Maximization of Nonlinear Autocorrelation for Blind Source Separation of Non-stationary Complex Signals

Abstract

Blind source separation of complex-valued signals has been a vital issue especially in the field of digital communication signal processing. This paper proposes a novel method based on nonlinear autocorrelation to solve the problem. Relying on the temporal structure with nonlinear autocorrelation of the signals, the method has a potential capability of extracting non-stationary complex sources with Gaussian or non-Gaussian distribution. Most traditional methods would fail in separating this kind of sources. We also analyze the stability conditions of the method in theory. Numerical simulations on artificial complex Gaussian data and orthogonal frequency division multiplexing sources corroborate the validity and efficiency of the proposed method. Moreover, with respect to classical methods, including cumulant-based approach using the non-stationarity of variance and complexity pursuit, our method offers equally good results with lower computational cost and better robustness. Finally, experiments for the separation of real communication signals illustrate that our method has good prospects in real-world applications.

Appendices

Appendix 1

1.1 Proof of $\arg \left( {\beta _1^*} \right) =-\theta$ in (14)

By the definition of $\beta _1 =\frac{\partial J(\mathbf{q}_1 )}{\partial q_1^*}$, we can obtain

$$\begin{aligned} \beta _1&= \sum _k {E\left\{ {g\left( {e^{-j\theta }s_1 } \right) G\left( {e^{j\theta }s_{1,\tau _k }^*} \right) s_1 } \right\} } E\left\{ {G\left( {e^{-j\theta }s_1 } \right) ^{*}G\left( {e^{j\theta }s_{1,\tau _k }^*} \right) ^{*}} \right\} \nonumber \\&\quad +\,E\left\{ {G\left( {e^{-j\theta }s_1 } \right) G\left( {e^{j\theta }s_{1,\tau _k }^*} \right) } \right\} E\left\{ {G\left( {e^{-j\theta }s_1 } \right) ^{*}g\left( {e^{j\theta }s_{1,\tau _k }^*} \right) ^{*}s_{1,\tau _k } } \right\} \nonumber \\ \end{aligned}$$

(21)

According to the approximation of $\hbox {G}(\cdot$) in Theorem, we have

$$\begin{aligned} \beta _1&= \sum _k {E\left\{ {\sum _{l=1}^\infty {lc_l \left( {e^{-j\theta }s_1 } \right) ^{l-1}} \sum _{l=0}^\infty {c_l \left( {e^{j\theta }s_{1,\tau _k }^*} \right) ^{l}} s_1 } \right\} }\nonumber \\&\times \, E\left\{ {\sum _{l=0}^\infty {c_l \left( {e^{j\theta }s_1^*} \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {e^{-j\theta }s_{1,\tau _k } } \right) ^{l}} } \right\} \nonumber \\&+\,E\left\{ {\sum _{l=0}^\infty {c_l \left( {e^{-j\theta }s_1 } \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {e^{j\theta }s_{1,\tau _k }^*} \right) ^{l}} } \right\} \nonumber \\&\times \, E\left\{ {\sum _{l=0}^\infty {c_l \left( {e^{j\theta }s_1^*} \right) ^{l}} \sum _{l=1}^\infty {lc_l \left( {e^{-j\theta }s_{1,\tau _k } } \right) ^{l-1}} s_{1,\tau _k } } \right\} \end{aligned}$$

(22)

We multiply $\beta _{1}$ by $e^{-j\theta }$ and define $h_1 =e^{-j\theta }s_1, h_2 =e^{j\theta }s_{1,\tau _k }^*$, then

$$\begin{aligned} e^{-j\theta }\beta _1&= \sum _k {E\left\{ {\sum _{l=1}^\infty {lc_l \left( {h_1 } \right) ^{l-1}} \sum _{l=0}^\infty {c_l \left( {h_2 } \right) ^{l}} h_1 } \right\} } E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1^*} \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2^*} \right) ^{l}} } \right\} \nonumber \\&+\,E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1 } \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2 } \right) ^{l}} } \right\} E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1^*} \right) ^{l}} \sum _{l=1}^\infty {lc_l \left( {h_2^*} \right) ^{l-1}} h_2^*} \right\} \nonumber \\&= \sum _k {E\left\{ {\sum _{l=1}^\infty {lc_l \left( {h_1 } \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2 } \right) ^{l}} } \right\} } E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1^*} \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2^*} \right) ^{l}} } \right\} \nonumber \\&+\,E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1 } \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2 } \right) ^{l}} } \right\} E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1^*} \right) ^{l}} \sum _{l=1}^\infty {lc_l \left( {h_2^*} \right) ^{l}} } \right\} \nonumber \\&= \sum _k {2Re\left( {E\left\{ {\sum _{l=1}^\infty {lc_l \left( {h_1 } \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2 } \right) ^{l}} } \right\} E\left\{ {\sum _{l=0}^\infty {c_l \left( {h_1^*} \right) ^{l}} \sum _{l=0}^\infty {c_l \left( {h_2^*} \right) ^{l}} } \right\} } \right) }\nonumber \\ \end{aligned}$$

(23)

Therefore, we can observe that $e^{-j\theta }\beta _{1}$ is actually real-valued number, which completely proves $\arg \left( {\beta _1^*} \right) =-\theta$.

