The bistability of higher-order differences of discrete periodic signals

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The bistability of higher-order differences of discrete periodic signals

Ay Shahverdian, Rp Agarwal, Rb Benosman

To cite this version:

Ay Shahverdian, Rp Agarwal, Rb Benosman. The bistability of higher-order differences of discrete

periodic signals. Advances in Difference Equations, Hindawi Publishin Corporation / SpringerOpen,

2014, 2014 (1), pp.1-9. �10.1186/1687-1847-2014-60�. �hal-01158530�

R E S E A R C H Open Access

The bistability of higher-order differences of discrete periodic signals

AY Shahverdian^1*, RP Agarwal²and RB Benosman³

*Correspondence: svrdn@yerphi.am

1Institute for Informatics and Automation Problems, National Academy of Sciences of Armenia, Paruir Sevak St. 1, Yerevan, 0014, Armenia

Full list of author information is available at the end of the article

Abstract

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

Keywords: higher-order differences; discrete dynamical systems; discrete-time signals

1 Introduction

This paper advances an approach in dynamical systems [–] which is based on consid- ering 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 con- tained 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 characteristicγ introduced in [,

] demonstrates a strong correlation with the Lyapunov exponent [] used in determin- istic chaos (the definition ofγ can be found in [, ]; see [, ] for its modification and discussions). Another possible generalization could relate to fractional analysis: it would be interesting to define the fractional analog ofγ and apply it to chaos discrimination in fractional dynamical systems (such systems are studied,e.g., in [–]).

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 Sec- tion  are considered.

The content of this paper is the following. In Section  our main Theorem  is formu- lated. This theorem establishes the property of bistability for higher-order absolute dif- ferences 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 abso- lute differences; if applicable (e.g., as for the case of arbitrary periodic signals), this method

©2014 Shahverdian et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Shahverdian et al.Advances in Difference Equations2014, 2014:60 Page 2 of 9 http://www.advancesindifferenceequations.com/content/2014/1/60

is more effective than the traditional ones used in communication theory. The proof of Theorem , based on a version of notion of peak sets originated in Banach algebras and approximation theory, is presented in Section .

2 Main results

In the difference analysis a given time series (or an orbit of a dynamical system)Xis de- composed into two constituents-the sign and magnitude components. Thesign compo- nentS reflects the alternation in monotony (increase/decrease) of higher-order absolute differences taken from successive terms ofXand does not depend on their exact values.

Themagnitude(orheight)componentH consists of these absolute differences and does not depend on the sign distribution.

Let us proceed to formal definitions. Let

X= (f,f, . . . ,fn, . . .) ()

be an infinite bounded numerical sequence-this can be some time series, discrete-time signal, experimental data (long enough), or an orbit of iterates of some map. For natural mwe considermth-orderabsolute differences

H_n^()=fn, H_n^(m+)=H_n+^(m)–H_n^(m) (m≥,n≥) ()

and define themthabsolute difference sequenceasH_m= (H_n^(m))^∞_n=. Furthermore, we define

Sm=

δ^(m)_ ,δ^(m)_ , . . . ,δ_n^(m), . . .

, whereδ_n^(m)=

⎧⎨

⎩

+, H_n+^(m)≥Hn^(m),

–, H_n+^(m)<Hn^(m). () The matrices S ={δ_n^(m):m≥,n≥}and H ={Hn^(m):m≥,n≥}are called S- and H- componentsofX, the sequencesSmandHmare calledmth S- and H-componentsofXand denotedSm=Sm(X) andHm=Hm(X). In what follows instead ofHn^(m)we use the notation H_n^m(omitted brackets).

The next proposition (see also []) states thatH_n^mcan be written as some linear forms of terms ofX. This is a consequence of Eq. () and the proof, which can be conducted by the induction method, is straightforward (omitted). TheN,Z, andRdenote the collections of natural, integer, and real numbers, respectively; _m

i

(= _i!(m–i)!^m! ) denotes the binomial coefficients.

