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Detection of harmonic signals from chaotic interference by empirical mode decomposition
Chaos, Solitons and Fractals 30 (2006) 930–935 www.elsevier.com/locate/chaos
Detection of harmonic signals from chaotic interference by empirical mode decomposition H.G. Li *, G. Meng
*
State Key Laboratory of Vibration, Shock & Noise, Shanghai Jiao Tong University, Shanghai 200030, PR China Accepted 30 August 2005
Abstract An empirical mode decomposition (EMD) approach to the harmonic signal extraction from chaotic interference is proposed. Based on the EMD and the concept that any signal is composed of a series of simple intrinsic modes, the chaotic interference signal is decomposed to a series of intrinsic mode functions (IMFs), among which one IMF is the recovered harmonic signal. In this study, harmonic signals are contaminated with a chaotic interference signal which is generated by a Duffing oscillator, and the simulation results show that the harmonic signals can be effectively recovered from the contaminated signals by the EMD approach. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction A number of scholars have investigated the chaotic signal processing, because numerous observable chaotic signals can be detected in real-world data such as sea clutter signals, EMG signals and electrocardiograph signals, and chaotic signals also have some special functions such as secure communication and electronic countermeasures [1–4]. Obviously, the harmonic component extraction from a chaotic signal is very important in theory and application. In this research field, the phase space volume method is introduced to estimate the coefficients of an autoregressive spectrum [5], the detection of a small target in sea clutter is investigated by means of neural network method [6]. The use of nonlinear dynamic (NLD) forecasting is considered to extract messages from chaotic communication systems [7]. Base on the geometry of chaotic interference, a method for signal extraction from received data contaminated with strong chaotic interference is proposed [8]. A new nonlinear technique, referred to as empirical mode decomposition (EMD), has recently been pioneered by Huang et al. [9], it was proved to be remarkably more effective than other signal processing methods for nonstationary signals [10,11]. In signal processing, time scale and energy associated with time scale are two of the most important parameters of signal. Since empirical mode decomposition is based on the local characteristic time scale of the data, it is able to decompose complex signals to a collection of intrinsic mode functions (IMFs). Based on and derived from the data, these IMFs can serve as the basis of that expansion which can be linear or nonlinear as dictated by the data, and it is complete and almost orthogonal. Most important of all, it is adaptive. *
Corresponding authors. E-mail addresses: [email protected] (H.G. Li), [email protected] (G. Meng).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.174
H.G. Li, G. Meng / Chaos, Solitons and Fractals 30 (2006) 930–935
931
Based on empirical mode decomposition and concept that any signals are composed of a series of the simple intrinsic modes, the contaminated signals can also be decomposed a collection of IMFs, the purpose of the paper is to propose a new approach of the harmonic signals extraction from chaotic interference. The harmonic signals are contaminated with different chaotic signals generated by a Duffing oscillator with different parameters, and recovered by means of empirical mode decomposition. The simulation results show that the method is satisfied.
