  We investigate the problem of a monostatic pulse-Doppler radar transceiver
trying to detect targets, sparsely populated in the radar's unambiguous
time-frequency region. Several past works employ compressed sensing (CS)
algorithms to this type of problem, but either do not address sample rate
reduction, impose constraints on the radar transmitter, propose CS recovery
methods with prohibitive dictionary size, or perform poorly in noisy
conditions. Here we describe a sub-Nyquist sampling and recovery approach
called Doppler focusing which addresses all of these problems: it performs low
rate sampling and digital processing, imposes no restrictions on the
transmitter, and uses a CS dictionary with size which does not increase with
increasing number of pulses P. Furthermore, in the presence of noise, Doppler
focusing enjoys an SNR increase which scales linearly with P, obtaining good
detection performance even at SNRs as low as -25dB. The recovery is based on
the Xampling framework, which allows reducing the number of samples needed to
accurately represent the signal, directly in the analog-to-digital conversion
process. After sampling, the entire digital recovery process is performed on
the low rate samples without having to return to the Nyquist rate. Finally, our
approach can be implemented in hardware using a previously suggested Xampling
prototype.
