This work reports the potential of first-order, non-autonomous chaotic circuits for bistatic radar applications. Unlike most chaotic systems, 1st order chaotic systems offer closed-form analytic solutions that aid in designing simple matched filters. In this work, a signal generated by a 1st chaotic oscillator is transmitted towards both the receiver and the target, enabling the use of this waveform for two purposes. First, the waveform serves to synchronize the bistatic radar receiver. Second, the waveform assists in acquiring an estimate of the target’s range. For the first time, we show that two 1st order chaotic circuits can be synchronized using a simple resistive coupling. The cross-correlation between the two synchronized circuits is of high quality, exhibiting a narrow main lobe width and low sidelobe levels. Consequently, these 1st order systems can generate high-range resolution profiles in bistatic configurations. Lastly, analytical expressions show that the cross-ambiguity function between the echo received from the target and synchronized waveforms yields a near thumb-tack shape, emphasizing the value of a noise-like waveform for radar-ranging applications.
The linear frequency modulated (LFM) waveforms for medical imaging have been explored previously. Although the chaotic waveforms are used for radar applications, their benefits for medical imaging applications are not adequately analyzed. In this work, we propose using chaos for microwave medical imaging. Firstly, we consider waveforms generated from two chaotic systems: the Lang-Kobayashi and the Lorenz. Through auto-correlation analysis, we show that these waveforms possess good medical imaging properties. Then, we model the received signal from a prototype of the body tissue consisting of multiple layers (media). This received signal incorporates the transmission and reflection coefficients which are a function of the intrinsic impedance of the media. Lastly, the received signal is cross-correlated with the transmitted signal, i.e., the matched filtering operation. The resultant sharp correlations peaks serve as input to the inversion algorithm that estimates the media's intrinsic impedance, which can further be used to assess the healthy/unhealthy nature of the body part.
Chaotic FM signals are an attractive choice for secure communications and bistatic radar imaging in a shared electromagnetic spectrum. In this work, we propose an approach that incorporates a system architecture with a chaotic system with three state variables, one of which is selected for transmission with an embedded audio message. The sum of the chaotic state variable and audio message feeds a voltage-controlled oscillator that generates a wideband FM signal useful for radar target imaging. Using a receiver equipped with a chaotic synchronization stage, we show that the message is effectively extracted if it is embedded in any of a family of FM signals generated via dissipative chaotic flows. Furthermore, our results suggest that this family of FM signals provides the bandwidth necessary to resolve target components in range and Doppler as determined by their bistatic radar ambiguity surface.
KEYWORDS: Signal to noise ratio, Fermium, Frequency modulation, Receivers, Radar, Transmitters, Oscillators, Modulation, Linear filtering, Interference (communication)
We propose a scheme for bistatic radar that uses a chaotic system to generate a wideband FM signal that is reconstructed at the receiver via a conventional phase lock loop. The setup for the bistatic radar includes a 3 state variable drive oscillator at the transmitter and a response oscillator at the receiver. The challenge is in synchronizing the response oscillator of the radar receiver utilizing a scaled version of the transmitted signal sr(t, x) = αst(t, x) where x is one of three driver oscillator state variables and α is the scaling factor that accounts for antenna gain, system losses, and space propagation. For FM, we also assume that the instantaneous frequency of the received signal, xs, is a scaled version of the Lorenz variable x. Since this additional scaling factor may not be known a priori, the response oscillator must be able to accept the scaled version of x as an input. Thus, to achieve synchronization we utilize a generalized projective synchronization technique that introduces a controller term –μe where μ is a control factor and e is the difference between the response state variable xs and a scaled x. Since demodulation of sr(t) is required to reconstruct the chaotic state variable x, the phase lock loop imposes a limit on the minimum error e. We verify through simulations that, once synchronization is achieved, the short-time correlation of x and xs is high and that the self-noise in the correlation is negligible over long periods of time.
