Principles of Modern Radar - Volume 2 1891121537

250 CHAPTER 6 Spotlight Synthetic Aperture RadarTo obtain closed-form results, we use expressions for the mean and variance of g(θ),which are derived in a 2003 paper by Richards [34] to be: μ g = exp{−σκ 2/2} and σ g 2 =1 − μ 2 g . We can use these to find the change in ISLR relative to the ideal ISLR, which isdefined as the increased integrated sidelobe energy (SLE) divided by the main lobe energy(MLE).ISLR ≈ μ2 g SLE 0 + σ 2 g /δxμ 2 g MLE 0= ISLR 0 + σ 2 g /δxμ 2 g MLE 0(6.40)We can simplify this expression further by approximating the main lobe energy as arectangle whose height is equal to the height of the magnitude-squared main lobe andwhose width is the −3 dB resolution. The former is equal to δx −2 while the latter isapproximated by δx. Thus, we find MLE 0 ≈ δx −1 andISLR = ISLR 0 + σ 2 gμ 2 g(6.41)where the second term represents the additional ISLR due to the phase noise term.Substituting in the expressions for the mean and variance of g(t), this is found to beISLR = exp{σκ 2 }−1. The corrupted ISLR can be used directly in equation (6.29) topredict the loss of contrast due to the random phase error.Figure 6-21 shows the effect of the image-degrading phase errors on actual SAR data.The imagery was collected by the U.S. Air Force Research Laboratory [15]. The primaryfeature in the scene is parking lot. Beside it is a road extending roughly vertically in theimage.6.8 AUTOFOCUSWe see from the previous section that synthetic aperture radars are highly sensitive tophase errors. For example, we find from Figure 6-18 that a two-way quadratic phase errorof λ/2 can effectively destroy the image, and we would like the two-way error to be lessthan λ/4 (or λ/8 one-way) to maintain good focus. If we assume the radar operates at Xband (λ = 3 cm), we find that we must know the platform motion to less than 4 mm overthe collection time, which can be as long as several seconds or a minute.The phase errors must be reduced to tolerable levels to obtain well-focused imagery,and there are two general methods for doing so. The first line of defense, so to speak, isto estimate the motion of the antenna phase center as accurately as possible using a GPSaidedinertial navigation system (INS). However, even high-performance devices may beincapable of measuring all motions and the resulting SAR image is very likely to containresidual error causing it to be at least slightly defocused.The second approach for mitigating phase error is to employ autofocus algorithms afterthe motion compensated data has been used to form an image. The term autofocus implesthat these techniques operate directly on the SAR image to estimate phase errors. Generallyspeaking, autofocus routines require as input only the complex imagery and only basicknowledge of the radar system parameters. Spotlight-mode autofocus algorithms exploit