The inference of adaptive notch filter is described in [25, 26]

The inference of adaptive notch filter is described in [25, 26]. The diagram of adaptive noise cancelling is shown in Figure 2. The system is an N-stage tapped delay line (TDL). The weight of the filter is updated according to the following equations:yk=wkTxk,��k=dk?yk,wk+1=wk+?��kxk,(1)where x is the reference input, d is the desired response, y is the output of the filter, w is the weight of the filter, ? is the adaptation constant, and k is the time index. As described in [26], the response from E(z) to Y(z) includes two parts. In practical applications, it is feasible to make the time-varying component to be insignificant (��/N �� 0) by changing the values of N and setting �� as follows:��=sin(NwrT)sin(wrT),(2)where wr is the frequency of the interference. If the reference input is considered to be the following form:x=Ccos(wrT+��),(3)the transfer function of adaptive notch filter can be expressed as follows:H(z)=z2?2zcos(wrT)+1z2?2(1?N?C2/4)zcos(wrT)+(1?N?C2/2).(4)Therefore, the parameter N can be set to the fixed value as described above. It can be seen that the above-mentioned filter is very flexible and can be adjusted using the adaption constants ? and C to provide the desired bandwidth and depth of a suitable notch filter.Figure 2The diagram of adaptive noise cancelling.4. Independent Component AnalysisAfter applying the notch filter, the main step used is ICA. First, the ��standard�� ICA is described. ICA can be briefly explained using a simple example of separating two source signals s1(t) and s2(t) that were mixed by an unknown linear process. Two different linear mixtures, x1(t) and x2(t), are given as follows:x1(t)=c11s1+c12s2,x2(t)=c21s1+c22s2,(5)where c11, c12, c21, and c22 are unknown coefficients. The objective of the problem is to recover the signal s1(t) and s2(t) from mixture signals x1(t) and x2(t) without knowing any prior information about the source signals s1(t) and s2(t) and the mixing process (i.e., c11, c12, c21, and c22), except that s1(t) and s2(t) are statistically independent. In the generalized case, where there are more latent sources and more mixture of signals, the formal definition of ICA is i��[1,n],(6)where si(t) is called latent?as follows:xi(t)=ci1s1+ci2s2+?+cinsn, source, xi(t) is the mixture signal, cij is the mixing coefficient between xi(t) and sj(t), and n is the number of latent sources and mixture signals. The above formulation can be expressed as the following matrix form:X=Cn��n?S,(7)where X is the matrix of mixture signals, in which each column is one mixture signal; S is the matrix of latent signals, in which each column is one latent signal; and Cn��n is the matrix for mixing coefficients.The feasibility of solving the ICA problem lies in the condition that the latent sources are independent of each other.