## Radar Imaging - Week 1

Margaret Cheney
Rensselaer Polytechnic Institute
www.rpi.edu/~cheney/

The lectures will begin with an overview of the idea of radar imaging, including some beautiful images to provide motivation. The radar system architecture is discussed, along with a mathematical description of what each part of the system does to the transmitted or received signal. The processing of the received signal already involves interesting mathematical ideas to pull the signal out of the noise.

Next there will be a discussion of how the electromagnetic waves propagate after they leave the antenna. The propagation of electromagnetic waves is governed by Maxwell's equations; the audience will learn why a scalar wave equation is appropriate for most analysis of radar scattering. The analysis of radar signals will be provided first with a simple one-dimensional wave propagation model. The audience will learn from this that radar can measure both target range and target velocity, but that there is an uncertainty principle that both range and velocity cannot be measured simultaneously with arbitrary accuracy. The theory based on the one-dimensional model is sufficient to understand some of the basic ideas behind high-resolution imaging, and understanding this point of view is important for communicating with radar engineers.

Next, a fully three-dimensional scalar wave propagation model will be introduced, and the basics of scattering theory will be discussed. The three-dimensional model leads to a formula for the radar signal that includes the transmitted waveform, the reflectivity of the target, and geometrical spreading factors. This model will be used to explain ISAR (Inverse Synthetic Aperture Radar) imaging, which is generally used for airborne targets. The audience will learn that ISAR imaging commonly reduces to a multidimensional Fourier transform. A movie will be shown of ISAR images.

For targets on the ground, it is important to analyze the radiation from the antenna. The discussion of antennas will begin with a slide show of the many different forms of antennas. For the analysis of radiation from antennas, the full vector Maxwell's equations will be used; in particular, the vector potential formulation will be introduced and used to derive a formula relating the current density on the antenna to the far-field radiation pattern. Examples will be included to show how typical antenna beam patterns arise.

With three-dimensional wave propagation and antenna beam patterns now understood, the lectures will address spotlight-mode SAR. The audience will discover the similarity between ISAR and spotlight-mode SAR. Next, strip-map mode SAR will be addressed. Here the imaging process is more complicated and depends on the fact that the map from target reflectivity to radar signal is a FIO (Fourier Integral Operator). The audience will learn how to construct a parametrix (approximate inverse) for the FIO; applying this parametrix to the radar signal results in a strip-map mode SAR image. Properties of this image follow from the properties of the FIO; the necessary notions from microlocal analysis will be introduced.

The lectures will end with a survey of the state of the art and of some of the areas of active research.