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Additional resources for Adaptive Filtering Applications
In the presence of uncorrelated disturbance, v(n ), the error signal picked-up by the error microphone is given as eo (n ) = e(n ) + v(n ) = [ d(n ) − ys (n )] + v(n ), (38) and hence, the update equation for FxLMS algorithm for W (z) can be written as w (n + 1) = w (n ) + μ w e(n )xˆ s (n ) + μ w v(n )xˆ s (n ). (39) It is evident that the adaptation is perturbed by the uncorrelated noise component v(n ), and as shown in (Sun & Kuo, 2007), the steady-state performance of the FxLMS algorithm will be degraded signiﬁcantly.
Both ANC ﬁlters W (z) and B (z) are adapted by FxLMS algorithms. The residual error signal eo (n ) is given in Eq. (41) and as explained earlier, the ﬁrst term is desired error signal for the adaptation of W (z) and second term is desired error signal for B (z). To achieve cancellation [ideally eo (n ) = 0], W (z) needs to be excited by the input correlated with d(n ) [the reference signal x (n ) is indeed that input], and B (z) needs to be excited by the input correlated with v(n ) [such input is not available directly and needs to be generated internally].
16)) with in [ c1 , c2 ] as that of x (n ), and neglects the tail beyond [ c1 , c2 ]. The stability of Sun’s algorithms depends heavily on appropriate choice of [ c1 , c2 ]. We have extended this idea, that instead of ignoring, the peaky samples are replaced by the thresholding values c1 and c2 . Effectively, this algorithm adds a saturation nonlinearity in the reference and error signal paths. Thus, the performance of this algorithm also depends on the parameters c1 and c2 . In order to overcome this difﬁculty of choosing appropriate thresholding parameters, we propose an FxLMS algorithm that does not use modiﬁed reference and/or error signals, and hence does not require selection of the thresholding parameters [ c1 , c2 ].