In our development of a chaotic oscillator technology to produce high-quality communication signals, we have found a novel method for limiting the out-of-band spectral power from chaotic oscillators. This development is an important breakthrough that has allowed us to make a major step toward a commercially viable technology

Glenn, Chance

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A process for synthesizing bandlimited chaotic waveforms for digital signal transmission

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Rochester Institute of TechnologyRIT Scholar WorksArticles2003A process for synthesizing bandlimited chaoticwaveforms for digital signal transmissionChance GlennFollow this and additional works at: http://scholarworks.rit.edu/articleThis Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorizedadministrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu.Recommended CitationGlenn, Chance, "A process for synthesizing bandlimited chaotic waveforms for digital signal transmission" (2003). Accessed fromhttp://scholarworks.rit.edu/article/446 Abstract-- In our development of a chaotic oscillatortechnology to produce high-quality communication signals, wehave found a novel method for limiting the out-of-bandspectral power from chaotic oscillators. This development isan important breakthrough that has allowed us to make amajor step toward a commercially viable technology.Index Terms--chaos, chaotic, communications, transmission,digital, wireless, bandlimiting, bandwidth, filtering, symbolicdynamics, binaryI. INTRODUCTIONIn a practical wireless communication system, the signaltransmitted from the antenna must be limited to a finiterange of frequencies. Such signals are known asbandlimited signals, and if the range of frequencies does notinclude dc (zero frequency) they are called passbandsignals. Typically, passband signals are bandlimited signalsthat occupy a small percentage bandwidth about a center orcarrier frequency. Multiple signals can then be transmittedby using passbands that do not overlap, then separating thesignals at the receivers by filtering all but the passband ofinterest. This method of sending multiple signals, known asfrequency-division multiplexing, is the basis for manymultiple user systems in use today. When many users canaccess a system on either a fixed or flexible frequencydivision plan, the method is known as frequency-divisionmultiple access. Although some methods, such as CDMA(code division multiple access) do not rely upon frequencydivision for signal separation, the signals are still limited toa passband defined by the FCC1.If the signals generated by chaotic systems are to be used incommercially viable systems, then the transmitted signalmust be bandlimited in some way. One obvious and often-used way to do this is to generate a signal that hasconsiderable out-of-band spectral power, and then filter thesignal to remove most of the power that lies outside of thedesired frequency band. This method is, however, costly inmany ways. Filters that can remove out-of-band energywell while passing the signal in band with minimaldistortion are expensive and take up space. The poweroutside of the band is also wasted, and must be dissipated to                                                            Chance M. Glenn is President and Director of Technology Developmentfor NextWave Technologies, Inc. , in Columbia, MD.  (www.nextwv.com), and a Ph.D. candidate at The Johns Hopkins University.heat. This wasted power translates to a higher requiredtransmitter power and faster battery drain.In our development effort to produce commercially viablechaotic oscillator technology, we have discovered aparticularly simple and effective way to limit the out-of-band radiation from a chaotic oscillator. This methodcauses the oscillator itself to produce a signal withbandwidth-constrained signal power, and thus there is noneed for a filter or waste of power. In doing so we areapplying a principle that may be used for more generalsignal shaping or spectral shaping.The idea behind our bandlimited chaotic oscillation (BCO)synthesis method is based on our segment hopping methodof oscillator control2. In segment hopping, a digital sourceproduces an analog waveform that is used to guide thetransmit oscillator. The guide signal is an analog copy of asignal that could be produced by the transmit oscillatoritself, except that it follows a pre-defined symbol sequencethat contains the digital information being transmitted. Inthis scheme, the transmit oscillator is acting as an amplifierfor the guide signal, because the output of the transmittercan be much higher in power than the power drawn fromthe guide source.II. SYNTHESIS/ANALYSISA prototypical chaotic oscillator used to produce signalsuseful for digital communication is the Lorenz system3.  