Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997

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Commun Nonlinear Sci Numer Simulat

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Denoising signals corrupted by chaotic noise

G. Manjunath a,*, S. Sivaji Ganesh b,1, G.V. Anand c

a Institut National des Sciences Appliquées de Toulouse-DGEI, 135 Avenue de Rangueil, 31077 Toulouse, Franceb L’equipe d’analyse numérique, La MUSE, Université Jean Monnet, 23 rue Paul Michelon, 42023 Saint-Etienne, Francec Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e i n f o

Article history:Received 25 August 2009Received in revised form 11 January 2010Accepted 14 January 2010Available online 18 January 2010

Keywords:Dynamical systemsChaosTopological dynamicsNonlinear signal processingDenoising

1007-5704/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cnsns.2010.01.015

* Corresponding author.E-mail addresses: manju.iisc@gmail.com (G. Man

1 Present address: Department of Mathematics, IIT

a b s t r a c t

We consider the problem of signal estimation where the observed time series is modeled asyi ¼ xi þ si with fxig being an orbit of a chaotic self-map on a compact subset of Rd and fsiga sequence in Rd converging to zero. This model is motivated by experimental results in theliterature where the ocean ambient noise and the ocean clutter are found to be chaotic. Mak-ing use of observations up to time n, we propose an estimate of si for i < n and show that itapproaches si as n!1 for typical asymptotic behaviors of orbits.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The study of reconstructing the states of a chaotic dynamical system from measurements corrupted by noise is well stud-ied (e.g., [3,12,13,18,21]). In this paper, we present a different method to address the related problem of reconstructing asignal corrupted by chaotic noise.

Examples of chaotic noise arise in the context of signal detection and estimation in the ocean. Time series representing (i)ambient acoustic noise in ocean, and (ii) clutter induced by an electromagnetic pulse directed at the ocean surface, exhibitchaos [6,8,9,19]. In these situations, the signal consists of the acoustic disturbance caused by a moving object (such as ship orsubmarine) and the radar pulse scattered by a ship respectively. These practical considerations lead us to consider the fol-lowing model for the observed time series fyig (similar to that considered in [8]):

yi ¼ xi þ si; i 2 Z; ð1Þ

where the sequence fxig is an orbit of a continuous self-map f, (i.e., xiþ1 ¼ f ðxiÞ;8i 2 Z) of a compact subset of Rd (for finite d)that exhibits chaos in the sense of Devaney [5]. The sequence si 2 Rd is bounded and is also such that si ¼ 0 for all negative iand si ! 0 as i!1. We call such a sequence as a fading sequence. The problem considered in this paper is to get an estimateof the fading signal si given the observations yi.

Since in (1), reconstructing xi from yi is mathematically equivalent to reconstructing si, the problem in this paper hassome similarities with the classical denoising problem of dynamical systems (e.g., [3,12,13,18,21]). In this classical problem,a standard model employed is yi ¼ xi þ ei, where xi is a time series obtained from a dynamical system and ei is a sequence of

. All rights reserved.

junath), siva@math.iitb.ac.in (S. Sivaji Ganesh), anandgv@ece.iisc.ernet.in (G.V. Anand).Bombay, Mumbai-76, India.

G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997 3989

bounded independent random variables with zero mean (mainly intended to model the observational error), and the task isto reconstruct the states xi from yi.

In this paper, we extend the work on denoising a deterministic time series by Lalley [13], Lalley and Nobel [15] by extend-ing their algorithms. The hypothesis in [13,15] that the maps are expansive (see Section 2.2 for definition) is a feature of asmall class of dynamical systems, prominently, hyperbolic diffeomorphisms [10]. The practical significance of using ourdenoising algorithm rather than those in [13,15] can be explained through the following observations:

(P1) Large dimensional real world chaotic systems only roughly tend to behave like continuous time uniformly hyperbolicsystems. Uniformly hyperbolic systems are very restricted and special examples of chaotic systems which satisfy otherdefinitions of chaos such as Devaney’s. Uniformly hyperbolic systems have a rich mathematical theory behind themcompared to many other chaotic systems, but serve as poor models of real world chaotic systems. To assume that theambient ocean noise xi is generated from a hyperbolic system cannot be considered a good approximation. In the par-ticular case of ocean clutter, evidence has been provided in [9] that one of the Lyapunov exponents of the chaotic sys-tem is consistently zero and the system is dissipative. This is not a feature of a uniformly hyperbolic system.

(P2) Even when the uniform hyperbolicity assumption is made, for modeling a physical system, a hyperbolic system evolv-ing in continuous time is more relevant than a discrete time hyperbolic system. Hence a time series from a uniformlyhyperbolic system should be considered to be obtained from a time-T map of a hyperbolic flow and not from a discretetime hyperbolic system. The time-T maps arise when adjacent elements of fxig are separated by a uniform distance ofT time units.

(P3) It is well-known that if ut is a hyperbolic flow on a compact space, the map uT (i.e., the time-T map of the flow) is nei-ther hyperbolic nor expansive [10] (but satisfies chaos in the sense of Devaney). The reader may note that non-expan-siveness of uT is since the trajectories of two near-by points on the central manifold of the flow ut are close-by for ever.

(P4) The hypothesis to assume a chaotic time series is expansive as used in [13,15] to our denoising problem in hand isunreasonable first due to (P1) — this is since expansiveness is a feature of special systems such as hyperbolic systemsand not of general chaotic systems. The expansiveness hypothesis loses much more weight due to (P3). Hence it ispractically relevant to assume an hypothesis on the time series xi that is satisfied by a general chaotic map whichis not necessarily hyperbolic.

(P5) From the above points it is clear that the hypothesis of [13,15] is not applicable to xi generated by many classes ofchaotic systems. In this paper, we also show that the algorithm in [13,15] is inadequate to reconstruct si from yi whenxi is generated by non-expansive chaotic maps (see Section 3.1).

(P6) To adopt a relevant hypothesis for the time series satisfied by many chaotic systems including that of a time-T map ofa hyperbolic flow, we employ the intra-orbit separation property (see Definition 2.1) which is satisfied by all typicalorbits of many chaotic systems including all those that satisfy the definition of Devaney’s chaos.

