Slowness araising in the discrete radon transform:A multirate system approach to beamforming

Time-domain beamforming with an array of space-time samples is equivalent to performing a discrete Radon transform (DRT). The result is a function of slowness (inverse velocity) and time intercept. As a function of slowness, beamforming decimates each channel in proportion to its distance from the beamformer reference channel. This viewpoint allows the slowness bandwidth to be expressed as a function of the array spatial locations and the normalized temporal bandwidth of the received waveforms. These relationships determine the required temporal sample rate increase at each channel when interpolation beam-forming is used to avoid slowness aliasing. The conclusions regarding the slowness sampling density are applicable as well to frequency domain beamforming, where the beams must also be computed at discrete values of slowness. A polyphase filter structure is shown to be particularly suited to interpolation beamforming. Examples are presented showing the slowness aliasing that results if either the temporal bandwidths or the array spatial aperture are too large. The beamformer output energy averaged over a time window is shown to require twice the slowness sampling rate as the beamformer output itself.