Principles of Modern Radar - Volume 2 1891121537

2.5 Stepped Frequency Waveforms 69Inserting (2.129) into equation (2.112)(ˆx (n) = exp j2π ( f 0 + nf ) 2 (R )0 − nvT )cand expanding terms(ˆx (n) = exp j2π 2R ) (0f 0 exp j2π 2R ) (0cc nf exp − j2π 2nvTcn = 0,...,(N − 1) (2.130)f 0)exp(− j2π 2n2 vTc(2.131)The first exponential in (2.131) is a constant phase term. The second exponential containsa linear phase term associated with the range to the target. Both of these terms are presentin the zero Doppler case (in equation (2.112)), and the remaining two exponentials are adirect result of the relative motion between the radar and the target. A Doppler shift is alsoimparted to the carrier frequency, but the associated phase contribution is small comparedwith the values defined in equation (2.131) and is therefore ignored.The third term in equation (2.131) introduces range-Doppler coupling with the peakof the response shifted in range byf)r shift = vTf 0f(2.132)Normalizing the range shift in equation (2.132) by the waveform’s range resolution yields¯r shift = 2NvT f 0c(2.133)which expresses the shift in terms of range resolution cells. The fourth exponential isquadratic in n producing a spreading in the compressed response. The maximum spreadoccurs when n = N orr spread = NvT (2.134)Normalizing the spread in equation (2.134) by the waveform’s range resolution yields¯r spread = 2vN2 T f(2.135)cThe spreading term produces in a loss in coherent gain and thus a loss in SNR. Therange-Doppler response of a stepped frequency waveform is very similar to an LFMwaveform with exception that the response aliases when the Doppler shift causes a rangedisplacement greater than c/2f .For a single target, the effects of Doppler may be mitigated if an estimate of the target’sradial velocity is ascertained via tracking or some other means. A range-independentcorrection factor may be applied to the samples prior to applying a DFT:(2n ˆvTx correct (n) = exp j2πc) (f 0 expj2π 2n2 ˆvTc)f(2.136)where ˆv is an estimate of the radial velocity. The correction factor is multiplied by themeasured samples in equation (2.131) to remove the undesired terms.