An optimal algorithm is presented for tracking the

phase of a slowly modulating signal by means of digital sampling of its sign. Error bounds and a numerical illustration are given

Bonnet, C.

Partington, J.R.

Sorine, M.

English

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Digital phase-locked loops tracked by a relay sensor

This is a repository copy of Digital phase-locked loops tracked by a relay sensor .White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/701/Article:Bonnet, C., Partington, J.R. and Sorine, M. (1999) Digital phase-locked loops tracked by a relay sensor. IEEE Transactions on Communications, 47 (5). pp. 667-669. ISSN 0090-6778 https://doi.org/10.1109/26.768757eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 5, MAY 1999 667Digital Phase-Locked Loops Tracked by a Relay SensorCatherine Bonnet, Jonathan R. Partington, and Michel SorineAbstract—An optimal algorithm is presented for tracking thephase of a slowly modulating signal by means of digital samplingof its sign. Error bounds and a numerical illustration are given.Index Terms— Demodulation, digital control, optimal algo-rithms, phase-locked loop, sampling.I. INTRODUCTIONDIGITAL phase-locked loops have been much studiedsince the 1970’s [3]–[7], [10], since they are used incommunications and system design in order to track thephase of an incoming signal. In the classical theory, thegiven data consist of actual (possibly corrupted) measurementsof the signal at appropriate sampling intervals. However,in some recent applications (e.g., [1] and [2]), it has beenfound necessary to consider approximately sinusoidal signalswhose values are only accessible after passing through a relaysensor—that is, the given data are now the sign (1) of theincoming signal. In this letter, we shall provide an algorithmfor tracking the phase changes of such a signal which, beingcentral in the sense of Information-Based Complexity theory[8], [9], possesses certain optimality properties. What thismeans is that at theth sampling instant, we know that theset of possible phase errors is an interval; to minimize theworst-case error, we must choose the midpoint of that intervalas our new estimate of the phase error.II. A PHASE-TRACKING ALGORITHMWe shall suppose that an approximately sinusoidal signal isgiven bywhere is its amplitude, its nominal frequency, andits slowly-varying phase carrying the information. We shallsuppose that is continuous as in the case of phase-coherentor frequency modulation. Demodulation will be possible underthe additional condition(1)where is sufficiently small. The measurements ofaredetected by a relay sensor, which provides us with a sequencePaper approved by M. Luise, the Editor for Sychronization of the IEEECommunications Society. Manuscript received December 5, 1997; revisedNovember 2, 1998.C. Bonnet and M. Sorine are with the INRIA Rocquencourt, Domainede Voluceau, 78153 Le Chesnay cedex, France (e-mail: {Catherine.Bonnet;Michel.Sorine}@inria.fr).J. R. Partington is with the University of Leeds, School of Mathematics,Leeds LS2 9JT, U.K. (e-mail: J.R.Partington@leeds.ac.uk).Publisher Item Identifier S 0090-6778(99)03907-0.of values, given by(2)where is a sequence of sampling times, and adisturbance that is small in magnitude, that is,withAs usual,ifif .We make no assumption about sign other than that itreturns a value in the intervalWe shall present an algorithm that is optimal when wessume a bounded-error model of noise; it should be remarkedhere that it will also have almost-optimality properties inthe case when the noise has a symmetrical distribution (e.g.,Gaussian), because it will provide an interpolatory algorithmi the sense of [8] and [9]. Moreover, if there is some absolutebound that is satisfied with high probability by the noise, thenthis value can be used to tune the algorithm. The performancein the Gaussian case will be illustrated later by means of asimulation.Further, we set and thesampling interval. is the control input of the samplingprocess. It will be determined in feedback-law form using theapriori informations and measurementsTo maximize the amount of information obtained from the signof we should sample near its zeros. As in [6], we attemptto track positive-going zero crossings of Such a closed-loop system is called a digital phased-locked loop. Note thatcondition (1) ensures that the positive-going zero crossings arewell defined. In fact, the th zero, is the unique solution ofLet(3)denote the true, unknown, phase error at theth sample. Thephase error can be interpreted as the controlled state, and(3) and (2) as the state-space equations:(4)where and are theinput and output errors.The amplitude of thea