http://www.eurasip.org/Proceedings/Ext/SampTA2013/papers/p193-bidegaray-fesquet.pdfInternational audienceWe address the problem of quantifying the number of samples that can be obtained through a level crossing sampling procedure for applications to mobile systems. We specially investigate the link between the smoothness properties of the signal and the number of samples, both from a theoretical and a numerical point of view

Bidégaray-Fesquet, Brigitte

Clausel, Marianne

Publication in the conference proceedings of SampTA, Bremen, Germany, 201

Bidegaray-Fesquet, Brigitte

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Level crossing sampling of strongly monoHölder functions

Hal - Université Grenoble Alpes

Level crossing samplingof strongly monoHo¨lder functionsBrigitte Bidegaray-Fesquetand Marianne ClauselUniv. Grenoble Alpes, LJK, BP 53, 38041 Grenoble Cedex 9, FRANCEEmail: Brigitte.Bidegaray@imag.fr and Marianne.Clausel@imag.frAbstract—We address the problem of quantifying the numberof samples that can be obtained through a level crossing samplingprocedure for applications to mobile systems. We speciallyinvestigate the link between the smoothness properties of thesignal and the number of samples, both from a theoretical anda numerical point of view.I. INTRODUCTIONAn important issue in the design of mobile systems is toincrease their autonomy and/or reduce their size and weight.This can be achieved by reducing their power consumptionby processing signal with a smaller number of samples. Fora large class of signals, especially sporadic signals, non-uniform sampling leads to a reduced number of samples,compared to a Nyquist sampling [7], [9], [10], [13]. A wayto obtain such samples is to use specific system architectures(e.g. event-driven). These architectures take samples each timesome specific event occurs, typically specific voltage levelsare crossed. We can therefore design simple analog circuits,with low power consumption, to acquire information, possiblyat high speed. Here we consider a system where amplitudesare selected thanks a M -bit asynchronous analog-to-digitalconverter (AADC) and 2M levels are predefined in the voltagerange.In this article we want to understand on which signal charac-teristics the number of samples depend. An intuitive look at theproblem indicates that the more the signal is oscillating locallythe higher the number of samples is. The number of samplesat the neighborhood of some point may then be related to thelocal smoothness of the signal, that is to its so–called Ho¨lderregularity. To put in evidence this relationship, we consider toymodels of signals whose smoothness properties are perfectlyknown at each point. We then perform numerical simulationsand link the sample reduction rate with this regularity. Thenext step, which will be the purpose of a forthcoming paper,will then be to consider signals whose regularity may changefrom point to point such as multifractional or multifractalsignals. We then intend to apply our results to biologicalsignals such as EEG signals or fMRI data which are well-known to be both highly irregular and non stationary signals,and which provide interesting ranges of application for non-uniform signal processing.II. ALGORITHMIn the event-driven systems, the signals are not sampledat totally arbitrary times. Indeed there are local clocks thatmeasure the time elapsed since the previous sample was taken.Therefore we can consider that the samples are taken at somemultiples of some basis time tb. The mathematical algorithmthat is used to mimic the AADC is the following:• Step 1: generate uniform samples on [0, 1] with samplinginterval tb;• Step 2: for each sample replace the amplitude by thevalue of the level