Page 282 - Fundamentals of Radar Signal Processing

its peak at F and a Rayleigh resolution of 1/τ Hz. The processor will be able to
b
resolve scatterers whose beat frequencies are at least ΔF = 1/τ Hz apart. The
b
time-delay spacing that gives this frequency separation satisfies

(4.112)

The corresponding range separation is then the usual result for range resolution

(4.113)

If the reference oscillator sweep is not lengthened as discussed above to fully
overlap the echo from scatterers at any location in the range window, the range
resolution will be degraded. Specifically, the duration τ′ of the beat frequency
will be less than τ seconds for scatterers at any delay other than the center of the
window due to the incomplete overlap. The Rayleigh resolution of the Fourier
transform of that scatterer’s beat frequency will increase to a value 1/τ′ > 1/τ,

causing the range resolution of Eq. (4.113) to increase proportionately. The
processing gain will similarly be reduced from the ideal factor of βτ.
To consider the effect of Doppler shift on the stretch processor, replace
in Eq. (4.107) with

(4.114)

Repeating the previous analysis, Eq. (4.108) becomes

(4.115)

Equation (4.115) shows that the effect of a Doppler shift is to increase the beat
frequency F by F Hz. Since beat frequency is mapped to differential range by
D
b
the stretch processor according to δR = –cF τ/2β, this implies a measured range
b
b
shift of

(4.116)

which is the same range-Doppler coupling relationship obtained previously.