Principles of Modern Radar - Volume 2 1891121537

300 CHAPTER 7 Stripmap SAR33 cycles per m. For a 1 m cross-range resolution, (7.3) and (7.4) suggest the along-tracksampling interval must be no larger than 0.5 m. If that corresponds to critical samplingthen there will be signal content at the highest spatial frequencies of −1 cycle/m to +1cycle/m. In this example there is a one-to-two orders-of-magnitude difference in spatialfrequency. Assuming 2ωc ≫ k u , the radical inside the brackets in (7.90) can be replacedwith the usual binomial approximation as√ ( )ω ω 2c/2 − − ku c/22 = ωc/2 − ωc/21 − ku2 ( )√ ω 2c/2∼= ωc/2 − ωc/2 + ku2 ( ) ω2c/2∼= ck2 u4ωSo, we denote the first approximation to the PSR by a single apostropheψ PSR∼ = ψ′PSR = cr 0k 2 u4ω(7.91)(7.92)Comparing (7.92) to (7.31), we see this approximation to the PSR in RDA is equivalentto the RMC term in DBS-AD-RMC, an interesting bridge over the gap between RDA andDBS.The second approximation comes from making a narrowband assumption; that is, thewaveform bandwidth is much less than the carrier frequency. Define a center frequencyω 0 and deviation from the carrier ω such that ω = ω 0 + ω. Then (7.92) becomesψ ′ PSR =cr 0 k 2 u4 (ω 0 + ω)(7.93)By assuming ω ≪ ω 0 and applying the binomial approximation, the denominator in(7.93) reduces to(ω 0 + ω) −1 = 1 (1 + ω ) −1ω 0 ω 0∼= 1 (1 − ω )ω 0 ω 0∼= 1 ω 0− ωω 2 0(7.94)We substitute the narrowband formulation into (7.93) and denote this second approximationby a double apostrophe:ψ ′ PSR ∼ = ψ ′′PSR = cr 0k 2 u4[ 1− ω ]ω 0ω 2 0(7.95)