Efficient Optimization of Reference-Based Negentropy for Noncircular Sources in Complex ICA

Abstract

Bingham proposed a complex fast independent component analysis (c-FastICA) algorithm to approximate the nengentropy of circular sources using nonlinear functions. Novey proposed extending the work of Bingham using information from a pseudo-covariance matrix for noncircular sources, particularly for sub-Gaussian noncircular signals such as binary phase-shift keying signals. Based on this work, in the present paper we propose a new reference-based contrast function by introducing reference signals into the negentropy, upon which an efficient optimization FastICA algorithm is derived for noncircular sources. This new approach is similar to Novey’s nc-FastICA algorithm, but differs in that it is much more efficient in terms of the computational speed, which is significantly notable with a large number of samples. In this study, the local stability of our reference-based negentropy is analyzed and the derivation of our new algorithm is described in detail. Simulations conducted to demonstrate the performance and effectiveness of our method are also described.

Appendix

First, we make orthogonal changes to coordinates ${\mathbf{p}} = {{\mathbf{A}}^H}{\mathbf{w}}$ and ${\mathbf{q}} = {{\mathbf{A}}^H}{\mathbf{v}}$ under the condition that reference signals update following objective signals, which results in the contrast function $I\left( {{\mathbf{p}},{\mathbf{q}}} \right) = E\left\{ {G\left( {y{z^ * }} \right) } \right\}$, where $y = {{\mathbf{w}}^H}{\mathbf{x}} = {{\mathbf{p}}^H}{\mathbf{s}}$ and $z = {{\mathbf{v}}^H}{\mathbf{x}} = {{\mathbf{q}}^H}{\mathbf{s}}$. Without a loss of generality, we assume an optimal solution for ${s_1}$ at ${{\mathbf{p}}_1} = {\left[ {{e^{j\theta }},0, \ldots ,0} \right] ^T}$, where $\theta$ points toward the direction of the principal component of ${s_1}$, as shown in [24].

Second, similar to the approach in [25], we seek a Taylor series expansion of I around the optimal solution ${{\mathbf{p}}_1}$, and obtain the gradient and Hessian matrices in the complex domain by differentiating with respect to ${\mathbf{p}}$ while treating ${{\mathbf{q}}^ * }$ as a constant. Similar to (18), we use (3)–(6), and the complex gradient defined in [29] to obtain the gradient as

$$\begin{aligned} \widetilde{\nabla }_{\mathbf{p}}^ * I = E\left( {\begin{array}{c} {\frac{{\partial I}}{{\partial p_1^ * }}}\\ {\frac{{\partial I}}{{\partial {p_1}}}}\\ \vdots \\ {\frac{{\partial I}}{{\partial p_N^ * }}}\\ {\frac{{\partial I}}{{\partial {p_N}}}} \end{array}} \right) = E\left( {\begin{array}{c} {\frac{{\partial I}}{{\partial p_1^ * }}}\\ {\frac{{\partial I}}{{\partial {q_1}}}}\\ \vdots \\ {\frac{{\partial I}}{{\partial p_N^ * }}}\\ {\frac{{\partial I}}{{\partial {q_N}}}} \end{array}} \right) = E\left( {\begin{array}{c} {E\left\{ {g\left( {y{z^ * }} \right) {z^ * }{s_1}} \right\} } \\ {E\left\{ {g\left( {y{z^ * }} \right) ys_1^ * } \right\} } \\ \vdots \\ {E\left\{ {g\left( {y{z^ * }} \right) {z^ * }{s_N}} \right\} }\\ {E\left\{ {g\left( {y{z^ * }} \right) ys_N^ * } \right\} } \end{array}} \right) \end{aligned}$$

