The bistability of higher-order differences of discrete periodic signals
© Shahverdian et al.; licensee Springer. 2014
Received: 27 November 2013
Accepted: 23 January 2014
Published: 7 February 2014
A theorem on higher-order absolute differences taken from successive terms of bounded sequences is proved. This theorem establishes the property of bistability of such difference series and suggests a method for converting periodic discrete-time signals into the binary digital form based only on the computation of absolute differences.
MSC:37E05, 37M10, 94A12.
This paper advances an approach in dynamical systems [1–5] which is based on considering higher-order differences taken from the orbits of a given system. Such an approach is motivated by the observation that some natural systems process the information contained in the signals’ difference structure. For example, it is claimed in brain theory that the visual cortex ‘responds to contrast rather than the uniform luminosity’ and the higher differences up to fourth order in these problems are considered .
The difference method reveals some new aspects in analyzing arbitrary time series and discrete dynamical systems. For instance, the difference characteristic γ introduced in [2, 3] demonstrates a strong correlation with the Lyapunov exponent  used in deterministic chaos (the definition of γ can be found in [2, 3]; see [8, 9] for its modification and discussions). Another possible generalization could relate to fractional analysis: it would be interesting to define the fractional analog of γ and apply it to chaos discrimination in fractional dynamical systems (such systems are studied, e.g., in [10–12]).
In the present paper another new aspect of the difference analysis, the bistability of higher-order differences taken from a bounded time series (discrete-time signals) is proved. Some possible applications to digital communication and signal processing in Section 2 are considered.
The content of this paper is the following. In Section 2 our main Theorem 1 is formulated. This theorem establishes the property of bistability for higher-order absolute differences taken from discrete-time signals. In the sense of applications it asserts that some systems operating on the basis of their inputs’ higher-order differences should rather be classified over digital ones. The theorem suggests a method for converting some discrete (and sampled analog) signals into a binary digital form based only on computation of absolute differences; if applicable (e.g., as for the case of arbitrary periodic signals), this method is more effective than the traditional ones used in communication theory. The proof of Theorem 1, based on a version of notion of peak sets originated in Banach algebras and approximation theory, is presented in Section 3.
2 Main results
In the difference analysis a given time series (or an orbit of a dynamical system) X is decomposed into two constituents-the sign and magnitude components. The sign component S reflects the alternation in monotony (increase/decrease) of higher-order absolute differences taken from successive terms of X and does not depend on their exact values. The magnitude (or height) component H consists of these absolute differences and does not depend on the sign distribution.
The matrices and are called S- and H-components of X, the sequences and are called m th S- and H-components of X and denoted and . In what follows instead of we use the notation (omitted brackets).
The next proposition (see also ) states that can be written as some linear forms of terms of X. This is a consequence of Eq. (2) and the proof, which can be conducted by the induction method, is straightforward (omitted). The ℕ, ℤ, and ℝ denote the collections of natural, integer, and real numbers, respectively; denotes the binomial coefficients.
which is the so-called δ-transform (of the sequence X) studied in the Hausdorff transforms of divergent series . This particular sign distribution appears also in definition of completely monotonic functions (see, e.g., the Bernstein theorem in ).
A finite subsequence of is called a segment of X of length k, ; for two functions of natural argument and , differed from 0 we denote if as . For a number we denote and its entire and fractional parts, respectively.
Definition 1 Let the numbers and be given. A sequence is called μ-binary if for every ; X is called -binary if there exists μ-binary such that . A collection of sequences is called μ-binary (-binary) if is μ-binary (-binary) for all large enough m; is called asymptotically μ-binary if for arbitrary it is -binary.
The -binary sequences are μ-binary sequences, -binary sequences are the sequences upper bounded by ε. For the case of finite sequences X of length n the above-given definitions are analogous, replacing in Eq. (7) and in Definition 1 the symbol ∞ by and n, respectively.
The proof of next proposition is straightforward (and is omitted).
Proposition 2 A collection of sequences is asymptotically μ-binary if and only if there exists a μ-binary collection such that as .