just below;• Step 3: decimate the samples so as to keep only one (thelast) sample when consecutive samples have the sameamplitude.III. MATHEMATICAL INTERPRETATIONUp to some time re-scaling we suppose that the precisionof the local clock that measures time delays is tb = 2−j , forj ∈ N. At best we only know the function f by its samplesat times k2−j , k ∈ Z. At scale 2−j , we define the intervalsIj,k = [k2−j , (k + 1)2−j [.A. Faber–Schauder hierarchical basisWe define the Faber–Schauder hierarchical basis as definedin [5]. Let Vj be the space of continuous functions, whichare affine on intervals Ij,k, k ∈ Z. We can uniquely definethe linear interpolation fj of f at scale 2−j by fj(k2−j) =f(k2−j), for all k ∈ Z. Let ϕ(x) = max{0, 1−|x|}. A naturalbasis for Vj is given by the functions ϕj,k = ϕ(2j ·−k), k ∈ Z.In this basis, we have the unique representationfj =∑k∈Zf(k2−j)ϕj,k.B. InterpretationWe now suppose that f is compactly supported in [0, 1]. Inthe previous notations we will only need k = 0, . . . , 2j − 1.We assume that levels are uniformly spaced by some quan-tum 2−M . The level crossing algorithm can be described asfollows.Step 1: approximation in Vj . We only know fj(k2−j) =f(k2−j).Proceedings of the 10th International Conference on Sampling Theory and Applications193Step 2: level crossing. We denote bxc the integer part of x,namely bxc = inf{n ∈ N, x ≤ n}. We then definef˜j =∑k∈Z2−M⌊2Mf(k2−j)⌋ϕj,k.Step 3: decimation of f˜j . We only keep a subsequence ofk = 0, . . . , 2j − 1, defined by induction: k0 = 0 andki+1 = min{k ≥ 1 + ki/⌊2Mf(k2−j)⌋ 6= ⌊2Mf(ki2−j)⌋}.To be able to reconstruct f˜j , we only store the couples (δti, ai)where δti = (ki − ki−1)2−j , i ≥ 1 is the delay since the lastsample, and ai = 2−M⌊2Mf(ki2−j)⌋) is the amplitude ofthe sample. There is no approximation in Step 3, we only donot store redundant data.IV. MATHEMATICAL PROPERTIESWe now want to illustrate through numerical experimentsthat the properties of our algorithm can be related to smooth-ness properties of the sampled signal.A. MonoHo¨lderian functionsDefinition 1 (Ho¨lder space Cα): Let α ∈ (0, 1). The func-tion f belongs to the Ho¨lder space Cα([0, 1]) if there exists aconstant C such that for all (x, y) ∈ [0, 1]2,|f(x)− f(y)| ≤ C|x− y|α .The following definition has been introduced in [12].Definition 2 (Anti-Ho¨lderian functions): Let α ∈ (0, 1).The function f is said to be uniformly anti-Ho¨lderian withexponent α, if there exists a constant C such that for all(x, y) ∈ [0, 1]2,sup(u,v)∈[x,y]2|f(u)− f(v)| ≥ C|x− y|α .The set of uniformly anti-Ho¨lderian functions is denotedIα([0, 1]).Definition 3: Let α ∈ (0, 1). If the function f both belongsto Cα([0, 1]) and Iα([0, 1]) then f is said to be monoHo¨lderianwith exponent α.B. Approximation propertiesAs already mentioned, only step 1 and 2 lead to approxima-tions. If f ∈ Cα([0, 1]), it is well–known [6], [11] that thereexists a constant C (which depends on f but not on the scalej) such that‖f − fj‖L∞ ≤ C2−jα,whereas, if the function f is assumed to be uniformlymonoHo¨lderian, one deduces from [4] that there exists aconstant C (which depends on f but not on the scale j) suchthat for any  > 0‖f − fj‖L∞ ≥ Cj−(2α+)2−jα.Note that the last condition is much weaker than uniform anti–Ho¨lderianity (see [4]) since it involves the modulus of continu-ity of f − fj on the whole interval [0, 1], whereas oscillationsof uniformly anti-Ho¨lderian functions can be bounded frombelow at any point. The approximation made at step 2 clearlydoes not depend on the regularity of function f , and we have‖fj − f˜j‖L∞ ≤ 2−M .C. Theoretical number of samples in the case of a monotonousfunctionIf f is a monoHo¨lderian function with exponent α, bydefinition there exists C1, C2 > 0 and for any scale j ≥ 0and 0 ≤ k ≤ 2j − 1C12−jα ≤ sup(u,v)∈[k/2j ,(k+1)/2j ]2|f(u)− f(v)| ≤ C22−jα.If the function is additionally supposed to be monotonous, wehave further thatsup(u,v)∈[k/2j ,(k+1)/2j ]2|f(u)−f(v)| = |f((k+1)/2j)−f(k/2j)|.HenceC12j(1−α) ≤ |f(1)− f(0)|=2j−1∑k=0|f(k + 12j)− f( k2j)| ≤ C22j(1−α).Such a signal crosses equi-spaced levels with quantum 2−Mat most C2M+(1−α)j times. With our algorithm, we also takeat most 2j samples (since we use the initial samples). Forlarge values of M (or small values of α), we indeed keepalmost all the 2j samples. Otherwise we can expect somereduction of the number of samples. For C = 1, the thresholdis M ' αj. Observe that the proof is based on the fact, thatin the monotonous, we can estimate in a very simple way theoscillationssup(u,v)∈[k/2j ,(k+1)/2j ]2|f(u)− f(v)|of the function. Of course in the general case, the situationcan be much more complicated. Nevertheless, generic resultsin the sense of prevalence as stated in [2] are expected to hold.In what follows, we illustrate through numerical simulationswhat may happens.D. Numerical simulations1) Test functions: We test level crossing on two toy models:sample paths of fractional Brownian motion BH and theWeierstrass function WH . Here H ∈]0, 1[ is called the Hurstindex. In each of these two cases, the smoothness propertiesof the function are well–known and related to the Hurst index.The fractional Brownian motion (fBm) BH is the uniqueGaussian H–self-similar process with stationary increments.It can be defined from its covariance functionE[BH(x)BH(y)]= 12(|x|2H + |y|2H − |x− y|2H)for all (x, y) ∈ [0, 1]2. For H = 1/2, we recover theclassical Brownian motion. We recall that the sample pathsof fBm are well-known to be almost surely continuous.Further, the Hurst index H of fBm is directly related to theroughness of its sample paths. More precisely, almost surely,Proceedings of the 10th International Conference on Sampling Theory and Applications194Fig. 1. Three realizations of fractional Brownian motions for H = 0.5 (blueplot), H = 0.7 (red plot), H = 0.9 (green plot)Fig. 2. The Weierstrass function for H = 0.5 (blue plot), H = 0.7 (redplot), H = 0.9 (green plot)BH ∈ CH−ε([0, 1])∩IH+ε([0, 1]) (classical law of the iteratedlogarithm). Roughly speaking, a.s. for all (x, y) ∈ [0, 1]2,sup(u,v)∈[x,y]2 |BH(u)−BH(v)| ∼ |x−y|H . Figure 1 presentsthree realizations of sample paths of fractional Brownianmotions for H = 0.5, H = 0.7, H = 0.9 and 1024 samples(j = 10).We also use the Weierstrass function defined asWH(x) =∞∑j=02−jH cos(2jx).The Weierstrass function WH is a classical example ofmonoHo¨lderian function with exponent H as proved in [8].Hence for all (x, y) ∈ [0, 1]2, sup(u,v)∈[x,y]2 |WH(u) −WH(v)| ∼ |x − y|H . Figure 2 present some graphs ofWeierstrass functions for H = 0.5, H = 0.7, H = 0.9 and1024 samples (j = 10).Fig. 3. Average number n of samples in terms of the Hurst number in thelog scale (log2(n) is represented on the y-axis). Four cases are plotted (solidlines) corresponding to j = 10 and 13 and M = 4 and 5. The dotted linescorrespond to the worst case M + (1−H)j.M = 4 M = 5j = 10 0.4 0.5j = 13 ∼ 0.3 ∼ 0.4TABLE I“CRITICAL” VALUES OF THE HURST NUMBER.2) Test cases: The tests are performed within the SPASSMatlab toolbox [1]. To generate fractional Brownian motions,we make use of the genFBMJFC.m function [3].We use two values of j (10 and 13) and two values ofM (4 and 5). These small values of M are sufficient for mostmobile applications. Our output is the number of samples afterdecimation (Step 3). For the fractional Brownian motion, weperform 1000 realizations and average the number of samplesobtained for each realization to obtain an average number n.We perform