Principles of Modern Radar - Volume 2 1891121537

6.9 Summary and Further Reading 253the peak of the correlation, x, is found and then related to the QPE coefficient viaβ = xL SA(6.42)where L SA is the length of the synthetic aperture. The estimated β is used to form theconjugate of the true phase error function, which is in turn applied at each range of thedata prior to Fourier transforming to the image domain.The phase difference algorithm is very similar to another autofocus technique knownas map drift (MD) [4,5]. The primary difference between the two is that MD performs thecross-correlation on the real-valued intensity subaperture imagery, whereas PDA crosscorrelatesthe complex-valued imagery. Furthermore, PDA averages over all range binswhile MD operates only on those ranges with very bright scatterers, typically a smallpercentage of the total. This latter difference means that MD is often implemented in aniterative fashion, while PDA can achieve the same result in a single iteration.6.8.2 Phase Gradient AutofocusPhase gradient autofocus (PGA) is a widely used technique for the nonparametric estimationof phase errors. It is particularly well suited to the estimation of high-frequency errors,but is capable of dealing with some amount of low-frequency error as well. The PGA algorithmis outlined in Figure 6-23 and begins with the complex image. For each range bin,the cross-range signal is circularly shifted so that the brightest pixel is placed at x = 0.The resulting line of bright pixels is windowed in cross-range to extract the corrupted PSRand its sidelobe structure. The remaining cross-range values are set to zero.The windowed data are then Fourier transformed in the cross-range dimension. Recallthat this operation takes the data from a domain where the ideal PSR is convolved withthe phase error spectrum and places it in one where the ideal phase history is multipliedby the phase error function. Since the center of the PSR is close to x = 0, its phasehistory is approximately constant, meaning that any phase terms present are due to noiseand the phase error we wish to estimate. For each range, this estimation is accomplishedby finite differencing the phase history in the angle (or slow-time) dimension and thenre-integrating it. This step eliminates any bias and enables us to average the data over allranges to obtain the desired phase error estimate.The conjugate of the estimated error is then multiplied against the data to correctit, and the data are then Fourier transformed back into the image domain. The processis repeated until a stopping criteria is reached. Since PGA targets high-frequency phaseerrors, a good choice of stopping criteria is the root mean square value of the estimatedphase error.6.9 SUMMARY AND FURTHER READINGSynthetic aperture radar imaging relies heavily on concepts from Fourier analysis. Weintroduced spotlight SAR by reviewing a few key properties such as the Fourier transformrelationship between the rect and sinc functions as well as the transform of a shiftedfunction s(t −t 0 ). The property stating that delta functions in one domain Fourier transforminto plane waves in the other domain is used to model the terrain as a continuum of