# Appendix A

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```					                                                                                        Appendix A

Appendix A

Initial Mean Convergence Derivation

Assuming that in the initial adaptation, e(n) is correlated to X(n) that the principle of

orthogonality is not application at the initial stage (E{e(n)x(n) ≠ 0). Since x(n) and e(n) are
Gaussian distributed, the Gaussian Moment Factoring Theorem is used to simplify the higher
order moment of e3(n) [Haykin 1991 and McGee 1971]:

E{V(n+1)} = E{V(n)} + γE{e3(n)X(n)}.                            (A.1)

Using the Gaussian moment factoring theorem:

E{e3(n)X(n)} = 2[E{e2(n)}E{e(n)X(n)}].                          (A.2)
Replacing (A.2) to (A.1):

E{V(n+1)} = E{V(n)} + 2γ[E{e2(n)}E{X(n)e(n)}],                         (A.3)

Since, e(n) = η(n) - XT(n)V(n),

E{V(n+1)} = E{V(n)} + 2γ[ E{e2(n)}E{X(n)[η(n) - XT(n)V(n)]}].                  (A.4)

Again using Assumption B, that E{X(n)η(n)} = 0, equation (A.3) is simplified to:

E{V(n+1)} = E{V(n)} - 2γE{e2(n)}E{X(n)XT(n)}E{V(n)},                                   (A.5)

|
E{V(n+1)} = [ I - 2γE{e2(n)R]E{V(n)} ||W(n)||<<||W*||.                         (A.6)

(A.5) represents the initial adaptation of the mean weight error difference equation.

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Appendix B

Publications

1. M.K. Chan, C.F.N. Cowan, “Normalised Least Mean Fourth (XE-NLMF)
Algorithm with Faster Convergence,” IEEE International Conference on Acoustics,
Speech, and Signal Processing ICASSP2001, Student Forum, Salt Lake City (US),
May 2001.

2. M.K. Chan, C.F.N. Cowan, “A Normalisation Technique For The Least Mean
Fourth Algorithm In System Identification,” Irish Signal and System Conference
ISSC2001, pp. 141-146, Maynooth (Ireland), June 2001.

3’. M.K. Chan, V. Stewart, C.F.N. Cowan, “A Parallel Adaptation for Speech Echo
Cancellation,” Irish Signal and System Conference ISSC2002, pp. 201-205, Cork
(Ireland), June 2002.

4. M.K. Chan, V. Stewart, C.F.N. Cowan, “Multiple Adaptive Filtering for Speech
Echo Cancellation,” IEE Non-linear and Non-Gaussian Signal Processing Workshop
N2SP, Peebles (Scotland), July 2002.

5’. M.K. Chan, C.F.N. Cowan, “Using A Normalised LMF Algorithm For Channel
Equalisation With Co-Channel Interference,” XI European Signal Processing
Conference Eusipco2002, Toulouse (France), Vol. 2, pp. 49-51, September 2002.

6. M.K. Chan, V. Stewart, C.F.N. Cowan, “"Multiple Adaptive Filtering Scheme For
Non-Gaussian Speech Signal in Echo Cancellation,” submitted to: IEE Proceedings -
Vision, Image and Signal Processing, Special issue on ‘Non-linear and Non-Gaussian
Signal Processing’, 2002.

’ reprinted at back of the thesis.

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Eusipco 2002

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