In prior work, we showed that any one of the state variables of the Lorenz chaotic flow can be used effectively as the
instantaneous frequency of an FM signal. We further investigated a method to improve chaotic-wideband FM
signals for high resolution radar applications by introducing a compression factor to the Lorenz flow equations and
by varying two control parameters, namely ρ and β, to substantially increase the bandwidth of the signal. In this
paper, we obtain an empirical quadratic relationship between these two control parameters that yields a high
Lyapunov exponent which allows the Lorenz flow to quickly diverge from its initial state. This, in turn, results in
an FM signal with an agile center frequency that is also chaotic. A time-frequency analysis of the FM signal shows
that variable time-bandwidth products of the order of 105 and wide bandwidths of approximately 10 GHz are
achievable over short segments of the signal. Next, we compute the average ambiguity function for a large number
of short segments of the signal with positive range-Doppler coupling. The resulting ambiguity surface is shaped as a
set of mountain ridges that align with multiple range-Doppler coupling lines with low self-noise surrounding the
peak response. Similar results are achieved for segments of the signal with negative range-Doppler coupling. The
characteristics of the ambiguity surface are directly attributed to the frequency agility of the FM signal which could
be potentially used to counteract electronic counter measures aimed at traditional chirp radars.
KEYWORDS: Target detection, Detection and tracking algorithms, General packet radio service, Wavelets, Ground penetrating radar, Radio propagation, Signal processing, Wave propagation, Data modeling, Dielectrics
Ground Penetrating Radars (GPR) process electromagnetic reflections from subsurface interfaces
to characterize the subsurface and detect buried targets. Our objective is to test an inversion
algorithm that calculates the intrinsic impedance of subsurface media when the signal transmitted
is modeled as the first or second derivative of a large bandwidth Gaussian pulse. For this
purpose we model the subsurface as a transmission line with multiple segments, each having
different propagating velocities and characteristic impedances. We simulate the propagation and
reflection of the pulse from multilayered lossless and lossy media, and process the received
signal with a rectifier and filter subsystem to estimate the impulse response. We then run the
impulse response through the inversion algorithm in order to calculate the relative permittivity of
each subsurface layer. We show that the algorithm is able to detect targets using the primary
reflections, even though secondary reflections are sometimes required to maintain inversion
stability. We also demonstrate the importance of compensating for geometric spreading losses
and conductivity losses to accurately characterize each substrate layer and target. Such
compensation is not trivial in experimental data where electronic range delays can be arbitrary,
transmitted pulses often deviate from the theoretical models, and limited resolution can cause
ambiguity in the range of the targets.
KEYWORDS: Fermium, Frequency modulation, Signal generators, Complex systems, Time-frequency analysis, Control systems, Nonlinear optics, Information operations, Electroluminescence, Interference (communication)
In previous work, we constructed wideband FM signals for high range resolution applications using the non-linear
Lorenz system, which has a set of three state variables and three control parameters. The FM signals were generated
using any one of the three state variables as the instantaneous frequency which was then controlled by adjusting the
values of the parameters in the chaotic regime. We now determine the spectral characteristics of the Lorenz FM signal
and compare the spectral characteristics to those of a similar FM signal based on the Lang-Kobayashi system. We show
that for either chaotic system, the local linearity of the attractor yields an FM signal with a distinct chirp behavior.
Irrespective of the statistical independence of the chaotic flow samples, we show that the chaotic FM signal follows
Woodward's theorem in the sense that the spectrum of the FM signal follows the shape of the probability density
function of the state variable. The chirp rate of the FM signal can be controlled through a time-scale parameter that
compresses or expands the chaotic flow. As the chaotic flow evolves in time, so does the spectrum of the corresponding
FM signal, which experiences changes in center frequency and bandwidth. We show that segments of the signal with a
high chirp rate can be significantly compressed to achieve high range-Doppler resolution. The ability to change the
center frequency and the shape of the spectrum is interpreted as added frequency agility.
We propose a novel approach to generate chaotic Frequency Modulated (FM) signals with potential applications in highresolution
radar imaging. The technique relies on the output of an n-dimensional (n>2) non-linear system that exhibits
chaotic behavior. For simplicity, we have chosen the Lorenz system which has a set of three state variables x, y and z,
and three control parameters ρ, β, and σ. FM signals are generated using any one of the state variables as the
instantaneous frequency by varying the values of ρ and β. The obtained FM signal is ergodic and stationary and the
time samples exhibit an invariant probability density function. The corresponding pseudo-phase orbits reveal themselves
as a strange attractor that may take on the shape of a Mobius strip depending on the time evolution of the signal. A timefrequency
analysis of the signal shows that the spectrum is centered on a time-dependent carrier frequency. Thus, the
FM signal has a high time-bandwidth product similar to that of a chirp. However, the carrier frequency continuously
shifts in a linear or quadratic pattern over a finite frequency range. A desirable feature of the signal is that the width of
its autocorrelation's mainlobe approaches the reciprocal of the bandwidth. Furthermore, simulations show that the
average of the time autocorrelation falls quickly and is void of sidelobes.
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