Switched phased array systems operating at high sample rates generate large amounts of data during measurements of radio channels, but many scenarios contain only few multipath components. Compressed Sensing suggests in these cases Nyquist-rate samples are wasteful in terms of data size. It has been observed that structural parameters, i.e., time of flight and angle of arrival, of the propagation paths are temporally consistent, as they vary slowly in time. Hence, we propose a local temporally consistent signal model that includes delay and angle of arrival, and also their time-derivatives, coherently connecting multiple radio channel snapshots. This allows to use a cyclic compression scheme consisting of a few compression matrices that extract mutually incoherent information from adjacent snapshots. Last, we present an algorithm to extract specular multipath components from these compressive measurements.We verify our findings on simulated data and real measurements. On simulated data we observe a good agreement of the estimates with the available ground-truth and show that the proposed cyclic compression scheme improves estimation accuracy. On measured data, we compare the estimates from data obtained at Nyquist rate to data compressed to 17% of the original size and find good agreement as well.