this for values of the Hurst number H in the]0, 1[ range and obtain the curves in Figure 3. We also plot thenumber of samples computed in the worst case (monotonousfunction i.e. maximum total variation) for C = 1: 2M+(1−H)j .The plots are given in the semi-log scale: log2(n) and M +(1 − H)j. This allows to distinguish the two regimes belowsome value of the Hurst number H ∼ M/j the algorithmmore or less keeps all the original samples, above this valuethe decimation is efficient and yields a significant reductionof the number of samples.For the different curves these “critical” values of H aregiven in Table I.We perform the same tests on the Weierstrass function. Theplots associated to fBm are much more regular because thereare obtained by an averaging procedure. We expect that thecritical value of M holds in an asymptotical way. Our resultsare then expected to improve when j tends to infinity.Proceedings of the 10th International Conference on Sampling Theory and Applications195Fig. 4. Number of samples in terms of the Hurst number in the log scale(log2(n) is represented on the y-axis). Four cases are plotted (solid lines)corresponding to j = 10 and 13 and M = 4 and 5. The dotted linescorrespond to the worst case M + (1−H)j.V. CONCLUSIONWe have first shown numerically that there is strong rela-tionship between the smoothness properties of a signal and thenumber of samples that can be obtained by the crossing levelalgorithm presented in this paper. We also proved this result inthe case of monotonous monoHo¨lderian functions. We intentto address this problem in more general cases. It will be thepurpose of a forthcoming paper.ACKNOWLEDGMENTThis work has been supported by the MathSTIC ProjectOASIS of the Grenoble University. The authors want to thankJean-Franc¸ois Coeurjolly for fruitful discussions.REFERENCES[1] B. Bide´garay-Fesquet and L. Fesquet, Signal Processing for ASyn-chronous Systems toolbox, Matlab toolbox, 2011.[2] M. Clausel and S. Nicolay, Some prevalent results about stronglymonoHo¨lder functions, Nonlinearity, 23(9), 2101–2116, 2010.[3] J.-F. Coeurjolly, Simulation and identification of the fractional Brownianmotion: a bibliographical and comparative study, Journal of StatisticalSoftware, 5(7), 1–53, 2000.[4] M. Clausel, S. Nicolay, A wavelet characterization for the upper globalHo¨lder index., to appear in J. Fourier Anal. Appl.2013.[5] A. Cohen, Wavelet Methods in Numerical analysis, Handbook of Numer-ical Analysis, Vol. VII, Elsevier, 2000.[6] R.A. DeVore, B. Jawerth, B.J. Lucier, Image Compression ThroughWavelet Transform Coding, IEEE Trans. Inf. Theory 38(2):719–746,1992.[7] K. Guan and A.C. Singer, Opportunistic sampling by level-crossing,In IEEE International Conference on Acoustics, Speech and SignalProcessing, ICASSP 2007, pages 1513–1516, Honolulu, Hawai’i, USA,April 2007. IEEE.[8] G.H. Hardy, Weierstrass’s non differentiable function, Trans. Amer. Math.Soc., 17, pp.301–325 (1916).[9] J.W. Mark and T.D. Todd, A nonuniform sampling approach to datacompression, IEEE Trans. Commun., 29(1):24–32, 1981.[10] F.A. Marvasti, Nonuniform Sampling. Theory and Practice, InformationTechnology: Transmission, Processing and Storage. Springer, 2001.[11] Y. Meyer, Ondelettes et ope´rateurs, Hermann, 1990.[12] C. Tricot, Courbes et dimension fractale, Springer (1992).[13] N. Sayiner, H.V. Sorensen, and T.R. Viswanathan, A level-crossing sam-pling scheme for A/D conversion, IEEE Trans. Circ. Syst. II, 43(4):335–339, 1996.Proceedings of the 10th International Conference on Sampling Theory and Applications196

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Level crossing sampling of strongly monoHölder functions

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