Page 297 - Fundamentals of Radar Signal Processing

It follows that applying a K-point DFT to the slow-time sequence

implements K filters, each matched to a different incremental delay δt. Thus, the
DFT of the slow-time data within a single range bin for a stepped frequency
waveform is a map of echo amplitude versus incremental range within that
coarse range bin.
The DTFT of an M-point sinusoid has a Rayleigh frequency resolution of
Δf = 1/M cycles per sample. Using the scaling between f and t from Eq. (4.129),

the corresponding time resolution is Δt = 1/M · ΔF seconds; the range
resolution is therefore

(4.131)

where β is the total stepped bandwidth M · ΔF. Thus, the linearly stepped

frequency waveform achieves the same range resolution as a single pulse of
bandwidth β. If a K-point DFT is used to process the slow-time data the DFT
output will provide range measurements at intervals of

(4.132)

Since K ≥ M normally, the DFT output provides echo amplitude samples at

intervals equal to or less than the range resolution. This fine-resolution
8
reflectivity map is often called a high resolution range profile or just a range
profile.
The total bandwidth β of the stepped frequency waveform is determined by
the desired range resolution. It can be realized by various combinations of the
number of frequency steps M and the step size ΔF. To determine how to choose

these parameters, note that the DTFT of the slow-time data is periodic in ω with
period 2π radians per sample. Because the DTFT peak is at ω = 2πΔFδt, the
p
range profile is periodic in δt with period 1/ΔF. This periodicity establishes the
required coarse range bin spacing. Specifically, avoiding range ambiguities in
the range profile requires c/2ΔF > L, where L is the maximum target length of
t
t
interest. Once ΔF is chosen, M is selected to span the bandwidth required to
provide the desired fine range resolution. The DFT range profile then
effectively breaks each relatively large coarse range bin (c/2ΔF meters) into M
fine-resolution range bins (c/2β meters) sampled at K points within the coarse
range bin. If K = M the range sample spacing equals the range resolution. If K >
M the range profile is oversampled compared to the resolution by the factor
K/M. The pulse length τ is chosen to balance straddle losses and range
ambiguities. Recall that the single-pulse matched filter output s (t) is 2τ seconds
p
long. Choosing τ < 1/2ΔF means that s (t) will be no more than 1/ΔF seconds
p