通信工程文献翻译——在现代通信，控制和信号处理领域，信号几乎总是嵌入噪声，信号也是随机的。 那么如何估计实际信号是非常重要的。 诸如维纳滤波器和卡尔曼滤波器之类的经典随机滤波器不能估计来自非平稳噪声的实际信号。 传统的自适应滤波算法（如LMS，RLS等）的性能也非常有限。 我们必须找到可以更准确地估计实际信号的其他自适应随机滤波算法。 我们已经知道卡尔曼滤波器已被证明是最好的线性滤波器，我们应该引入非线性滤波算法。 本文提出了一种新的非线性滤波算法，引入了薛定谔方程（SWE）。

众所周知，量子力学是微观世界中最好的物理理论，它引入了一些基本假设，即所有物质的不连续能量和薛定谔波动方程。 这些假设根本不能被数学证明。 在量子力学中，微物体的演化可以通过与宏观低速世界中的牛顿第二定律相同的薛定谔波动方程来描述。 我们也知道电子信号是大量电子传递的结果，这是一个微观现象，所以它们的随机演化可以用薛定谔方程来表示。如果我们可以正确地调制薛定谔波方程的势场，我们可以准确地描述电子的演化，然后我们可以精确地描述信号，这是循环量子神经网络（RQNN）的原理。 现在我们的问题是如何适当地调整潜在的领域。

Bucy表示，随机过滤问题的每一个解决方案涉及在观察系统的状态空间上计算时变概率密度函数（PDF），在本文中为x。 在，Dawes给出了RQNN的原始理论架构，这是一个参数雪崩随机过滤器。 在Behera等人 通过引入最大似然估计（MLE）来改进Dawes0的架构。 在中，实际使用。 本文提出了一种新的Behera0神经网络，可以确保系统始终稳定于任何参数。 本文采用循环神经网络对其进行调制，并通过无监督学习算法更新了神经网络的权重，该算法是经典Hebbian学习算法的一个变体。

本文的其余部分分为6个部分。 第1节描述了薛定谔波动方程的物理意义和RQNN的概念框架。第2节描述了RQNN的原理。 第3节描述了整个RQNN系统的数值实现。 第4节介绍如何选择参数。 第5节描述了仿真结果和讨论。第6节总结了本文。

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In modern communication, control and signal processing fields, the signals are almost always embedded with noises and the signal is also stochastic. So how to estimate the actual signal is very important. The classical stochastic filter such as the Wiener filter and Kalman filter cannot estimate the actual signal from the non-stationary noises . The performance of the traditional adaptive filtering algorithm, such as the LMS, RLS and etc., is also very limited. We must find other adaptive stochastic filtering algorithm that can estimate the actual signal much more accurate. As we already know that the Kalman filter has been proved the best linear filter, we should introduce nonlinear filtering algorithm. This paper provides a new nonlinear filtering algorithm by introducing the Schr¨odinger wave equation (SWE).

As we all know that the quantum mechanics is the best physical theory in the micro world for its introducing some basic assumptions, i.e., the discontinuous energy of everything and the Schr¨odinger wave equation. These assumptions could not be proved just by the math until now. In quantum mechanics, the evolution of a micro object can be described by the Schr¨odinger wave equation which is the same as the Newton0 s second law in the macro low speed world. We also know that electronic signal is the results of a large number of electron transfers, which is a micro phenomenon, so their stochastic evolution could be described by the Schr¨odinger wave equation. If we can modulate the potential field of the Schr¨odinger wave equation properly, we can describe the evolution of the electronic accurately, then we can describe the signal accurately, which is the principle of the recurrent quantum neural network (RQNN). Now our problem is how to modulate the potential field properly.

Bucy said that every solution to a stochastic filtering problem involves the computation of a time varying probability density function (PDF) on the state space of the observed system, which in this paper is the x. In , Dawes gave the original theoretical architecture of the RQNN which was a parametric avalanche stochastic filter. In, Behera et al. improved Dawes0 s architecture by introducing the maximum likelihood estimate (MLE) to it. In [10?13], it was used in practice. This......