Generation of ultrawideband phase chaos in the decimeter band

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ISSN 1064-2269, Journal of Communications Technology and Electronics, 2009, Vol. 54, No. 6, pp. 675–684. © Pleiades Publishing, Inc., 2009.Original Russian Text © A.S. Dmitriev, A.V. Kletsov, L.V. Kuz’min, 2009, published in Radiotekhnika i Elektronika, 2009, Vol. 54, No. 6, pp. 709–718.

INTRODUCTION

At present, ultrawideband signals are intensivelyused for data transmission [1–3]. Dynamic chaos is oneof promising types of ultrawideband signals. For exam-ple, it is used in the direct chaotic communication cir-cuit proposed at the Institute of Radio Engineering andElectronics, Russian Academy of Sciences [4–7]. Inthis circuit, chaotic radio pulses are used as a data car-rier. The direct chaotic scheme of wireless data trans-mission is included in an international standard of ultra-wideband local-area wireless communication as anoptional solution [8].

At present, transistor chaos generators based onself-oscillation systems with a small number of degreesof freedom are used as sources of chaotic oscillations[9–12]. This technology has been well developed in therecent years [10, 12]. It allows development of compacttransceivers [13] and can be implemented in the form ofmonolithic integrated microcircuits [14].

Transistor chaos sources generate amplitude chaos:oscillations whose amplitude irregularly varies withtime. However, for a number of applications, in partic-ular, for application of chaotic radio pulses to deter-mine the position of a transceiver, it is more preferableto use phase, rather than amplitude, chaos, i.e., signalswhose amplitude in a pulse is practically constant,while the phase varies chaotically.

At least two approaches can be used to obtain phasechaos: (i) modulation of the phase (frequency) of ahigh-frequency harmonic signal generated by a volt-age-controlled oscillator (VCO) by an amplitude-cha-otic signal with the use of a phase (frequency) modula-tor and (ii) application of a phase-locked loop (PLL)operating in the chaotic mode. In the latter case, chaotic

signal

ϕ

(

t

)

is generated in the feedback loop of thephase system. This signal then enters the PLL input andprovides chaotic phase deviation of its oscillations

X

(

t

) =

A

sin

(

ω

0

t

+

ϕ

(

t

)), (1)

where

A

is the amplitude of phase-chaotic oscillations,

ω

0

is the frequency of free oscillations in the VCO, and

t

is time.This paper is devoted to the theoretical and experi-

mental investigation of the possibility of obtainingwideband phase chaos in the decimeter frequency bandusing a PLL.

Conventionally, in the analysis of PLL systems, thephase dynamics of oscillations obtained at the VCOoutput is of interest [15–19]. However, from the point ofview of application of a PLL as the source of high-fre-quency signals—data carriers—high-frequency oscilla-tions at the VCO output, rather than the phase of theseoscillations, are of interest. In this paper, primary attentionis paid to the investigation of these oscillations.

1. MATHEMATICAL MODEL

From the point of view of efficient use of frequencyresources in wireless communication systems, it is nec-essary to form chaotic signals with a power spectrumclose to uniform in a given frequency band. It wasshown in [20] that, for obtaining wideband (ultrawide-band) VCO oscillations with chaotic phase and a spec-tral density uniform in a frequency band, it is necessaryto implement phase chaos of the vibrational–rotationaltype in the PLL.

For formation of such oscillations, a PLL with thephase dynamics described by the third-order ordinarydifferential equation

Generation of Ultrawideband Phase Chaosin the Decimeter Band

A. S. Dmitriev, A. V. Kletsov, and L. V. Kuz’min

Received April 24, 2008

Abstract

—The problem of obtaining ultrawideband phase chaos in the decimeter band with the use of a phase-locked loop is considered. The mathematical simulation of a third-order phase-locked loop is performed withaccount for the real characteristics of the phase detector, the signal frequency divider in the phase-locked loop,and the voltage-controlled oscillator. This simulation allows determination of the parameters of a prototype ofa phase-locked loop operating in the chaotic generation mode. Based on the obtained results and available cir-cuit technology, a prototype of a phase-locked loop generating ultrawideband phase-chaotic oscillations with auniform power spectrum in a frequency range from 700 to 1300 MHz is developed and tested.

