Principles of Modern Radar - Volume 2 1891121537

7.5 Range Migration Algorithm 315example. The scatterers in the L-band image in Figure 7-57b are focused and comparewell with those in the exact Stolt image in Figure 7-46b except for slightly higher downrangesidelobes on the scatterers at the upper left and lower right. The Stolt approximationwas appropriate here.7.5.6 Chirp Scaling AlgorithmAnother option for approximating the Stolt interpolation in RMA is found in the chirpscaling algorithm (CSA). CSA may be understood by revisiting RDA [12]. Recall twokinds of RDA were developed:1. Matched filtering with the exact PSR from a reference down-range via a phase multiplicationin the two-dimensional frequency (k u ,ω) domain. Cross-range focusing iserror-free at the reference range but gets increasing worse as one moves away from thereference range.2. Matched filtering with an approximation to the PSR, implemented as a phase modulationand time shifts in the (k u ,t) domain. Focusing is not perfect anywhere (becauseapproximations were made to the PSR to get to this implementation), but the spatialvariation of the PSR over down-range may be accounted for to a certain extent byallowing the phase and shift functions to vary over time delay t.An intriguing amalgamation of these two procedures can be imagined as follows:1. Match filter with the exact PSR via a phase multiplication in (k u ,ω), as in (1) in theprevious list.2. Transform to (k u ,t) and compensate for residual errors in the responses of scatterersdisplaced from the reference range through differential time shifts. By differential wemean with respect to the reference range. In other words, no time shift is appliedat the reference range but shifts become nonzero up-range and down-range from thereference.This procedure realizes error-free focusing at the reference range, like RDA using theexact PSR, and improves focusing away from the reference range, like the approximationto RDA.The typical CSA implementation realizes these stages in a reversed order:1. Transform to (k u ,t) and apply differential time shifts to give PSRs up-range and downrangeforms more closely matched to the PSR at the reference range.2. Transform to (k u ,ω) and apply the matched filter for the PSR at the reference range.Then the SAR image is formed with a 2-D IFFT as in Figure 7-35.The differential time shifts in the first step need to be defined. Bulk time shifts t Shiftfor the approximation to RDA are found in the final term of (7.100)t Shift = c2 ku28ω02 t (7.108)For the reference range r 0 the time delay is t 0 = r 0 /(c/2), and the resulting time shift ist Shift (t 0 ) = c2 ku28ω02 t 0 (7.109)