Recently there have been several attempts to construct a Markov information source based on chaotic dynamics of the PLM (piecewise-linear-monotonic) onto maps. Study, however, soon informs us that Kalman's 1956 embedding of a Markov chain is to be highly appreciated. In this paper Kalman's procedure for embedding a prescribed Markov chain into chaotic dynamics of the PLM onto map is revisited and improved by using the PLM onto map with the minimum number of subintervals.

It is shown that the previously reported time series analysis using a finite element approximation to the Perron-Frobenius operator can be used to investigate periodic chaos in one-dimensional discrete dynamical systems as well as mixing chaos.

A new statistical test has been recently presented for determining whether a binary sequence into which a real-valued sequence to be tested is transformed precisely mimics Bernoulli trials B (p, q) with probabilities of 0 and of 1, p and q, or not. This paper gives a theoretical test based on such a stringent test and shows its usefulness. This method uses the ensemble average technique under the assumption that the pseudorandom-number generator is mixing with respect to an absolutely continuous measure. The existence of such a measure permits us to theoretically calculate the ensemble average of several statistics by using the Perron-Frobenius integral operator. Furthermore, this operator releases us from cumbersome and tedious procedures to calculate multivariate distributions, in connection with several statistical tests. Three kinds of tests, the runs test, poker test, and serial correlation test are presented. The Galerkin approximation to the operator on a suitable functional space is also introduced which provides a finite dimensional matrix (referred to as a Galerkin-approximated matrix of the Perron-Frobenius operator). The largest eigenvalue of the matrix, nearly equal to 1, corresponds to the existence of the measure. Each theoretical value of three tests for B (p, q) shows that the magnitude of the second largest eigenvalue plays an important role in determing randomness of the sequence generated by the generation.

We define a d-matched digraph and propose a recursive procedure for designing an optimal d-matched digraph without bidirectional edges. The digraph represents an optimal highly structured system which is a special class of self-diagnosable systems and identifies all of the faulty units independently and locally in O(|E|) time complexity. The procedure is straightforward and gives a system flexible in network connections. Hence the procedure is applicable to real systems such as the Internet or cooperative robotic systems which change their topology dynamically.

Binary sequences with good correlation properties are required for a variety of engineering applications. We previously proposed simple methods to generate binary sequences based on chaotic nonlinear maps. In this paper, statistical properties of chaotic binary sequences generated by Chebyshev maps are discussed. We explicitly evaluate the correlation functions by means of the ensemble–average technique based on the Perron–Frobenius (P–F) operator. As a consequence, we can confirm an important role of the P–F operator in evaluating statistics of chaos by means of the ensemble-average technique.

The quadratic Liénard equation, that is, the Liénard differential equation in which the coefficients of and x are polynomials of degree two in x is studied with respect to its periodic solutions of small amplitude. The equation can represent a hard or a soft oscillation depending on parameter values. The existence of periodic solutions is proved and a simple formula for amplitude is given.

A simple method is given for obtaining new families of pseudonoise (PN) sequences based on chaotic non-linear maps. Such families are worse than the Gold and the Kasami families in terms of maximum correlation values. Nevertheless, such a method has several advantages: the generation is easy, and various families with an arbitrary family size and sequence period can be obtained primarily because non-linear maps have several parameters to be secret keys for communications security. Hence these sequences are good candidates of spreading sequences for CDMA.

We consider the problem of constructing nonlinear dynamical systems that realize an arbitrary prescribed tree sources. We give a construction of dynamical systems by using piecewise-linear maps. Furthermore, we examine the obtained dynamical system to show that the structure of the memory of tree sources is characterized with some geometrical property of the constructed dynamical systems. Using a similar method, we also construct a dynamical system generating an arbitrary prescribed reverse tree source and show that the obtained dynamical system has some interesting geometrical property explicitly reflecting the tree structure of the memory of the reverse tree source.

Code acquisition performance in the Direct-Sequence Code-Division Multiple-Access (DS/CDMA) communication system is strongly related to the quality of the communication systems. The performance is assessed by (i) code acquisition time; (ii) precision; and (iii) complexity for implementation. This paper applies the method of maximum likelihood (ML) to estimation of propagation delay in DS/CDMA communications, and proposes a low-complexity method for code acquisition. First, a DS/CDMA system model and properties of outputs with a passive matched-filter receiver are reviewed, and a statistical problem in code acquisition is mentioned. Second, an error-controllable code acquisition method based on the maximum likelihood is discussed. Third, a low-complexity ML code acquisition method is proposed. It is shown that the code acquisition time with the low-complexity method is about 1.5 times longer than that with the original ML method, e.g. 13 data periods under 4.96 dB.

