Principles of Modern Radar - Volume 2 1891121537

234 CHAPTER 6 Spotlight Synthetic Aperture Radar6.4.4 Cross-Range ResolutionWe have discussed the cross-range (along-track) sampling requirements for SAR. Nowwe turn to finding the cross-range resolution achievable by the radar. We will later makethe connection to our understanding of basic signal processing, finding that the the crossrangeresolution is inversely proportional to the spatial bandwidth observed by the radar.For the moment, we can take a more traditional approach for determining the cross-rangeresolution of the SAR image by treating the synthetic aperture as real aperture of lengthL SA and employing the usual argument that cross-range resolution is determined by theproduct of the range to the target and the beamwidth. For a SAR, this virtual beamwidthis θ SAR ≈ λ/2L SA . The factor of two improvement in the denominator arises because thesynthetic aperture is energized piecewise, as compared with a real aperture that transmitsand receives on the entire aperture simultaneously. This difference provides the syntheticaperture with additional phase information that makes it twice as effective as a real apertureof the same length. Continuing this argument, we find that the cross-range resolution canbe written asδx = Rθ SAR ≈ Rλ =λ(6.26)2L SA 2θ intwhere θ int is the synthetic aperture integration angle. This formula holds for all SAR modesas long as θ int is interpreted as the relative rotation between the radar and the target. Forinverse SAR, both the target and the platform may move at the same time, making theintegration angle less straightforward to compute [20,21]. Regardless of mode, angulardiversity is the key to SAR resolution in the cross-range dimension.6.5 IMAGE RECONSTRUCTIONIn this chapter we focus exclusively on the polar format algorithm for image reconstructionfor two reasons: First, PFA is the most common technique currently in practice. Second, anunderstanding of the operation of PFA is extremely useful for understanding the principlesof interferometry and coherent change detection that appear in the next chapter.When describing the polar format algorithm, it’s perhaps best to begin with the punchlineand work backward: under the tomographic paradigm, a radar using deramp-onreceivemeasures directly the spectrum of the scene reflectivity—not the reflectivity itself.The complication is that these measurements are made in a polar coordinate system, whilethe fast Fourier transform (FFT) requires regularly spaced samples on a Cartesian grid.The key functionality of the PFA is the frequency-domain interpolation that takes themeasured data from its native polar representation onto a Cartesian form to which the FFTis applied to obtain the SAR image.6.5.1 The Tomographic ParadigmThe reflected signal observed at any instant of time t = 2R/c comes from the superpositionof all reflections from scatterers at a distance R from the radar. The radiated wavefrontsare spherical, but we observe that the transmitted energy is not uniformly distributedthroughout the entire 4π steradians comprising the sphere enclosing the radar. By designthe antenna’s directivity pattern concentrates the radiated energy into a small solid angle,effectively illuminating only a limited region of the earth. If the scene diameter D is