Research-based Fisher Information Gain Problem Of Weak Signal Processing

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.