2. EMD basics At any given time, the data may involve more than one oscillatory mode, each mode, for linear or nonlinear, has same numbers of extrema and zero crossings, and only one extremum between the successive zero crossings. These modes should all be orthogonal to each other for a linear decomposition. Thus, an arbitrary signal can be decomposed to a collection of IMFs. An intrinsic mode function (IMF) is a function that satisfies two conditions [9]: (1) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. Comparing with simple monotone function, an IMF is a simple vibrating mode. Given a signal x(t), the effective algorithm of EMD can be summarized as follows: (1) identify all extrema of x(t), connect all the local maxima a cubic spline line as the upper envelope; (2) connect all the local minima a cubic spline line as the upper envelope, the upper and lower envelopes should cover all the data between them; (3) the mean of upper and lower envelopes is designated as m1, and the difference between the data and m1 is the first component, h1, i.e. hðtÞ ¼ xðtÞ mðtÞ
ð1Þ
if h1 is an IMF, h1 is the first component of x(t). (4) if h1 is not an IMF, h1 is treated as the original data, continue the step (1), (2) and (3), get the mean of upper and lower envelopes, which is designated as m11, if h11 = h1 m11 is still not an IMF, continue the steps (1)–(3), until the first component h1k is an IMF, and designated as c1 = h1k. c1 is the first IMF component of x(t); (5) separate c1 from the rest of the data by r1 ðtÞ ¼ xðtÞ c1 ðtÞ
ð2Þ
Since the residue, r1, still contains information of longer period components, it is treated as the new data and subjected to the same sifting process as described above, get the second IMF component of x(t) designated as c2, the above procedure can be repeated to get nth IMF component until the residue, rn becomes a monotonic function from which no more IMF can be extracted. Thus, we achieved a decomposition of the data x(t) into n-empirical modes, and a residue rn, where ci, i = 1 n, contain different component of the signal from high to low frequency bands respectively. Frequency components in each band are different to other bands. The residue rn can be either the mean trend of signal x(t).
3. Simulation results and analysis The signal used in this paper is from the Duffing equation. The Duffing oscillator is one of the most common types of nonlinear oscillators and has many typical nonlinear phenomena [12,13], and can be written as follows: €x þ c_x x20 x þ dx3 ¼ P cosðxtÞ
ð3Þ
where, x0 is natural frequency of the oscillator, c is damping, d is nonlinear parameter, P and x are amplitude and frequency of external exciting force respectively. Chaotic or periodic responses of the Duffing oscillator in the planes defined by the frequency–amplitude of the external periodic excitation have been traced, and Fig. 1 displays the relevant
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Fig. 1. Black regions depict the domains where chaotic behaviors of the Duffing oscillator are possible (c = 0.05, = 0.2, d = 1, x(0) = 1.0 and x_ ð0Þ ¼ 0:0) and Grey and white colors corresponds to the periodic motion of the oscillator.
domains. A harmonic signal yðtÞ ¼ A sinðxh tÞ is added to the chaotic signal. In this paper, the sampled displacement signals x(k) is obtained using the classical fourth order Runge–Kutta algorithm and the initial conditions are x(0) = 1.0 and x_ ð0Þ ¼ 0:0. So, the contaminated signal z(k) is given by zðkÞ ¼ yðkÞ þ xðkÞ
ð4Þ
Fig. 2. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
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The contaminated signal z(k) can be decomposed to a finite of intrinsic mode functions (IMFs), and one of the IMFs is a recovered harmonic signal, whose frequency and phase angle equal to that of the original harmonic signal y(k). To illustrate the decomposition procedure, we use three time series data collected from Eq. (4). The first original chaotic signal was generated by a Duffing system with following parameters: c = 0.05, x20 ¼ 0:2, d = 1, x = 1.0 and P = 27.5, and the largest Lyapunov exponent is 0.084. The original harmonic signal was generated with A = 1.0 and xh = 0.5. Sampling interval Dt = p/400, the number of receiving signal is N = 20,000. The original chaotic signal, the original harmonic signal and the contaminated signal are shown in Fig. 2(a)–(c) respectively. Comparing with original chaotic signal, original harmonic signal is rather small. The harmonic signal can not be observed from the waveform of the contaminated signal z(k). Using empirical mode decomposition, the contaminated signal z(k) can be decomposed to a series of intrinsic mode functions (IMFs), and c3, one of the IMFs, is the recovered harmonic signals. The comparison of original harmonic signal and recovered harmonic signal is shown in Fig. 2(d), where