Efficient Variant of Noncircular Complex FastICA Algorithm for the Blind Source Separation of Digital Communication Signals

Abstract

In this paper, an improved version of the noncircular complex FastICA (nc-FastICA) algorithm is proposed for the separation of digital communication signals. Compared with the original nc-FastICA algorithm, the proposed algorithm is asymptotically efficient for digital communication signals, i.e., its estimation error can be made much smaller by adaptively choosing the approximate optimal nonlinear function. Thus, the proposed algorithm can have a significantly improved performance for the separation of digital communication signals. Simulations confirm the efficiency of the proposed algorithm.

Appendix: Proof of Theorem 1

As shown in [20], the cost function of nc-FastICA algorithm

$$\begin{aligned} J(\mathbf{w})=E\left\{ {G\left( \left| {\mathbf{w}^\mathrm{H}\mathbf{Vz}} \right| ^{\mathbf{2}}\right) } \right\} \end{aligned}$$

(9)

where $G:{\mathbb {R}}^{+}\cup \{0\}\rightarrow {\mathbb {R}}$ is a smooth even function, $\mathbf{V}$ is the whitening matrix and $\mathbf{w}\in {\mathbb {C}}^{N}$ with $| \mathbf{w}|=1$, i.e., $E({|{\mathbf{w}^\mathrm{H}\mathbf{Vz}}|^{2}})=E( {|{{\bar{\mathbf{w}}}^\mathrm{H}\mathbf{z}}|^{2}})=1$, ${\bar{\mathbf{w}}}$ is a column of the demixing matrix.

By making the orthogonal change of coordinates $\mathbf{q}=\mathbf{A}^\mathrm{H}{\bar{\mathbf{w}}}=\mathbf{A}^\mathrm{H}\mathbf{V}^\mathrm{H}\mathbf{w}$, the Karush–Kuhn–Tucker condition for the constraint optimization problem (9) becomes

$$\begin{aligned} {\partial E\left\{ {G\left( {\left| {\mathbf{q}^\mathrm{H}\mathbf{s}} \right| ^{\mathbf{2}}} \right) } \right\} }\bigg /{\partial \mathbf{q}}=\lambda {\partial E\left( {\left| {\mathbf{q}^\mathrm{H}\mathbf{s}} \right| ^{2}} \right) }\bigg /{\partial \mathbf{q}} \end{aligned}$$

(10)

i.e.,

$$\begin{aligned} \sum _t {\mathbf{s}(t)\left( {\mathbf{q}^\mathrm{H}\mathbf{s}(t)} \right) ^{*}g\left( {\left| {\mathbf{q}^\mathrm{H}\mathbf{s}(t)} \right| ^{2}} \right) } =\lambda \sum _t {\mathbf{s}(t)\mathbf{s}^\mathrm{H}(t)\mathbf{q}} \end{aligned}$$

(11)

where $t$ is the sample index and has been dropped for simplicity in the following expressions, $T$ is the sample size, $\lambda$ is the Lagrange multiplier.

Without loss of generality, we assume $\mathbf{q}$ is an approximation optimal solution for $s_1$, then $\lambda =E[{| {s_1 } |^{2}g({| {s_1}|^{2}})}]$. By using first-order Taylor expression and ignoring the perturbation to the first component, we can derive

$$\begin{aligned}&\mathbf{s}\left( {\mathbf{q}^\mathrm{H}\mathbf{s}} \right) ^{*}g\left( {\left| {\mathbf{q}^\mathrm{H}\mathbf{s}} \right| ^{2}} \right) \nonumber \\&\quad \approx \mathbf{s}\left\{ {\left( {s_1^*+\mathbf{s}_{-1}^\mathrm{H} \mathbf{q}_{-1} } \right) \left[ {g\left( {\left| {s_1 } \right| ^{2}} \right) +g^{\prime }\left( {\left| {s_1 } \right| ^{2}} \right) \left( {\left| {s_1 } \right| ^{2}\mathbf{s}_{-1}^\mathrm{H} \mathbf{q}_{-1} +s_1^{*2} \mathbf{q}_{-1}^\mathrm{H} \mathbf{s}_{-1} } \right) } \right] } \right\} \nonumber \\&\quad \approx \mathbf{s}\left\{ {s_1^*g\left( {\left| {s_1 } \right| ^{2}} \right) \!+\!\left[ {g\left( {\left| {s_1 } \right| ^{2}} \right) \!+\!g^{\prime }\left( {\left| {s_1 } \right| ^{2}} \right) \left| {s_1 } \right| ^{2}} \right] \mathbf{s}_{-1}^\mathrm{H} \mathbf{q}_{-1} +g^{\prime }\left( {\left| {s_1 } \right| ^{2}} \right) s_1^{*2} \mathbf{s}_{-1}^T \mathbf{q}_{-1}^*} \right\} \end{aligned}$$

(12)

$$\begin{aligned}&\lambda \mathbf{ss}^\mathrm{H} \mathbf{q}=\mathbf{s}\left( {\lambda s_1^*+\mathbf{s}_{-1}^\mathrm{H} \mathbf{q}_{-1} } \right) \end{aligned}$$

(13)

where $\mathbf{s}_{-1}$ denotes $\mathbf{s}$ without its first component, $\mathbf{q}_{-1}$ denotes $\mathbf{q}$ without its first component.

