Abstract

The aim of the study is to propose an adaptive algorithm using convex combinational approach to have both fast convergence and less steady state error simultaneously. For this purpose, we have used two affine projection adaptive filters with complementary nature (both in step size and projection order) as the component filters. The first component filter has high projection order and large step size which makes it to have fast convergence at the cost of more steady state error. The second component filter has slow convergence and less steady state error due to the selection of small step size and projection order. Both are combined using convex combiner so as to have best final output with fast convergence and less steady state error. Each of the component filters are updated using their own error signals and stochastic gradient approach is used to update the convex combiner so as to have minimum overall error. By using energy conservation argument, analytical treatment of the combination stage is made in stationary environment. It is found that during initial stage the proposed scheme converges to the fast filter which has good convergence later it converges to either of the two (whichever has less steady state error) and towards the end, the final output converges to slow filter which is superior in lesser steady state error. Experimental results proved that the proposed algorithm has adopted the best features of the component filters.

Keywords:

Affine projection adaptive filter, convex combination, echo cancellation, energy conservation argument, excess mean square error , steady state error analysis,

References

1. Arenas-Garcia, J., A.R. Figueiras-Vidal and A.H. Sayed, 2005a. Steady state performance of convex combinations of adaptive filters. Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP'05), 4: iv33-iv36.
   CrossRef    
2. Arenas-Garcia, J., V. Gomez-Verdejo and A.R. Figueiras-Vidal, 2005b. New algorithms for improved adaptive convex combination of LMS transversal filters. IEEE T. Instrum. Meas., 54(6): 2239-2249.
   CrossRef    
3. Arenas-Garcia, J., A.R. Figueiras-Vidal and A.H. Sayed, 2006. Mean-square performance of a convex combination of two adaptive filters. IEEE T. Signal Proces., 54(3): 1078-1090.
   
4. Azpicueta-Ruiz, L.A., A.R. Figueiras-Vidal and J. Arenas-Garcia, 2008. A normalized adaptation scheme for the convex combination of two adaptive filters. Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP'08), pp: 3301-3304.
   CrossRef    
5. Azpicueta-Ruiz, L.A., M. Zeller, A.R. Figueiras-Vidal, J. Arenas-Garcia and W. Kellermann, 2011. Adaptive combination of Volterra kernels and its application to nonlinear acoustic echo cancellation. IEEE T. Audio Speech, 19(1): 97-110.
   CrossRef    
6. Choi, J.H., S.H. Kim and S.W. Kim, 2014. Adaptive combination of affine projection and NLMS algorithms. Signal Process., 100: 64-70.
   CrossRef    
7. Das, B.K. and M. Chakraborty, 2014. Sparse adaptive filtering by an adaptive convex combination of the LMS and the ZA-LMS algorithms. IEEE T. Circuits-I, 61(5): 1499-1507.
   
8. Haiquan, Z., Y. Yi, G. Shibin, Z. Xiangping and H. Zhengyou, 2014. Memory proportionate APA with individual activation factors for acoustic echo cancellation. IEEE T. Audio Speech, 22(6).
   
9. Haykin, S., 2002. Adaptive Filter Theory. 4th Edn., Prentice-Hall, Upper Saddle River, NJ.
   PMid:12487794    
10. He, X., R.A. Goubran and P.X. Liu, 2014. A novel sub-band adaptive filtering for acoustic echo cancellation based on empirical mode decomposition algorithm. Int. J. Speech Technol., 17(1): 37-42.
    CrossRef    
11. Kim, K.H., Y.S. Choi, S.E. Kim and W.J. Song, 2008. Convex combination of affine projection filters with individual regularization. Proceeding of the 23rd International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC).
    
12. Kozat, S.S., A.T. Erdogan, A.C. Singer and A.H. Sayed, 2010. Steady-state MSE performance analysis of mixture approaches to adaptive filtering. IEEE T. Signal Proces., 58(8): 4050-4063.
    
13. Kwang-Hoon, K.I.M., C. Young-Seok, K. Seong-Eun and S. Woo-Jin, 2014. A low-complexity complementary pair affine projection adaptive filter. IEICE T. Fund. Electr., 97(10): 2074-2078.
    
14. Martinez-Ramon, M., J. Arenas-Garcia, A. Navia-Vazquez and A.R. Figueiras-Vidal, 2002. An adaptive combination of adaptive filters for plant identification. Proceeding of the 14th International Conference on Digital Signal Processing, (DSP’02), 2: 1195-1198.
    CrossRef    
15. Naylor, P.A., J. Cui and M. Brookes, 2006. Adaptive algorithms for sparse echo cancellation. Signal Process., 86(6): 1182-1192.
    CrossRef    
16. Paleologu, C., J. Benesty and S. Ciochin, 2008. A variable step-size affine projection algorithm designed for acoustic echo cancellation. IEEE T. Audio Speech, 16(8).
    
17. Radhika, S. and S. Arumugam, 2012. A survey on the different adaptive algorithms used in adaptive filters. Int. J. Eng. Res. Technol., 1(9).
    
18. Sayed, A.H., 2003. Fundamentals of Adaptive Filtering. Wiley, New York.
    
19. Shin, H.C. and A.H. Sayed, 2004. Mean-square performance of a family of affine projection algorithms. IEEE T. Signal Proces., 52(1): 90-102.
    
20. Zhang, Y. and J.A. Chambers, 2006. Convex combination of adaptive filters for a variable tap-length LMS algorithm. IEEE Signal Proc. Let., 13(10): 628-631.
    CrossRef    
21. Zhao, H. and Y. Yu, 2014. Improved efficient proportionate affine projection algorithm based on l 0-norm for sparse system identification. J. Eng., 1(1).
    CrossRef    

Competing interests

The authors have no competing interests.

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