We now outline briefly the method of signal separation proposed in this paper. We fix an increasing function m : N! N

satisfying mðnÞ ! 1;n�mðnÞ ! 1 as n!1. We then create an estimate si;n of each siði 6 nÞ making use of observationsy�mðnÞ; . . . ; y0; . . . ; yn and then show that jsi;n � sij ! 0 as n!1 (here, j j denotes the Euclidean norm). The idea of creatingan estimate is based on exploiting topological properties of the orbit fxig and is termed as ‘denoising algorithm’. The mainresult of the paper is Theorem 3.1 which concerns the proof of convergence of the denoising algorithm when fxig is a sep-arating orbit (see Section 3.2) of any continuous self-map of a compact subspace of Rd. This result is significant owing to thefollowing considerations : (i) all dense orbits of chaotic maps on compact subsets of Rd are separating orbits (see Proposition2.1), (ii) dense orbits are the most likely behavior one would expect to observe in practice as a chaotic time series and, more-over, the set of points having a dense orbit under a chaotic map is residual (residual sets are topologically big), (iii) in theconventional signal processing methodology, the problem of signal recovery becomes difficult when the signal is weak,i.e., when jsij is much smaller than jxij; our approach deviates from classical estimation procedures [11] and we show thatsignal recovery is possible even when the signal is weak.

The rest of the paper is organized as follows. In Section 2, we elaborate on the hypothesis of intra-orbit separation for thechaotic time series used by us. We then discuss the hypotheses used in Lalley’s papers on the map and establish their inad-equacy to our problem. The main result of this paper is presented in Section 3, where the algorithm is presented and its con-vergence is proved (Theorem 3.1). Finally in Section 4 a few numerical results of the signal separation problem are presentedbesides some practical view points.

2. Preliminaries

2.1. The relevance of the model

We first turn to relate the qualitative indicators in the model to a physical situation. In a practical scenario, a measuringinstrument is switched on perpetually to record the observations yi. The first instance at which si is non-zero is labeled as s0.This labeling is for theoretical purposes and it is important to note that, in practice, the first instance at which si is non-zero isnot known. We also assume that jsij is bounded above by a value dependent on the map f and that the signal fades or decays

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to zero, i.e., jsij ! 0 as i!1. This decaying assumption on the noise is justified in the two different situations in which ourmodel is considered. The two contexts are (i). xi represents ocean clutter as in Haykin and Li’s [8], a target’s presence gen-erates si, and its motion away from the receiver leads si to fade to zero, and (ii). xi represents ocean ambient noise, the acous-tical disturbances caused by an enemy ship or a submarine generates si and is usually of a very short duration.

2.1.1. Assumptions and conventionsA dynamical system refers to a pair ðX; f Þwhere f is a continuous self-map of a metric space ðX; dÞwithout isolated points.

The open ball in ðX; dÞ centered at x 2 X with radius r > 0 is denoted by BxðrÞ. The standard topology on Rd is generated by theEuclidean metric and subsets of Rd are endowed with the subspace topology.

By a forward orbit of a point x, we mean a sequence OþðxÞ ¼ fx0; x1; x2; . . .g, where x0 ¼ x and f ðxnÞ ¼ xnþ1 for all n P 0.Given a surjective map f : X ! X, an orbit of x 2 X is a bi-infinite sequence OðxÞ ¼ f. . . ; x�1; x0; x1; . . .g such that

f ðxnÞ ¼ xnþ1 for n 2 Z. We denote fx�1; x�2; . . .g (resp. fx0; x1; . . .g) as O�ðxÞ (resp. OþðxÞ) and refer to it as a backward orbit(resp. the forward orbit) of x0.

The set of all limit points of the forward orbit OþðxÞ is denoted by xðx; f Þ and is referred to as the x-limit set of the point xor its orbit. For any given backward orbit O�ðxÞ of x, the set of its limit points is called its a-limit set. A point x is said to be atransitive point if its forward orbit OþðxÞ is dense in X.

A point x0 2 X is said to be a periodic point of a map f if f nðx0Þ ¼ x0 for some n P 1 and the smallest such n is called itsperiod. In particular, when n ¼ 1; x0 is called a fixed point of the map f. A set in a topological space is said to be residual if itscomplement is of first Baire-Category.

If r is a positive real number, the symbol brc denotes the largest integer less than or equal to r.For convenience we use the notation j j to denote the cardinality of a set if its argument is a finite set, in addition to the

usual Euclidean norm in Rd. The inner product between two vectors u; v in Rd is denoted by u � v .A map f : X ! X is said to be topologically transitive (transitive in brief) if for any two non-empty open subsets U;V of X,

there exists an integer n P 0 such that f nðUÞ has at least one element in common with V. A map f : X ! X is said to exhibitsensitive dependence on initial conditions (SDIC) (sensitivity in brief) if there exists a constant d > 0 such that for every x 2 Xand for any neighborhood U of x there exists a y 2 U and an n P 0 such that dðf nðxÞ; f nðyÞÞ > d. A map f : X ! X is said to bechaotic in the sense of Devaney [5] if the map f has the following three properties : (i) f is SDIC, (ii) f is topologically transitive,and (iii) the set of periodic points is dense in X.

2.2. Expansiveness and separating orbits

Given an invertible map f : X ! X and a s > 0, for all x 2 X denote Nonsepf ðx; sÞ ¼ fy 2 X : dðf nðxÞ; f nðyÞÞ 6 s for alln 2 Zgand its complement in X by Sepf ðx; sÞ. If f is noninvertible, the sets Nonsepf ðx; sÞ and Sepf ðx; sÞ are analogously defined as be-fore by replacing the phrase ‘‘n 2 Z” with ‘‘n P 0”.

We recall (e.g., [15,20]) that an invertible (resp. not necessarily invertible) map f : X ! X is said to be expansive (resp. for-ward expansive) if there exists a d > 0 such that for any two distinct points x; y 2 X, there exists an integer (resp. a non-neg-ative integer) n ¼ nðx; yÞ satisfying dðf nðxÞ; f nðyÞÞ > d (such a d is know as an expansive constant of the map f). In other wordsf is expansive if there exists a d > 0 such that for every x 2 X the set Nonsepf ðx; dÞ is a singleton, in which case it is the sin-gleton set consisting of x.

