  We consider a synthetic aperture radar (SAR) system that uses
ultra-narrowband continuous waveforms (CW) as an illumination source. Such a
system has many practical advantages, such as the use of relatively simple,
low-cost and low-power transmitters, and in some cases, using the transmitters
of opportunity, such as TV, radio stations. Additionally, ultra-narrowband CW
signals are suitable for motion estimation due to their ability to acquire high
resolution Doppler information. In this paper, we present a novel synthetic
aperture imaging method for moving targets using a bi-static SAR system
transmitting ultra-narrowband continuous waveforms. Our method exploits the
high Doppler resolution provided by ultra-narrowband CW signals to image both
the scene reflectivity and to determine the velocity of multiple moving
targets. Starting from the first principle, we develop a novel forward model
based on the temporal Doppler induced by the movement of antennas and moving
targets. We form the reflectivity image of the scene and estimate the motion
parameters using a filtered-backprojection technique combined with a contrast
optimization method. Analysis of the point spread function of our image
formation method shows that reflectivity images are focused when the motion
parameters are estimated correctly. We present analysis of the velocity
resolution and the resolution of reconstructed reflectivity images. We analyze
the error between the correct and reconstructed position of targets due to
errors in velocity estimation. Extensive numerical simulations demonstrate the
performance of our method and validate the theoretical results.