Appendix 2

1.1 Proof of Lemma 1

Since $a_j =\frac{\partial ^{2}J(\mathbf{q}_i )}{\partial q_j^*\partial q_j }$ ($\forall j\ne i)$, where $J(\mathbf{q}_{i})$ denotes the cost function $J(\mathbf{q})=\sum \limits _k {\left| {E\left\{ {G\left( {y(t)} \right) G(y(t-\tau _k )^{*})} \right\} } \right| ^{2}}$ at the stable point $\mathbf{q}_i =(0,\ldots , e^{j\theta },\ldots , 0)^{T}$ pointing in the direction of the principal components of $s_{i}$, we can obtain

$$\begin{aligned} a_j&= \sum _k {E\left\{ {g\left( {e^{-j\theta }s_i } \right) g\left( {e^{j\theta }s_{i,\tau _k }^*} \right) s_j s_{j,\tau _k }^*} \right\} } E\left\{ {G\left( {e^{-j\theta }s_i } \right) ^{*}G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}} \right\} \nonumber \\&+\,\left| {E\left\{ {g\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) s_j } \right\} } \right| ^{2} \nonumber \\&+\,\left| {E\left\{ {G\left( {e^{-j\theta }s_i } \right) g\left( {e^{j\theta }s_{i,\tau _k }^*} \right) s_{j,\tau _k }^*} \right\} } \right| ^{2}\nonumber \\&+\,E\left\{ {G\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) } \right\} E\left\{ {g\left( {e^{-j\theta }s_i } \right) ^{*}g\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}s_{j,\tau _k } s_j^*} \right\} \nonumber \\ \end{aligned}$$

(24)

Careful inspection reveals that the second-order derivative $a_j$ is also real-valued number. Taking into account the assumption of Lemma 1 and the zero mean assumption of the source signals, we can get $a_j =0$. Therefore, from the Theorem, we have the further stability condition: $\left| {\frac{\partial J(\mathbf{q}_i )}{\partial q_i^*}} \right| -\left| {\frac{\partial ^{2}J(\mathbf{q}_i )}{\partial q_j^*\partial q_j^*}} \right| >0$.

1.2 Proof of Lemma 2

From the definition of $b_j =\frac{\partial ^{2}J(\mathbf{q}_i )}{\partial q_j^*\partial q_j^*}$ ($\forall j\ne i)$ and the assumptions of Lemma 1, we have

$$\begin{aligned} b_j&= \sum _k {E\left\{ {g{\prime }\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) s_j^2 } \right\} } E\left\{ {G\left( {e^{-j\theta }s_i } \right) ^{*}G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}} \right\} \nonumber \\&+\,2E\left\{ {g\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) s_j } \right\} E\left\{ {G\left( {e^{-j\theta }s_i } \right) ^{*}g\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}s_{j,\tau _k } } \right\} \nonumber \\&+\,E\left\{ {G\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) } \right\} E\left\{ {G\left( {e^{-j\theta }s_i } \right) ^{*}g{\prime }\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}s_{j,\tau _k } s_j^*} \right\} \nonumber \\&= \sum _k {E\left\{ {s_j^2 } \right\} E\left\{ {g{\prime }\left( {e^{-j\theta }s_i } \right) G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) } \right\} } \times E\left\{ {G\left( {e^{-j\theta }s_i } \right) ^{*}G\left( {e^{j\theta }s_{i,\tau _k }^*} \right) ^{*}} \right\} \nonumber \\ \end{aligned}$$

(25)

where $\hbox {g}^{\prime }(\cdot )$ is the derivative of $\hbox {g}(\cdot$). As we know, if sources $\left\{ {s_j } \right\}$ are circular (rotation invariant), the pseudo-covariance $E\left\{ {s_j^2 } \right\} =0$, which means that $b_j =0$. Hence, the stability conditions are given by $\left| {\frac{\partial J(\mathbf{q}_i )}{\partial q_i^*}} \right| \ne 0$ for circular sources and $\left| {\frac{\partial J(\mathbf{q}_i )}{\partial q_i^*}} \right| -\left| {\frac{\partial ^{2}J(\mathbf{q}_i )}{\partial q_j^*\partial q_j^*}} \right| >0$ for non-circular sources ($E\left\{ {s_j^2 } \right\} \ne 0)$.