  This article is written to serve as an introduction and survey of imaging
with synthetic aperture radar (SAR). The reader will benefit from having some
familiarity with harmonic analysis, electromagnetic radiation, and inverse
problems. After an overview of the SAR problem and some main concepts, the SAR
imaging problem is contextualized in terms of classical harmonic analysis.
Within this context, we consider partial Fourier sums of off-centered Fourier
data and correspondingly the convolutional kernels resulting from conventional
SAR image formation techniques. Following this, we revisit imaging of random
complex signals from frequency data as in SAR, providing simpler derivations of
some previous results and extending these ideas to the continuous setting.
These concepts are tied in with the derived convolutional kernels, and it is
deduced how good an image approximation is when it is obtained from only a
small band of high frequency Fourier coefficients. Finally, regularization
methods are presented to improve the quality of SAR images. Corresponding
MATLAB software is made available for reproducibility of most figures and to
facilitate further exploration of the methods presented here.