Expansiveness is a very strong property compared to SDIC. In other words, expansiveness implies SDIC and the converse isnot true. For example if the map f is not invertible on a connected space X then the map fails to be locally injective at least at asingle point p 2 X. Then for any d-neighborhood of p, there exists distinct pair x; y in this neighborhood of p such that f ðxÞ ¼ f ðyÞ.Hence y 2 Nonsepf ðx; dÞ and the map is not expansive. But noninvertible maps like the logistic or the tent map satisfy SDIC [5].

Definition 2.1. Let f : X ! X be any map on a metric space ðX; dÞ and c be a positive real number. An orbit OðxÞ of f is said tobe a separating orbit with an instability constant c if for every pair of distinct points xl; xj 2 OþðxÞ, there exists an integer n P 0such that dðf nðxlÞ; f nðxjÞÞ > c.

A map f : X ! X is said to be uniformly rigid [7] if there exists a subsequence of ðf nÞ that converges uniformly to the iden-tity map on X. In other words, infnP0supxdðf nðxÞ; xÞÞ ¼ 0 : It is a result in [7] that on an infinite compact metric space if f isboth transitive and uniformly rigid then f cannot exhibit SDIC. Hence transitive maps which exhibit SDIC (a feature of chaos)are not uniformly rigid.

For topologically transitive maps on compact spaces, points having dense orbits are abundant. In fact, they form a residualset [22]. This justifies our assumption that the observed chaotic orbit is expected to be a dense orbit in practice. The follow-ing result shows that all dense orbits of a chaotic map f are separating orbits with an instability constant depending only on f.

Proposition 2.1. Let f be a continuous self-map of a compact metric space ðX; dÞ which is not uniformly rigid. If xj and xl are anytwo distinct points on a given dense orbit, then there exists a cðf Þ > 0 such that lim supn!1dðf nðxjÞ; f nðxlÞÞ > cðf Þ. In particular,there exists an nðxj; xlÞP 0 such that dðf nðxjÞ; f nðxlÞÞ > cðf Þ.

Proof. Let xj and xl lie on a dense orbit. Without loss of generality let l > j. Define i :¼ l� j. Since the map f is not uniformlyrigid, there exists a cðf Þ > 0 such that infnsupxdðf nðxÞ; xÞÞð Þ > cðf Þ. Since X is compact, the value supxdðf iðxÞ; xÞ is attained at

G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997 3991

some point pi 2 X. Thus we have, dðpi; fiðpiÞÞP infnsupxdðf nðxÞ; xÞÞ. As consequence, we get dðpi; f

iðpiÞÞ > cðf Þ. Since xj is atransitive point, there exists a subsequence ff nmðxjÞg1m¼1 which converges to pi as m!1. Note thatf nm ðxlÞ ¼ f nm ðf iðxjÞÞ ¼ f iðf nm ðxjÞÞ. By continuity of f, we have

limm!1

f nm ðxlÞ ¼ limm!1

f iðf nm ðxjÞÞ ¼ f iðpiÞ:

As a consequence, we have limm!1dðf nm ðxlÞ; f iðf nm ðxjÞÞÞ ¼ dðpi; fiðpiÞÞ. Since dðpi; f

iðpiÞÞ > cðf Þ, by the definition of limits, thereexists a natural number M such that dðf nmðxlÞ; f nm ðxjÞÞ > cðf Þ for all m P M. This implies that lim supn!1dðf nðxjÞ;f nðxlÞÞ > cðf Þ. h

Since transitive maps which exhibit SDIC are not uniformly rigid, by Proposition 2.1, all maps exhibiting chaos in thesense of Devaney have their dense orbits as separating orbits. Another example of a class of maps which are not uniformlyrigid and transitive are time-T maps of hyperbolic flows – time-T maps are partially hyperbolic maps and they are topolog-ically transitive and exhibit SDIC [10]. We use the notion of a separating orbit as a fundamental hypothesis in our denoisingproblem.

The following result is helpful in explaining the heuristics of the denoising algorithm in this paper:

Proposition 2.2. Let f : X ! X be a continuous map where ðX; dÞ is a compact metric space. Let fxmg and fymg be two sequencesin X such that xm ! x and ym ! y as m!1 and xm 2 Sepf ðym; sÞ for every m. Denote ssðxm; ymÞ :¼ minðjnj : dðf nðxmÞ; f nðymÞÞ iff is invertible and ssðxm; ymÞ :¼ minðn P 0 : dðf nðxmÞ; f nðymÞÞ if f is not invertible. If ssðxm; ymÞð Þm is an unbounded sequence,then x 2 Nonsepf ðy; sÞ.

Proof. We prove that if x 2 Sepf ðy; sÞ, then the sequence of natural numbers fssðxm; ymÞg is bounded. By definition ofx 2 Sepf ðy; sÞ, there exists a integer n such that dðf nðxÞ; f nðyÞÞ > s. Let � > 0 be such that dðf nðxÞ; f nðyÞÞ ¼ 3�þ s. By the uni-form continuity of the map f n, there exists dnð�Þ > 0 such that for all u;v 2 X; dðu;vÞ < dn, we have dðf nðuÞ; f nðvÞÞ < �. By tri-angle inequality, we have

s < dðf nðxÞ; f nðyÞÞ 6 dðf nðxÞ; f nðwÞÞ þ dðf nðwÞ; f nðzÞÞ þ dðf nðzÞ; f nðyÞÞ:

Thus it follows that dðf nðwÞ; f nðzÞÞ > s for all w and z satisfying dðw; xÞ < dn and dðz; yÞ < dn respectively. Hence ssðw; zÞ 6 n.Since xm ! x and ym ! y as m!1, it follows that there exists m0 such that ssðxm; ymÞ 6 n for m P m0. Let N1 be the max-imum of the numbers ssðxm; ymÞ, m ¼ 1;2; . . . ;m0. Then we have, ssðxm; ymÞ 6 maxfn;N1g. h

3. Denoising algorithm and its motivation

Let OðxÞ ¼ f. . . ; x�2; x�1; x0; x1; x2; . . .g be a separating orbit of a surjective map on a compact subspace of Rd with an insta-bility constant s. Let the observed time series yi be a bi-directional Rd valued sequence,

yi ¼ xi þ si; i 2 Z; ð2Þ

where si is a sequence in Rd such that si ¼ 0 for all i < 0 and jsij < C < s=4 for all i > 0. Further let jsij ! 0 as i!1.