posterioriregulation error is given bywhere is the central minimax estimateof given0090–6778/99$10.00 1999 IEEE668 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 5, MAY 1999Our procedure is to choose in order to minimize thea priori worst-case phase error. Doing so, we shall obtain asa byproduct the central (minimax) estimate ofknowingTheorem II.1: Suppose thatThen, under the conditions above, the worst-case optimalphase estimates and the control are given by(5)(6)(7)and the worst-case bounds on and areand (8)where(9)Asymptotically, the estimation and phase errors satisfy(10)(11)Proof: We suppose that, for lies within aninterval indeed, we shall see that we can takefor each Assuming this for themoment, then given the additional information we knowthatfrom which we conclude thator for . The caseis slightly different, and our initial data allow usto conclude only that we may assume that, . We denote by the radiusof the interval All points of this interval are possible. Thusour central (minimax) estimate of is nowfor (12)and . To compensate for this, the optimal choicefor is given by(13)Therefore, knowing that lies in an interval of lengthwill lie in a similar interval,symmetrized about 0, and expanded by the output errorThe error is bounded in modulus by andthus we can take(14)for , whereas(15)We shall now show inductively thatfor all However, it is not necessary for ourpurposes that decrease monotonically. Note thatAs we have inductively thatfor each Further, we haveinductively that(16)by virtue of the hypothesis on We deduce thatso thatand (14) can be rewrittenAt this point, note that thatis, the control is admissible. Due to (3), the central estimateof at time satisfies(17)leading to (8). Now to obtain the second formula in (6), weuse (17) to obtain(18)which simplifies, using (4) and the definition of, to give(19)and then the formula follows using (13).Let be given by and forThen clearly for eachBONNET et al.: DIGITAL PHASE-LOCKED LOOPS 669Fig. 1.Fig. 2.The solution to a recurrence relation of the formwith isIn this case,andAs we have and, asymptotically,as asserted in (10). Likewise (11) now follows from (12). Wehave also proved the remark below.Remark II.1: The feedback law from to isaffine, but not time-invariant, since earlier measurements willnecessarily have more effect as the sampling method locksonto the correct phase. Asymptotically, however, it is approx-imately time-invariant:Fig. 3.III. EXAMPLEWe take andTo show that the method still performs very well in thepresence of Gaussian disturbances, the simulation makes useof such noise, distributed according to the lawFig. 1 shows the theoretical bound on the phase error;and Fig. 2 shows the sampling intervalThe phase error bound quickly settles down to avalue of approximately 0.27, which is in accordance with thepredicted worst-case valueLikewise, the sampling intervals are close to the true period,Fig. 3 shows the actual phase error, which is notavailable to the algorithm, but which is in practice smaller thanthe predicted worst-case bound. We remark that the algorithmis very robust, in that it rapidly compensates for any outlier.REFERENCES[1] K. Aouchiche, P.-A. Bliman, and M. Sorine, “P.I. control of periodicoscillations of relay systems,” inProc. Conf. Chaotic Oscillatory Syst.,St. Petersburg, 1997.[2] C. Bonnet, J. R. Partington, and M. Sorine, “Robust control and trackingof a delay system with discontinuous nonlinearity in the feedback,”Int.J. Contr., preprint, 1997.[3] S. C. Gupta, “Phase-locked loops,”Proc. IEEE, vol. 63, pp. 291–306,1975.[4] G. C. Hsieh and J. C. Hung, “Phase-locked loop techniques—A survey,”IEEE Trans. Indust. Electron., vol. 43, pp. 609–615, 1996.[5] W. C. Lindsey and C. M. Chie, “A survey of digital phase-locked loops,”Proc. IEEE, vol. 69, pp. 410–431, 1981.[6] H. C. Osborne, “Stability analysis of anN th power digital phase-lockedloop—Part I: First-order DPLL,”IEEE Trans. Automat. Contr., vol. 28,pp. 1343–1354, 1980.[7] , “Stability analysis of anN th power digital phase-lockedloop—Part II: Second- and third-order DPLL’s,”IEEE Trans. Automat.Contr., vol. 28, pp. 1355–1364, 1980.[8] J. R. Partington,Interpolation, Identification and Sampling. London,U.K.: Oxford University Press, 1997.[9] J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski,Information-Based Complexity. New York: Academic, 1988.[10] A. Weinberg and B. Liu, “Discrete-time analyzes of nonuniform sam-pling first- and second-order digital phase lock loops,”IEEE Trans.Commun., vol. 22, pp. 123–137, 1974.

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Digital phase-locked loops tracked by a relay sensor

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