In this paper we study the large deviation property for chaotic binary sequences generated by one-dimensional maps displaying chaos and thresholds functions. We deal with the case when nonlinear maps are the r-adic maps. The large deviation theory for dynamical systems is useful for investigating this problem.

In traditional system-level fault diagnosis in the presence of intermittent faults, intermittent faults can escape detection by fault free units. This paper focuses on a system under hybrid fault situations where all the intermittent faults either pass or fail all the tests of the fault free units which test them. A new diagnosable system is introduced where no syndrome from a hybrid fault situation is identical to one from a special class of hybrid fault situations whenever for every pair of allowable fault sets, there is some difference between two sets of permanent faults as their subsets. Although the faults whose the intermittent nature is rather restrictive than the previous one are considered in this paper, the new diagnosability is shown to be equivalent to the previously known t/γ/τ-diagnosability. This indicates that there is a distinct tradeoff between the intermittent nature of the faults and the unique diagnosability of permanent faults in a hybrid fault situation.

We propose a post-filter (digital filter applied after the correlator) to reduce multiple-access interference (MAI) in the correlator output in asynchronous communications. Optimum filter coefficients are derived for Markov and i.i.d. codes. It is shown that post-filter is not needed for Markov case. Variance of MAI is reduced in i.i.d. codes and it becomes equal to that of Markov codes; thus, both will have the same bit error rate (BER) performance. This post-filter reduces level of MAI in the correlator output for Gold codes as well.

A detector based on calculation of a posteriori probability is proposed for code acquisition in singleuser direct sequence code division multiple access (DS/CDMA) systems. Available information is used for decision making, unlike conventional methods which only use a part of it. Although this increases the overhead in terms of additional memory and computational complexity, significant performance improvements are achieved. The frame work is extended to multiuser systems and again mean acquisition time/correct acquisition probability performance is superior to the conventional systems although computational complexity is high. An approximate multiuser method with significantly less complexity is also derived.

The one-dimensional map represented by the Bernoulli shift is a candidate of true random number generators. However, no computer with finite-word-length can precisely realize the Bernoulli shift. The M-sequence is shown to be one of finite-word-length approximations to the Bernoulli shift in the sense that the many-to-many map generated by M-sequence is the same as the finite-word-length realization of the iterated Bernoulli shift.

There are several attempts to generate chaotic binary sequences by using one-dimensional maps. From the standpoint of engineering applications, it is necessary to evaluate statistical properties of sample sequences of finite length. In this paper we attempt to evaluate the statistics of chaotic binary sequences of finite length. The large deviation theory for dynamical systems is useful for investigating this problem.

In statistical tests, the uniform distribution property is frequently requested. Two well-known procedures are discussed for transforming nonuniform good random sequences into uniform ones. Results of a recently proposed randomness tests are also shown to be invariant under these transformations.

A review is presented of the definitions of 'chaos' in the discrete system, the diagnosing methods of chaotic systems, and examples of engineering and/or biological chaos. First, enumerating physically intuitive pictures of one-dimensional chaos shows that there are many possible definitions of 'chaos' and that the 'observable chaos' is an important concept. Important roles of the Frobenius-Perron operator are discussed in theoretically studying statistical quantities of a completely chaotic orbit. In order to measure chaos, several quantities of a strange attractor are listed. Some of chaotic maps are shown to be applicable to a pseudorandom number generator. To examine biological chaos, macroscopic analyses and microscopic ones well be reviewed.

An efficient algorithm is given for systematically calculating several statistics such as the invariant measure, the Kolmogorov-Sinai entropy, the autocorrelation function and the power spectrum of chaos in one-dimensional discrete dynamical systems defined by a map. The method is based on the Galerkin approximation to the Frobenius-Perron integral operator. Several numerical examples demonstrate that the proposed method can give approximations with high accuracy to the statistics of various one-dimensional chaos with the absolutely continuous invariant measure under the map.