We describe below our denoising algorithm which is an extension of Lalley’s algorithm [13].

Denoising Algorithm� Fix a monotonic function h : N! N such that hðkÞ < k, and k� hðkÞ ! 1 and hðkÞ ! 1 as k!1.

� Let k 2 N and k < n. For each l ¼ 0; . . . ;n� k define the index set

Anðl; kÞ :¼ fj : j–l; 0 6 j 6 n� k; maxi:jij6k

jyjþi � ylþij 6 s� 2Cg: ð3Þ

� For each j 2 Anðl; kÞ, define a sum of squares in a left window by

zjðn; l; kÞ :¼X

�k6i6�hðkÞjyjþi � ylþij

2: ð4Þ

� Fix b 2 ð0;1Þ and let M ¼ bjAnðl; kÞjbc.� Let j1; j2; . . . ; jM be a labeling of elements of Anðl; kÞ having the property:

zj1 ðn; l; kÞ 6 zj2 ðn; l; kÞ 6 � � � 6 zjM ðn; l; kÞ:

� Define the set Bnðl; kÞ :¼ fj1; j2; . . . ; jMg:� If Bnðl; kÞ is non-empty then define

xl;n;k :¼ 1jBnðl; kÞj

Xj2Bnðl;kÞ

yj: ð5Þ

� Define sl;n;k :¼ yl � xl;n;k, an estimate of sl. h

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Remark 3.1. The algorithm proposed here is computationally simpler than the Denoising Algorithm W proposed by Lalley

[14]. A naive implementation has a running time complexity of Oðn3Þ which suggests that for large data sets, such animplementation is very expensive. The naive implementation that we are talking about does not even take into account thesimple information that j 2 Anðl; kÞ implies l 2 Anðj; kÞ. Significant reductions in the running time complexity can be made bytaking into account such symmetries. Details of efficient and also approximate implementations where the running timecomplexity can be reduced up to Oðn log nÞ can be found in Lalley [14].

3.1. Motivation behind extending Lalley’s algorithm

3.1.1. Inadequacy of Lalley’s algorithmIf Lalley’s algorithm were to be used as in [13,15], then the estimate of xl denoted by xl;n;k would be defined as

xl;n;k :¼ 1jAnðl;kÞj

Pj2Anðl;kÞyj, where xi is generated by an expansive map with an expansivity constant s. By writing yj ¼ xj þ sj

in this estimate and using triangle inequality, we can easily arrive at

jxl � xl;n;kj 6P

j2Anðl;kÞjxl � xjjjAnðl; kÞj

þjP

j2Anðl;kÞsjjjAnðl; kÞj

: ð6Þ

(Pt1). When some recurrence properties on the point xl are assumed it is ensured that for every given k; jAnðl; kÞj tends toinfinity as n increases (Lemma 3.1). When such a growth of jAnðl; kÞj occurs, since sj ! 0 as j!1, the second sumin the RHS of (6) converges to zero as n!1 for every k.

(Pt2). If j 2 Anðl; kÞ then by triangle inequality we obtain dðxjþi; xlþiÞ 6 s for jij 6 k since jsij < C. Hence ssðxl; xjÞ > k. Notehenceforth, that k!1 only if n!1, and hence whenever we refer to the limit k!1, it is implicitly assumedthat n!1. Thus ssðxl; xjÞ > k whenever j 2 Anðl; kÞ. Hence if jn;k 2 Anðl; kÞ then the limit set of the double-indexedsequence fxjn;kg as k!1 belongs to Nonsepf ðxl; sÞ (by Proposition 2.2). Since for an expansive map,Nonsepf ðxl; sÞ ¼ fxlg, as k!1 we see that the LHS of (6) converges to zero. This is the basis for convergence ofLalley’s algorithm when expansive maps are used.

(Pt3). For non-expansive maps, the set Nonsepf ðxl; sÞ is not necessarily equal to fxlg, hence the convergence of thealgorithm cannot be guaranteed.

3.1.2. Modification of Lalley’s algorithmSome comments on the modified algorithm presented by us are in order.

(Pt4). The set Anðl; kÞ has two types of indices: one mn;k 2 Anðl; kÞ for which limk!1xmn;k¼ z; z–xl and the other

jn;k 2 Anðl; kÞ for which limk!1xjn;k ¼ xl. By discarding the indices of the first type while averaging, we can getthe convergence of the denoising error in (6). This is the motivation to consider a new index set Bnðl; kÞ asexplained next.

(Pt5). We writeP�k6i6�hðkÞjyrn;kþi � ylþij

2 ¼P�k6i6�hðkÞjxrn;kþi � xlþij2 þ

P�k6i6�hðkÞ�k, where each �k involves at least one

element belonging to the sequence fsig. For k sufficiently large, it can be shown that (see step (iv) and step (vi)of Lemma 3.4) the limit of the second term as n increases tends to zero in view of si � 0 for i < 0 and si ! 0 as

i!1. Hence for large k, limn!1P�k6i6�hðkÞðjyrn;kþi � ylþij

2 � jxrn;kþi � xlþij2Þ ¼ 0.

(Pt6). The key to convergence of the algorithm is in recognizing the dichotomy between the asymptotic behaviors (as

k!1) ofP�k6i6�hðkÞjxjn;kþi � xlþij2 and

P�k6i6�hðkÞjxmn;kþi � xlþij2. Since xmn;k

! z–xl and xjn;k ! xl, we can expect

by continuity of the underlying map thatP�k6i6�hðkÞjxjn;kþi � xlþij2 <

P�k6i6�hðkÞjxmn;kþi � xlþij2 for all sufficiently

large k (see step (iii) and step (v) of Lemma 3.4). In view of (Pt5), for large n and k, the quantityP�k6i6�hðkÞjyjn;kþi � ylþij

2 will be smaller thanP�k6i6�hðkÞjymn;kþi � ylþij

2. The valueP�k6i6�hðkÞjyjn;kþi � ylþij

2 is indeed

the one in (4) to determine indices of Bnðl; kÞ from Anðl; kÞ. The end effect is that as k increases, we tend to pickindices jn;k rather than mn;k from Anðl; kÞ while determining Bnðl; kÞ. Since we use Bnðl; kÞ in determining our esti-mate in (5), the denoising error can be expected to tend to zero as n!1 for large k.