Itwas the oscillator first used by Hayes4 to introduce thenotion of controlling symbolic dynamics, a process toencode digital information in the oscillations of a chaoticsystem. The Lorenz system is described by a threedimensional system of equations having the form:where σ, ρ and β are parameters that Lorenz orignally set to10, 28, and 8/3 respectively. The state-space attractordefined by these equations takes on a double-lobedstructure which lends itself nicely to a binary symbolpartition. Fig. 1 is an example of a two-dimensionalprojection of the solutions of the Lorenz equations, showingA Process for Synthesizing Bandlimited ChaoticWaveforms for Digital Signal TransmissionChance M. Glennzxyzxzyxyxyxβρσσ−=−−=−=!!!Bandlimitedsegment-hoppingsourceContinuous-timeLorenz Oscillatory[n;t]synchronizationx(t)y(t)z(t)Figure 4. Block diagram of the implementation of abandlimited digital signal synthesizer.10001100001010…binary sequencestate-variables x and y. Two Poincaré surfaces are placed inpositions piercing the focal points and extending outwardbeyond the boundaries of attractor.The time varying state-variable, x(t), is a bipolar waveformwell-suited for baseband transmission of digitalinformation.  Fig. 2 is an example of the waveform in it’sgeneralized time coordinates, and Fig. 3 a plot of thefrequency content of this signal with respect to it’s averagecycle frequency.   The average cycle frequency is thereciprocal of the average cycle time. The cycle time isdefined as the time it takes a point on the attractor to travelfrom one Poincaré surface to the other, or back to itself.The average cycle time is the mean time, in dimensionlessunits, calculated by integrating the system with a fixedintegration step and collecting  thousands of  surfacecrossings.  Tavg was found to be about 1.7, with a varianceof about 1.8.Fig. 4 is a block diagram of the implementation of abandlimited chaotic signal source.  A binary sequence is fedinto a discrete-time, bandlimited, segment-hopping source.We will describe this source in the next section.  The outputsignal, y[n;t], is a sampled waveform used as a guide signalto synchronize a continuous-time Lorenz oscillator to it. Asingle state-variable synchronization method is used to lockthe oscillator to the dynamics, thus to the embedded digitalsequence of the guide signal.Since Pecora and Carroll first introduced the concept ofsynchronization of chaotic oscillator circuits5 manymethods have been developed and realized.  The followingmathematical model works extremely well:30 35 40 45 50 55 60-20-15-10-505101520normalized time (sec)x(t)Figure 2. Time dependent solution of the Lorenzequations for state coordinate x(t).0 2 4 6 8 10 12 14 16 18 20-100-90-80-70-60-50-40-30-20-100average cycle frequencyX (dB)Figure 3. Frequency content of the statecoordinate x(t) with respect to the averagecycle frequency.( )zxyzRytnyxzyxyxyxβρσσ−=−+−−=−=!!!];[Figure 1. Two-dimensional projection of the solutionsto the Lorenz equations showing state coordinates xand y and the binary partitioning of the state-space.-20 -15 -10 -5 0 5 10 15 20-30-20-100102030xy 1 00 1 2 3 4 5 6 7 8 9 10-70-60-50-40-30-20-100average cycle frequencyY (dB)Figure 5. The result of the low-pass filtering of the state-variable of y(t).unfilteredfilteredHere R is a coupling factor. This method works well fromthe standpoint of simplicity and ultimately from a circuitrealization perspective. There have been circuit realizationsproposed for the Lorenz equations6. In such circuits thiscoupling method is simply a resistive feed of a voltageacross, or current into, a single arm.III. SEGMENT HOPPINGNow we turn our attention to the synthesis of the guidesignal, y[n;t]. Two new concepts are introduced here.1. Continuous, digitally encoded chaos waveforms can begenerated by piecing together the proper signalsegments.2. These segments can be bandlimited prior to storage,and can impress its characteristics upon a continuoustime oscillator via synchronization.The sequencing of stored segments in memory in some pre-defined order is commonly used in