Robust extraction of specific signals with time structure (Part 2)

3、 The theoretical analysis

In this section , We consider the effects of finite samples . This problem is mainly caused by Bermejo[4] From the perspective of higher-order statistics . Here it is , We discuss the problem of two source signals from the perspective of second-order statistics .

use $V$ Represents the pre whitening matrix , use $A$ Represents an unknown mixing matrix , be $VA$ It is orthogonal. . therefore , function (2) become
$$\begin{gathered} J(w)=w^{T}E\{(VA)s(k)s(k-{\tau }^{\ast }{)}^T(VA{)}^T\}w=q^T{R}_s({\tau }^{\ast })q \\ (8) \end{gathered}$$
among $q=w^{T}VA$,$R_s({\tau }^{\ast })=E\{s(k)s(k-{\tau }^{\ast }{)}^T\}$, therefore , Maximize (2) And maximize (8) In constraining $||q|{|}^2=1$ Lower equivalence . Due to the influence of the above limited samples ,$R_s({\tau }^{\ast })\ne I$

When there are two source signals ,(8) The maximum value of can be expressed as the maximum value of
$$\begin{gathered} J(q_1,q_2)=a{q}_1^2+b{q}_2^2+c{q}_1{q}_2 \\ (9) \end{gathered}$$
stay $q_1^2+q_2^2=1$ Constrained by , among $q=[q_1,q_2{]}^T,a=E\{s_1(k)s_1(k-{\tau }^{\ast })\},b=E\{s_2(k)s_2(k-{\tau }^{\ast })\},c=E\{s_1(k)s_2(k-{\tau }^{\ast })\}+E\{s_2(k)s_1(k-{\tau }^{\ast })\}$, among $s_1$ Yes, the required period is ${\tau }^{\ast }$ The signal . Usually we have $a>0$ and $a>b$.

If $c=0$, Deduce and calculate $s_1$ and $s_2$ The cross-correlation value of is $0$（ Ideal situation ）,（9） For optimal solution $q_1=\pm 1,q_2=0$, The proposed signal is $y(k)=w^{T}x(k)=q^{T}s(k)=q_1{s}_1(k)+q_2{s}_2(k)=\pm s_1(k)$ obviously , under these circumstances , The desired source signal is perfectly extracted .

But it's actually $c\ne 0$. about $c>0$,(9) The solution of is
$$\begin{gathered} \{\begin{array}{c}{q}_1=\pm \frac{h+\sqrt{h^2+1}}{\sqrt{1+(h+\sqrt{h^2+1}{)}^2}}\\ q_1=\pm \frac{1}{\sqrt{1+(h+\sqrt{h^2+1}{)}^2}}\end{array} \\ (10) \end{gathered}$$
among $h=(a-b)/c$, about $c<0$, The solution is
$$\begin{gathered} \{\begin{array}{c}{q}_1=\pm \frac{h-\sqrt{h^2+1}}{\sqrt{1+(h-\sqrt{h^2+1}{)}^2}}\\ q_1=\pm \frac{1}{\sqrt{1+(h-\sqrt{h^2+1}{)}^2}}\end{array} \\ (11) \end{gathered}$$
because $q=w^{T}VA$ Is a global vector , Therefore, a better extraction performance index is given
$$\begin{gathered} P{I}_1=\frac{1}{N-1}(\sum _{i=1}^N\frac{q_i^2}{ma{x}_i{q}_i^2}-1) \\ (12) \end{gathered}$$
For any vector $q=[q_1....q_N{]}^T$ The value range is $[0,1]$,$P{I}_1$ The smaller it is , Better extraction performance . In the case of two source signals , Without losing generality , We consider the $c>0$ The situation of , among
$$\begin{gathered} P{I}_1=(\frac{a-b}{c}+{\sqrt{(\frac{a-b}{c}{)}^2+1}}^{-2} \\ (13) \end{gathered}$$
therefore , To improve extraction performance , We should improve $a$ Value , Reduce $b$ and $|c|$ Value ( combination $c<0$ Result ). Now consider the modified objective function (7), It is equivalent to
$$\begin{gathered} \widetilde{J(w)}=\frac{1}{P}{q}^T(\sum _{i=1}^{P}E\{s(k)s(k-i{\tau }^{\ast })\})q \\ (14) \end{gathered}$$
among $\tilde{a}={\sum }_{i=1}^{P}E\{s_1(k)s_1(k-i{\tau }^{\ast })\}/P,\tilde{b}={\sum }_{i=1}^{P}E\{s_2(k)s_2(k-i{\tau }^{\ast })\}/P,\tilde{c}={\sum }_{i=1}^{P}E\{s_1(k)s_2(k-i{\tau }^{\ast })+s_2(k)s_1(k-i{\tau }^{\ast })\}/P$, Be careful ${\tau }^{\ast }$ yes $s_1$ fundamental frequency .

therefore , be relative to (9) Medium $a,b,c$, $\tilde{b},|\tilde{c}|$ Generally with P The increase of has a rapid downward trend , and $\tilde{a}$ Tends to be constant ( Or descend at a relatively slow rate ). therefore , The algorithm of this paper (6) Of $P{I}_1$ Tends to be less than the algorithm (5) Of $P{I}_1$, This means that the extraction quality has been improved , This will be verified by the following simulation .

4、 Computer simulation

4.3、 Experiment with real ECG data

5、 Conclusion

In this paper, a fast robust source extraction algorithm based on eigenvalue decomposition of multiple delay covariance matrices is proposed . The effectiveness and stability of the method are verified by theoretical analysis and simulation . Clearly see , The proposed algorithm involves principal component analysis (PCA). So much about PCA Result [6,11,8] Can be used to improve the algorithm , This is our future work .

It is worth noting that , Using eigenvalue decomposition to develop the algorithm is BSE and BSS One of the development trends of . In order to obtain satisfactory results , Previous algorithms [10] The delay covariance matrix using a large number of observations , such as 500 Matrix , Thus, the efficiency of the algorithm is reduced . This situation also occurs in the algorithm based on joint diagonalization [3,5]. However , This paper points out that , Using prior information can greatly reduce the number of matrices used . For another article on this topic [13,9].