A Novel Method for Complex-Valued Signals in Independent Component Analysis Framework

Abstract

This paper deals with the separation problem of complex-valued signals in the independent component analysis (ICA) framework, where sources are linearly and instantaneously mixed. Inspired by the recently proposed reference-based contrast criteria based on kurtosis, a new contrast function is put forward by introducing the reference-based scheme to negentropy, based on which a novel fast fixed-point (FastICA) algorithm is proposed. This method is similar in spirit to the classical negentropy-based FastICA algorithm, but differs in the fact that it is much more efficient than the latter in terms of computational speed, which is significantly striking with large number of samples. Furthermore, compared with the kurtosis-based FastICA algorithms, this method is more robust against unexpected outliers, which is particularly obvious when the sample size is small. The local consistent property of this new negentropic contrast function is analyzed in detail, together with the derivation of this novel algorithm presented. Performance analysis and comparison are investigated through computer simulations and realistic experiments, for which a simple wireless communication system with two transmitting and receiving antennas is constructed.

Appendix

First, we denote a gradient operator by $\nabla$ and partial gradient operators by ${\nabla _1}$ (respectively, ${\nabla _2}$) with respect to the first (respectively, second) parameter. More precisely, $\nabla J(\mathbf{{w}})$ is the vector composed of all partial derivatives of $J(\mathbf{{w}})$ whereas ${\nabla _1}I(\mathbf{{w}},\mathbf{{v}})$ (respectively, ${\nabla _2}I(\mathbf{{w}},\mathbf{{v}})$) is the vector of partial derivatives of $I(\mathbf{{w}},\mathbf{{v}})$ with respect to $\mathbf{{w}}$ (respectively, $\mathbf{{v}}$ ).

Make the orthogonal change of coordinates $\mathbf{{p}} = {\mathbf{{\mathrm{A}}}^H}\mathbf{{w}}$ and $\mathbf{{q}} = {\mathbf{{\mathrm{A}}}^H}\mathbf{{v}}$, giving $I(\mathbf{{p}},\mathbf{{q}}) = E\left\{ {G(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })} \right\}$. When $\mathbf{{w}}$ coincides with one of the rows of ${\mathbf{{\mathrm{A}}}^{ - 1}}$, we have $\mathbf{{p}} = {\left( {0, \ldots ,p,0, \ldots ,0} \right) ^T}$ and the corresponding $\mathbf{{q}} = {\left( {0, \ldots ,q,0, \ldots ,0} \right) ^T}$ updating following $\mathbf{{w}}$. Also, $\mathbf{{p}}$ and $\mathbf{{q}}$ are N-dimensional complex-valued column vectors, denoted by ${\left[ {{p_1},{p_2}, \ldots ,{p_N}} \right] ^T}$ and ${\left[ {{q_1},{q_2}, \ldots ,{q_N}} \right] ^T}$, respectively. In the following we shall analyze the stability of such $\mathbf{{p}}$.

We now search for a Taylor expansion of $I$ in the extrema. The partial gradient of $I$ with respect to $\mathbf{{p}}$ is

$$\begin{aligned} {\nabla _1}I(\mathbf{{p}},\mathbf{{q}}) = \left( \begin{array}{c} {\frac{{\partial I(\mathbf{{p}},\mathbf{{q}})}}{{\partial {p_{1r}}}}} \\ {\frac{{\partial I(\mathbf{{p}},\mathbf{{q}})}}{{\partial {p_{1i}}}}} \\ \vdots \\ {\frac{{\partial I(\mathbf{{p}},\mathbf{{q}})}}{{\partial {p_{Nr}}}}} \\ {\frac{{\partial I(\mathbf{{p}},\mathbf{{q}})}}{{\partial {p_{Ni}}}}} \\ \end{array} \right) = \left( \begin{array}{l} {E\left\{ ({\Delta _1}{s_{1r}} + {\Delta _2}{s_{1i}})g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ {E\left\{ ({\Delta _1}{s_{1i}} - {\Delta _2}{s_{1r}})g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ \vdots \\ {E\left\{ ({\Delta _1}{s_{Nr}} + {\Delta _2}{s_{Ni}})g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ {E\left\{ ({\Delta _1}{s_{Ni}} - {\Delta _2}{s_{Nr}})g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ \end{array} \right) \end{aligned}$$

