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An object illuminated by a source produces a scattered signal; this signal depends upon both the source and the physical properties of the object. The problem of deducing coordinates, shape and/or certain physical properties of the object from the measurements of the returned signal is an inverse problem called echo imaging. The problem of echo imaging arises in medical imaging, remote sensing (radar; sonar; geophysical exploration), and non-destructive testing. In this paper, we address the problem of imaging an object form its returned signals using a phased array. Our approach is to exploit the array's various radiation patterns and the recordable portion of the returned signal's spectrum to generate the data base for this echo imaging system. Rapid steering of a phased array's radiation patterns can be achieved electronically. These steered waves can be utilized to synthesize waves with varying angles of propagation. In this case, the recorded returned signal for each direction of propagation can be viewed as data obtained by a bistatic array configuration. We first formulate the imaging problem for a plane wave source in a bistatic configuration. We utilize the two-way propagation time and amplitude of the returned signal to relate the object's properties, reflectivity function and coordinates, to the measured data (system modeling). This relationship is the basis for deducing the object's reflectivity function from the recorded data (inverse problem). We then extend these results for an arbitrary radiation pattern and synthesized radiation patterns generated by an array capable of beam steering in cross-range. We show that the recorded returned signals can be related to the spatial frequency contents of the reflectivity function. We also show that these array processing principles can be utilized to formulate a system model and inversion for synthetic aperture radar (SAR) imaging that incorporates wavefront curvature.
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