Indirect Time Series Analysis of Chaos and Its Applications to Designs of 1/f Noise Generators Using SC Circuits

一维离散混沌的时间序列分析有两种方法。其中之一是“时间平均技术”，我们评估某些统计的样本长时间轨迹与一些初始值;另一个是“整体平均技术”的基础上的绝对连续不变的测量的地图，这被称为“间接方法”。这种间接方法有望在理论上理解混沌方面发挥重要作用。事实上，Perron-Frobenius（PF）算子允许我们从理论上计算几个统计量的集合平均值。然而，这个算子，记为P_r，不能直接计算，因为它的无限维。本文在适当的函数空间上，基于算子PP_r的Galerkin逼近，给出了系统计算几种统计量的有效算法。此外，我们还提出了用于产生1/f噪声的开关电容（SC）电路，该电路用Procaccia-Schuster型映射来模拟动力系统。实值序列）进行了研究。然而，最近，我们给出了简单的方法来获得二进制序列从混沌轨迹。在扩频系统和密码系统中，要求二进制序列具有良好的相关特性。在这项研究中，我们从理论上评估的统计混沌二进制序列的整体平均技术的基础上PF算子。结果表明，这种混沌二进制序列具有良好的相关特性。此外，我们从理论上评估了混沌实值和二进制序列的高阶统计量。从而证明了由Chebyshev映射产生的混沌序列具有很好的统计特性。

There are two kinds of time series analysis for one-dimensional discrete chaos. One of them is the "time-average technique" in which we evaluate certain statistics of a sample long-time trajectory with some initial value ; the other one is the "ensemble-average techinque" based on an absolutely continuous invariant measure of the map which is referred to as the "indirect method". Such an indirect method is expected to play an important role in theoretically understanding chaos. In fact, the Perron-Frobenius (PF) operator permits us to theoretically calculate the ensemble average of several statistics. However, this operator, denoted by P_r, cannot be calculated directly because of its infinite dimensionality. In this research, we have given an efficient algorithm for systematically calculating several statistics, which is based on the Galerkin approximation to the operator PP_r, on a suitable function space. Furthermore, we have presented switched capacitor (SC) circuits for generation of 1/f noise which simulate dynamical systems with Procaccia-Schuster-type maps.Usually, statistics of a chaotic trajectory itself (i.e., real-valued sequence) have been investigated. Recently, however, we give simple methods to obtain binary sequences from chaotic trajectories. In spread spectrum systems and cryptosystems, binary sequences with good correlation properties are required. In this research, we have theoretically evaluated statistics of chaotic binary sequences by the ensemble-average technique based on the PF operator. It has been shown that such chaotic binary sequences have good correlation properties. Moreover, we have theoretically evaluated high-order statistics of chaotic real-valued and binary sequences. Thus we have showed that chaotic sequences generated by the Chebyshev maps have quite good statistical properties.