Nonlinear equation and dynamics of tracking loop in spread spectrum communication
Naoki Kimura, Teruo Miyashita, Chikara Sato · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1992
Abstract This paper aims at the representation and analysis of the mathematical model for the tracking of a receiver in the spread spectrum communication using the deterministic differential equation. The tracking system is based on the 2Δ‐type baseband delay‐lock loop using the pseudorandom waveform by M‐sequence. Since the M‐sequence has the periodicity, the correlator output is approximated by the Fourier series expansion with the power series coefficients and is treated as a function of the time and the phase difference between the pseudorandom waveforms. By those approaches, the basic equation for the tracking system is derived as a second‐order nonautonomous nonlinear differential equation for the phase difference, including the nonlinear restoring force, the forced input, and the parameter excitation. A computer simulation is applied to the derived nonlinear differential equation, and it is verified that the synchronization process is represented clearly as a trajectory on the phase plane. The behavior in the neighborhood of the phase difference zero is analyzed, and it is shown that the delay‐lock loop may become unstable, resulting in the one‐half subharmonic oscillation. The Poincaré map is constructed by a computer simulation to verify the fore‐mentioned property.