3.2. Main result on the convergence of the algorithm

Definition 3.1. Let v denote the characteristic function, i.e., vAðxÞ ¼ 1 if x 2 A and 0 otherwise. Let OðxÞ ¼ f. . . ; x�1;

x0; x1; x2; . . .g be an orbit of a surjective map and a 2 ð0;1Þ. The point x is said to be a-moderately recurrent if for anyneighborhood U of any xi 2 OðxÞ; lim infn!1

1na

Pn�1i¼0 vUðf iðxÞÞ > 1:

It is trivial to verify that if xl is a-moderately recurrent then every xj 2 OðxÞ is also a-moderately recurrent. The followingremark tells us that the hypothesis of the a-moderately recurrent assumption is a very elementary consequence of ergodicproperty satisfied by a chaotic map. Further, we have made the definition of ‘a-moderately recurrent point’ taking cues fromresults in quantitative recurrence in [1,2].

G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997 3993

Remark 3.2 (Characterization of a-moderately recurrent points). Let ðX;B;lÞ be a probability space and f : X ! X be a selfmap. We know from Birkhoff’s ergodic theorem (e.g., [22,4]) that when f is ergodic and invariant with respect to l, thenlim infn!1

1n

Pn�1i¼0 vUðf iðxÞÞ exist for all x except for a set of measure zero and is equal to lðUÞ for every U 2 B. Suppose if the

support of l [4, p.11] is the whole of X, then for every non-empty open set U;lðUÞ > 0. Hence clearly for almost all x (withrespect to l), lim infn!1

1na

Pn�1i¼0 vUðf iðxÞÞ ¼ 1, where a 2 ð0;1Þ. Thus these points are a-moderately recurrent. All typical

examples of transitive maps on a compact space have an ergodic invariant measure whose support is the whole space X [10,Section 4.1] and hence the set of a-moderately recurrent points have full measure with respect to such a measure. There aresome very special examples of transitive maps which do not have any ergodic measure with full support [10, Section 4.1].But these maps are not of interest to us as they are not chaotic in the sense of Devaney because they do not have periodicpoints.

Finally before stating our theorem on the convergence of the algorithm, we mention a convention that will be followedthroughout. From now on, a double-indexed sequence fjn;kg with index n; k have meaning only for k < n.

For notational convenience we present our results for the estimation errors jxl � xl;n;kj rather than jsl � sl;n;kj.

Theroem 3.1. Let f be a continuous surjective self-map of a compact subspace of R. Let OðxÞ ¼ f. . . ; x�1; x0; x1; . . .g be anonperiodic separating orbit of the map f with an instability constant s. Let yi and si be as defined in (2), and let xl;n;k be as definedin Eq. (5). Then if xl 2 OþðxÞ is an a-moderately recurrent point with a > b where b is as in the algorithm, then

limk!1

lim supn!1

jxl � xl;n;kj ¼ 0: ð7Þ

Remark 3.3. The interpretation of the result in (7) follows from the definition of the two limits. To enunciate it for a readerinclined in a practical application, (7) means ‘‘ given any � > 0, there exist numbers K and NðkÞ, such that for all windowlengths k > K and observations n > NðkÞ the estimation error jxl � xl;n;kj is less than �”.

The definition of the following subset of Anðl; kÞ helps us in simplifying the presentation of proof of Theorem 3.1.

Qnðl; kÞ :¼ fj : j 2 Anðl; kÞ and jxlþi � xjþij 6 s�4Ck for all� k 6 i 6 0g: ð8Þ

We have so far defined the sets Anðl; kÞ; Gn;Mðl; kÞ and Qnðl; kÞ. But it is not clear if these sets are non-empty and we have thefollowing result that asserts positively the same under some conditions on the orbit of x.

Lemma 3.1. Let OðxÞ be as in the hypothesis of Theorem 3.1 and let xl be as in statement of Theorem 3.1, that is, xl 2 OþðxÞ an a-moderately recurrent point. Then jQnðl; kÞjP na for n sufficiently large.

Proof

Step (i) A simple consequence of xl 2 OþðxÞ being a a-moderately recurrent point is O�ðxÞ# xðx0; f Þ. Let xj 2 OþðxÞ andxj–xl. By continuity of f, there exists a neighborhood U containing xl�k such that if xj�k 2 U, thenjxlþi � xjþij 6 s�4C

k for all �k 6 i 6 0 and jxlþi � xjþij 6 s� 4Cfor all1 6 i 6 k. We know that O�ðxÞ# xðx0; f Þ. Thus itfollows by continuity that xj�k 2 xðx0; f Þ irrespective of whether xj�k is in OþðxÞ or O�ðxÞ. Therefore, by nonper-iodicity of OþðxÞ there exists an infinite set of points belonging to OþðxÞ contained in U. Now, using the fact thatsi < C, by triangle inequality we get jxlþi � xjþij 6 s� 4C which would in turn imply jylþi � yjþij 6 s� 2C for all�k 6 i 6 k. Hence by the definition of Anðl; kÞ (see (3)) it follows that j belongs to Anðl; kÞ. Hence jQnðl; kÞj ! 1 asn!1.

Step (ii) In the same vein for an a-moderately recurrent point xl, for every given k, the cardinality of the setfx0; x1; . . . xng \ U exceeds na when n is sufficiently large. Hence, jQnðl; kÞjP na for n sufficiently large. h

Thanks to Lemma 3.1, one can construct special sequences of indices as illustrated below.