arbitrary waveformsynthesis. What is new and not obvious about segmenthopping is that an arbitrary controlled trajectory of adeterministic dynamical system can be generated, eventhough the system produces continuous and non-repeatingtrajectories in state space under the action of deterministicdifferential equations. This signal can furthermore be madeto carry an arbitrary sequence of digital symbolsrepresenting encoded data. Thus the difference between thismethod and arbitrary waveform synthesis is that thismethod allows for the production of a signal carryingarbitrary data that appears to have been produced by theaction of differential equations. This is achieved by puttingout segments that follow the desired symbol sequence whilesatisfying the grammar of the oscillator. The theoreticalbasis for this method of signal synthesis is the idea fromergodic theory that chaos can be approximated to anarbitrary degree of accuracy by completely deterministicmappings in state space intermediated by completelyrandom choices7.For example, in an 8-bit encoding there are 256 signalpieces, or segments, that can be put together to form anydesired binary symbol sequence. The segments are assignednumbers from 0 to 255 according to the bit sequence theyinitiate.In a physical implementation, a segment hopping systemcan be stored in a static memory device such as a ROM orEPROM and clocked out according to input the bit-sequence.A more detailed description of the segment hopping processwill be published shortly. In the following section weconsider a complete computer model of the BCO synthesismethod.IV. COMPUTER MODEL RESULTSThe first, important step is to determine the amount ofbandlimiting that can be achieved. Any amount of filteringwill cause some distortion to the waveforms. It is thenimportant that the binary encoding remain preserved, andthat the guide signal remain capable of synchronizing thecontinuous-time oscillator.Given the determination of the average cycle time, Tavg, asstated earlier, we applied a smooth, low-pass filter to theLorenz oscillations having a cutoff-frequency at twice theaverage bit rate. With the filtering applied we stored thesegments which source the appropriate 8-bit symbolsequences. Figs. 5 and 6 below show the filtercharacteristics and the response of the frequency spectraand the resultant attractor.-15 -10 -5 0 5 10 15-15-10-5051015x (filtered)y (filtered)Figure 6. Two-dimensional projection of a filteredLorenz oscillation using the low-pass filter characteristicshown by its effect on the y spectrum in fig. 5.5 10 15 20 25 30-20-15-10-505101520normalized time (sec)x(t)Figure 7. Typical Lorenz oscillation producing thebinary sequence,B = 11001101100010101111000110101100011110011011000101011110001101011000110 1 2 3 4 5 6 7 8 9 10-70-60-50-40-30-20-100average cycle frequencyX (dB)Figure 8. Frequency spectra of the x(t) from theLorenz equations, a filtered version, and the output ofa BCO system.x(t)x(t) filteredx(t) from BCO5 10 15 20 25 30-15-10-5051015normalized time (sec)bandlimited x(t)11001101100010101111000110101100011Figure 9. BCO synthesized transmit signal. Note thatthe binary encoding is preserved.The filtering is applied to all three state coordinates in orderto produce a new attractor. It is with this new attractor thatthe symbolic dynamics of the system are determined andthe associated segments are produced. Even though onlyone state-coordinate is used as a guide signal, namely y, theentire attractor must be transformed in order to synthesizethe segments needed for segment hopping control.In the following example we will encode and transmit thebinary sequence,B = 11001101100010101111000110101100011.Fig. 7 shows what the desired output waveform x(t) wouldbe for a typical Lorenz oscillator.Given the bandlimited segments, we now can synthesize theguide signal, y[n].  Fig. 8 is a plot of the frequencyspectrum of the original, unfiltered transmit signal x(t), theresult if x(t) was filtered using the low-pass filter described,and the frequency spectrum of the new transmit signal fromthe BCO synthesis method.Note the dramatic reduction in frequency content.Particularly, there is nearly a 20 dB reduction at 3.5 timesthe cycle frequency and a 30 dB reduction at 6 times thecycle frequency.Earlier we stated that it is important that binary encodingremain preserved and that synthesized BCO guide signal becapable of synchronizing a Lorenz oscillator to it. Figs. 9and 10 demonstrate this.What is remarkable is how “flexible” the Lorenz oscillatorcan be. This is an important realization not only for thiswork but as we continue to work towards synthesizingsignals even more compatible with traditionalcommunication signal formats and standards.  