AAS-MTNOGA algorithm for per-tone equalisation in DMT-based systems. We describe the tap-weight estimated PTEQ vector **p**ˆ *<sup>m</sup>*(*k*) for *m* ∈ *M* of *M*-combining tones. The mixed-tone cost function is demonstrated as the sum of mixed-tone weight estimated errors of adjacent tones. With the method of adaptive step-size approach and the normalised orthogonal gradient adaptive algorithm, two of low complexity adaptive step-size mechanisms can be achieved for per-tone equalisation based on the mixed-tone criterion. The derivation and analysis of two low complexity adaptive step-size schemes are presented. The adaptation of mean square mixed-tone errors (MSE) and excess mean square mixed-tone errors (EMSE) curves of proposed MAS-MTNOGA and AAS-MTNOGA algorithms are shown to converge rapidly to steady-state condition in the simulated channel. According to simulation results, the proposed algorithms can provide the good performance and are appeared to be robust in AWGN and NEXT channel in comparison with the fixed step-size algorithm of MTNOGA algorithm.

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[7] K.V.Acker, G.Leus, M.Moonen, O.van de Wiel and T.Pollet, "Per Tone Equalization for DMT-based Systems", *IEEE Transactions on Communications*, vol. 49, no. 1, pp. 109-119,

Adaptive Step-Size Orthogonal Gradient-Based Per-Tone Equalisation in Discrete Multitone Systems

[8] S.Sitjongsataporn and P.Yuvapoositanon, "Bit Rate Maximising Per-Tone Equalisation with Adaptive Implementation for DMT-based Systems", *EURASIP Journal on Advances in Signal Processing*, vol. 2009, Article ID 380560, 13 pages, 2009.

[9] S.Sitjongsataporn and P.Yuvapoositanon, "A Mixed-Tone RLS Algorithm with Orthogonal Projection for Per-Tone DMT Equalisation", in *Proc. IEEE International Midwest Symposium on Circuits and Systems (MWSCAS)*, Knoxville, USA, pp. 942-945,

[10] J.A.Apolinario Jr., R.G.Alves, P.S.R.Diniz and M.N.Swamy, "Filtered Gradient Algorithm ´ Applied to a Subband Adaptive Filter Structure", in *Proc. IEEE International Conference*

[11] S.Sitjongsataporn and P.Yuvapoositanon, "Mixed-Tone Normalised Orthogonal Gradient Adaptive Per-Tone DMT Equalisation", in *Proc. IEEE International Conference on Electrical Engineering*/*Electronics, Computer, Telecommunications and Information Technology*

[12] S.Sitjongsataporn and P.Yuvapoositanon, "Low Complexity Adaptive Step-size Filtered Gradient-based Per-Tone DMT Equalisation", in *Proc. IEEE International Symposium on*

[13] P.K.Pandey and M.Moonen, "Resource Allocation in ADSL Variable Length Per-Tone Equalizers", *IEEE Transanctions on Signal Processing*, vol. 56, no. 5, May 2008.

[15] L.Wang, Y.Cai and R.C.de Lamare, "Low-Complexity Adaptive Step-Size Constrained Constant Modulus SG-based Algorithms for Blind Adaptive Beamforming", in *Proc. IEEE International Conference Acoustics, Speech, and Signal Processing (ICASSP)*, pp.

[16] L.Wang, R.C.de Lamare and Y.Cai, "Low-Complexity Adaptive Step-Size Constrained Constant Modulus SG Algorithms for Adaptive Beamforming", *Signal Processing*, vol.89,

[17] J.S.Lim, "New Adaptive Filtering Algorithm Based on an Orthogonal Projection of Gradient Vectors", *IEEE Signal Processing Letters*, vol. 7, no. 11, pp. 314-316, Nov. 2000.

[18] R.C.de Lamare and R.Sampaio-Neto, "Low-Complexity Variable Step-Size Mechanisms for Stochastic Gradient Algorithms in Mimimum Variance CDMA Receivers", *IEEE*

[19] Y.Cai and R.C.de Lamare, "Low-complexity Variable Step-Size Mechanism for Code-Constrained Constant Modulus Stochastic Gradient Algorithms Applied to

*Transactions on Signal Processing*, vol. 54, pp. 2302-2317, Jun. 2006.

*Acoustics, Speech, and Signal Processing (ICASSP)*, vol.6, pp. 3705-3708, May 2001.

*(ECTI-CON)*, Pattaya, Thailand, pp. 1151-1154, May 2009.

*Circuits and Systems (ISCAS)*, Paris, France, pp. 2526-2529, May 2010.

[14] G.Strang, *Linear Algebra and Its Applications*, Harcourt Brace Jovanovich, 1988.

Jan. 2001.

Aug. 2008.

2593-2596, 2008.

pp. 2503-2513, 2009.

doi:10.1155/2009/380560.