Lemma 3.2. With the hypothesis as in statement of Theorem 3.1, let xl 2 OþðxÞ. Then there exists a double-indexed sequence j0n;kwhich satisfies the following: (i) for every fixed k, j0n;k 2Qnðl;kÞnBnðl;kÞ for sufficiently large n, (ii) for every fixed k, limn!1j0n;k¼1,(iii) for every fixed k, limn!1jxl�xj0n;k

j¼0.

Proof. Proof of (i). By Lemma 3.1, we have jQ nðl; kÞj > na > 2nb þ 1 P 2bjAnðl; kÞjbc þ 1 for sufficiently large n. Hencenk :¼ minfn : jQnðl; kÞj > nag is well-defined. Note that nk is non-decreasing in k and limk!1 nk ¼ 1. We now choose asequence ðki;nki

Þ such that k1 ¼ 1, ki !1, nki!1 as i!1, and nki

< nkiþ1. For each i, define the set

Si :¼ Q nkiðl; kiÞ n Bnðl; kÞ. By the choice of nki

, the set Si has at least na

2 þ 1 elements since jBnðl; kÞj 6 bna

2 c.We now define the double-indexed sequence j0n;k by defining the sequence fj0n;kgn2N for every fixed k. Let ki0 be the smallest

integer with the property ki0 P k. For k < n < nki0, define j0n;k to be any arbitrary admissible index, i.e., any index having values

in 0;1; . . . ;n� k (it does not matter the way these indices are defined as our results are of asymptotic nature). We nowproceed to define j0n;k for n 2 ½nki0

;nki0þ1Þ by choosing an index from Si. Since Si0 has at least na

2 þ 1 elements and Bnðl; kÞ alwayshas at most bna=2c elements, we define j0n;k to be any element belonging to the set Si0 n Bnðl; kÞ.

3994 G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997

Since Qnðl; kÞ# Qnþ1ðl; kÞ, we have j0n;k 2 Qnðl; kÞ. In a similar way by considering an appropriate index in the set Si0þ1, wedefine j0n;k for n 2 ½nki0þ1

;nki0þ2Þ and so on. This completes the definition of j0n;k for all admissible values of k and n. By this

construction, the sequence j0n;k satisfies (i) of the statement of the lemma.Proof of (ii). In view of the construction of the sequence j0n;k described above, it is enough to prove that limi!1ti is infinity

whenever ti 2 Si. Let ftig be any such sequence. We prove that lim inf i!1ti is infinity. If it were finite, then it would followthat there exists an index, say i0, that belongs to Qnki

ðl; kiÞ for infinitely many i. Recalling the definition of the set Q nkiðl; kiÞ, we

obtain jxl � xti0j ¼ 0, a contradiction to the fact that OþðxÞ is not a periodic orbit. This proves (ii).

Proof of (iii). The assertion (iii) is a consequence of the definition of the indicesfj0n;kg and the fact that Si � Qnkiðl; kiÞ for every i. h

Lemma 3.3. Let OþðxÞ be as in the hypothesis of Theorem 3.1 and let xl 2 OþðxÞ. If jn;k 2 Anðl; kÞ, then lim infk!1 lim infn!1jn;k ¼ 1. In particular, the result holds if jn;k 2 Bnðl; kÞ.

Proof

Step (i). Suppose that lim infk!1 lim infn!1 jn;k is finite. Then there exists an infinite subset J of N and an integer t such thatfor every k 2 J, there exists an infinite subset Fk of N such that jn;k ¼ t for all n 2 Fk. That is, t 2 Anðl; kÞ for all k 2 Jand for all n 2 Fk.

Step (ii). Since xt ; xl 2 OðxÞ and OðxÞ is a separating orbit, there exists an mðl; tÞ 2 N such that jxtþm � xlþmj > s.Step (iii). Since J is an infinite subset of natural numbers, there exists a k0 2 J with k0 > m. Since t 2 Anðl; k0Þ, we also have

jytþm � ylþmj 6 s� 2C by the definition of the set Anðl; k0Þ. Using the definition of yi (see (2) ) and by applying tri-angle inequality, in view of Step(ii) above, we get

s� jstþm � slþmj < jxtþm � xlþmj � jstþm � slþmj 6 jytþm � ylþmj 6 s� 2C;

which is absurd as jstþm � slþmj 6 2C. This means that there exists no k0 2 J with k0 > m which further implies thatthe set J is a finite set. Thus we conclude that lim infk!1 lim infn!1 jn;k is not finite. This proves the result ifjn;k 2 Anðl; kÞ. The second part is a consequence in view of the inclusion Bnðl; kÞ# Anðl; kÞ. h

Lemma 3.4. Let OðxÞ be as in the hypothesis of Theorem 3.1 and let xl be as in statement of Theorem 3.1, that is, xl 2 OþðxÞ an a-moderately recurrent point. Consider any sequence of elements jn;k in fBnðl; kÞg such that for every k, limn!1jn;k ¼ 1. Then

limk!1

lim supn!1

jxl � xjn;k j ¼ 0: ð9Þ

Proof

Step (i) Note that proving (9) is equivalent to proving: lim supk!1 lim supn!1 jxl � xjn;k j ¼ 0: Suppose, if possible, that thisequality does not hold. Using the fact that lim sup of a sequence is a limit point of the sequence, we get an infinitesubset of natural numbers E and a sequence of infinite subsets of natural numbers Fr indexed by r 2 E such thatlim supk!1 lim supn!1 jxl � xjn;k j ¼ limr2E limnr

p2Fr jxl � xjnrp ;rj: Further, the sets Fr and E can be chosen such that the

following limits exist

qr :¼ limnr

p2Fr

xjnrp ;r; q :¼ lim

r2Eqr : ð10Þ

Thus, by our assumption, we have

limr2E

limnr

p2Fr

jxl � xjnrp ;rj–0: ð11Þ

Step (ii) Now let zjn;k :¼P�k6i6�hðkÞjyjn;kþi � ylþij

2;wjn;k :¼

P�k6i6�hðkÞjxjn;kþi � xlþij2; and ujn;k :¼

P�k6i6�hðkÞjsjn;kþi � slþij2þP

�k6i6�hðkÞ2ðxjn;kþi � xlþiÞ � ðsjn;kþi � slþiÞ: Also let zj0n;k;wj0n;k

and uj0n;kbe defined analogously, for xj0n;k

where j0n;k is asin Lemma 3.2. Note that for each k, jn;k 2 Bnðl; kÞ and j0n;k R Bnðl; kÞ when n is sufficiently large. Hence, for each kwe get the inequality zjn;k 6 zj0n;k