(30)

where the middle term holds because ${\mathbf{q}}$ updates following ${\mathbf{p}}$ such that the values of $\frac{{\partial I}}{{\partial {p_i}}}$ and $\frac{{\partial I}}{{\partial {q_i}}}$ are the same. In addition, the Hessian can be written as

$$\begin{aligned} {\widetilde{\mathbf{H}}_{\mathbf{p}}}I = E\left\{ {\frac{{{\partial ^2}I}}{{\partial {{\widetilde{\mathbf{p}}}^ * }\partial {{\widetilde{\mathbf{p}}}^T}}}} \right\} = E\left\{ {\left[ {\begin{array}{ccccc} {\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial p_N^ * }}}\\ {\frac{{{\partial ^2}I}}{{\partial {p_1}\partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {p_1}\partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial {p_1}\partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {p_1}\partial p_N^ * }}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial p_N^ * }}}\\ {\frac{{{\partial ^2}I}}{{\partial {p_N}\partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {p_N}\partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial {p_N}\partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {p_N}\partial p_N^ * }}} \end{array}} \right] } \right\} \end{aligned}$$

(31)

and similar to (19), we reformulate the Hessian in (31) as

$$\begin{aligned} {\widetilde{\mathbf{H}}_{\mathbf{p}}}I= & {} E\left\{ {\left[ {\begin{array}{ccccc} {\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_1^ * \partial p_N^ * }}} \\ {\frac{{{\partial ^2}I}}{{\partial {q_1}\partial {q_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {q_1}\partial q_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial {q_1}\partial {q_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {q_1}\partial q_N^ * }}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial {p_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial p_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial {p_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial p_N^ * \partial p_N^ * }}} \\ {\frac{{{\partial ^2}I}}{{\partial {q_N}\partial {q_1}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {q_N}\partial q_1^ * }}}&{} \ldots &{}{\frac{{{\partial ^2}I}}{{\partial {q_N}\partial {q_N}}}}&{}{\frac{{{\partial ^2}I}}{{\partial {q_N}\partial q_N^ * }}} \end{array}} \right] } \right\} \nonumber \\= & {} E\left\{ {\left[ {\begin{array}{ccccc} 0&{}{s_1^2c}&{} \ldots &{}0&{}{{s_1}{s_N}c}\\ {s{{_1^ * }^2}d}&{}0&{} \ldots &{}{s_1^ * s_N^ * d}&{}0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0&{}{{s_N}{s_1}c}&{} \ldots &{}0&{}{s_N^2c}\\ {s_N^ * s_1^ * d}&{}0&{} \ldots &{}{s{{_N^ * }^2}d}&{}0 \end{array}} \right] } \right\} \end{aligned}$$

(32)

Here, $c = g'\left( {y{z^ * }} \right) {z^ * }^2$ and $d = g'\left( {y{z^ * }} \right) {y^2}$.

Evaluating the gradient (30) and Hessian (32) at ${{\mathbf{p}}_1}$ and ${{\mathbf{q}}_1}$, we get

$$\begin{aligned} {\widetilde{\nabla }_{\mathbf{p}}}I\left( {{{\mathbf{p}}_1}} \right) = E\left( {\begin{array}{c} {E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} {e^{ - j\theta }}} \\ {E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} {e^{j\theta }}} \\ 0 \\ \vdots \\ 0 \end{array}} \right) \end{aligned}$$

(33)

and

$$\begin{aligned} {\widetilde{\mathbf{H}}_{\mathbf{p}}}I\left( {{{\mathbf{p}}_1}} \right) = E\left\{ {\left[ {\begin{array}{cccc} {{{\mathbf{B}}_1}}&{}0&{} \ldots &{}0\\ 0&{}{{{\mathbf{B}}_2}}&{} \ldots &{} \vdots \\ \vdots &{} \vdots &{} \ddots &{}0\\ 0&{} \ldots &{}0&{}{{{\mathbf{B}}_N}} \end{array}} \right] } \right\} \end{aligned}$$

(34)

where ${{\mathbf{B}}_1} = \left( {\begin{array}{cc} 0&{}{E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} {e^{j2\theta }}} \\ {E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} {e^{ - j2\theta }}}&{}0 \end{array}} \right)$ and ${{\mathbf{B}}_i} = \left( {\begin{array}{cc} 0 &{} {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s{{_1^ * }^2}} \right\} {e^{j2\theta }}} \\ {E\left\{ {s{{_i^ * }^2}} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) {s_1}^2} \right\} {e^{ - j2\theta }}}&{}0 \end{array}} \right)$.