Proposition  Let X= (fn)^∞_n=and m≥be arbitrary.Then the mth-order absolute differ- ences H_n^mcan be represented as

H_n^m=k_m,n^()f_n+k^()_m,nf_n++· · ·+k_m,n^(m)f_n+m, () where the coefficients km,n⁽ⁱ⁾ ∈Zare constructed recurrently:for p≥

k_,n^()= –k_,n^()= –δ_n^(), k_p+,n⁽ⁱ⁾ =

⎧⎪

⎪⎨

⎪⎪

⎩

–δn^(p)k⁽ⁱ⁾p,n, i= , δ^(p)_n (k^(i–)_p,n+–kp,n⁽ⁱ⁾), ≤i≤p, δ^(p)n k^(i–)_p,n+, i=p+ .

()

One can see thatk⁽ⁱ⁾m,nin Eq. () depend only on S-componentsS, . . . ,SmofX. The rule () is a generalization of the additive scheme for constructing the Pascal triangle of bino- mial coefficients. It can be proved (we omit this) that form,n≥

m

i=

k_m,n⁽ⁱ⁾ =  and k_m,n⁽ⁱ⁾ ≤ m

i

. ()

The equality sign here is assumed whenδ^(m)_n = (–)^m for alln≥; for this case, () gains the form

H_n^m=

m

i=

(–)ⁱ m

i

fn+i

which is the so-called δ-transform (of the sequenceX) studied in the Hausdorff trans- forms of divergent series []. This particular sign distribution appears also in definition ofcompletely monotonic functions(see,e.g., the Bernstein theorem in []).

In this paper we are interested in the H-components of bounded sequences (discrete- time signals). To formulate our main theorem, Theorem , we present Definition  - namely in the context of this definition the (physics-related) term “bistability” in Section  is un- derstood. In what follows, for boundedX= (xi)^∞_i=we consider thesup-norm:

X=sup

|xi|: ≤i<∞

. ()

A finite subsequenceh= (xi+, . . . ,xi+k) ofX= (xi)^∞_i=is called a segment ofXof lengthk,

|h|=k; for two functions of natural argumenta(m) andb(m), differed from  we denote a(m)∼b(m) ifa(m)/b(m)→ asm→ ∞. For a numberx∈Rwe denote [x] and{x}its entire and fractional parts, respectively.

Definition  Let the numbers ≤μ<  and ≤ε<  be given. A sequenceX= (xi)^∞_i=

is calledμ-binaryifxi∈ {,μ}for everyi≥;Xis called (μ,ε)-binaryif there existsμ- binaryZ= (zi)^∞_i= such thatX–Z ≤ε. A collection of sequences (Xm)_m≥ is calledμ- binary ((μ,ε)-binary) ifXm isμ-binary ((μ,ε)-binary) for all large enoughm; (Xm)_m≥is calledasymptoticallyμ-binaryif for arbitraryε>  it is (μ,ε)-binary.

The (μ, )-binary sequences areμ-binary sequences, (,ε)-binary sequences are the se- quences upper bounded byε. For the case of finite sequencesXof lengthnthe above-given definitions are analogous, replacing in Eq. () and in Definition  the symbol∞byn+  andn, respectively.

The proof of next proposition is straightforward (and is omitted).

Proposition  A collection of sequences(Xm)_m≥is asymptoticallyμ-binary if and only if there exists aμ-binary collection(Z_m)_m≥such thatX_m–Z_m →as m→ ∞.

Theorem  is the main result of this paper. We define some notions involved in its for- mulation. ForX= (fn)^∞_n=andHm=Hm(X) we consider

μ_m=Hm and μ= lim

m→∞μ_m (μ≥). ()

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For periodicXthesupremumin Eq. () can be replaced bymaximum. We note that Eq.

() yields the result that for everyXwe haveμ_m+≤μ_m(m≥). Due to this monotony of μ_m, for arbitraryXone and only one of next two options (a) and (b) is possible:

(a) μ=μ_m for all large enoughm, (b) μ<μ_m form= , , . . . . () We define thepeak sets P_m,

P_m=

n≥ :H_n^m≥μ

() (it follows from Eq. () thatP_m=∅form≥), modifying the corresponding notion from theory of Banach algebras (e.g., []). LetPmbe numbered in increasing order of its entries,

Pm=

n^()_m,n^()_m, . . .

and denote dm=maxn^(i+)_m –n⁽ⁱ⁾_m:i≥

. ()

In item (b) of Theorem  the following restrictions onP_mare imposed:

≤d_m≤m, lim

m→∞(m–d_m) =∞. ()

