A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation

Abstract

This note on backprojection for spotlight synthetic aperture radar image formation is mainly pedagogic in purpose and is intended to be accessible. The presentation 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 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 paper 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.

Brief Analysis of Aliasing in Polar and Keystone Formats for Spotlight Synthetic Aperture Radar

“…Other waveforms are suitable and stepped frequency or stepped sine signals [8,9] will be considered herein for simplicity and clarity of exposition since they, too, provide discrete Fourier plane samples. Similar principles apply to other waveforms with perhaps some modifications in post-processing [10].…”

“…(Reference [10] contains a full development from first principles of these basic results and others mentioned herein.) An important concept is that a harmonic illumination of the ground patch with transmitter signal e j ωt samples its Fourier transform G (k) at G 2kx wherex = (cos θ, sin θ) is a unit vector in the direction of the x and k x axes.…”

“…Spotlight SAR functions to place many such samples of G (k) over a limited region of the k plane and then to find an estimate of the bandlimited ground patch by Fourier inversion, either by the so-called direct Fourier inversion also known as the polar format algorithm (PFA) [2] or by backprojection [11,10]. For example, backprojecting (2) reconstructs the phasor version of field (1); performing such backprojections from all θ and all ω reconstructs a Dirac impulse at x s .…”

A Note on Fourier Transform Conventions Used in Wave Analyses

“…Also in the radar setting, the transmitted, outgoing, wave can be described as propagating in the increasing direction of a coordinate axis, a forward propagating wave. But the reflected, returning, wave, typically of more interest, is then usually a backward propagating wave along the same coordinate axis [10]. In this situation, both terms of the propagation exponent have the same sign, e i (ωt +k•x) , and both the favored forms (4) and (5) can be used for time-frequency transforms and (6) and (7) for space-wavenumber transforms.…”