Wireless Communication of a Chaotic Waveform by William Abell A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama August 1, 2015 Keywords: Exactly Solvable Chaotic Oscillator, Analog Modulation, Communication Systems Copyright 2015 by William Abell Approved by Robert N. Dean, Associate Professor of Electrical and Computer Engineering Lloyd S. Riggs, Professor of Electrical and Computer Engineering Michael C. Hamilton, Assistant Professor of Electrical and Computer Engineering Abstract Over the past couple of decades, mathematical equations have been developed that successfully model natural phenomena that occur in various fields such as chemistry, biology, and fluid mechanics. These equations, however simple, produce complicated solutions with occasional unpredictable behavior. Analysis and research of these models and other models that exhibit such response has been deemed “chaos theory”. One of the requirements for a system to be considered chaotic is that the system must be sensitive to initial conditions. The term “chaos theory” implies that an exact outcome is incomprehensible, leading to a common assumption that an analytic solution is unattainable for chaotic systems. This expectation has since been refuted as an exact solution has been derived for some chaotic systems. This unlocked the potential for a chaotic system to be implemented in practical applications. One such novel technique has arisen in the form of an electronic circuit. This circuit has been designed to oscillate in a chaotic manner and possess an exact solution that can be calculated. Due to sensitivity intrinsic to all chaotic systems, small perturbations can be used to control the chaotic oscillation. Because the chaotic oscillation can be controlled and its response determined, a circuit can be employed as a form of a modulator in encoding and encryption in communication systems. Common communication systems, subsystems and circuits are discussed along with analog modulation and demodulation techniques. A transmitting and receiving is circuit are detailed that successfully presents the transmission of a chaotic waveform in a wireless medium. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 From Academia to Application . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Exactly Solvable Chaotic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 Chaotic Hybrid System . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Angle Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Demodulation and Demodulators . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Frequency Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Heterodyne and Homodyne Receivers . . . . . . . . . . . . . . . . . . 25 2.3.2 Superheterodyne Receiver . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Communication Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.1 Electrically Small Antennas . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Resonant Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 iii 3.1.3 Broadband Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.4 Aperture Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.1 Passive Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Active Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.3 Switching Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Local Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.1 Variable Frequency Oscillators . . . . . . . . . . . . . . . . . . . . . . 41 3.4.2 Crystal-Controlled Oscillators . . . . . . . . . . . . . . . . . . . . . . 41 4 Communication System Requirements and Components . . . . . . . . . . . . . . 43 4.1 Communication Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Communication System Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 AM Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 AM Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3 Printed Circuit Board Implementation . . . . . . . . . . . . . . . . . . . . . 60 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A Diode Double-balanced Mixer Research (Done) . . . . . . . . . . . . . . . . . . 74 iv List of Figures 1.1 Waveform segment illustrating the hybrid dynamics [16]. . . . . . . . . . . . . . 4 1.2 Successive maxima return map [16]. . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 The top plot represents an arbitrary message signal within the frequency domain. The middle plot depicts how the message is duplicated and shifted, with half of its power centered on the carrier frequency, fc. Using a bandpass filter the desired information, highlighted in blue, is isolated so that it can be transmitted, illustrated by the bottom plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Plots of Bessel functions of the first kind [23]. . . . . . . . . . . . . . . . . . . . 19 2.3 DiscreteamplitudespectraofanFMsignal,normalizedwithrespecttothecarrier amplitude, for the case of sinusoidal modulation of fixed frequency and varying amplitude. Only the spectra for positive frequencies are shown [23]. . . . . . . . 21 2.4 DiscreteamplitudespectraofanFMsignal,normalizedwithrespecttothecarrier amplitude, for the case of sinusoidal modulation of varying frequency and fixed amplitude. Only the spectra for positive frequencies are shown [23]. . . . . . . . 22 2.5 The top plot represents the AM wave within the frequency domain. The middle plot depicts the location of the data after mixing the AM wave with a local oscillator that is identical to that used to create the AM wave. The bottom plot is the resulting spectral density of the signal after passing through a low pass filter. 24 2.6 Schematic diagram of homodyne receiver [24] . . . . . . . . . . . . . . . . . . . 26 v 2.7 Block diagram of a superheterodyne receiver [24] . . . . . . . . . . . . . . . . . 28 3.1 Examples of electrically small antennas . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Examples of resonant antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 An example of broadband antennas . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Examples of aperture antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Schematic diagram of a double balanced mixer . . . . . . . . . . . . . . . . . . 36 3.6 Schematic diagram of a SSB mixer . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Schematic diagram of an image-rejection mixer . . . . . . . . . . . . . . . . . . 37 3.8 Schematic diagram of a termination-insensitive mixer . . . . . . . . . . . . . . . 38 3.9 Schematic diagram of a push-pull dual gate FET mixer [24] . . . . . . . . . . . 39 3.10 Schematic diagram of a double balanced FET mixer [24] . . . . . . . . . . . . . 39 5.1 Block diagram of the transmitting circuit. . . . . . . . . . . . . . . . . . . . . . 46 5.2 A spectral depiction of the test sinusoidal wave and its harmonics. . . . . . . . . 47 5.3 The carrier signal and its harmonics as seen in the frequency domain. . . . . . . 48 5.4 The input signal is adjusted to include a DC offset and adjusted to approximately -3dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 The spectral content of the amplified local oscillator signal. . . . . . . . . . . . 50 5.6 The top plot is the S parameter of the BPF, showing the level of attenuation 21 as a function of frequency. The middle and bottom plot depicts the change in the AM waveform’s magnitude due to the BPF. . . . . . . . . . . . . . . . . . . 52 vi 5.7 Patch antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.8 Block diagram of the receiving circuit. . . . . . . . . . . . . . . . . . . . . . . . 54 5.9 Schematic of diode detection circuit. . . . . . . . . . . . . . . . . . . . . . . . . 55 5.10 (Top) The envelope detector circuit in the form of a KiCAD schematic. (Bottom) The envelope detector circuit implemented on a one-sided PCB. . . . . . . . . . 57 5.11 The transmitter design with the (1) voltage controlled oscillator, (2) low noise amplifier, (3) frequency mixer, (4) band-pass filter and the (5) envelope detector circuit shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.12 Acting as the last step in the demodulation process, a low pass filter is used to recover the original signal. The top plot is the S parameter of the LPF, showing 21 the level of attenuation as a function of frequency. The middle and bottom plots depicts the effect of the LPF on the demodulated signal. . . . . . . . . . . . . . 59 5.13 The power supply circuit in KiCad Eeschema. . . . . . . . . . . . . . . . . . . . 60 5.14 The printed circuit boards designed in KiCad PCB Editor. . . . . . . . . . . . . 61 5.15 The transmitting (top) and receiving (bottom) circuits mounted to their respec- tive printed circuit boards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.16 A chaotic signal of 1.6 MHz in the time domain . . . . . . . . . . . . . . . . . . 63 5.17 Spectral content of the 1.6 MHz chaotic signal . . . . . . . . . . . . . . . . . . . 63 5.18 The spectral content of an AM waveform at three different frequency spans: 50 MHz (top), 10 MHz (middle) and 5 MHz (bottom). The signal was created by mixing a 1.6 MHz chaotic signal with a +17.3 dBm carrier signal at 2.3 GHz . . 64 vii 5.19 Thetopplotshowsthechaoticwaveformfeddirectlyintothetransmitter’smixer. The waveform in the lower plot is produced from the output of the receiver. . . 65 7.1 Block diagram of a phasing-type SSB transmitter. . . . . . . . . . . . . . . . . . 69 A.1 Double-balanced mixer [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 A.2 The mixers spectral output due at various input levels with a local oscillator of +17.3 dBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.3 The power level of the carrier, both sidebands and the primary harmonic asso- ciated from each sidebands resulting from an input magnitude from -23 dBm to +17 dBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 viii List of Tables 3.1 Antenna Performance Parameters [26] . . . . . . . . . . . . . . . . . . . . . . . 30 5.1 Themixersoutputpowerwithrespecttovariousinputpowerlevels. Themessage is a sine wave with no DC offset. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ix Chapter 1 Introduction In 1926, Gilbert Vernam documents one of the first attempts at secure communications by combining plain text with pseudorandom code to create a ciphertext [27]. Decades later this process was revisited by independent research groups from which the first chaotic circuit was realized. Created by Leon Chua, the Chua oscillator was able to prove that synchro- nized chaos is, in fact, a possible and viable means of secure communication and various applications [1–7]. 1.1 Chaos Theory Chaos theory is the study of deterministic nonlinear dynamical systems exhibiting un- stable and aperiodic behavior. An unstable system is unable to achieve and operate in a steady state that can endure small perturbations. Aperiodic behavior is described as the recurrence of variables in an irregular manner. Therefore a system classified as unstable and aperiodic describes a system that is unable to repeat and reacts to the presence of small disturbances. While complex, simple mathematical systems have been shown to produce unstable aperiodic behavior making dynamical systems of interest to chaos theory. Differentiable dynamical systems are prevalent, characterized by consistent changes in variables. The evolution of these variables can be expressed by a set of differential equations, occasionally referred to as “evolution equations”. For dynamical systems expressed mathe- matically, analytical solutions can be generated to produce immediate, future and, in some cases, paststatesofsystem. Thisprovidesastraightforwardmodelofthetimedependencyof actualsystems. Untilrecently, aclosed-formsolutiontononlineardynamicalsystemsseemed unattainable or at the very least impractical. During this time, much research drifted away 1