Thirdly, we make a small perturbation ${{\varepsilon }} = {\left[ {\begin{array}{cccc} {{\varepsilon _1}}&{{\varepsilon _2}}&\ldots&{{\varepsilon _N}} \end{array}} \right] ^T}$ around the optimal solution ${{\mathbf{p}}_1}$ using the complex Taylor series expansion shown in [29] as

$$\begin{aligned} I\left( {{{\mathbf{p}}_1} + {{\varepsilon }},{{\mathbf{q}}_1}} \right)= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) + {\widetilde{{\varepsilon }}^T}{\widetilde{\nabla }_{\mathbf{p}}}I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) + \frac{1}{2}{\widetilde{{\varepsilon }}^H}{\widetilde{\mathbf{H}}_{\mathbf{p}}}I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) \widetilde{{\varepsilon }} + o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \nonumber \\= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) + E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} \left( {{\varepsilon _1}{e^{ - j\theta }} + \varepsilon _1^ * {e^{j\theta }}} \right) \nonumber \\&+\, \frac{1}{2}E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} \nonumber \\&\times \left( {{\varepsilon _1}^2{e^{ - j2\theta }} + \varepsilon _1^{ * 2}{e^{j2\theta }}} \right) \nonumber \\&+\, \frac{1}{2}\sum \limits _{i > 1}^N {\left( \begin{array}{l} \varepsilon _i^2E\left\{ {s_i^{ * 2}} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) {s_1}^2} \right\} {e^{ - j2\theta }} \\ + \varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }} \end{array} \right) } + o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \end{aligned}$$

(35)

Under the constraint $\left\| {\mathbf{p}} \right\| = 1$, and using ${\left\| {{{\mathbf{p}}_1} + {{\varepsilon }}} \right\| ^2} = 1 + {\varepsilon _1}{e^{ - j\theta }} + \varepsilon _1^ * {e^{j\theta }} + {\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } ^2}$, we have

$$\begin{aligned}&{\varepsilon _1}{e^{ - j\theta }} + \varepsilon _1^* {e^{j\theta }} = - {\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } ^2} \Rightarrow {\left( {{\varepsilon _1}{e^{ - j\theta }} + \varepsilon _1^ * {e^{j\theta }}} \right) ^2} = {\left( { - {{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) ^2} \nonumber \\&\quad \Rightarrow {\varepsilon _1}^2{e^{ - j2\theta }} + 2{\left| {{\varepsilon _1}} \right| ^2} + \varepsilon _1^{ * 2}{e^{j2\theta }} = {\left( {{{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) ^2} \nonumber \\&\quad \Rightarrow {\varepsilon _1}^2{e^{ - j2\theta }} + \varepsilon _1^{ * 2}{e^{j2\theta }} = {\left( {{{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) ^2} - 2{\left| {{\varepsilon _1}} \right| ^2} \end{aligned}$$

(36)

Substituting (36) into (35), we obtain

$$\begin{aligned} I\left( {{{\mathbf{p}}_1} + {{\varepsilon }},{{\mathbf{q}}_1}} \right)= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) - \sum \nolimits _{i = 1}^N {{{\left| {{\varepsilon _i}} \right| }^2}E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} } \nonumber \\&+\, \frac{1}{2}E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} \left( {{{\left( {{{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) }^2} - 2{{\left| {{\varepsilon _1}} \right| }^2}} \right) \nonumber \\&+\, \frac{1}{2}\sum \limits _{i > 1}^N {\left( \begin{array}{l} {\varepsilon _i}^2E\left\{ {s_i^{ * 2}} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) {s_1}^2} \right\} {e^{ - j2\theta }} \\ + \varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }} \end{array} \right) } + o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \nonumber \\= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) - {\left| {{\varepsilon _1}} \right| ^2}\left( {E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} + E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} } \right) \nonumber \\&+\, \frac{1}{2}{\left( {{{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) ^2}E\left\{ {{{\left| {{s_1}} \right| }^4}g'\left( {{{\left| {{s_1}} \right| }^2}} \right) } \right\} \nonumber \\&- \sum \limits _{i > 1}^N {{{\left| {{\varepsilon _i}} \right| }^2}E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} } \nonumber \\&+ \frac{1}{2}\sum \limits _{i > 1}^N {\left( \begin{array}{l} {\varepsilon _i}^2E\left\{ {s_i^{ * 2}} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) {s_1}^2} \right\} {e^{ - j2\theta }}\\ + \varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }} \end{array} \right) } + o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \end{aligned}$$