Excluding the first component in (11) and taking (12) and (13) into consideration, we obtain

$$\begin{aligned}&\frac{1}{\sqrt{T}}\sum _t {\mathbf{s}_{-1} \left[ {s_1^*g\left( {\left| {s_1 } \right| ^{2}} \right) -\lambda s_1^*} \right] } \nonumber \\&\quad =\frac{1}{T}\sum _t {\mathbf{s}_{-1} \mathbf{s}_{-1}^H \left[ {\lambda -g\left( {\left| {s_1 } \right| ^{2}} \right) -\left| {s_1 } \right| ^{2}g^{\prime }\left( {\left| {s_1 } \right| ^{2}} \right) } \right] \mathbf{q}_{-1} \sqrt{T}} \nonumber \\&\qquad +\,\frac{1}{T}\sum _t {\mathbf{s}_{-1} \mathbf{s}_{-1}^T \left[ {-s_1^{*2} g^{\prime }\left( {\left| {s_1 } \right| ^{2}} \right) } \right] \mathbf{q}_{-1}^*\sqrt{T}} \end{aligned}$$

(14)

Moreover, (14) can be further written in a simple form of

$$\begin{aligned} \mathbf{u}=\mathbf{V}_1 {\hat{\mathbf{q}}}_{-1} \sqrt{T}+\mathbf{V}_2 {\hat{\mathbf{q}}}_{-1}^*\sqrt{T} \end{aligned}$$

(15)

where $\mathbf{V}_1 \rightarrow E[ {| {s_1 } |^{2}g({| {s_1 }|^{2}})-g( {| {s_1 }|^{2}})-| {s_1 } |^{2}g^{\prime }( {| {s_1 }|^{2}})} ]\mathbf{I}_{N-1}$, $\mathbf{I}_{N-1}$ is a $(N-1)\times (N-1)$ dimensional identity matrix, $\mathbf{V}_2 \rightarrow -E[ {s_1^{*2} g^{\prime }( {| {s_1 }|^{2}} )} ]E( {\mathbf{s}_{-1} \mathbf{s}_{-1}^T})$, u converges to a normal distribution variable with zero mean, covariance $\mathbf{V}_3 =\{ {E[ {| {s_1 } |^{2}g^{2}( {| {s_1 }|^{2}})} ]-E^{2}[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]} \}\mathbf{I}_{N-1}$ and pseudo-covariance $\mathbf{V}_4 = \{ {E[ {s_1^{*2} g^{2}( {| {s_1 } |^{2}} )} ]+E[ {s_1^{*2} } ]E^{2}[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]-2E[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]E[ {s_1^{*2} g( {| {s_1 } |^{2}} )} ]} \}E( {\mathbf{s}_{-1} \mathbf{s}_{-1}^T } )$.

For simplicity, we assume that all the sources belong to the same modulation. Then, $\mathbf{V}_2 \rightarrow -E[ {s_1^{*2} g^{\prime }( {| {s_1 } |^{2}} )} ]E( {s_1^2 } )\mathbf{I}_{N-1}$, $\mathbf{V}_4 =\{ E[ {s_1^{*2} g^{2}( {| {s_1 } |^{2}} )} ]+E[ {s_1^{*2} } ] E^{2}[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )}]$ $-2E[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]E[ {s_1^{*2} g( {| {s_1 } |^{2}} )} ] \}E( {s_1^2 } )\mathbf{I}_{N-1}$.

Therefore, the asymptotic variance of ${\hat{\mathbf{q}}}_{-1}$ is

$$\begin{aligned} \mathbf{V}=\frac{1}{\left( {a_1^2 -a_2^2 } \right) ^{2}}\left[ {\left( {a_1^2 +a_2^2 } \right) a_3 -a_1 a_2 \left( {a_4 +a_4^*} \right) } \right] \mathbf{I}_{N-1} \end{aligned}$$

(16)

where $a_1 =E[{| {s_1 } |^{2}g( {| {s_1 } |^{2}} )\!-\!g( {| {s_1 } |^{2}} )-| {s_1 } |^{2}g^{\prime }( {| {s_1 } |^{2}} )} ]$, $a_2 \!=\!-E[ {s_1^{*2} g^{\prime }( {| {s_1 } |^{2}} )} ]E( {s_1^2 })$, $a_3 =E[ {| {s_1 } |^{2}g^{2}( {| {s_1 } |^{2}} )} ]-E^{2}[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]$, $a_4 =\{ E[ {s_1^{*2} g^{2}( {| {s_1 } |^{2}} )} ]+E[ {s_1^{*2} } ]E^{2}[ {| {s_1 } |^{2}}{g( {| {s_1 } |^{2}} )}]-2E[ {| {s_1 } |^{2}g( {| {s_1 } |^{2}} )} ]E[ {s_1^{*2} g( {| {s_1 } |^{2}} )} ]\}E( {s_1^2 })$.

Thus, (3) can be easily obtained since $\mathbf{q}_{-1} =\mathbf{B}\bar{\mathbf{w}}$, where $\mathbf{B}$ is the conjugate transpose of $\mathbf{A}$ without its first row.