(36)

where

$$\begin{aligned}&{\Delta _1} = {q_{1r}}{s_{1r}} + {q_{1i}}{s_{1i}} + \cdots + {q_{Nr}}{s_{Nr}} + {q_{Ni}}{s_{Ni}}\end{aligned}$$

(37)

$$\begin{aligned}&{\Delta _2} = {q_{1r}}{s_{1i}} - {q_{1i}}{s_{1r}} + \cdots + {q_{Nr}}{s_{Ni}} - {q_{Ni}}{s_{Nr}} \end{aligned}$$

(38)

The Hessian of $I$ with respect to $\mathbf{{p}}$ is

$$\begin{aligned} {\nabla _1}^2I(\mathbf{{p}},\mathbf{{q}}) = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} {\frac{{\partial {h_{R1}}}}{{\partial {p_{1r}}}}} &{} {\frac{{\partial {h_{R1}}}}{{\partial {p_{1i}}}}} &{} \cdots &{} {\frac{{\partial {h_{R1}}}}{{\partial {p_{Nr}}}}} &{} {\frac{{\partial {h_{R1}}}}{{\partial {p_{Ni}}}}} \\ {\frac{{\partial {h_{I1}}}}{{\partial {p_{1r}}}}} &{} {\frac{{\partial {h_{I1}}}}{{\partial {p_{1i}}}}} &{} \cdots &{} {\frac{{\partial {h_{I1}}}}{{\partial {p_{Nr}}}}} &{} {\frac{{\partial {h_{I1}}}}{{\partial {p_{Ni}}}}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {\frac{{\partial {h_{RN}}}}{{\partial {p_{1r}}}}} &{} {\frac{{\partial {h_{RN}}}}{{\partial {p_{1i}}}}} &{} \cdots &{} {\frac{{\partial {h_{RN}}}}{{\partial {p_{Nr}}}}} &{} {\frac{{\partial {h_{RN}}}}{{\partial {p_{Ni}}}}} \\ {\frac{{\partial {h_{IN}}}}{{\partial {p_{1r}}}}} &{} {\frac{{\partial {h_{IN}}}}{{\partial {p_{1i}}}}} &{} \cdots &{} {\frac{{\partial {h_{IN}}}}{{\partial {p_{Nr}}}}} &{} {\frac{{\partial {h_{IN}}}}{{\partial {p_{Ni}}}}} \\ \end{array} \right) \end{aligned}$$

(39)

where we define

$$\begin{aligned} {h_{Rj}}&= E\left\{ ({\Delta _1}{s_{jr}} + {\Delta _2}{s_{ji}})g(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} {j = 1,2, \ldots ,N}\end{aligned}$$

(40)

$$\begin{aligned} {h_{Ij}}&= E\left\{ ({\Delta _1}{s_{ji}} - {\Delta _2}{s_{jr}})g(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} {j = 1,2, \ldots ,N} \end{aligned}$$

(41)

Then in (39), we have

$$\begin{aligned}&\frac{{\partial {h_{Rj}}}}{{\partial {p_{nr}}}} = E\left\{ ({\Delta _1}{s_{jr}} + {\Delta _2}{s_{ji}})({\Delta _1}{s_{nr}} + {\Delta _2}{s_{ni}})g'(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} \end{aligned}$$

(42)

$$\begin{aligned}&\frac{{\partial {h_{Rj}}}}{{\partial {p_{ni}}}} = E\left\{ ({\Delta _1}{s_{jr}} + {\Delta _2}{s_{ji}})({\Delta _1}{s_{ni}} - {\Delta _2}{s_{nr}})g'(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} \end{aligned}$$

(43)

$$\begin{aligned}&\frac{{\partial {h_{Ij}}}}{{\partial {p_{nr}}}} = E\left\{ ({\Delta _1}{s_{ji}} - {\Delta _2}{s_{jr}})({\Delta _1}{s_{nr}} + {\Delta _2}{s_{ni}})g'(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} \end{aligned}$$

(44)

$$\begin{aligned}&\frac{{\partial {h_{Ij}}}}{{\partial {p_{ni}}}} = E\left\{ ({\Delta _1}{s_{ji}} - {\Delta _2}{s_{jr}})({\Delta _1}{s_{Ni}} - {\Delta _2}{s_{Nr}})g'(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\right\} \end{aligned}$$

(45)

where $n = 1,2, \ldots ,N, j = 1,2, \ldots ,N$ for (42) to (45).

