  The properties of relativistic jets, their interaction with the environment,
and their emission of radiation can be self-consistently studied by using
particle-in-cell (PIC) numerical simulations. Using three-dimensional (3D),
relativistic PIC simulations, we present the first self-consistently calculated
synthetic spectra of head-on and off-axis emission from electrons accelerated
in cylindrical, relativistic plasma jets containing an initial toroidal
magnetic field. The jet particles are initially accelerated during the linear
stage of growing plasma instabilities, which are the Weibel instability (WI),
kinetic Kelvin-Helmholtz instability (kKHI), and mushroom instability (MI). In
the nonlinear stage, these instabilities are dissipated and generate turbulent
magnetic fields which accelerate particles further. We calculate the synthetic
spectra by tracing a large number of jet electrons in the nonlinear stage, near
the jet head where the magnetic fields are turbulent. Our results show the
basic properties of jitter-like radiation emitted by relativistic electrons
when they travel through a magnetized plasma with the plasma waves driven by
kinetic instabilities (WI, kKHI, and MI) growing into the nonlinear regime. At
low frequencies, the slope of the spectrum is ~ 0.94, which is similar to that
of the jitter radiation. The results are relevant to active galactic
nuclei/blazars and gamma-ray burst jet emission and set the ground for future
studies on synthetic spectra from relativistic jets.
