Experimental system used to produce chaotic frequency-modulated microwave signals. The system uses a conventional microwave VCO with ahomodyne microwave phase discriminator to produce a sinusoidal nonlinearity. The output is then fed back to the input through a time-delayed integrator which is implemented on a field-programmable gate array (FPGA).
Experimentally measured relationship between the input v(t) and output w(t) for the self-homodyne phase discriminator comprising a 10 ns microwave delay line and mixer. The solid curve indicates the best-fit sinusoidal function. From these measurements, one can determine the two constants A = 0.2 V and , as shown.
Phase portraits of the system recorded experimentally (top) and simulated numerically (bottom) as the normalized feedback gain R is varied.
Bifurcation diagrams for the system with were constructed using the Poincare section dx/dt = 0. The experimental data were recorded using an 8-bit oscilloscope while the simulation results were obtained using MATLAB.
(a) Measured baseband power spectrum of the signal entering the VCO, v(t), and (b) corresponding microwave spectrum of the resulting frequency-modulated signal. Both measurements were taken with R = 4.176, which produces chaotic dynamics. The resolution bandwidth (RBW) was 30 Hz and 2 MHz for the baseband and microwave spectra, respectively, and both spectra were normalized relative to their maximum values.
Time traces and time delay embedding plots of the system using Boolean nonlinearity at different values of normalized feedback gain R.
Bifurcation diagram of the system utilizing the Boolean nonlinearity when R is varied from 2.75 to 4.5. (a) Experiment, (b)–(d) Simulation.
Summary of experimental parameters.
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