PACS numbers: 05.45.Vx, 07.57.Hm

DOI:

10.1134/S1064226909060096

DYNAMIC CHAOS IN RADIOPHYSICS AND ELECTRONICS

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Vol. 54

No. 6

2009

DMITRIEV et al.

(2)

or the equivalent system of three first-order differentialequations,

(3)

can be used.

Here,

γ

,

µ

,

ε

are the parameters;

F

(

ϕ

)

is the nonlinearfunction describing the characteristic of the phasedetector (PD); and

y

and

z

are the phase variables.

The dynamics of such PLLs was studied in detail forthe sinusoidal nonlinear PD characteristic

F

(

ϕ

) =

sin

(

ϕ

)

[17–19]. However, in modern commercial high-fre-quency PLLs, the characteristic of the phase detectorrealized on the

XOR

element is a piecewise linear func-tion of the form

(4)

rather than a sinusoid. It should also be mentioned thatthese PLLs contain frequency dividers. The structure ofsuch a PLL system is shown in Fig. 1. If a second-orderlow-pass filter (LPF) is used in the phase locked loop,

µϕ εϕ ϕ+ + γ F ϕ( )–=

ϕ y, =

y z, =

µz γ F ϕ( )– y– εz,–=

FXOR ϕ( ) 2 π --- ϕ π

2---– ⎝ ⎠

⎛ ⎞ , 0 ϕ π , < <

2 π --- 3 π 2

------ ϕ – ⎝ ⎠⎛ ⎞ , π ϕ 2 π , < <

=

the practical PD characteristic (see Section 3) has theform

(5)

The dimensionless parameters of Eqs. (2) and (3) andthe initial PLL parameters are related as follows [21]:

(6a)

(6b)

(6c)

Here,

Ω

l

is the locking frequency range;

R

,

L

,

C

arethe LPF resistance, inductance, and capacitance,respectively;

m

and

n

are the integer numbers corre-sponding to division of the oscillation frequencies ofthe VCO and the reference generator; and

ω

RG

is theoscillation frequency of the reference generator.

Thus, the mathematical model for investigation ofthe phase dynamics in the considered device based on acommercial PLL system is described by Eqs. (2) and(3) with nonlinear function (5) and the parametersdetermined from relations (6), which take into accountthe discrete nature of parameter

γ

.

2. NUMERICAL SIMULATION

Application of a PD with sinusoidal characteristichas shown [17–19] that, in a PLL system described byformulas (2) and (3) with large dissipation in the sec-ond-order filter, chaotic oscillations appear in smallranges of the parameter values of the PLL system.Therefore, the first problem in the analysis of dynamicsof a PLL system with nonlinearity (5) of the phasedetector is the elucidation of the possibility of extend-ing the regions of chaotic generation and improvementof the characteristics of chaotic signals without consid-erable complication of the PLL system.

Let us show that this problem can be solved viareduction of dissipation in the second-order LPF. Fig-ure 2a shows the bifurcation diagram of ae PLL systemwith nonlinear characteristic (5) for the case of largedissipation ( ε = 1). Similarly to the case of sinusoidalcharacteristic, only narrow chaos regions are observed.The situation changes considerably if the dissipation isseveral times lower (Fig. 2b,

ε

= 0.35). In this case, thechaos formation regions in the parameter space areextended and are dominant in the whole region of vari-ation of parameter

γ

.

FXOR ϕ( ) 2 π --- ϕ π

2---– ⎝ ⎠

⎛ ⎞ , 0 ϕ π , < <

0, π ϕ 2 π . < <

=

µΩl

m-----⎝ ⎠

⎛ ⎞2

LC,=

εΩl

n-----RC,=

γ nΩl-----

ωRG

m---------

ω0

n------–⎝ ⎠

⎛ ⎞ .=

RGFD1

PDFD2

x

(

t

)

VCO LPF CA

y

(

t

)

Fig. 1.

Block diagram of a PLL: VCO is the voltage-con-trolled oscillator, PD is the phase detector, RG is the refer-ence frequency generator, CA is the controlled amplifier,LPF is the low-pass filter, FD1 is the frequency divider forthe signal of the reference generator with division coeffi-cient

m

, FD2 is the frequency divider for the signal of thevoltage controlled generator with division coefficient

n

;

x

(

t

)

is the VCO output signal, and

y

(

t

)

is the LPF output signal.