We havebeen able to demonstrate in the lab in prototypes thatphysical chaotic oscillators can be made to produce signals-15 -10 -5 0 5 10 15-15-10-5051015y (sync)x (sync)Figure 10. BCO synthesized attractor projection.that have constant timing intervals between Poincarésurface crossings by simply synchronizing it to anartificially synthesized waveform having those properties.This timing regularization is of utmost importance for usein commercially viable communication systems because itmakes accurate clock timing recovery is possible.Combined with bandwith compression, as outlined here, wehave solved some of the most critical technologicalproblems for using chaotic oscillators in commercialsystems.V. CONCLUSIONWhat we have described here in a simple example is anenabling technology. In order to use chaotic systems andprocesses in commercial digital communication systems,the waveforms must be restricted in bandwidth in acontrollable way. The focus must shift to the design ofwaveforms and sources, and away from the use of existingeasily-constructed oscillators.Our BCO synthesis technique is a simple, efficient,effective way to generate baseband signals for wirelesscommunication.  Some chaotic oscillators, such as theColpitt’s circuit8 and the double scroll oscillator9, producesignals with very different characteristics.  However, thistechnique is a general procedure applicable to a variety ofchaotic oscillators.Another area of technology development that we arefocusing on is in the development of oscillators that aremore ideally suited for given communication channels.Although the Lorenz system has excellent properties forbinary baseband digital signaling in white noise, there maybe oscillations even more suitable that can produced withefficient circuitry.Since Ott, Grebogi, and Yorke’s initial formalism forcontrolling chaotic processes using small perturbations10,there has been a consistent march toward development oftechnological applications.  Arguably, the application tocommunications technology has held the most promise andhas been the source of the most activity. Beyond simpleapplications is the realm of commercially viableapplications that have characteristics that make chaoticdynamics technology attractive to the fast-moving world ofthe telecommunications industry.Some of he improvement areas that make chaotic dynamicstechnology attractive are increases in efficiency, reductionof system complexity, increase in transmission ranges anddigital transmission data rates.VI. ACKNOWLEDGEMENTSThe author would like to acknowledge the input andsupport of Dr. Scott T. Hayes in the ongoing developmentof these concepts.This work was supported by funding from the Office ofNaval Research (ONR) and from the Defense AdvancedResearch Projects Agency (DARPA).VII. REFERENCES[1] Theodore S. Rappaport, Wireless Communications: Principles andPractice, Prentice Hall, Inc., New Jersey, 1996.[2] Scott T. Hayes, Chance M. Glenn, Syncrodyne Amplification,Phase II Final Report, NextWave Technologies, Contract # N00014-00-C-0044, Office of Naval Research, 2000.[3] Edward N. Lorenz, Deterministic Nonperiodic Flow, Journal ofthe Atmospheric Sciences, vol. 20, March 1963.[4] S. Hayes,  C. Grebogi, E. Ott, Phys. Rev. Lett. 70, 3031 (1993).[5] L. M. Pecora and T. L. Carroll, Synchronization in ChaoticSystems, Phys. Rev. Lett. 64, 821 (1990).[6] R. Tokunaga, M. Komuro, T. Matsumoto, L.O. Chua, ‘LorenzAttractor’ From an Electrical Circuit with Uncoupled ContinousPiecewise-Linear Resistor, International Journal of Circuit Theoryand Applications, Vol. 17, 71-85 (1989).[7] Athanasios Papoulis, Probability, Random Variables, andStochastic Processes, 3rd ed., McGraw-Hill, Inc., New York, 1991.[8] Martin J. Hasler, Electrical Circuits with Chaotic Behavior,Proceeding sof the IEEE, vol. 75, no. 8, August 1987.[9] T. Matsumoto, L. O. Chua, M. Komuro, The Double Scroll, IEEETrans. CAS-32, no. 8 (1985), 797-818.[10] E. Ott, C. Grebogi, J. A. Yorke, Phys. Rev. Lett. 64, 1196(1990).

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A process for synthesizing bandlimited chaotic waveforms for digital signal transmission

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