Thedmcharacterize the density ofPmin natural seriesN. The first condition in () means thatPmis dense enough inN(every segment ofNof lengthmcontains a point ofPm); the second relation implies that the set

E={m–dm:m∈N}, E⊆N ()

is infinite. Ifn∈E, then there is somem≥ suchn=m–dm; we denote such anmas m=m_n; hence, one can consider a function of the natural argument,

ϕ:N→N, defined as ϕ(n) =mn–dmn. ()

For example, ifdm= [m/] (entire part of half ofm), then () is satisfied and one can assignϕ(n) = [n/]; another example is

d_m=T(= const.), and then ϕ(n) =n–T. ()

In these examples,Ecoincides withN; Eq. () holds,e.g., forT-periodicX.

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 setPm

is, the denser is the asymptotically binaryH_m.

Theorem  Let X= (fn)^∞_n=,Hm=Hm(X),andμ_mandμbe defined by Eq. ().Then the following statements,provided that a finite collection(whose cardinality is smaller than dm)of entries of Hmcan be excluded,are true:

(a)For every m∈Nthere exists a finite segment h_m⊂H_msuch that|h_m| ∼m as m→ ∞ and the collection(hm)^∞_m=is asymptoticallyμ-binary.

(b)If the peak sets()satisfy Eq. ()then the collection(Hn)_n∈E,where E is defined by Eq. (),is asymptoticallyμ-binary;more precisely,for every n∈E the H_nis(μ,ε_n)-binary withε_n=μ_ϕ(n)–μwhereϕis defined by Eq. ().

(c)Let X be periodic with a period T≥.Then the collection(Hm)^∞_m=is eitherμ-binary or asymptoticallyμ-binary.More precisely,if Eq. (a)holds, (Hm)^∞_m=isμ-binary,and if Eq.

(b)holds, (Hm)^∞_m=is asymptoticallyμ-binary:for every m>T the Hm is(μ,ε_m)-binary withε_m=μ_m–T–μ.

The condition Eq. () on peak sets in this theorem is imposed on the difference series Hm, but not immediately onX, which leads to certain difficulties to decide whether for a givenXits higher order difference series possess the asymptotic bistability. Theorem (c) asserts that this property always holds for arbitrary periodicX.

In the next corollary both options from Theorem (c) are presented (below, the real numbersf, . . . ,fTare called H-independentif for arbitraryλ, . . . ,λT∈Zfor whichλ+

· · ·+λ_T= , the relation_T

i=λ_ifi=  implies that all theλ_iare zero):

Corollary  Let X = (f_n)^∞_n= be periodic with a period T≥,H_m=H_m(X),and letμbe defined by Eq. ().Then if f, . . . ,fTare rational numbers then the collection(Hm)^∞_m=isμ- binary,and if f, . . . ,fTareH-independent then(Hm)^∞_m=is asymptoticallyμ-binary.

Let us discuss the reason why a bistability, considered in Theorem , is emerged. De- spite theX(the lineH_in the matrix H) can be arbitrary, Eq. () sets a strong interrelation between the entries of every triplet {H_n^m+,H_n^m,H_n+^m }of the matrix H. Namely such in- trinsic 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  we exploit the following situation: if the absolute difference (=H_n^m+) of two positive quantities (H_n^mandH_n+^m ) is close to their up- per 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 the- ory, see,e.g., the Bishop-de Leeuw theorem ([], Ch. ), Bishop’s “^_–^_”-criterion ([], Ch. II, Comments) and Mergelyan’s earlier work [] on rational approximations where a binary-valued functionm(ζ) is of the main interest.

Before discussing on applications of Theorem  we note that a given signalX= (f_n)^∞_n=

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 originalXby givenS, . . . ,Sm,Hm

and the first termsf, . . . ,fmofX(or, by first terms ofH_^mdenoted now asH_^k= [f, . . . ,fk],

≤k≤m).

Proposition  Let m≥be given,let H= (hn)^∞_n=be some infinite sequence,hn≥and Sk= (sk,n)^∞_n=, sk,n=±, ≤k≤m be some infinite binary sequences. Let X= (fn)^∞_n= be defined as follows:f, . . . ,fmare arbitrary and for every natural n≥

fn=f+

m

k=

[f, . . . ,fk]

n–k+

i=

Bn,i,k–+

n–m

i=

hiBn,i,m–, ()

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wherej

i= if j<i and Bn,i,p∈Zare constructed recurrently:for p≥ Bn,i,=s,i, Bn,i,p=sp,i

n–p

j=i+

Bn,j,p–.

