基于Fisher信息量的弱信号处理增益问题研究

【作者】 马玉梅；

【导师】 段法兵；

【作者基本信息】 青岛大学 ， 系统理论， 2014， 博士

【摘要】 随机共振是研究一些非线性系统中噪声的积极建设性作用的一类物理现象。本文在深入研究随机共振和循环平稳理论的基础上,在弱周期信号条件下,利用信噪比和费舍尔信息量进一步研究随机共振现象,并且找到了二者之间的关联。费舍尔信息量能够描述几个重要非线性处理过程中的性能；一个局部最优处理器能够获得最大输出输入信噪比,最大输出输入信噪比增益是由标准噪声分布的费舍尔信息量给定的,并且最大信噪比增益是静态非线性元素组成的阵列的信噪比增益的上限。在本论文中,又进一步的对静态和动态非线性系统的随机共振现象进行了对比。论文的主要研究成果如下：1.最初费舍尔信息量是作为参数估计的性能指标。我们将它扩展并且表明费舍尔信息量能够描述几个重要非线性处理过程中的性能。对于加性白噪声中的弱信号,费舍尔信息量能决定如下四个方面：(i)周期信号的最大输出信噪比；(ii)信号检测的最优渐近性能；(iii)信号传输的最优互相关系数；(iv)无偏估计值的最小均方差。通过费舍尔信息量不等式,这个统一的结论用于建立通过噪声改善随机共振是否可行的条件。2.通过噪声概率密度和噪声强度能精确地决定一个局部最优处理器,并且局部最优处理器的输出输入信噪比增益是由标准噪声分布的费舍尔信息量给定的。基于这个关联,我们发现对于局部最优处理器,能够获得比一任意大的信噪比增益。对于随机共振,考虑向已知信号中加入额外噪声时,我们证明了通过费舍尔信息量不等式,和新噪声完全匹配的更新的局部最优处理器,不能改进输出信噪比以超过无额外噪声时所对应的初始值。这个结果印证了一个以前只对高斯噪声存在的定理。此外,在参数不可调处理器的情况下,比如由噪声概率密度描述的局部最优处理器的结构不能完全适应噪声强度时,表明了可以恢复随机共振的一般条件,通过添加额外声提高输出信噪比的可能性来证明。3.研究了为传输在加性白噪声中的弱周期信号,由任意的静态非线性元素组成的非耦合并联阵列的输出输入信噪比增益。在小信号的限制条件下,推导出信噪比增益的一个渐近表达式。并且证明了对任意给定的非线性系统和噪声环境,信噪比增益是关于阵列大小的单调递增的函数。由局部最优非线性系统所对应的信噪比增益,是静态非线性元素组成的阵列的信噪比增益的上限。在局部最优非线性系统中,随机共振不能发生,也就是说,在阵列中加入内部噪声不能改善信噪比增益。然而,在一个由次优但易实现的阈值非线性系统组成的阵列中,我们证明了随机共振发生的可行性,也证明了对于各种内部噪声分布,信噪比增益大于一的可能性。4.利用输出信噪比作为测量方法,比较了静态和动态非线性系统的随机共振现象。对于给定的含噪弱周期信号,通过调谐内部噪声强度,静态和动态非线性并联阵列都能提高输出信噪比。静态非线性系统容易实现,而动态非线性系统有较多参数需要调整,存在不能利用内部噪声的有利作用的风险。并且外部噪声是非高斯类型时,可以观察到动态非线性系统是优于静态非线性系统,可以获得一个更好的输出信噪比,证明了加入额外白噪声以提高输出信噪比的可能性。更多还原

【Abstract】 Stochastic resonance (SR) is a nonlinear phenomenon where the transmission of a coherent signal by certain nonlinear systems can be improved by the addition of noise. On the basis of stochastic resonance and cyclostationary theory, in the small-signal limit we find the relationship between signal-to-noise (SNR) and fisher information and extend the stochatic resonance theoretical framework. We show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing a weak periodic signal in additive white noise, a locally optimal processor (LOP) achieves the maximal output SNR. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. The main results of the thesis are summarized as follows:1. The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (ⅰ) the maximum output signal-to-noise ratio for a periodic signal;(ⅱ) the optimum asymptotic efficacy for signal detection;(ⅲ) the best cross-correlation coefficient for signal transmission; and (ⅳ) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.2. For processing a weak periodic signal in additive white noise, a LOP achieves the maximal output SNR. In general, such a LOP is precisely determined by the noise probability density and also by the noise level. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. Based on this connection, we find that an arbitrarily large SNR gain, for a LOP, can be achieved ranging from the minimal value of unity upwards. For stochastic resonance, when considering adding extra noise to the original signal, we here demonstrate via the appropriate Fisher information inequality that the updated LOP fully matched to the new noise, is unable to improve the output SNR above its original value with no extra noise. This result generalizes a proof that existed previously only for Gaussian noise. Furthermore, in the situation of non-adjustable processors, for instance when the structure of the LOP as prescribed by the noise probability density is not fully adaptable to the noise level, we show general conditions where stochastic resonance can be recovered, manifested by the possibility of adding extra noise to enhance the output SNR.3. We study the output-input SNR gain of an uncoupled parallel array of static, yet arbitrary, nonlinear elements for transmitting a weak periodic signal in additive white noise. In the small-signal limit, an explicit expression for the SNR gain is derived. It serves to prove that the SNR gain is always a monotonically increasing function of the array size for any given nonlinearity and noisy environment. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. With locally optimal nonlinearity, it is demonstrated that stochastic resonance cannot occur, i.e. adding internal noise into the array never improves the SNR gain. However, in an array of suboptimal but easily implemented threshold nonlinearities, we show the feasibility of situations where stochastic resonance occurs, and also the possibility of the SNR gain exceeding unity for a wide range of input noise distributions.4. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. For a given noisy periodic signal, parallel arrays of both static and dynamical nonlinearities can enhance the output SNR by optimally tuning the internal noise level. The static nonlinearity is easily implementable, while the dynamical nonlinearity has more parameters to be tuned, at the risk of not exploiting the beneficial role of internal noise terms. More interestingly, as the input signal buried in external Laplacian noise, it is observed that the dynamical nonlinearity is superior to the static nonlinearity in obtaining a better output SNR.更多还原