The rotational Doppler shift (RDS) is typically measured by illuminating a rotating target with a laser prepared in a simple, known orbital angular momentum (OAM) superposition. We establish theoretically and experimentally that detecting the rotational Doppler shift does not require the incident light to have a well-defined OAM spectrum but instead requires well-defined correlations within the OAM spectrum. We demonstrate measurement of the rotational Doppler shift using spatially incoherent light.