the dotted line is original harmonic signal y(k). Comparing with original harmonic signal y(k), the amplitude of recovered harmonic signal c3 decreases, but the frequency and phase angle do not change. Fig. 2(d) also shows that the edge effects can be occur at both the beginning and end of the recovered harmonic signal. If the edges of the contaminated signal are lifted untreated, the cubic spline fittings will introduce large perturbations into the data which can propagate and eventually corrupt the signal [10]. In this paper, a number of local maxima (minima) at the beginning and end of the contaminated signal are introduced to eliminate the edge effects, and the satisfactory recovered signal is obtained. The second original chaotic signal was generated by same Duffing system with following parameters: c = 0.05, x20 ¼ 0:2, d = 1, x = 1.1 and P = 10.0, and the largest Lyapunov exponent is 0.121. The original harmonic signal was generated by y(t) = A sin(xht) with A = 1.0 and xh = 0.3. Sampling interval Dt = p/400, the number of receiving signal is N = 20,000. The original chaotic signal x(k), the rather small original harmonic signal y(k) and the contaminated signal z(k) are shown in Fig. 3(a)–(c) respectively. The harmonic signal can not be observed from the waveform of the contaminated signal z(k). the contaminated signal z(k) can be decomposed to a series of intrinsic mode functions (IMFs) by means of empirical mode decomposition, and c3, one of the IMFs, is the recovered harmonic signals. The comparison of original harmonic signal and recovered harmonic signal is shown in Fig. 3(d), where the dotted line
Fig. 3. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
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is original harmonic signal y(k). Comparing with original harmonic signal y(k), the only change of recovered signal c3 is amplitude. According to empirical mode decomposition, all extrema of signal should be identified, and all the local maxima (minima) are connected by a cubic spline line as the upper (low) envelope. The shorter sampling interval, the more local extrema. In following numerical simulation, the Duffing oscillator has parameters of c = 0.05, x20 ¼ 0:2, d = 1, x = 1.0 and P = 27.5. The harmonic signal is generated with parameter A = 1.0 and xh = 0.3. Sampling interval Dt = p/200, the number of receiving signal is N = 20,000. The numerical results of original chaotic signal x(k), the small original harmonic signal y(k) and the contaminated signal z(k) are shown in Fig. 4(a)–(c) respectively, and the harmonic signal can not be observed directly from the waveform of the contaminated signal z(k). Fig. 4(d) shows the trajectories of the third intrinsic mode function c3 (the solid line) and the original harmonic signal y (the dotted line). In this simulation, the numerical results are rather satisfactory except the beginning and the end of the recovered signal c3.
Fig. 4. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
Fig. 5. The spectrum diagrams of intrinsic mode functions c1 (solid black line), c2 (dotted line) and c3 (solid grey line) decomposed from (a) the first contaminated signal (b) the third contaminated signal.
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The comparison between the original harmonic signal y and the recovered harmonic signal c3 in Fig. 2(d) shows that the amplitude of first c3 is smaller, the Fig. 4(d) shows the recovered harmonic signal is really very good. The spectrum diagrams of intrinsic mode functions c1, c2 and c3 decomposed from the first contaminated signal and the third contaminated signal are shown in Fig. 5. The second IMF c2, as it is displayed in Fig. 5(a), centered at 0.16 Hz, and covered from 0.07 to 0.3 Hz. The frequency of the original harmonic signal is about 0.08 Hz, therefore, the amplitude of the recovered signal c3 is smaller than that of the original harmonic signal. In Fig. 5(b), the frequency of the recovered signal c3 is about 0.048 Hz, the c1 and c2 do not contain component at this frequency, therefore, the recovered harmonic signal is very good.
4. Conclusion In this paper, a method for harmonic signal extraction from chaotic interference is proposed based on the empirical mode decomposition. The harmonic signal is contaminated with a chaotic signal generated by a Duffing oscillator. The simulation results for the chaotic interference signals were analyzed by applying the empirical mode decomposition to obtain the intrinsic mode functions (IMFs), among which one IMF is the recovered harmonic signal. This approach is a simple signal processing tool to use and provides reasonably results.
Acknowledgements The authors thank Professor He Jihuan for valuable discussion. This work was supported by the National Natural Science Foundations of China (No. 50335030, No. 10325209, No. 10502032 and No. 50375094).