Without loss of generality, we assume to analyze the stability of the points $\mathbf{{p}} = p{\mathbf{{e}}_\mathbf{{1}}} = {(p,0, \ldots ,0)^T}$ and the corresponding $\mathbf{{q}} = q{\mathbf{{e}}_\mathbf{{1}}} = {(q,0, \cdots ,0)^T}$, which relates to $\mathbf{{w}} = p{\mathbf{{a}}_\mathbf{{1}}}$. Due to the normalization of $\mathbf{{w}}$ and $\mathbf{{v}}$ after each one-dimensional optimization, we have ${\left| {{\mathbf{{p}}^H}\mathbf{{s}}} \right| ^2} = {\left| {{\mathbf{{q}}^H}\mathbf{{s}}} \right| ^2} = {\left| {{s_1}} \right| ^2}$ and ${\left| \mathbf{{p}} \right| ^2} = {\left| \mathbf{{q}} \right| ^2} = 1$. Evaluating the gradient in (36) at points $\mathbf{{p}} = p{\mathbf{{e}}_\mathbf{{1}}}$ and $\mathbf{{q}} = q{\mathbf{{e}}_\mathbf{{1}}}$, under the assumptions A1 and A2, we have

$$\begin{aligned} {\nabla _1}I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) = \left( {\begin{array}{c} {{p_r}E\left\{ {{\left| {{s_1}} \right| }^2}g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ {{p_i}E\left\{ {{\left| {{s_1}} \right| }^2}g(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } \\ 0 \\ \vdots \\ 0 \\ \end{array}} \right) = \left( \begin{array}{c} {{p_r}E\left\{ {{\left| {{s_1}} \right| }^2}g({{\left| {{s_1}} \right| }^2}\right\} } \\ {{p_i}E\left\{ {{\left| {{s_1}} \right| }^2}g({{\left| {{s_1}} \right| }^2}\right\} } \\ 0 \\ \vdots \\ 0 \\ \end{array} \right) \nonumber \\ \end{aligned}$$

(46)

Similarly, considering (39), we have

$$\begin{aligned}&{\nabla _1}^2I(p{e_1},q{e_1}) \nonumber \\&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} {{p_r}^2E\left\{ {{\left| {{s_1}} \right| }^4}g'(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } &{} {{p_r}{p_i}E\left\{ {{\left| {{s_1}} \right| }^4}g'(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } &{} 0 &{} \cdots &{} 0 \\ {{p_r}{p_i}E\left\{ {{\left| {{s_1}} \right| }^4}g'(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } &{} {{p_i}^2E\left\{ {{\left| {{s_1}} \right| }^4}g'(({\mathbf{{p}}^H}\mathbf{{s}}){{({\mathbf{{q}}^H}\mathbf{{s}})}^ * })\right\} } &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} \alpha &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} \alpha \\ \end{array} \right) \nonumber \\&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} {{p_r}^2E\left\{ {{\left| {{s_1}} \right| }^4}g'({{\left| {{s_1}} \right| }^2})\right\} } &{} {{p_r}{p_i}E\left\{ {{\left| {{s_1}} \right| }^4}g'({{\left| {{s_1}} \right| }^2})\right\} } &{} 0 &{} \cdots &{} 0 \\ {{p_r}{p_i}E\left\{ {{\left| {{s_1}} \right| }^4}g'({{\left| {{s_1}} \right| }^2})\right\} } &{} {{p_i}^2E\left\{ {{\left| {{s_1}} \right| }^4}g'({{\left| {{s_1}} \right| }^2})\right\} } &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} \alpha &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} \alpha \\ \end{array} \right) \end{aligned}$$

(47)

where $\alpha = E\left\{ {\left| {{s_1}} \right| ^2}g'(({\mathbf{{p}}^H}\mathbf{{s}}){({\mathbf{{q}}^H}\mathbf{{s}})^ * })\} = E\{ {\left| {{s_1}} \right| ^2}g'({\left| {{s_1}} \right| ^2})\right\}$

