The Method Of Tuning Parameters To Realize Stochastic Resonance And Its Application To Signal Processing

In this thesis, we investigate the phenomenon of parameter-induced stochastic resonance (PSR) from the points of its mechanism in mechanics and its application to signal processing. The major contents are summarized as follows: a) Study the PSR theories and methods with different noise cases, b) Work out an applicable calculation method of optimizing parameters, c) Analyze the influence of the shape and density of the input signal spectrum on signal processing with PSR. d) Finally, we find that, when the height of the spectrum is used to measure the system output, the method of PSR in processing multi-frequency analog signal is more excellent than the method of a general linear filter. These works are important and significative for the development of PSR theory and its application to nonlinear signal processing.An important question of PSR is how to tune system parameters to realize SR with fixed noise intensity. In fact, it is a parameter optimization problem with signal-to-noise ratio (SNR) gain as an objective function under the condition λ1 - const (i.e., the system response speed is prescript). The system response speed is a very important quantity in the method of tuning system parameters and its calculation precision can affect the performance of the optimized system. In Chapter 3, we present a Hermite interpolating method to evaluate its values. Compared with the previous polynomial interpolating method, this method can avoid ill-conditioned matrices in the process of the calculation and obtain more accurate values. To solve this optimization problem, we put forward two techniques: 1) Compensate the deviation from the prescript system response speed in each step; 2) Take the curve length increment on the parameter plane as the step size in searching the maxima] SNR gain. By above methods, we get an effective and fast parameter optimized arithmetic. The arithmetic can be used to design the optimal nonlinear systems. It also builds the foundation for its application to adaptive signal processing.For an analog signal spoiled by a Lorentzian noise, it is approximate to a constant in a short time slice if the system response speed is high enough, and the signal varying is slow. In this sense, the ac-signal can be viewed as the connections of dc signals with different levels. We put forward the method of weighted mean of magnitudes with system response speeds, by which an appropriate magnitude for the input multi-frequency signal can be obtained. This magnitude is needed in the process of optimizing system parameters. The mean steady SNR gain is defined, which is combined with the system response speed to measure the whole gain of multi-frequency signal. And deduce the system response speed in the presence of colored noise, which is an important part of the PSR method. Furthermore, by an example, we investigate that this method (with colored noise) can be applied to sampling signal processing.In terms of the nonlinear characteristic of a system, when treating a multi-frequency analog signal, the distortion of the output waveform is unable to avoid. However, for digital signals, the waveform of the output signal is not considered any more, but only the decision according to decision levels should be made, which is simpler. Therefore, taking the PSR system as a receiver of digital signals is a subject that is worth studying. In Chapter 4, with the method of an expansion of eigenfunctions of the FPK, the dynamic solution (i.e., dynamic probability density) is obtained. By this dynamic probability density, we define the dynamic bit error rate (BER) to characterize the output of the nonlinear system with a baseband binary pulse amplitude modulated (PAM) signal. The dynamic BER is closer to the actual one than the quasi-steady-state BER, especially at low system response speeds. Because this BER is dynamic, it is not necessary to give the system response speed before system parameters are tuned. Hence, we can directly optimize the nonlinear system by minimizing the dynamic BER, which avoids the error when choosing the prescript system response speed (because this speed may not be optimal while the speed in the minimal dynamic BER must be optimal). Moreover, we extend the dynamic BER to the case of the input quaternary PAM signal, and optimize the signal levels by this BER in order to get the optimal effect for the signal transmission by nonlinear systems.The aperiodic SR (ASR) with multiplicative noise is discussed too, which completes the theory of PSR and makes it more applicable. Firstly, the model in the presence of multiplicative and additive colored noise with colored cross-correlation is introduced. Then the dynamic BER is defined by the nonstationary solution of the FPE. Because the output probability densities of the nonlinear system are unsymmetrical, the decision level of the binary signal is decided by minimizing the dynamic BER. Finally, we take this minimal dynamic BER as a quantity to optimize in the ASR problem via tuning the system parameters. Since the signal and noise are correlated, the theory may be edifying and instructive for water signal processing.The above theories are about a general potential function, i.e., U(x) = -jX +\x . In Chapter 7, we put forward some other potential functions, and realize PSR in corresponding nonlinear systems. By comparing the performances of nonlinear systems, we try to find more effective systems for signal processing.Finally, the effect of the input signal spectrum distribution on outputs of PSR systems is investigated. For a digital signal with a continuous spectrum, we find the optimal BER smaller when the signal spectrum is higher and narrower. And for an analog signal with a discrete spectrum, the SNR gain is lower and the performance of the output is worse, while the signal spectrum is thicker or lower. These results are instructive for choosing a proper signal as the input of the PSR system. Besides, for analog signals, we find that, as viewed from the analysis of thepower spectral density, the PSR system has the advantage of a linear filter. If this advantage is still kept when the frequency of the input signal is high, the application of the PSR method will be promising, and it is worth studying.