References [1] Moez F, Bruno R, Guillaume G, Maxime C. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons & Fractals 2003;18(4):881–90. [2] Lei M, Wang ZZ, Feng ZJ. Detecting nonlinearity of action surface EMG signal. Phys Lett A 2001;290(5–6):297–303. [3] Li ZG, Xu DL. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22(2):477–81. [4] Bai EW, Lonngren KE, Ahmet U. Secure communication via multiple parameter modulation in a delayed chaotic system. Chaos, Solitons & Fractals 2005;23(3):1071–6. [5] Leung H, Huang XP. Parameter estimation in chaotic noise. IEEE Trans Sig Proc 1996;44:2456–63. [6] Haykin S, Li XB. Detection of signals in chaos. Proc IEEE 1995;83:94–122. [7] Short KM. Signal extraction from chaotic communications. Int J Bifurcat Chaos 1997;7:1579–97. [8] Wang FP, Guo JB, Wang ZJ, Xiao DC, Li MT. Harmonic signal extraction from strong chaotic interference. Acta Phys Sin 2001;50:1019–23 [in Chinese]. [9] Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series. Proc R Soc Lond A 1998;54(4):903–95. [10] Lai YC, Ye N. Recent developments in chaotic time series analysis. Int J Bifurcat Chaos 2003;13:1383–422. [11] Balocchi R, Menicucci D, et al. Deriving the respiratory sinus arrhythmia from the heartbeat time series using empirical mode decomposition. Chaos, Solitons & Fractals 2004;20(1):171–7. [12] He JH. Modified Lindsted-Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion. Int J Nonlinear Sci Numer Simul 2001;2(4):317–20. [13] Tan CA, Kang BS. Chaotic motions of a Duffing oscillator subjected to combined parametric and quasiperiodic excitation. Int J Nonlinear Sci Numer Simul 2001;2(4):353–64.
Detection of harmonic signals from chaotic interference by empirical mode decomposition H.G. Li *, G. Meng
*
State Key Laboratory of Vibration, Shock & Noise, Shanghai Jiao Tong University, Shanghai 200030, PR China Accepted 30 August 2005
Abstract An empirical mode decomposition (EMD) approach to the harmonic signal extraction from chaotic interference is proposed. Based on the EMD and the concept that any signal is composed of a series of simple intrinsic modes, the chaotic interference signal is decomposed to a series of intrinsic mode functions (IMFs), among which one IMF is the recovered harmonic signal. In this study, harmonic signals are contaminated with a chaotic interference signal which is generated by a Duffing oscillator, and the simulation results show that the harmonic signals can be effectively recovered from the contaminated signals by the EMD approach. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction A number of scholars have investigated the chaotic signal processing, because numerous observable chaotic signals can be detected in real-world data such as sea clutter signals, EMG signals and electrocardiograph signals, and chaotic signals also have some special functions such as secure communication and electronic countermeasures [1–4]. Obviously, the harmonic component extraction from a chaotic signal is very important in theory and application. In this research field, the phase space volume method is introduced to estimate the coefficients of an autoregressive spectrum [5], the detection of a small target in sea clutter is investigated by means of neural network method [6]. The use of nonlinear dynamic (NLD) forecasting is considered to extract messages from chaotic communication systems [7]. Base on the geometry of chaotic interference, a method for signal extraction from received data contaminated with strong chaotic interference is proposed [8]. A new nonlinear technique, referred to as empirical mode decomposition (EMD), has recently been pioneered by Huang et al. [9], it was proved to be remarkably more effective than other signal processing methods for nonstationary signals [10,11]. In signal processing, time scale and energy associated with time scale are two of the most important parameters of signal. Since empirical mode decomposition is based on the local characteristic time scale of the data, it is able to decompose complex signals to a collection of intrinsic mode functions (IMFs). Based on and derived from the data, these IMFs can serve as the basis of that expansion which can be linear or nonlinear as dictated by the data, and it is complete and almost orthogonal. Most important of all, it is adaptive. *
Corresponding authors. E-mail addresses: [email protected] (H.G. Li), [email protected] (G. Meng).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.174
H.G. Li, G. Meng / Chaos, Solitons and Fractals 30 (2006) 930–935
931
Based on empirical mode decomposition and concept that any signals are composed of a series of the simple intrinsic modes, the contaminated signals can also be decomposed a collection of IMFs, the purpose of the paper is to propose a new approach of the harmonic signals extraction from chaotic interference. The harmonic signals are contaminated with different chaotic signals generated by a Duffing oscillator with different parameters, and recovered by means of empirical mode decomposition. The simulation results show that the method is satisfied.