Then for all≤k≤m the relations

S_k(X) =S_k and H_m(X) =H ()

hold.

Let us outline some possible applications of the above-presented theory to signal pro- cessing. Theorem  suggests a method for converting the discrete signals into the binary ones based only on the computation of differences. For processing the signalsX= (fn)^∞_n=

for whichμ_mconverges toμfast enough, this method can be far more effective than the ones which deal with replacingfnby their binary codes. Namely, we claim that, when ap- plicable (e.g., ifXis 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  (and using Proposition ) a givenXis 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 .

To illustrate this, we consider the periodic case. Let a signalX= (f_n)^∞_n=be periodic; by us- ing the binary symbols it should be transmitted (it suffices to transmit the periodf_, . . . ,f_T).

The traditional method, where each of thefiis replaced by a binary code of some lengthq, requiresqTbinary digits (bits) to be transmitted. By the difference method, this transmis- sion task is solved as follows. Letmbe a minimal number for which the sequenceHmcan be treated as a binary one (theε_mfrom Theorem  is small enough). According to Propo- sition , to transmitXone should transmitm(q+T+ ) bits ((m+ )qof them to transmit f, . . . ,fmandμby usual method,mTto transmitS, . . . ,Sm, andmbits to transmitHm). If m<T, the ratio

m(q+T+ ) qT

can be smaller than , i.e., the difference method indeed is more effective;e.g., iffi=i (≤i≤T), thenm=  and the above-mentioned ratio is indeed small ifqandTare large.

3 Proof of Theorem 1

The proof of Theorem  is based on the following lemma, where the notion of peak sets defined by Eq. () is essentially used.

Lemma  Let X= (f_n)^∞_n=,H_m=H_m(X),μ_mandμbe defined by Eq. (),and for some m,n≥

,the inequality

H_n^m≥μ ()

holds.Then for every≤k≤m and every≤i≤k one of the two relations()or(), μ≤H_n+i^m–k≤μ_m–k and ≤H_n+i+^m–k ≤μ_m–k–μ, () μ≤H_n+i+^m–k ≤μm–k and ≤H_n+i^m–k≤μm–k–μ, () holds;that means that for every≤k≤m the finite sequence

H_n^m–k,H_n+^m–k, . . . ,H_n+k^m–k ()

is(μ,ε)-binary withε=μ_m–k–μ.

Proof The proof is conducted by the induction method with respect to the variablek,

≤k≤m– . Let us first prove the lemma for the casek= . From definition ofμ_mand μ, we have

μ≤H_n^m≤μ_m, that is, by Eq. (), μ≤H_n^m––H_n+^m–≤μ_m. () Let us assume thatH_n^m–≥H_n+^m–(for the contrary caseH_n^m–≤H_n+^m–the proof is analo- gous); that is, the second relation in Eq. () can be written as

μ≤H_n^m––H_n+^m–≤μ_m. ()

From the left-hand side of () we obtainμ≤H_n^m–, and from the definition in Eq. () it follows thatH_n^m–≤μ_m–, that is,μ≤H_n^m–≤μ_m–. Furthermore, from Eq. () one obtains

H_n+^m–≤H_n^m––μ≤μ_m––μ

and, hence, Lemma  for the casek=  is proved.

To prove the lemma for arbitrary ≤k≤m, we apply the induction method: we assume that one of the relations () or () for some ≤k≤m–  holds, and we prove that it holds also whenkis substituted byk+ . Let us assume that Eq. () holds (for the case of Eq. () the proof is analogous). By Eq. ()

H_n+i^m–k=H_n+i^m–(k+)–H_n+i+^m–(k+)

and let us assume that the difference here is positive (if it is negative, the proof is the same),

H_n+i^m–k=H_n+i^m–(k+)–H_n+i+^m–(k+). ()

From () we haveH_n+i^m–(k+)≥μ. We also have H_n+i^m–(k+)≤μ_m–(k+)=μ+ (μm–(k+)–μ)

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and, hence, the first relation in Eq. (), wherekis substituted byk+ , is proved. Let us now consider the second relation in (). Eqs. () and () give

H_n+i+^m–(k+)=H_n+i^m–(k+)–H_n+i^m–k≤μ_m–(k+)–μ=ε_m–(k+)

and Eqs. () and (), wherekis substituted byk+ , are proved. Lemma  is proved.