Now we make a small perturbation $\mathbf{{\varepsilon }} = {({\varepsilon _{1r}},{\varepsilon _{1i}}, \ldots ,{\varepsilon _{Nr}},{\varepsilon _{Ni}})^T}$ and evaluate the Taylor expansion of $I$ :

$$\begin{aligned}&I(p{\mathbf{{e}}_\mathbf{{1}}} + \mathbf{{\varepsilon }},q{\mathbf{{e}}_\mathbf{{1}}}) \nonumber \\&= I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) + {\mathbf{{\varepsilon }}^T}{\nabla _1}I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) + \frac{1}{2}{\mathbf{{\varepsilon }}^T}{\nabla _1}^2I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}})\mathbf{{\varepsilon }} + o({\left\| \mathbf{{\varepsilon }} \right\| ^2}) \nonumber \\&= I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) + ({\varepsilon _{1r}}{p_r} + {\varepsilon _{1i}}{p_i})E\left\{ {\left| {{s_1}} \right| ^2}g({\left| {{s_1}} \right| ^2})\right\} + \frac{1}{2}\Bigg ({p_r}^2{\varepsilon _{1r}}^2E\left\{ {\left| {{s_1}} \right| ^4}g'({\left| {{s_1}} \right| ^2})\right\} \nonumber \\&\quad +\, 2{p_r}{p_i}{\varepsilon _{1r}}{\varepsilon _{1i}}E\left\{ {\left| {{s_1}} \right| ^4}g'({\left| {{s_1}} \right| ^2})\right\} + {p_i}^2{\varepsilon _{1i}}^2E\left\{ {\left| {{s_1}} \right| ^4}g'({\left| {{s_1}} \right| ^2})\right\} \Bigg ) \nonumber \\&\quad +\, E\left\{ {\left| {{s_1}} \right| ^2}g'({\left| {{s_1}} \right| ^2})\right\} \sum \limits _{j = 2}^N {({\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2)} + o({\left\| \varepsilon \right\| ^2}) \end{aligned}$$

(48)

Due to the constraint $\left\| \mathbf{{w}} \right\| = 1$, thus we have $\left\| {p{\mathbf{{e}}_\mathbf{{1}}} + \mathbf{{\varepsilon }}} \right\| = 1$. We get

$$\begin{aligned}&{({\varepsilon _{1r}} + {p_r})^2} + {({\varepsilon _{1i}} + {p_i})^2} + \sum \limits _{j = 2}^N {\left( {\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2\right) } = 1 \nonumber \\&2({\varepsilon _{1r}}{p_r} + {\varepsilon _{1i}}{p_i}) = - \sum \limits _{j = 1}^N {\left( {\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2\right) } \end{aligned}$$

(49)

Using this, we have

$$\begin{aligned}&I(p{\mathbf{{e}}_\mathbf{{1}}} + \mathbf{{\varepsilon }},q{\mathbf{{e}}_\mathbf{{1}}}) \nonumber \\&\quad \! = I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) \!-\! \frac{1}{2}E\left\{ {\left| {{s_1}} \right| ^2}g({\left| {{s_1}} \right| ^2})\right\} \sum \limits _{j = 1}^N {({\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2)} \!+\! \frac{1}{2}\Bigg (\!\!\! -\! \frac{1}{2}\sum \limits _{j = 1}^N {({\varepsilon _{jr}}^2 \!+\! {\varepsilon _{ji}}^2)\!{\Bigg )^2}} \nonumber \\&\qquad \times E\left\{ {\left| {{s_1}} \right| ^4}g'({\left| {{s_1}} \right| ^2})\right\} + \left( E\left\{ {\left| {{s_1}} \right| ^2}g'({\left| {{s_1}} \right| ^2})\right\} \right) \sum \limits _{j = 2}^N {({\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2)} + o({\left\| \varepsilon \right\| ^2}) \nonumber \\&\quad = I(p{\mathbf{{e}}_\mathbf{{1}}},q{\mathbf{{e}}_\mathbf{{1}}}) - \frac{1}{2}E\left\{ {\left| {{s_1}} \right| ^2}g({\left| {{s_1}} \right| ^2})\right\} \sum \limits _{j = 1}^N {({\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2)}\nonumber \\&\qquad +\, E\left\{ {\left| {{s_1}} \right| ^2}g'({\left| {{s_1}} \right| ^2})\right\} \sum \limits _{j = 2}^N {({\varepsilon _{jr}}^2 + {\varepsilon _{ji}}^2)} + o({\left\| \varepsilon \right\| ^2}) \end{aligned}$$

(50)

Finally, Theorem 1 has been proved.