when n is sufficiently large.Step (iii) Claim : lim infnr

p2Fr wjnrp ;r> 0 for all sufficiently large r 2 E. Thanks to (11) and in view of (10), we have q–xl. Let

r 2 E be large enough such that qr–xl. Fix any such r. Since dðxl; qrÞ > 0, denoting fxjnrp ;rþig0 as the set of accumu-

lation points of fxjnrp ;rþigp, by continuity it follows that fxjnr

p ;rþig0 belongs to the ith pre-image of qr (denoted by f iðqrÞ)

for every i < 0. Owing to the continuity of f, the set f iðqrÞ is compact. As a consequence, due to qr–xl, we havedðxlþi; f iðqrÞÞ > 0 for every i. Using this fact in the definition of wjn;k in step (ii), the Claim follows.

Step (iv) Claim: lim infnrp2Fr zjnr

p ;rP lim infn zjn;r > 0 for all sufficiently large r 2 E. Recall that ujn;r :¼

P�r6i6�hðrÞj sjn;rþi � slþij2þP

�r6i6�hðrÞ2ðxjn;rþi � xlþiÞ � ðsjn;rþi � slþiÞ: Since the function h is increasing, there exists an r0 such thathðrÞ > l;8r P r0; and we thus have sl�hðrÞ ¼ 0. Since for every r, limn!1jn;r ¼ 1, we have sjn;r ! 0 as n!1 forevery r. Thus, for sufficiently large values of r, limn!1ujn;r ¼ 0. Recall that zjn;r ¼ wjn;r þ ujn;r . Hence,lim infn!1zjn;r > 0 using the fact that lim infn!1wjn;r > 0 from step (iii).

G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997 3995

Step (v) Claim : lim infn!1wj0n;r ¼ 0 for every r. By Lemma 3.2, for every r, limn!1xj0n;r¼ xl. This with the continuity of the

map yields lim infn!1wj0n;r¼ 0.

Step (vi) Claim : limn!1uj0n;r¼ 0 for all r sufficiently large. Recall that uj0n;r

:¼P�r6i6�hðrÞðsj0n;rþi � slþiÞ2 þ

P�r6i6�hðrÞ

2ðxj0n;rþi � xlþiÞðsj0n;rþi � slþiÞ: Expanding uj0n;rlike that in step (iv), and by exactly employing the same arguments

there to prove limn!1ujn;r ¼ 0, we have here, for r sufficiently large, limn!1uj0n;r¼ 0 follows.

Step (vii) Recall that zj0n;r¼ wj0n;r

þ uj0n;r. Hence, lim infn!1zj0n;r

P lim infn!1wj0n;rþ lim infn!1uj0n;r

. From step (vi), we have forsufficiently large values of r, limn!1uj0n;r

¼ 0. Therefore for sufficiently large r, we have lim infn!1zj0n;r¼

lim infn!1wj0n;rþ limn!1uj0n;r

: Thus from steps (v) and step (vi), lim infn!1zj0n;r ¼ 0 for r sufficiently large. From step(iv) we had lim infn!1zjn;r > 0 for all r sufficiently large. This means that for r sufficiently largelim infn!1ðzjn;r � zj0n;r

Þ > 0. This would imply that for each such r , except for a finite set of n, we have zjn;r > zj0n;r;

and hence for all these values of r and n, j0n;r 2 Bnðl; rÞ. This contradicts the fact that j0n;r R Bnðl; rÞ for n sufficientlylarge. h

Proof of Theorem 3.1: Proof of (7). From Eq. (5) and using triangle inequality we get

jxl � xl;n;kj 6P

j2Bnðl;kÞjxl � xjjjBnðl; kÞj

þP

j2Bnðl;kÞjsjjjBnðl; kÞj

: ð12Þ

Since xl is a-moderately recurrent we have jBnðl; kÞj ! 1 as n!1 for every k. As a consequence, there exists a sequence ofnatural numbers rn;k satisfying the following three properties:

(i) rn;k < maxfj : j 2 Bnðl; kÞg(ii) limn!1rn;k ¼ 1 for each k.

(iii) For every k, #fj:j2Bnðl;kÞ and j6rn;kgjBnðl;kÞj ! 0 as n!1.

Now define tn;k :¼ any index m 2 Bnðl; kÞ such that jxl � xmj ¼maxðj2Bnðl;kÞandj>rn;kÞjxl � xjj. The second term on the RHS of(12) can be written as

P

j2Bnðl;kÞjsjjjBnðl; kÞj

¼P

j2Bnðl;kÞ and j6rn;kjsjj

jBnðl; kÞjþP

j2Bnðl;kÞ and j>rn;kjsjj

jBnðl; kÞj6

Pj2Bnðl;kÞ and j6rn;k

jsjjjBnðl; kÞj

þ maxj2Bnðl;kÞ and j>rn;k

jsjj:P

The boundedness of jsjj and property (iii) of rn;k yields limn!1

j2Bn ðl;kÞ and j6rn;kjsj j

jBnðl;kÞj ¼ 0.Since limn!1rn;k ¼ 1 for each k and sj ! 0 as j!1, we have limn!1maxj2Bnðl;kÞ and j>rn;k

jsjj ¼ 0. Thus we have

limn!1

Pj2Bn ðl;kÞ

jsj jjBnðl;kÞj ¼ 0:

Now, the first term on the RHS of (12) can be written as

Pj2Bnðl;kÞjxl � xjjjBnðl; kÞj

¼P

j2Bnðl;kÞ and j6rn;kjxl � xjj

jBnðl; kÞjþP

j2Bnðl;kÞ and j>rn;kjxl � xjj

jBnðl; kÞj

6

Pj2Bnðl;kÞ and j6rn;k

jxl � xjjjBnðl; kÞj

þ maxj2Bnðl;kÞ and j>rn;k

jxl � xjj 6P

j2Bnðl;kÞ and j6rn;kjxl � xjj

jBnðl; kÞjþ jxl � xtn;k

j: ð13Þ

By property (iii) of rn;k, we have

limn!1

Pj2Bnðl;kÞ and j6rn;k

jxl � xjjjBnðl; kÞj

¼ 0: ð14Þ

Since limn!1rn;k ¼ 1, we have limn!1tn;k ¼ 1 for each k and hence tn;k satisfies the hypothesis of a sequence in Bnðl; kÞ in

Lemma 3.4. Now, by Lemma 3.4, we have limk!1lim supn!1

Pj2Bn ðl;kÞ

jxl�xj jjBnðl;kÞj ¼ 0; and this completes the proof of Theorem 3.1.