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GENERATION OF ULTRAWIDEBAND PHASE CHAOS IN THE DECIMETER BAND 677

max(

ϕ

)

1

0

–10 0.2 0.4 0.6 0.8

γ

(‡)

(b)max(

ϕ

)

1

0

–10 0.2 0.4 0.6 0.8

γ

2

Fig. 2.

Bifurcation diagram for parameter

γ. Here, µ = 3 and (a) ε = 1 and (b) ε = 0.35.

The analysis of the oscillation modes correspondingto the values of parameters µ, ε, γ that were used for con-struction of the diagram in Fig. 2b shows that, in themodel, there is vibrational chaos at small γ (Fig. 3a), wide

regions of vibrational–rotational chaos at γ ∈ (0.15, 0.65)(Fig. 3b), and rotational chaos at γ ~ 0.7 (Fig. 3c).

As was found in [20] and noted above, vibrational–rotational chaos is the most appropriate for application

.

.

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JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 54 No. 6 2009

DMITRIEV et al.

in wideband and ultrawideband communication sys-tems using chaotic radio pulses. This is exactly the casewhen the spectrum of the VCO output oscillation ishomogeneous in a ceratin frequency band. An exampleof this spectrum is shown in Fig. 4. The oscillation

spectrum at the LPF output is shown in Fig. 4a and thepower spectrum of the high-frequency phase chaos atthe VCO output is shown in Fig. 4b.

Based on the analysis of the bifurcation diagram(Fig. 2b), the division coefficients and the parameters ofthe second-order filter were chosen for experimentalstudies. The, operation of the PLL system was simulatedfor the chosen parameter values. Figure 5 shows the bifur-cation diagram obtained for these values. For convenienceof comparison with experiment, the VCO frequency divi-sion coefficient (m), which can be directly measured inexperiment via reprogramming of the PD chip, is used inconstruction of this diagram as the bifurcation parameter.

It can be seen from the presented diagram that, for thechosen conditions, the required modes of vibrational–rota-tional chaos can be expected in experiment at values ofdivision coefficient m ranging from 14 to 19.

3. EXPERIMENTAL MODEL

While VCOs and VCO-based PLL systems arewidely used in modern wireless communication tech-nology, the experiment on generation of chaos in a PLLsystem with a high-frequency VCO is a considerable engi-neering problem. Most of the PLL systems used in wire-less communication are designed for application in Wi-Fiand Wi-Max systems, cellular communication, etc. There-fore, many PLL systems are narrowband in principle, withan operating band of up to several tens of MHz. At thesame time, wideband and ultrawideband devices are mostinteresting for the purposes of chaotic communication.Besides, as a rule, modern PLLs are manufactured asmonolithic integrated circuits (or are included in such cir-cuits). These integrated circuits are designed for obtainingregular oscillations and cannot be used directly for obtain-ing chaotic modes.

For performing the experiments, it was necessary tocreate the prototype of a high-frequency PLL systemwith a second-order filter in the feedback loop usingspecialized highly integrated circuits for implementa-tion of a PLL structure that allows changes in the filterstructure. In view of these requirements, a PLL systemdesigned on a special board with the use of aV590ME01 VCO and an MB15E03 PD was chosen forprototyping. The circuit allowed changes in the struc-ture of the filter in the feedback loop. A first-order filterwas replaced with a second-order filter with the calcu-lated parameter values.

The created prototype consisted of a VCO, a PDmicrocircuit, a reference frequency generator (RG), acontrolled amplifier (CA) with variable gain, and anLPF. All these components were connected in the PLLloop in the order shown in Fig. 1. Thus, the signal fromthe VCO is supplied to the PD, where frequencies of theVCO and the RG signals are divided by predefinedcoefficients m and n.

ϕ

1

0

–1

–4 –2 0 2 ϕ

(‡)

(b)ϕ

1

0

–1

0 2 ϕ4

2

(c)ϕ

0 2 ϕ4

1.4

1.2

1.0

0.8

0.6

0.4

Fig. 3. Phase portrait in plane (ϕ, ) for µ = 3 and ε = 0.35:(a) vibrational regime, γ = 0.03; (b) vibrational–rotationalregime, γ = 0.3; and (c) rotational regime, γ = 0.704.

