Most of the cost functions of adaptive filtering algorithms include the square error, which depends on the current error signal. When the additive noise is impulsive, we can expect that the square error will be very large. By contrast, the cross error, which is the correlation of the error signal and its delay, may be very small. Based on this fact, we propose a new cost function called the mean square cross error for adaptive filters, and provide the mean value and mean square performance analysis in detail. Furthermore, we present a two-stage method to estimate the closed-form solutions for the proposed method, and generalize the two-stage method to estimate the closed-form solution of the information theoretic learning methods, including least mean fourth, maximum correntropy criterion, generalized maximum correntropy criterion, and minimum kernel risk-sensitive loss. The simulations of the adaptive solutions and closed-form solution show the effectivity of the new method.

自适应滤波算法的大多数成本函数都包含平方误差，平方误差取决于当前误差信号。当加性噪声是脉冲性的时，我们可以预期平方误差会非常大。相反，作为误差信号及其延迟的相关性的交叉误差可能很小。基于这一事实，我们提出了一种新的代价函数，称为自适应滤波器的均方交叉误差，并详细提供了均值和均方性能分析。此外，我们提出了一种两阶段方法来估计所提出方法的封闭形式解，并推广了两阶段方法来估计信息理论学习方法的封闭形式解，包括最小均值第四，最大熵准则，广义最大熵准则，和最小的内核风险敏感损失。自适应解和闭式解的仿真表明了该方法的有效性。