Page 353 - Fundamentals of Radar Signal Processing

The blind Doppler frequency that would be observed in an unstaggered
waveform with this average PRI is

(5.37)

Using Eqs. (5.34), (5.36) and (5.37) and noting that F T = 1 gives an
g g
expression for the first blind Doppler frequency of the staggered PRI system in
terms of the staggers {k } and the blind Doppler of the reference unstaggered
p
system

(5.38)

For example, a two-PRI system with a stagger ratio of 3:4 would have a first
blind Doppler that is 3.5 times that of a system using a fixed PRI equal to the
average of the two individual PRIs. If a third PRF is added to give the set of
staggers {3, 4, 5}, the first blind Doppler will be four times that of the

comparable unstaggered system.
These equations simplify if all of the staggers are indeed mutually prime. In
that event, the LCM of the set of inverse staggers {1/k } equals 1 (See Prob.
p
5.14). The blind Doppler shift [Eq. (5.34)] of the staggered system then equals
F and the factor by which the blind Doppler frequency is expanded relative to
g
the unstaggered case [Eq. (5.38)] is just the average of the staggers.
If a pure sinusoid Aexp(jΩt) is input to a linear time-invariant (LTI)
system, the output will be another pure sinusoid at the same frequency but with
possibly different amplitude and phase, Bexp(jΩt + ϕ). However, if a pure
sinusoid is sampled at nonuniform time intervals the resulting series of samples,
if interpreted as a conventional discrete-time sequence, will not be equivalent
to a uniformly sampled pure sinusoid at the appropriate frequency so that the

sampled signal will contain multiple frequency components. Any subsequent
processing, even though itself LTI, will still result in an output spectrum
containing multiple frequency components. Thus, a system utilizing nonuniform
time sampling is not LTI and the frequency response of a pulse-to-pulse
staggered system does not exist in a conventional sense. Instead, an approach
based on first principles can be used to explicitly compute the effect of a two-

pulse canceller on a complex sinusoid of arbitrary frequency and initial phase
for the MTI filter structure of interest. Repeating for each possible sinusoid
frequency, the effect of the combination of staggered sampling and MTI filtering
can be determined for targets of different Doppler shifts (Roy and Lowenschuss,