Principles of Modern Radar - Volume 2 1891121537

7.4 Range-Doppler Algorithms 301This form of the PSR phase termψ ′′PSR = cr 0k 2 u4ω 0− cr 0ku24ω02 ω (7.96)when substituted back into the PSRH (k u ,ω = ω 0 + ω; r 0 ) ≈ exp(j cr ) (0ku2 exp − j cr )0ku24ω 0 4ω02 ω(7.97)leads to two terms:1. The first is a quadratic phase function in k u . Because there is no dependence on RF,this modulation may be applied in either time or frequency.2. The second is a linear phase modulation over baseband frequency ω. Because multiplicationby a linear phase is equivalent to a shift in the Fourier domain, the secondterm may be implemented as a time shift of the data, the degree of shift a function ofspatial frequency.Therefore, we may write this version of the PSR as(H (k u ,t; r 0 ,ω 0 ) ≈ exp j cr ) (0ku2 ∗ t δ t − cr )0ku24ω 0 4ω02(7.98)This form of the PSR is implemented by Fourier transforming the data d(u,t) into thespatial-frequency domain D(k u ,t), time-shifting each k u bin by the second term in (7.98),phase modulating the data by the first term in (7.98), and inverse Fourier transformingfrom k u back to u, which is now x in the image.The time-delay shift in the second term in (7.98) is quadratic in k u . It serves as anexplicit straightening of the curved response of the PSR in (k u ,t), an operation reminiscentof DBS-AD-RMC seen in Figure 7-26.Note this version of RDA operates in the spatial-frequency–time-delay domain. As discussedearlier, spatial-frequency maps to Doppler frequency and time delay maps to slantrange. This algorithm, then, operates in the range-Doppler domain, hence the monikerthe range-Doppler algorithm.This approximation to RDA offers two implementation advantages over the fullmatched filter development:1. Fourier transforms from t to ω and back to t are avoided, while the time delays areeasily realized using low-order interpolation filters.2. Staying in the time domain makes it possible to vary the PSR with down-range. Indeed,the phase modulation can vary continuously with down-range. The time shift is moredifficult since a finite length of range extent is needed for interpolation. Range variationon the time shift is commonly realized by processing the data in small but discreteblocks, or subswaths, wherein each subswath sees a different time-shift function. Withdown-range variation the PSR becomes( ) ( )H (k u ,t; r,ω 0 ) ≈ exp j crk2 u∗ t δ t − crk2 u4ω 0 4ω02 (7.99)