1. Introduction

Many complex signals obtained from high-dimensional nonlinear dynamical systems are not purely simple chaotic signals. They may be contaminated by different components of individual chaotic signals or internal noise signals [1,2]. The extraction of each chaotic component from mixed signals is a very attractive and important research area for chaosbased communication systems. The correlation detection technique for the separation of a chaotic component from mixed chaotic signals is used for multiplexing communications using chaos such as code division multiple access (CDMA) [3–5]. The technique of dual synchronization of chaos is another method to separate mixed chaotic signals by using the dynamical stability of synchronization [6–12]. In this method parameter sensitivity of synchronization limits the number of mixed chaotic signals that can be separated. A different approach for separation of linearly superimposed uncorrelated signals has been proposed by using time delayed correlation functions [13]. The adaptation process for the inhibitory interactions in a neural network is used to solve the source separation problem.

Recently blind source separation by using independent component analysis (ICA) has been rapidly developed and applied for many complex signals [14,15]. ICA is a statistical estimation technique to separate mixed complex signals based on non-Gaussianity of the probability density function (PDF) of signals. The PDF of the sum of multiple source signals approaches the Gaussian distribution when compared with that of each source signal, because of the central limit theorem. Independent components can be extracted from the observed mixed signals by maximizing non-Gaussianity of the PDF of estimated signals. ICA has been successfully applied for auditory signal separation (the cocktail party effect), neural network without teaching data, analysis of electroencephalogram (EEG) data in the brain, signal processing in medical engineering, image processing [14,15]. However, no study on blind signal separation of chaotic laser signals by ICA has been reported.

In this study we experimentally demonstrate blind source separation of mixed optical chaotic signals by using ICA. We use Nd:YVO₄ microchip solid-state lasers with external modulation as chaotic sources. Chaotic signals are mixed together with randomly selected mixing ratio and ICA is applied to the mixed chaotic signals for blind source separation. We also use many chaotic signals and investigate the ability of ICA to separate a large number of mixed signals.

2. Experimental result of ICA

ICA is a signal processing technique whose goal is to express a set of random variables as linear combinations of statistically independent component variables [5,14,15]. Among algorithms for ICA, we adopt the FastICA algorithm, which is based on a fixed point scheme for finding a maximum of the non-Gaussianity [14,15].

To test blind source separation of mixed chaotic signals by using ICA, we use two Nd:YVO₄ microchip lasers [11,12] with external modulation. Two independent chaotic laser beams are combined at a beam splitter (BS in Fig. 1). The two outcoming beams from the beam splitter contain a portion of the two laser outputs with different mixing ratio: one beam contains R% of the laser 1 output and T% of the laser 2 output, and vice versa for the other beam, where R and T indicate the reflectivity and the transmittance of the beam splitter (R+T=100%). Note that no information on the values of R and T is required for blind source separation by ICA. The two mixed chaotic signals are detected as mixed signal 1 and 2 by using two photodiodes (PD _m1 and PD _m2 in Fig. 1), so that the incoherent sum of the laser intensities are observed. These two mixed chaotic signals are used for blind source separation by ICA. The chaotic source signals of the two microchip lasers are also detected for reference by using two other photodiodes (PD _s1 and PD _s2 in Fig. 1).

Figure 2 shows the experimental results of the blind source separation by using ICA. Figure 2(a) indicates the two mixed chaotic signals that are used for signal separation. The two separated signals are obtained from the two mixed chaotic signals by using ICA, and compared with the source signals, as shown in Figs. 2(b) and 2(c). The separated signals are almost identical to the source signals. Note that we made corrections of the signs, amplitudes, and the orders of the independent components in the separated signals by postprocessing.

 

Fig. 1. Experimental setup of Nd:YVO₄ microchip solid-state lasers with external modulation for blind source separation by ICA. BS, beam splitter; GP, glass plate; L, lens; M, mirror; and PD, photodiode.

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We quantitatively define the degree of successful signal separation as the cross correlation between the source and separated temporal waveforms normalized by the product of their standard deviations: i.e.,

where I _1,2 are the total intensities of the two temporal waveforms of source and separated signals, Ī _1,2 are their mean values, and σ _1,2 are their standard deviations. The angle brackets denote time averaging. The best signal separation is obtained at cross-correlation coefficient of C=1.

Figure 3 shows the correlation plots between one of the source signals and one of the separated signals. The corresponding pairs of the source and separated signals show very good correlation, i.e., the cross correlations of 0.980 and 0.952 for Figs. 3(a) and 3(d), respectively. However, the different pairs of signals show low correlation, i.e., the cross correlations of 0.0743 and -0.0433 for Figs. 3(b) and 3(c), respectively. Therefore we succeed blind signal separation of two mixed chaotic laser waveforms by ICA.