Proof of Theorem (a) LetX= (f_n)^∞_n=be given,Hm=Hm(X), andμ_mandμbe defined by Eq. (). It follows from definition ofμthat for everym≥ there is somen=nmsuch that H_n^m≥μ(cf.Eq. ()). Then by Lemma  for every ≤k≤mthe sequence

H_n^m–k,H_n+^m–k, . . . ,H_n+k^m–k

is (μ,μ_m–k–μ)-binary. Letk(m) be such thatk(m) <m, (m–k(m))→ ∞, andk(m)∼m asm→ ∞. Then by Lemma  the finite sequence

hm=

H_n^m–k(m)_m ,H_n^m–k(m)_m+ , . . . ,H_n^m–k(m)_m+k(m)

() with length|h_m|=k(m)+ is (μ,ε)-binary withε=μ_m–k(m)–μ. Sinceε→ and|h_m| → ∞ asm→ ∞, item (a) of the theorem is proved.

(b) As in Eq. (), with everyn⁽ⁱ⁾m ∈Pm(see Eq. ()) we associate a sequenceh⁽ⁱ⁾m. Since dm<mand (m–dm)→ ∞, one can assign in Eq. ()k(m) =dmwhat means that

∞ i=

h⁽ⁱ⁾_m =Hm–dm.

Since everyithh⁽ⁱ⁾_m is (μ,ε)-binary withε=μ_m–dm–μ, their unionH_m–dm is also (μ,ε)- binary with the sameε. Sinceε→ asm→ ∞, item (b) of the theorem is proved.

(c) The proof can be deduced from the previous item (b) and the example from Eq. ().

We present a more detailed proof. If Eq. (a) holds, the proof of the theorem followsab absurdo, if one takes into account Eq. () and the periodicity of everyHm. Let us consider the case when Eq. (b) holds. Since everyHm isT-periodic, to prove thatHm is (μ,ε)- binary it suffices to prove that for somenthe finite sequence

H_n^m,H_n+^m , . . . ,H_n+T^m ()

is (μ,ε)-binary. This statement follows from Lemma  if one considers the sequenceHm+T

and chooses somen≥ such thatH_n^m+T≥μ(cf.Eq. ()): indeed, then Lemma  yields the result that the finite sequence () is (μ,ε)-binary withε=μm–T–μ. Then due to the T-periodicity ofH_mwe see thatH_mis also (μ,ε)-binary. Item (c) of the theorem is proved.

Theorem  is proved.

Proof of Corollary To prove the first point of Corollary , we note that sinceXis peri- odic and thefiare rational, by considering the common denominator forf, . . . ,fTone can suppose thatXis a (periodic) sequence of natural numbers; then it follows that Eq. (a)

for suchXholds, and hence (Hm)_m≥isμ-binary. The second point of Corollary  follows from Eq. (). Indeed, due to the assumption on H-independence none of theH_n^mcan take the value , and hence by Theorem (c) (H_m)_m≥cannot beμ-binary and then Eq. (a) is impossible; then the alternative option Eq. (b) mentioned in Theorem (c) asserts that (Hm)m≥is asymptoticallyμ-binary. Corollary  is proved.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AYS participated in the most part of results formulations, their proofs, and the manuscript preparation. RPA participated in the preparation of proofs of results and drafted the manuscript. RBB participated in preparing of the proofs, discussing signal processing, and drafting the manuscript. All authors read and approved the final manuscript.

Author details

1Institute for Informatics and Automation Problems, National Academy of Sciences of Armenia, Paruir Sevak St. 1, Yerevan, 0014, Armenia. ²Department of Mathematics, Texas A&M University, Kingsville 700 University Blvd., Kingsville, TX 78363-8202, USA.³Vision and Natural Computation Group, Vision Institute, University Pierre et Marie Curie-Paris 6, 17 rue Moreau, Paris, 75012, France.

Acknowledgements

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.

Received: 27 November 2013 Accepted: 23 January 2014 Published: 7 February 2014

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doi:10.1186/1687-1847-2014-60

Cite this article as:Shahverdian et al.:The bistability of higher-order differences of discrete periodic signals.Advances in Difference Equations20142014:60.