We now make remarks on different aspects of the denoising algorithm. h

Remark 3.4 (Choice of k with n). It is obvious from Theorem 3.1 that a large k will ensure that the estimation error is small.However, as k increases, for jAnðl; kÞj to become larger, we need larger n. A very small value of jAnðl; kÞj would mean very fewobservations are averaged in defining the estimates. In Lalley’s paper [13] k ¼ oðlog nÞ is used. In practice, one has to make areasonable choice depending on the amount of observations available.

Remark 3.5 (Monotonicity of hðkÞ). The condition hðkÞ ! 1 as k!1 is used in step (iv) of Lemma 3.4. In particular, wechoose r0 > 1 such that hðrÞ > l for all r > r0. This implies that as l increases, hðrÞ has to be larger, and hence the demand for alarger r. This is undesirable from a practical point of view. However, the situation is not as bad as it appears to be since, inpractice, denoising a small segment of fsigmay be sufficient. Strategically, obtaining estimates of such a small finite segmentis often the crux of signal detection problems. Also for the Lemma 3.4 to hold the condition hðrÞ ! 1 as r !1 was asufficient condition and may not be a necessary one. Further discussion of this aspect can be found in Manjunath [16].

3996 G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997

4. Simulation results

Some practical issues that may arise while implementing the denoising algorithms are addressed in Manjunath [16,Chapter 6]. We only give here a sample of the simulation results obtained by using our algorithm. More extensive simulationresults may be found in Manjunath [16].

Consider the logistic map defined by f ðxÞ ¼ 4xð1� xÞ on the interval [0,1] and its fifth iterate f 5. In our simulation resultsfor obtaining the time series xi, we use an orbit of f 5 observed on the computer. It is known that this map is topologicallytransitive and exhibits SDIC. Hence f 5 is not unformly rigid and as a consequence of Proposition 2.1 this implies all denseorbits are separating orbits. Theoretically, supfjf 5ðxÞ � xj : x lies between two fixed pointsg gives an instability constant[17] for all dense orbits of f 5. Hence it may be verified that any number s < 1

4 ð2þffiffiffi2pÞ is an instability constant for all dense

orbits of f 5.Often, in a practical scenario only xi (or just yi) is available without the knowledge of the underlying map. In that case, the

quantity inf isupj–idðxi; xiþjÞ (or inf isupj–idðyi; yiþjÞ) is an estimate of an instability constant. More details on the estimation ofthis quantity are found in [16].

The signal si (for i P 0) used in the simulations is defined by si ¼ C sinðð i10 Þ

2) for 0 6 i < 500 and si ¼ Ci sinðð i

10 Þ2Þ for all

other i P 500 and C is taken to be 0.14 which is less than s=4. This signal si belongs to the class of ‘‘chirp signals” in signalprocessing literature, and they are observed when a sinusoidal signal is reflected from a moving target with an acceleration.It may be recalled from the denoising algorithm that hð�Þ is defined as a monotonically increasing function such th at bothhðkÞ ! 1 and k� hðkÞ ! 1 as k!1. We set hðkÞ ¼ k=2 while implementing the algorithm in the simulation. With thesedata, we obtain the estimate sl;n;k using the denoising algorithm.

A measure of the relative strengths of signal and noise(SNR) in the data yi ¼ xi þ si; i 2 J, is given by the input signal-to-noise ratio (in decibels) defined as

Fig

SNRin :¼ 10log10

Pi2Jjsij2Pi2Jjxij2

!:

The output SNR for the estimated signal with indices in the index set J is defined as

SNRout :¼ 10log10

Pi2Jjsij2P

i2J jsi � ^si;nj2

!:

We present the SNRin and SNRout for denoising the first 500 samples, i.e., for J above contains all integers i such thatð0 6 i < 500Þ. For C ¼ 0:4; SNRin ¼ �6:51dB.

Assume that xi is a-moderately recurrent for every a 2 ð0;1Þ (this is the case for typical orbits under the ergodic hypoth-esis). Recall from Lemma 3.1 that for every given a; jAnðl; kÞjP na (since Anðl; kÞ# Qnðl; kÞ) when n is sufficiently large. Hencefor large n, jBnðl; kÞj ¼ bjAnðl; kÞjbc. Thus, for a large b used in the algorithm, we obtain a larger cardinality of Bnðl; kÞ when n issufficiently large. An average xl;n;k (see (5)) obtained over a Bnðl; kÞ of larger cardinality is better based on statistical consid-erations. On the expected lines from the theory of the algorithm, in Fig. 1, larger values of n; k; b provide a better result,although an increase in k and b makes practical sense only when n is sufficiently large. Further, larger values of k; b implythat the computational complexity increases. Comparing the plots with different parameter values in Fig. 1, the results indi-cate that the descending order of influence of the parameters on the denoising error is k followed by b.

2 4 6 8 10 12 142

3

4

5

6

7

8

9

outp

ut S

NR

in d

ecib

els

k

data1data2data3data4

. 1. Plot of SNRout vs. k for various data: ðn;bÞ in data1, data2, data3 and data4, respectively are ð105;0:2Þ; ð105; 0:3Þ; ð106; 0:2Þ, and ð106; 0:3Þ.

G. Manjunath et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3988–3997 3997

5. Conclusions

We have used the notions of intra-orbit separation satisfied by all dense orbits of chaotic maps to reconstruct fading sig-nals in the presence of chaotic noise arising from ocean clutter or ambient ocean noise. The denoising algorithm is unique inthe sense that its convergence is proven.

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