2. EMD basics At any given time, the data may involve more than one oscillatory mode, each mode, for linear or nonlinear, has same numbers of extrema and zero crossings, and only one extremum between the successive zero crossings. These modes should all be orthogonal to each other for a linear decomposition. Thus, an arbitrary signal can be decomposed to a collection of IMFs. An intrinsic mode function (IMF) is a function that satisfies two conditions [9]: (1) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. Comparing with simple monotone function, an IMF is a simple vibrating mode. Given a signal x(t), the effective algorithm of EMD can be summarized as follows: (1) identify all extrema of x(t), connect all the local maxima a cubic spline line as the upper envelope; (2) connect all the local minima a cubic spline line as the upper envelope, the upper and lower envelopes should cover all the data between them; (3) the mean of upper and lower envelopes is designated as m1, and the difference between the data and m1 is the first component, h1, i.e. hðtÞ ¼ xðtÞ mðtÞ
ð1Þ
if h1 is an IMF, h1 is the first component of x(t). (4) if h1 is not an IMF, h1 is treated as the original data, continue the step (1), (2) and (3), get the mean of upper and lower envelopes, which is designated as m11, if h11 = h1 m11 is still not an IMF, continue the steps (1)–(3), until the first component h1k is an IMF, and designated as c1 = h1k. c1 is the first IMF component of x(t); (5) separate c1 from the rest of the data by r1 ðtÞ ¼ xðtÞ c1 ðtÞ
ð2Þ
Since the residue, r1, still contains information of longer period components, it is treated as the new data and subjected to the same sifting process as described above, get the second IMF component of x(t) designated as c2, the above procedure can be repeated to get nth IMF component until the residue, rn becomes a monotonic function from which no more IMF can be extracted. Thus, we achieved a decomposition of the data x(t) into n-empirical modes, and a residue rn, where ci, i = 1 n, contain different component of the signal from high to low frequency bands respectively. Frequency components in each band are different to other bands. The residue rn can be either the mean trend of signal x(t).
3. Simulation results and analysis The signal used in this paper is from the Duffing equation. The Duffing oscillator is one of the most common types of nonlinear oscillators and has many typical nonlinear phenomena [12,13], and can be written as follows: €x þ c_x x20 x þ dx3 ¼ P cosðxtÞ
ð3Þ
where, x0 is natural frequency of the oscillator, c is damping, d is nonlinear parameter, P and x are amplitude and frequency of external exciting force respectively. Chaotic or periodic responses of the Duffing oscillator in the planes defined by the frequency–amplitude of the external periodic excitation have been traced, and Fig. 1 displays the relevant
932
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Fig. 1. Black regions depict the domains where chaotic behaviors of the Duffing oscillator are possible (c = 0.05, = 0.2, d = 1, x(0) = 1.0 and x_ ð0Þ ¼ 0:0) and Grey and white colors corresponds to the periodic motion of the oscillator.
