A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation

This note on backprojection for spotlight synthetic aperture radar image formation is mainly pedagogic in purpose and is intended to be accessible. The presentation from first principles is elementary and detailed, beginning with the wave equation and melding wave notions with signal processing notions using a compact and consistent notation throughout. A reflection model is developed including a general expression for the receiver signal, which does not depend on a particular transmitted waveform. Then, the signal is specialized to monochromatic waves to show how waves and the Fourier transform fit together. In the end, the signal is once again generalized so that the theory works for any signal type. Backprojection is shown to reconstruct the wave field that was lost by sampling it at only one point, the receiving antenna. After specializing some details to the synthetic aperture radar geometry, the Projection Slice Theorem is introduced late, after an understanding of the underlying principles is obtained. Computational aspects are considered and it is seen that backprojection and direct Fourier inversion, also known as the polar format algorithm, are fundamentally the same, differing only in some implementation details, albeit significant ones, thus overturning the notion that backprojection is not a Fourier process. Those who might benefit from this article include people who have worked in this field and who seek a somewhat different point of view from the usual presentation, people in other fields who are unfamiliar with some of the engineering concepts involved, and signal processing engineers who appreciate a bit of wave theory.