Weak Sinusoid Signal Recovery Against Strong White Noise Based On Chaos Theory

1. IntroductionFor a long period, scholars have made great efforts in theoretical and experimental analysis in the field of weak signal detection (WSD). Hereinto, the spectrum analysis and wavelet transforms are both in common use. However, when the noise is too strong and the signal is too weak, both two methods will not work as they are supposed to. Recently, as the chaos theory being evolved, there has been some increasing interest in the approach of WSD based on chaos theory. Donald L. Birx has established some simple chaotic models to detect weak signals, but only put forward some experimental results. Prof. Qu Liangsheng has designed a chaotic oscillator using difference equations, and detected a weak sinusoid signal distinctly. Wang Guanyu has built another chaotic oscillator using Duffing equation, and impressively detected a weak sine wave under a SNR of -26dB. This is the beginning of WSD using chaotic systems. And now, the method of detecting a sine wave from the strong noise using chaotic systems has been molded by and large.There are still some clarifications about this method: 1) A sinusoid is considered as the simplest periodic signal, and if it can be detected, a more complicated one will also do as well. 2) The Duffing equation is in common use for this method; mostly, it may be improved for some special purpose. 3) The fundamental thought of the method is based on the sensitivity of a chaotic system when input a small periodic signal, i.e., phase shake-up. 4) Usually, a small sine signal drives the chaotic system, preparing for the noisy input. 5) There is no uniform estimation of sine parameters in this method, all the parameters are estimated separately, hereinto, the amplitude estimation works well, but the frequency and phase estimations are still in probing.Noticeably, C. M. Glenn and S. Hayes brought forward a WSD method concerned about the chaos control. Different from the methods mentioned above, it chooses a slow periodic trajectory from the chaotic attractor, and control the chaos toward this periodic trajectory. If input a small sine signal, which is also called the signal to detect, the system that is motioning along the periodic trajectory, will suddenly get back to chaos. Now input a small periodic perturbation into the system, while the periodic perturbation equals the signal to detect, the system turn into the slow periodic trajectory back. The method is a very helpful attempt using the chaos control.However, since it selects a slow periodic trajectory, as it will motioning locally, the system maybe sensitive to noise, too, and may retrieve both the signal and noise from the small perturbation. Also, it suggests the amplitude of perturbation should be direct ratio to the signal, that is one should get some pre-knowledge of the signal. Moreover, the overall system is not adaptive. For these reasons, the method is still limited in theoretical discussions.Consider those methods mentioned above, this paper will integrate the WSD via chaos and the chaos control. Establish a new adaptive control of chaotic Duffing system, which is very appropriate with the WSD (calling the system APWSD for short); the APWSD is based on the speed gradient method (SG method for short). The advantages of SG method are: fast convergence velocity, excellent system stability and easy gain parameters setup.2. APWSD DesignTo design an APWSD, two essential aspects should be considered below: 1)It should be sensitive to signal but not to noise, that means, a weak signal in strong noise should be detected at least. 2) The time-domain waveform of the signal should be recovered via the adaptive control mechanism.For the 1st aspect, the WVD by chaotic Duffing system would be proper. And for the 2nd one, one should establish a controlled Duffing system which has the same initial conditions with the Duffing system for detection, and via the adaptive control mechanism, the controlled Duffing system would approach the Duffing system for detection by time. The result of this convergence motion is, the output of the feedback controller would be a convergence of the signal, and the signal will be retrieved.To avoid the APWSD recover the noise adding to the signal as well, the output of the feedback controller should be an estimation of the signal in some meaning, but not the signal-self directly. The result for this deed is: the APWSD would be a well-done nonlinear adaptive filter as well. In this meaning, the choice of SG method for the design of APWSD is considerable. This is because the SG method is with both fast convergence velocity and excellent system stability. Since the SG method is selected for the adaptive control algorithm design, the output of the feedback controller would be an estimation of the signal in the Lyapunov stability meaning. The simulations without noise illuminate the validity of APWSD.3. Sine Wave Recovery based on APWSDBased on the APWSD system, this paper puts forward novel method on the recovery of weak sine wave against strong noise.The APWSD system, which is a well-done nonlinear adaptive filter as well, would prop the following filtering algorithms well. As above said, the output of the feedback controller would be an estimation of the signal in the Lyapunov stability meaning. To obtain smoother waveform and higher output SNR, one should add another adaptive filter following the system output. This paper chooses a continuous-time adaptive notch filter (ANF).The advantages of the ANF are: different from those discrete ANF, it is a differential system, and can filter a sine wave in continuous time-domain; No need for any pre-knowledge of the sine signal, it would estimate the frequency of the signal accurately; easy parameters setup; low system complexity and good system stability.In this paper, a weak sine signal under a SNR of -20dB, WGN, is recovered. Hereinto, while the frequency of the signal is known, the waveform will be retrieved excellent, and the output SNR is 20.2dB; while the frequency is unknown, the system will estimate the frequency accurately and also output a good waveform with SNR of 16.2dB. The simulations illuminate the weak sine signal recovery system is very effective.