Page 155 - Fundamentals of Radar Signal Processing
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processing operations depends critically on accurately modeling or estimating
the phase history of the collected data. Examples include pulse compression,
adaptive interference cancellation, and imaging.
2.6.3 Measuring Doppler Shift: Spatial Doppler
The Doppler shifts observed in radar are too small to be measured from a single
pulse echo in most cases. In Chap. 7 it will be seen that a lower bound on the
standard deviation of the error in measuring the frequency of a complex sinusoid
with unknown amplitude, frequency, and phase using a discrete Fourier
transform (DFT) and an observation of length T seconds at an integrated SNR
obs
in the DFT of χ is Hz. Applying this to measuring Doppler, this
value must be much less than the Doppler shift if that shift is to be measured
with reasonable precision, leading to a requirement that . Even
for a rather high Doppler shift of 10 kHz and a very good SNR of 30 dB (χ =
1000), T must be much larger than 123 μs. To measure the Doppler shift with
obs
a single pulse would therefore require pulse lengths greater than 1 ms, much
longer than the sub-millisecond (usually less than 100 μs) pulse lengths typically
used. For a 1-kHz Doppler shift and 20-dB SNR, a pulse longer than 10 ms
would be needed. For this reason, most radars do not measure Doppler shift on
an intrapulse basis, although a few designed for very high speed targets
(satellites and missiles) and using very long pulses can do so.
The long observation time needed can be obtained by using multiple
pulses. Suppose a series of M distinct pulses of duration τ are transmitted
beginning at times t = mT, where T is the pulse repetition interval (PRI). The
m
mth transmitted pulse and received echo (using the quasi-stationary assumption)
are
(2.105)
(2.106)
After demodulation, the baseband received signal is
(2.107)
where k′ includes the exp(–jϕ ) term. Assume each baseband pulse echo is
0
sampled 2R /c seconds after transmission, corresponding to a range R . Also
s
s