In these examples, E coincides with ℕ; Eq. (15) holds, e.g., for T-periodic X.
The next theorem establishes the bistability of higher-order absolute differences taken from discrete signals. In addition, it asserts (descriptively) that the denser the peak set is, the denser is the asymptotically binary .
For every there exists a finite segment such that as and the collection is asymptotically μ-binary.
If the peak sets (10) satisfy Eq. (12) then the collection , where E is defined by Eq. (13), is asymptotically μ-binary; more precisely, for every the is -binary with where φ is defined by Eq. (14).
Let X be periodic with a period . Then the collection is either μ-binary or asymptotically μ-binary. More precisely, if Eq. (9a) holds, is μ-binary, and if Eq. (9b) holds, is asymptotically μ-binary: for every the is -binary with .
The condition Eq. (12) on peak sets in this theorem is imposed on the difference series , but not immediately on X, which leads to certain difficulties to decide whether for a given X its higher order difference series possess the asymptotic bistability. Theorem 1(c) asserts that this property always holds for arbitrary periodic X.
In the next corollary both options from Theorem 1(c) are presented (below, the real numbers are called H-independent if for arbitrary for which , the relation implies that all the are zero):
Corollary 1 Let be periodic with a period , , and let μ be defined by Eq. (8). Then if are rational numbers then the collection is μ-binary, and if are H-independent then is asymptotically μ-binary.
Let us discuss the reason why a bistability, considered in Theorem 1, is emerged. Despite the X (the line in the matrix H) can be arbitrary, Eq. (2) sets a strong interrelation between the entries of every triplet of the matrix H. Namely such intrinsic local restrictions are usually the reason of some special features (in our case this is the bistability) of a given object (in our case this is the matrix H); e.g., the maximum principle for harmonic functions (both classical and discrete  versions) is due to their local mean value property. For proving Theorem 1 we exploit the following situation: if the absolute difference () of two positive quantities ( and ) is close to their upper bound, then one of them should also be close to this bound while another one should be close to zero. Similar situations arise also in Banach algebras and approximation theory, see, e.g., the Bishop-de Leeuw theorem (, Ch. 8), Bishop’s “”-criterion (, Ch. II, Comments) and Mergelyan’s earlier work  on rational approximations where a binary-valued function is of the main interest.
Before discussing on applications of Theorem 1 we note that a given signal can be completely restored by its first entries and components S and H. Namely, the next proposition (see also ; the proof is by the induction method and is omitted) provides us with analytical expressions for the computation of the original X by given , and the first terms of X (or, by first terms of denoted now as , ).
Let us outline some possible applications of the above-presented theory to signal processing. Theorem 1 suggests a method for converting the discrete signals into the binary ones based only on the computation of differences. For processing the signals for which converges to μ fast enough, this method can be far more effective than the ones which deal with replacing by their binary codes. Namely, we claim that, when applicable (e.g., if X is periodic), the difference converting method can reduce the data to be transmitted and can increase significantly the transmission speed. One can suggest the following scheme for processing (not only the transmission) the discrete-time signals by the digital systems: by Theorem 1 (and using Proposition 1) a given X is converted into a binary form, which is then processed as a digital signal, and then the obtained (binary) signal is deconverted (provided that a number of S-components and some finite set of the resulting signal entries are given) by Proposition 3.
can be smaller than 1, i.e., the difference method indeed is more effective; e.g., if (), then and the above-mentioned ratio is indeed small if q and T are large.
3 Proof of Theorem 1
The proof of Theorem 1 is based on the following lemma, where the notion of peak sets defined by Eq. (10) is essentially used.
is -binary with .
and, hence, Lemma 1 for the case is proved.
with length is -binary with . Since and as , item (a) of the theorem is proved.
