Principles of Modern Radar - Volume 2 1891121537

7.4 Range-Doppler Algorithms 291to the spatial-frequency domain withH(k u ,t; r o ) ≡∫ ∞−∞h(u,t; r o )e − jk uu du (7.58)Then the convolution over u is realized as the inverse Fourier transform over k u of theproduct of the data and PSRf (x = u,r 0 ) ==∫ ∞−∞∫ ∞−∞( d(u,t) ∗u h ∗ (−u,t; r 0 ) ) dt⎛⎝ 12π∫ ∞−∞⎞D(k u ,t)H ∗ (k u ,t; r o )e jkuu dk u⎠ dt(7.59)The order of integration in (7.59) may be reversed without errorf (x = u,r 0 ) = 1 ∫ ∞ ⎛∫ ∞⎞⎝ D (k u ,t) H ∗ (k u ,t; r o ) dt⎠ e jkuu dk u (7.60)2π−∞−∞By summing over time first the two-dimensional product is reduced to a one-dimensionalsignal, so the final inverse Fourier transform is a much more reasonable one-dimensionaloperation.This image formation method is known as the range stacking algorithm (RSA) [4]because it outputs one down-range set of cross-range bins at a time and stacks them next toone another, as Figure 7-33 suggests. The RSA procedure is summarized in flow diagramform in Figure 7-33.RSA is an exact image former in that it makes no approximations to the PSR, in contrastto the DBS family of techniques. It is more computationally efficient than backprojection,the brute-force pixel-by-pixel approach suggested by (7.51) and described in some detailin [1], because convolution is implemented in the frequency domain for one of the twodimensions. Finally, it fully accounts for the down-range variation in the PSR by explicitlybuilding a different matched filter for each down-range set of pixels.d(u, t)1-D FFT(u)D(k u , t)Loop over rf(x, r)Range r 0 D(ku , t)H*(k u , t; r 0 )Generate PSRh(u, t; r 0 )1-D FFT(u)H(k u , t; r 0 )×∫∞⋅dt−∞D(k u )H*(k u ; r 0 )1-D IFFT(k u )f(x, r 0 )FIGURE 7-33Signal flow diagramfor the rangestacking algorithm.