3. Separation ability for large number of mixed chaotic signals by ICA

To test the separation ability of ICA for many chaotic signals, we increase the number of chaotic signals used for the mixed signals. Due to our experimental limitation, we detect many independent chaotic source signals from one microchip laser at different times and store in a computer. These source signals are examined with a randomly selected mixing matrix and the mixed signals are generated numerically. We then apply FastICA for these mixed signals to test blind source separation.

Figure 4(a) shows the length of temporal waveform T _L required for successful signal separation as a function of the number of mixed signals N _m at different data-sampling frequencies f _s. Here we define the successful signal separation when the average value of the cross correlations between the source and corresponding separated signals is more than 0.95. Note that we succeed blind source separation of 100 mixed chaotic laser signals by ICA as shown in Fig. 4(a). It is found that the longer T _L is required with increase of N _m for all f _s. Moreover, the curves of T _L decrease and saturate as f _s is increased, (f _s>=50 MHz in Fig. 4(a)). Therefore, the minimum T _L required for successful blind source separation by ICA is almost proportional to N _m for large f _s (i.e., oversampled data).

 

Fig. 2. Experimental results of blind source separation of chaotic laser signals by using ICA. (a) Mixed signals 1 and 2, (b) separated signal 1 and source signal 1, and (c) separated signal 2 and source signal 2.

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Fig. 3. Correlation plots between one of the separated signals and one of the source signals. (a) Source signal 1 and separated signal 1, (b) source signal 1 and separated signal 2, (c) source signal 2 and separated signal 1, and (d) source signal 2 and separated signal 2.

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Fig. 4. (a) The length of temporal waveform T _L and (b) the number of data point N _d (=T _L f _s) required for successful signal separation as a function of the number of mixed signals N _m at different data-sampling frequencies f _s for the microchip lasers. The curves correspond to the condition at which the average value of cross correlations between the source and successfully separated signals is 0.95. The fundamental frequency of chaotic laser signals is 3.25 MHz. Solid black curve; f _s=100 MHz, dashed blue curve; f _s=50 MHz, dotted red curve; f _s=25 MHz, and dotted-dashed green curve; f _s=12.5 MHz.

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Figure 4(b) shows the number of data points N_d required for successful signal separation as a function of N _m at different f _s. Note that N _d is equal to T _L f _s. It is found that the curves of N _d decrease and saturate as f _s is decreased, (f _s<=50 MHz in Fig. 4(b)). Therefore, the minimum N _d required for successful blind source separation is almost proportional to N _m for small f _s (i.e., undersampled data).

From the results of Figs. 4(a) and 4(b), the optimal f _s for successful separation with the minimum T _L and N _d is 50 MHz (15.4 times faster than the fundamental frequency of chaos, 3.25 MHz). This condition indicates that roughly 200 periods of chaotic oscillations are required for the separation of 100 mixed chaotic signals at f _s=50 MHz.

 

Fig. 5. Probability density function (PDF) of the source, two-mixed, and 50-mixed signals obtained from the microchip lasers. Solid black curve; source signal, dashed blue curve; twomixed signal, dotted red curve; 50-mixed signal, and dotted-dashed green curve; Gaussian distribution for reference. The kurtosis is 7.858 for the source signal, 4.647 for the two mixed signal, and 0.0841 for 50 mixed signals.

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4. Probability density function of mixed chaotic signals

We measure the non-Gaussianity of mixed chaotic signals by using the PDFs. Figure 5 shows the PDFs of the source, two mixed and 50 mixed signals obtained from the chaotic laser signals. As the number of mixed signals is increased the PDFs approach the Gaussian distribution. We calculate the kurtosis of these PDFs, which is an indicator of non-Gaussianity. The kurtosis is 7.858 for the source signal, 4.647 for the two mixed signal, and 0.084 for 50 mixed signals. The kurtosis becomes close to zero as the number of mixed signals is increased. Therefore, non-Gaussianity of chaotic source signals is crucial for successful blind source separation of chaotic laser signals by ICA, as well as other mixed data sources [14,15].

Experimentally observed chaos in lasers does not satisfy the independent and identically distributed condition, and chaos has a time correlation in general. The PDF of chaotic laser signals has a non-Gaussian distribution, unlike quantum noise signals. The degree of non- Gaussianity depends on the type of chaotic laser sources and how to generate chaotic signals. Our method is applicable to other chaotic laser sources to separate mixed signals when the PDF of the chaotic signals has a non-Gaussian distribution.

5. Conclusion

We have experimentally demonstrated blind source separation of chaos generated in Nd:YVO₄ microchip solid-state lasers by using ICA. We have successfully obtained the source signals from the linearly mixed chaotic signals by using the FastICA algorithm. We have succeeded 100-signal separation of chaotic laser waveforms and found that longer temporal waveforms are required for the larger number of mixed signals. Non-Gaussianity of the PDF of chaotic signals is a crucial property to succeed blind source separation. The technique of blind source separation of chaos by ICA could be useful for multiplexing communication systems with chaotic codes.