Since every i th is -binary with , their union is also -binary with the same ε. Since as , item (b) of the theorem is proved.
is -binary. This statement follows from Lemma 1 if one considers the sequence and chooses some such that (cf. Eq. (18)): indeed, then Lemma 1 yields the result that the finite sequence (26) is -binary with . Then due to the T-periodicity of we see that is also -binary. Item (c) of the theorem is proved. Theorem 1 is proved. □
Proof of Corollary 1 To prove the first point of Corollary 1, we note that since X is periodic and the are rational, by considering the common denominator for one can suppose that X is a (periodic) sequence of natural numbers; then it follows that Eq. (9a) for such X holds, and hence is μ-binary. The second point of Corollary 1 follows from Eq. (6). Indeed, due to the assumption on H-independence none of the can take the value 0, and hence by Theorem 1(c) cannot be μ-binary and then Eq. (9a) is impossible; then the alternative option Eq. (9b) mentioned in Theorem 1(c) asserts that is asymptotically μ-binary. Corollary 1 is proved. □
AYS would like to thank Prof. H Niederreiter for his interest to this work and discussions. The authors thank the reviewers of the manuscript for their comments.
- Shahverdian AY, Apkarian AV: On irregular behavior of neuron spike trains. Fractals 1999, 7: 93-103. 10.1142/S0218348X99000116View ArticleMATHGoogle Scholar
- Shahverdian AY: The finite-difference method for analyzing one-dimensional nonlinear systems. Fractals 2000, 8: 49-65.MathSciNetMATHGoogle Scholar
- Shahverdian AY, Apkarian AV: A difference characteristic for one-dimensional nonlinear systems. Commun. Nonlinear Sci. & Comput. Simul. 2007, 12: 233-242. 10.1016/j.cnsns.2005.02.004MathSciNetView ArticleMATHGoogle Scholar
- Shahverdian AY: Minimal Lie algebra, fine limits, and dynamical systems. Rep. Armenian Natl. Acad. Sci. 2012, 112: 160-169.MathSciNetGoogle Scholar
- Shahverdian AY, Kilicman A, Benosman RB: Higher difference structure of some discrete processes. Adv. Differ. Equ. 2012., 2012: Article ID 202Google Scholar
- Miller KD 3. In Models of Neural Networks. Edited by: Domany E, Hemmen JL, Schulten K. Springer, New York; 1996.Google Scholar
- Schuster HG: Deterministic Chaos. Physik-Verlag, Weinheim; 1984.MATHGoogle Scholar
- Dai L, Wang G: Implementation of periodicity ratio in analyzing nonlinear dynamic systems: a comparison with Lyapunov exponent. J. Comput. Nonlinear Dyn. 2008, 3: 1006-1015.View ArticleGoogle Scholar
- Dai L: Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments. World Scientific, Singapore; 2008.View ArticleMATHGoogle Scholar
- Wu GC, Baleanu D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 2014, 75: 283-287. 10.1007/s11071-013-1065-7MathSciNetView ArticleMATHGoogle Scholar
- Wu GC, Baleanu D, Zeng SD: Discrete chaos of fractional sine and standard maps. Phys. Lett. A 2014, 378: 484-487. 10.1016/j.physleta.2013.12.010MathSciNetView ArticleMATHGoogle Scholar
- Tarasov VE, Edelman M: Fractional dissipative standard map. Chaos 2010., 20: Article ID 023127Google Scholar
- Hardy GH: Divergent Series. Clarendon, Oxford; 1949.MATHGoogle Scholar
- Phelps RP: Lectures on Choquet Theorems. Van Nostrand, Princeton; 1966.MATHGoogle Scholar
- Dynkin EB, Yushkevich AA: Theorems and Problems on Markov Processes. Nauka, Moscow; 1967.MATHGoogle Scholar
- Gamelin TW: Uniform Algebras. Prentice Hall, Englewood Cliffs; 1969.MATHGoogle Scholar
- Mergelyan SN: Some Results in the Theory of Uniform and Best Approximation by Polynomials and Rational Functions. Amer. Math. Soc., Providence; 1960. Appendix to the Russian Translation of the Walsh, JL: Interpolation and Approximation by Rational Functions in the Complex DomainsGoogle Scholar
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