二维最佳线性数字滤波器的设计原理

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L2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）=|Si（f,k）|2/{|Si（f,k）|2+λ|Ni（f,k）|2}。式中,|Si（f,k）|2及|Ni（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

DESIGNING PRINCIPLES OF A TWO-DIMENSIONAL OPTIMUM LINEAR DIGITAL FILTER

In order to discern slowly varrying weak anomalies on a background of noise field and to deal with problem like limiting the instability of the solution due to high-frequency magnification of errors in the series of calculations such as the downward-continuation of potential field,this paper discusses the designing principles of an optimum linear digital filter in the least square sense. This may be transformed into the mathematical problem,namely how to select the optimum filtering function in the space of the L2 linear normed function. It would be very complicated and difficult,if the problem is to be solved directly in the space domain. We found that it may be mathematically much simplier and more rigorous,if we should directly select the trnasfer function or the wave-number response of the optimum linear filter in the wave-number domain by using the method of isoperimetric problem in calculus of variations. In this way,the expression of the transfer function of the selected optimum linear filter is quite simple,namely,L（f,k）=|Si（f,k）|2/{|Si（f,k）|2+λ|Ni（f,k）|2}where|Si（f,k）|2 and |Ni（f,k）|2 express the energy spectra （or the power spectra） of the filter input signal and noise respectively; f,k are wave numbers on the x and y directions.In regard to the above-mentioned two types of the problem and the two related optimum linear filters,the expressions of L（f,k） are the same. They differ only in the conditions of selecting the parameter （λ）.After setting up the theoretical expression of the transfer function L（f,k） of the optimum linear filter,we should be able in the least square sense to examine various linear filtering methods,so far published in foreign and domestic literatures in solving the above-mentioned two types of the problems and to show that the optimum results of the linear filtering can be achieved theoretically for different signal and noise conditions. Thus,it provides theorectical criterion for designing two-dimensional linear digital filters.For the observed results of the harmonic functions of potential fields,the above theory can be applied easily to the designing of optimum linear digital filters,but only in the approximate manner.