domains. A harmonic signal yðtÞ ¼ A sinðxh tÞ is added to the chaotic signal. In this paper, the sampled displacement signals x(k) is obtained using the classical fourth order Runge–Kutta algorithm and the initial conditions are x(0) = 1.0 and x_ ð0Þ ¼ 0:0. So, the contaminated signal z(k) is given by zðkÞ ¼ yðkÞ þ xðkÞ
ð4Þ
Fig. 2. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
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The contaminated signal z(k) can be decomposed to a finite of intrinsic mode functions (IMFs), and one of the IMFs is a recovered harmonic signal, whose frequency and phase angle equal to that of the original harmonic signal y(k). To illustrate the decomposition procedure, we use three time series data collected from Eq. (4). The first original chaotic signal was generated by a Duffing system with following parameters: c = 0.05, x20 ¼ 0:2, d = 1, x = 1.0 and P = 27.5, and the largest Lyapunov exponent is 0.084. The original harmonic signal was generated with A = 1.0 and xh = 0.5. Sampling interval Dt = p/400, the number of receiving signal is N = 20,000. The original chaotic signal, the original harmonic signal and the contaminated signal are shown in Fig. 2(a)–(c) respectively. Comparing with original chaotic signal, original harmonic signal is rather small. The harmonic signal can not be observed from the waveform of the contaminated signal z(k). Using empirical mode decomposition, the contaminated signal z(k) can be decomposed to a series of intrinsic mode functions (IMFs), and c3, one of the IMFs, is the recovered harmonic signals. The comparison of original harmonic signal and recovered harmonic signal is shown in Fig. 2(d), where the dotted line is original harmonic signal y(k). Comparing with original harmonic signal y(k), the amplitude of recovered harmonic signal c3 decreases, but the frequency and phase angle do not change. Fig. 2(d) also shows that the edge effects can be occur at both the beginning and end of the recovered harmonic signal. If the edges of the contaminated signal are lifted untreated, the cubic spline fittings will introduce large perturbations into the data which can propagate and eventually corrupt the signal [10]. In this paper, a number of local maxima (minima) at the beginning and end of the contaminated signal are introduced to eliminate the edge effects, and the satisfactory recovered signal is obtained. The second original chaotic signal was generated by same Duffing system with following parameters: c = 0.05, x20 ¼ 0:2, d = 1, x = 1.1 and P = 10.0, and the largest Lyapunov exponent is 0.121. The original harmonic signal was generated by y(t) = A sin(xht) with A = 1.0 and xh = 0.3. Sampling interval Dt = p/400, the number of receiving signal is N = 20,000. The original chaotic signal x(k), the rather small original harmonic signal y(k) and the contaminated signal z(k) are shown in Fig. 3(a)–(c) respectively. The harmonic signal can not be observed from the waveform of the contaminated signal z(k). the contaminated signal z(k) can be decomposed to a series of intrinsic mode functions (IMFs) by means of empirical mode decomposition, and c3, one of the IMFs, is the recovered harmonic signals. The comparison of original harmonic signal and recovered harmonic signal is shown in Fig. 3(d), where the dotted line
Fig. 3. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
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is original harmonic signal y(k). Comparing with original harmonic signal y(k), the only change of recovered signal c3 is amplitude. According to empirical mode decomposition, all extrema of signal should be identified, and all the local maxima (minima) are connected by a cubic spline line as the upper (low) envelope. The shorter sampling interval, the more local extrema. In following numerical simulation, the Duffing oscillator has parameters of c = 0.05, x20 ¼ 0:2, d = 1, x = 1.0 and P = 27.5. The harmonic signal is generated with parameter A = 1.0 and xh = 0.3. Sampling interval Dt = p/200, the number of receiving signal is N = 20,000. The numerical results of original chaotic signal x(k), the small original harmonic signal y(k) and the contaminated signal z(k) are shown in Fig. 4(a)–(c) respectively, and the harmonic signal can not be observed directly from the waveform of the contaminated signal z(k). Fig. 4(d) shows the trajectories of the third intrinsic mode function c3 (the solid line) and the original harmonic signal y (the dotted line). In this simulation, the numerical results are rather satisfactory except the beginning and the end of the recovered signal c3.
