Conventional Synthetic Aperture Radar combines high range resolution waveforms collected from disparate directions/locations to form an image in range and cross-range. If the radar bandwidth is narrow, then range resolution will suffer and the overall image will be degraded. (This necessarily happens when the radar's carrier frequency is small, for instance.) There is, however, a complementary imaging mode in which very narrow frequency-domain pulses are collected by a platform in relative motion with the target and combined to form an image. Such systems rely on Doppler frequency shift measurements (instead of range information). For various practical reasons, this kind of imaging has not been well examined, but there are situations where the scheme is useful (in principle).
We develop the theory of radar imaging from data measured by a moving antenna emitting a single-frequency waveform. We show that, under a linearized (Born) scattering model, the signal at a given Doppler shift is due to a superposition of returns from stationary scatterers on a cone whose axis is the flight velocity vector. This cone reduces to a hyperbola when the scatterers are known to lie on a planar surface. In this case, reconstruction of the scatterer locations can be accomplished by a tomographic inversion in which the scattering density function is reconstructed from its integrals over hyperbolas. We give an approximate reconstruction formula and analyze the resolution of the resulting image. We provide a numerical shortcut and show results of numerical tests in a simple case.