复梯度算子及其在自适应信号处理中的应用

A Complex Gradient Operator and Its Application in Adaptive Signal Processing

自适应信号处理广泛应用于通信、雷达、导航、声纳等领域。但是 ,在自适应信号处理中经常会遇到一类以Z为复自变量的标量函数g(z,z )求极值或条件极值的问题。按照复变函数理论 ,求g(z,z )等于min(或max)的最优解 ,会遇到极大的困难 ,因为偏导数z /z不存在。定理给出了这类标量函数的复梯度算子的正确表达式。这对自适应信号处理具有重要意义 ,它能使问题的解决变得很简单。

Adaptive signal processing (ASP) is extensivly applied in communication, radar, navigation sonar, and so on. But the optimizing problem of a real scalar function g(z,z *) where z and its conjugate z * are complex variables frequently arises in adaptive signal processing. To obtain the optimum solution of the problem g(z,z *) equals to min ( or max ) is very difficult, because the partical differential z */z do not exist acording to the theory of complex variable function. The expression of a complex gradient operator is provided by the theorems in this paper. This expression has impotant significance for ASP and simplifies the problem which discribe above. Finally two typical examples appling these theorems to ASP are given.