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

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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.