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针对如何在干扰场的背景上区分出低缓异常,以及在位场的向下延拓一类计算中如何限制因误差的高频放大所导至的解的不稳定性等问题,本文探讨了在“最小二乘”意义下的最佳线性数字滤波器的设计原理,并将它转化为下述数学问题,即在L_2线性赋范函数空间中如何选取最佳滤波函数的问题。在空间域中直接解这个问题是十分复杂和困难的,我们发现在波数域中用变分法中的等周问题的解法直接选取最佳线性滤波器的传输函数(或波数响应),则在数学方法上既简单又严格。这样选取的最佳线性滤波器的传输函数L(f,k)其表达式也很简单,即L(f,k)=|S_i(f,k)|~2/{|S_i(f,k)|~2+λ|N_i(f,k)|~2}。式中,|S_i(f,k)|~2及|N_i(f,k)|~2分别代表滤波器输入端讯号和干扰的能谱(或功率谱),f、k分别代表x、y方向上的波数,λ为大于零的常数。 对上述两类问题以及相关的两种最佳线性滤波器而言,L(f,k)的表达式是相同的,而区别仅在于其参变量λ的选取条件不同而已。 有了最佳线性滤波器的传输函数L(f,k)的理论公式,就可以在最小二乘的意义下分析和评价国内外所发表的解决上述两类问题的各种线性滤波方法,并能指出在不同的讯号与干扰条件下,在理论上线性滤波可能达到的最佳效果,从而为设计二维线性数字滤波器时,提供一个理论上的准则。 对位?
Aiming at the problems of how to distinguish the low-lying anomaly in the background of the disturbing field and how to limit the instability of the solution caused by the high-frequency amplification of the error in the calculation of a downward extension of the field, The design principle of the best linear digital filter in the sense of “least squares” is transformed into the following mathematical problem: how to choose the best filter function in L 2 linear function space. It is very complicated and difficult to solve this problem directly in the space domain. We find that the transmission function (or wavenumber response) of the optimal linear filter is directly selected by the solution of the equipotential problems in the variational method in the wave number domain. Mathematical methods are simple and rigorous. The transfer function L (f, k) of the optimal linear filter thus selected is also very simple: L (f, k) = | S_i (f, k) | ~ 2 / {| S_i (f, k ) | ~ 2 + λ | N_i (f, k) | ~ 2}. In the formula, | S_i (f, k) | ~ 2 and | N_i (f, k) | ~ 2 represent the energy spectrum (or power spectrum) of the signal and interference at the input of the filter respectively. Wave number in the direction, λ is a constant greater than zero. The expressions of L (f, k) are the same for the above two types of problems and the two best linear filters involved, except that the selection condition of the parameter λ is different. With the theoretical formula of L (f, k), the transfer function L (f, k) of the best linear filter, we can analyze and evaluate the various linear filtering methods published in the world and abroad to solve the above two kinds of problems in the least square It can be pointed out that under different signal and interference conditions, theoretically the best effect that linear filtering can achieve, so as to provide a theoretical criterion for the design of two-dimensional linear digital filter. Right?