Fig. 4. The numerical simulation of a chaotic signal contaminating a harmonic signal. (a) Original chaotic signal. (b) Original harmonic signal. (c) The contaminated signal. (d) Original harmonic signal y (dotted line) and recovered harmonic signal c3 (solid line).
Fig. 5. The spectrum diagrams of intrinsic mode functions c1 (solid black line), c2 (dotted line) and c3 (solid grey line) decomposed from (a) the first contaminated signal (b) the third contaminated signal.
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The comparison between the original harmonic signal y and the recovered harmonic signal c3 in Fig. 2(d) shows that the amplitude of first c3 is smaller, the Fig. 4(d) shows the recovered harmonic signal is really very good. The spectrum diagrams of intrinsic mode functions c1, c2 and c3 decomposed from the first contaminated signal and the third contaminated signal are shown in Fig. 5. The second IMF c2, as it is displayed in Fig. 5(a), centered at 0.16 Hz, and covered from 0.07 to 0.3 Hz. The frequency of the original harmonic signal is about 0.08 Hz, therefore, the amplitude of the recovered signal c3 is smaller than that of the original harmonic signal. In Fig. 5(b), the frequency of the recovered signal c3 is about 0.048 Hz, the c1 and c2 do not contain component at this frequency, therefore, the recovered harmonic signal is very good.
4. Conclusion In this paper, a method for harmonic signal extraction from chaotic interference is proposed based on the empirical mode decomposition. The harmonic signal is contaminated with a chaotic signal generated by a Duffing oscillator. The simulation results for the chaotic interference signals were analyzed by applying the empirical mode decomposition to obtain the intrinsic mode functions (IMFs), among which one IMF is the recovered harmonic signal. This approach is a simple signal processing tool to use and provides reasonably results.
Acknowledgements The authors thank Professor He Jihuan for valuable discussion. This work was supported by the National Natural Science Foundations of China (No. 50335030, No. 10325209, No. 10502032 and No. 50375094).
References [1] Moez F, Bruno R, Guillaume G, Maxime C. Secure digital communication using discrete-time chaos synchronization. Chaos, Solitons & Fractals 2003;18(4):881–90. [2] Lei M, Wang ZZ, Feng ZJ. Detecting nonlinearity of action surface EMG signal. Phys Lett A 2001;290(5–6):297–303. [3] Li ZG, Xu DL. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22(2):477–81. [4] Bai EW, Lonngren KE, Ahmet U. Secure communication via multiple parameter modulation in a delayed chaotic system. Chaos, Solitons & Fractals 2005;23(3):1071–6. [5] Leung H, Huang XP. Parameter estimation in chaotic noise. IEEE Trans Sig Proc 1996;44:2456–63. [6] Haykin S, Li XB. Detection of signals in chaos. Proc IEEE 1995;83:94–122. [7] Short KM. Signal extraction from chaotic communications. Int J Bifurcat Chaos 1997;7:1579–97. [8] Wang FP, Guo JB, Wang ZJ, Xiao DC, Li MT. Harmonic signal extraction from strong chaotic interference. Acta Phys Sin 2001;50:1019–23 [in Chinese]. [9] Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series. Proc R Soc Lond A 1998;54(4):903–95. [10] Lai YC, Ye N. Recent developments in chaotic time series analysis. Int J Bifurcat Chaos 2003;13:1383–422. [11] Balocchi R, Menicucci D, et al. Deriving the respiratory sinus arrhythmia from the heartbeat time series using empirical mode decomposition. Chaos, Solitons & Fractals 2004;20(1):171–7. [12] He JH. Modified Lindsted-Poincare methods for some strongly nonlinear oscillations. Part III: Double series expansion. Int J Nonlinear Sci Numer Simul 2001;2(4):317–20. [13] Tan CA, Kang BS. Chaotic motions of a Duffing oscillator subjected to combined parametric and quasiperiodic excitation. Int J Nonlinear Sci Numer Simul 2001;2(4):353–64.