Principles of Modern Radar - Volume 2 1891121537

6.4 Sampling Requirements and Resolution 227support the bandwidth implied by the minimum and maximum frequencies correspondingto the ends of the desired range swath. We present the basic ideas behind deramp receptionin this section. More detail is found in Chapter 2 and specifically as applied to syntheticaperture imaging in [3,5,17].The transmitted LFM chirp signal is( ) tp(t) = rect exp { − j ( ω 0 t + π K c t 2)} (6.14)τ cwhere τ c is the transmitted chirp duration (s), B c is the chirp bandwidth (Hz), andK c = B c /τ c is the sweep rate (Hz/s). The reflected signal is described by convolvingthe transmitted pulse with the scene reflectivity, but we will consider only the return froma single range r:(s 0 (t) = f (r) · p t − 2r )(6.15)cwhere f (r) is the scene reflectivity associated with range r. A deramp receiver mixess 0 (t) with a delayed chirp having the same sweep rate as the transmitted chirp, but with alonger duration τ g , which we term the range gate. The result of mixing is(s(t) = f (r) · pt − 2rc)· p ∗ (( ) t − 2r0 /c= rect· f (r) · expτ c{ ( (ω 0· exp+ j( ) t − 2r0 /c= rectτ c{· exp − j( 2ct − 2r 0c· f (r)([ω 0 + 2π K ct − 2r )0c{ (− j(ω 0)+ π K c(t − 2r 0ct − 2r 0ct − 2rc) 2)})+ π K c(t − 2rc) 2)})](r − r 0 ) − 4π K cc 2 (r − r 0 ) 2 )}(6.16)The frequency of the reflected signal located at r after deramp processing is found bytaking the time derivative of the phase in (6.16):f deramp = ω deramp2π= 2K cc (r − r 0) (6.17)We see that each range maps into a unique frequency, or tone, that is proportional tothe range between the scatterer of interest and the deramp reference range r 0 . Thus,the reflectivity profile f (r) can be recovered by Fourier transforming s(t). The rangeresolution of our estimate of f (r) is inversely proportional to the time over which f derampis observed, with the constant of proportionality being (c/2K c ) −1 . This fact follows fromthe time–frequency scaling property of the Fourier transform (see Table 6-1). If the deramptime gate τ g is sufficiently long, the observation time of f deramp is equal to or greater thanthe pulse length. Thus, the range resolution is found to be c/2K c τ c = c/2B c .