ϕ



S, dB

0

–10

–20

–30

–40

–50

0 1 2 3 4 f, MHz

(a)

(b)S, dB

0

–10

–20

–30

–40

–50

0.95f, GHz

1.00 1.05 1.10 1.15 1.20

Fig. 4. Typical power spectra of (a) the LPF output signal and (b) the VCO output signal. Here, µ = 10.6, γ = 0.6, and ε = 0.35. Sym-bols S and f denote the spectral power density and the frequency, respectively.

The PD connection circuit assumes that the PD out-put signal controls the current source to which the inte-grating RC filter is connected. However, in the mathe-

matical model, it is assumed that the PD output is thesource of control voltage, rather than current, and thePD is followed by an RLC low-pass filter, rather than an

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DMITRIEV et al.

RC filter. Therefore, in the prototype, it was decided toreject the current source and supply the signal from oneof the logical outputs of the PD microchip. This con-nection of the PD results in formation of an output sig-nal similar to the signal that can be formed in the caseof application of a XOR element forming a rectangularpulse. The difference of the PD signal used in the modelfrom the PD signal operating according to the XOR dia-gram is the absence of half of the pulses when the phasedifference exceeds π. This feature is reflected in theshape of nonlinear PD characteristic (5).

The principle of operation of this PD is illustrated inFig. 6. At the instants determined by the fronts of thereference pulses, a duration-modulated pulse with aduration proportional to the phase difference of the RGand VCO signals is formed at the PD output. A periodicsequence of video pulses with a repetition frequencyequal to the divided RG frequency and the pulse ratioequal to the VCO signal phase is supplied from the PDoutput to the CA. After the CA, the sequence of videopulses amplified to the necessary amplitude is suppliedto the LPF. The variable signal with the added constantcomponent of a given value is supplied from the LPFoutput to the VCO control input. This signal is taken

from the loop at the LPF output and is supplied to theVCO input.

The physical configuration of the PLL device isshown in Fig. 7. This device contains a voltage-con-trolled generator with an operating frequency range of800–1300 MHz, a supply voltage of 5 V, and a con-sumption current of 30 mA. The control voltage variesfrom 0 to 9 V. The frequency of the reference generatoris 80 MHz. The PD microchip contains frequencydividers for the reference signal and VCO signal. TheVCO frequency divider consists of two independentfrequency dividers. The division coefficient of the firstdivider can be set equal to 64 or 128. In the prototype,the coefficient is set equal to 64. The second divider isadjustable and its division coefficient can be set equalto any number from 3 to 2047. In the prototype, thevalue of this coefficient is 3. The tuning range of the fre-quency divider for the RG signal is 3–16383. In theprototype, the value of this coefficient is 13. The set-tings of the dividers are written in the microchip via aserial digital interface.

The CA gain is controlled with a potentiometer inthe feedback loop. The voltage gain control range is0.7–2.

max(ϕ)

2.5

2.0

1.5

1.0

0.5

0

–0.59 10 11 12 13 14 15 16 17 18 m

·

Fig. 5. Bifurcation diagram for parameter m. Here, ε = 0.35 and µ = 10.6.



4. EXPERIMENTAL RESULTS

The variable parameter in the experiments was coef-ficient n of the frequency divider for the reference sig-nal; the value of this coefficient was varied by almost afactor of 2, from 11 to 19.

Unlike the investigation of the mathematical model forthe phase variable, the phase difference was not fixed inthe experiment. The derivative of the phase difference wasused as the VCO control signal. This derivative was mea-sured. Therefore, it was inconvenient to construct thephase portraits in coordinates (ϕ, ), and the phase por-traits constructed using experimental data were plotted incoordinates ( (t), (t – τ)), where delay τ was approxi-mately equal to 1/4 of the quasi-period.

In order to compare the experimental results withthe results of numerical simulation, phase portraitsobtained from numerical simulation were alsoexpressed in terms of these variables.

Typical experimental realizations of the phase deriv-ative, phase portraits, and the power spectra of the VCOoutput signal are shown in Fig. 8. Comparison of thephase portraits obtained in the experiment (Fig. 8c, 8d)and the phase portraits obtained from numerical simu-lation (Fig. 9) points to the qualitative agreement of thedynamics of the experimental and mathematical mod-els. It also follows from the comparison of these char-acteristics that, in the parameter ranges under study, thevibrational–rotational chaos for phase ϕ was observedin experiment.

