—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 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. Index Terms —backprojection, direct Fourier inversion, image processing, image reconstruction, keystone format, polar format, polar format algorithm, projections, projection-slice theorem, Radon transform, signal processing, spotlight synthetic aperture radar I. I NTRODUCTION SIMILARITIES between spotlight-mode synthetic aperture radar (SAR) and computerized tomography (CT) were illuminated in 1983 [ 1]. With the parallels drawn between the two ﬁelds, researchers in SAR were able to borrow and adapt image reconstruction ideas from CT and some existing SAR algorithms beneﬁted from the new perspective. Some of the closely related CT algorithms that have been borrowed and used extensively are called backprojection ,convolution- backprojection ,ﬁltered backprojection , or a related method called ﬁlter of backprojections . Fig. 1 shows a typical basic set-up and geometry of a spotlight SAR. The radar transmitter-receiver is mounted on Jerry Bauck is a researcher, consultant, and inventor in Tempe, Arizona, USA. E-mail: ieee@my last name.net. Accepted for publication on July 13, 2021. ©2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.a movable vehicle, typically an airplane or low earth orbiting satellite, which moves around a scene of interest on the ground. The radar’s antenna is pointed at the scene, or dwelled, as the aircraft moves, allowing the scene to be interrogated by a structured, designed, electromagnetic pulse which reﬂects from the ground patch and is then collected and processed by the receiver; the interrogation is repeated over a range of angles. An alternate and older form of SAR is stripmap mode whereby the antenna is not dwelled and thus illuminates a strip of the ground as the platform moves. While stripmap SARs image a larger area than spotlight SAR, the resolution is worse given similar system parameters. Variants include inverse SAR where the radar is considered to be stationary and the target rotates and bistatic or multistatic spotlight SAR whereby the one or more transmitters are in different locations than the one or more receivers. Other SAR modes include scanSAR, sliding spotlight, and interferometric SAR (InSAR). Some radars are able to switch between multiple modes. There are a number of classic and modern papers and books on SAR but most of them focus on stripmap mode and mention spotlight mode little if at all, for example [ 2], [3], [4], [5], [6]. Of those that do signiﬁcantly mention spotlight mode, most discuss mainly or only the polar format algorithm about which we shall have more to say later [ 7], [8], [9], [10], [11], [12], [13], and [ 14]. Works with a signiﬁcant orientation to backprojection are [ 1] which apparently inspired [ 15], plus [ 16], [17], [18], and [ 19]. Other