(37)

where the terms ${\left| {{\varepsilon _1}} \right| ^2}$ and ${\left( {{{\sum \nolimits _{i = 1}^N {\left| {{\varepsilon _i}} \right| } }^2}} \right) ^2}$ are of order $o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right)$ according to (36) and can be neglected, resulting in

$$\begin{aligned} I\left( {{{\mathbf{p}}_1} + {{\varepsilon }},{{\mathbf{q}}_1}} \right)= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) - \sum \limits _{i > 1}^N {{{\left| {{\varepsilon _i}} \right| }^2}E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} } \nonumber \\&+ \sum \limits _{i > 1}^N {{{\left| {{\varepsilon _i}} \right| }^2}\left| {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }}} \right| } \cos \left( {{\varphi _i}} \right) + o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \nonumber \\= & {} I\left( {{{\mathbf{p}}_1},{{\mathbf{q}}_1}} \right) + \sum \limits _{i > 1}^N {\left( \begin{array}{l} - E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} + \\ \left| {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }}} \right| \cos \left( {{\varphi _i}} \right) \end{array} \right) {{\left| {{\varepsilon _i}} \right| }^2}} \nonumber \\&+\, o\left( {{{\left\| {{\varepsilon }} \right\| }^2}} \right) \end{aligned}$$

(38)

where ${\varphi _i} = \arg \left( {\varepsilon _i^{ * 2}} \right) + \arg \left( {E\left\{ {s_i^2} \right\} } \right) + \arg \left( {E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }}} \right)$ is some arbitrary phase shift that results from

$$\begin{aligned}&{\varepsilon _i}^2E\left\{ {s_i^{ * 2}} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) {s_1}^2} \right\} {e^{ - j2\theta }} + \varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\} \nonumber \\&\quad = {\left( {\varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }}} \right) ^ * } + \left( {\varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\} } \right) \nonumber \\&\quad = 2\left| {\varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\} } \right| \cos \left( {{\varphi _i}} \right) \nonumber \\&\quad = 2\left| {\varepsilon _i^{ * 2}} \right| \left| {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\} } \right| \cos \left( {{\varphi _i}} \right) \nonumber \\&\quad = 2\left| {{\varepsilon _i}^2} \right| \left| {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\} } \right| \cos \left( {{\varphi _i}} \right) \end{aligned}$$

(39)

where the identity $\mathrm{Z} + {\mathrm{Z}^ * } = 2{\mathrm{Z}^R} = 2\left| \mathrm{Z} \right| \cos \left[ {\arg \left( \mathrm{Z} \right) } \right]$ is used, in which $\mathrm{Z} = \varepsilon _i^{ * 2}E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}{e^{j2\theta }}} \right\}$. The term $\cos \left( {{\varphi _i}} \right) \in \left[ { - 1,1} \right]$ for any perturbation ${\varepsilon _i}$, resulting in the conditions for a local minimum (respectively, maximum) as

$$\begin{aligned} - E\left\{ {g\left( {{{\left| {{s_1}} \right| }^2}} \right) {{\left| {{s_1}} \right| }^2}} \right\} \pm \left| {E\left\{ {s_i^2} \right\} E\left\{ {g'\left( {{{\left| {{s_1}} \right| }^2}} \right) s_1^{ * 2}} \right\} {e^{j2\theta }}} \right| > 0,\, \left( {resp. < 0} \right) \end{aligned}$$

(40)

which must be satisfied for each source ${s_i}$, where $i = \left[ {2,3, \ldots ,N} \right]$.

Finally, the local stability of our reference-based negentropy has been proved.