Fourier analysis is proposed as a tool for the recognition of non-diffusive features of heat pulse propagation in fusion plasmas. Application to a simulated ballistic sawtooth heat pulse shows that Fourier analysis can identify the plasma region (often well beyond the mixing radius) where the heat pulse is diffusive. In this region the initial temperature perturbation, as recorded by a diagnostic with moderate (≃ 1 kHz) time resolution, is negligible. Fourier analysis of an example of JET sawtooth heat pulse diagnosed with the new ECE heterodyne diagnostic is consistent with the results from the simulated ballistic heat pulse. It cannot, however, be used as evidence of a ballistic effect owing to the limited diagnostic time resolution. When both the perturbed (χep) and the power balance (χePB) diffusivity can be determined reliably in recent JET sawtoothing plasmas, their ratio is χep/χePB ≃ 2-4 in agreement with previous results