Let us consider the result of action of a signal (thephase derivative) on the VCO. The center of the fre-quency range of the VCO output oscillations is deter-mined by the constant bias voltage supplied to theVCO, and the width of this range is determined by theamplitude of the input signal, which is proportional tothe phase derivative. Although a controlled amplifier isinstalled at the output of the second-order filter, it is dif-ficult to vary the amplitude of the VCO control signalusing this amplified, because this change in the gainresults in a change in the phase-oscillation mode. Thiscircumstance should be taken into account in formationof a chaotic signal with a given power spectrum at theVCO output. In this series of experiments, the amplifierwas actually used just as a buffer, rather than for controlof the signal spectrum.

The experimental results on formation of the signalspectrum at the VCO output are shown in Figs. 8e and8f. It can be seen that, in complete agreement with thecalculations, it is possible to obtain the VCO output sig-nal with a power spectrum sufficiently uniform in thetuning band.

In principle, both the position of center of this spec-trum and the spectrum width can be controlled. Here, itis important to change the combination of parameters in

ϕ

ϕ ϕ

such a way that, on the one hand, frequency ω0 of theVCO free oscillations and the bandwidth of the VCOoutput chaotic signal are changed accordingly, and onthe other hand, the vibrational–rotational modes of cha-otic generation providing the required chaos quality are

1

2

3

t

t

t

Fig. 6. Signal waveforms at the input and output of thephase detector: (1) the RG reference pulses, (2) the VCOinput signal, and (3) the signal at the PD output.

Fig. 7. Physical configuration of the prototype PLL.

682


DMITRIEV et al.

V, V

8

4

0

–40 8 16 24 32 t, µs

(a) (b)

(c) (d)

(e) (f)

V, V

8

4

0

–4

0 8 16 24 32 t, µs

ϕ (t – τ)

7

6

5

4

3

2

1

0 ϕ (t)·

·

1 2 3 4 5 6 7

ϕ (t – τ)

7

6

5

4

3

2

1

0 ϕ (t)·

·

1 2 3 4 5 6 7

8

8

S, dB

–10

–30

–50

–70

0.6 0.8 1.0 1.2 1.4f, GHz

S, dB

–10

–30

–50

–70

0.6 0.8 1.0 1.2 1.4f, GHz

Fig. 8. Experiment. A fragment of the experimental realization of the LPF output signal (V is the voltage and t is time) for (a) m = 12and (b) m = 16. The power spectrum of the VCO output signal (S is the spectral power density and f is the frequency) for (c) m = 12and (d) m = 16. Phase portraits in plane ( (t), (t – τ)) for (e) m = 12 and (f) m = 16; τ = 250 ns.ϕ ϕ

preserved. The problems related to formation of chaoswith given characteristics require additional studies.

Another important parameter in formation of phasechaos is the rate of change of the signal frequency. For-mally, if the complete frequency scanning in the pulse

takes place, the signal band is assumed equal to thescanning band. In principle, slower scanning related tothe same band is also possible. In our case, the rate ofchange of the signal frequency is determined by the fre-quency band of the PLL signal, which was approxi-mately equal to 1 MHz.



ϕ(t – τ)

2

1

0

ϕ(t)·

·

0 1 2

(‡)

(b)ϕ(t – τ)

2

1

0

ϕ(t)·

·

0 1 2 3–2 –1

3

–1

–2

Fig. 9. Simulation. Phase portraits in plane (ϕ, ): (a) µ = 12 and (b) µ = 16; τ = 250 ns.ϕ

CONCLUSIONS

The studies have provided the theoretical substanti-ation of the possibility of generation in the decimeterwave band of phase chaos with a uniform power spec-

trum in some frequency band. The results obtained haveallowed the development of a prototype of a PLL sys-tem. The prototype has ensured generation of chaoticoscillations with predicted properties. The bifurcationphenomena describing transition to chaos in the math-

684


DMITRIEV et al.

ematical model for the phase (phase difference) agreewith the results obtained in the experiment. The powerspectrum of the VCO output signal is in the frequencyrange 650–1250 MHz. The frequency band of this sig-nal is about 600 MHz. Chaos at the VCO output hasphase nature, i.e., the signal phase varies chaotically,while the signal amplitude is constant.

Thus, it has been theoretically substantiated andexperimentally proved that the characteristics of theoutput signal optimal for a number of applications ofdirect chaotic communication systems can be imple-mented in PLL-based phase chaos generators by appro-priately choosing the parameter ranges.

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Generation of ultrawideband phase chaos in the decimeter band