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148 6 Signal Processing

We now run the adaptive ﬁ lter canc for 10 iterations and use the above
value of u.

[z,e,mer] = canc(yn1,yn2,0.0019,5,10);
The evolution of the mean-squared error

plot(mer)
illustrates the performance of the adaptive ﬁlter, although the chosen step

size u=0.0019 obviously leads to a relatively fast convergence. In most ex-
amples, a smaller step size decreases the rate of convergence, but increases

the quality of the ﬁnal result. We therefore reduce u by one order of magni-

tude and run the ﬁlter again with more iterations.
[z,e,mer] = canc(yn1,yn2,0.0001,5,20);
The plot of the mean-squared error against the iterations

plot(mer)
now convergences after around six iterations. We now compare the ﬁ lter
output with the original noise-free signal.

plot(x,y,'b',x,z,'r')
This plot shows that the noise level of the signal has been reduced dramati-
cally by the ﬁlter. Finally, the plot

plot(x,e,'r')
shows the noise extracted from the signal. In practice, the user should vary
the parameters u and l in order to obtain the optimum result.
The application of this algorithm has been demonstrated on duplicate
oxygen-isotope records from ocean sediments (Trauth 1998). The work by

Trauth (1998) illustrates the use of the modiﬁed LMS algorithm, but also

another type of adaptive ﬁlter, the recursive least-squares (RLS) algorithm
(Haykin 1991) in various environments.