Spectral analysis of radiation emitted transverse to laser propagation in laser-wakefield acceleration experiments shows broadband emission when electrons are accelerated to relativistic energies. The region over which emission occurs is short compared with the overall interaction length. The energy of the emission and location along the interaction length both vary with plasma density. A model for the radiation from self-trapped electrons indicates that the emission is a signature of the violent initial acceleration, and hence can be used as a diagnostic of the self-injection mechanism.