742 Encyclopedia of the Solar System
pulse would be dispersed in time delay, and the total extent
τTARGETof the distributionσ(τ) of echo power (in units of
radar cross section) would beD/cfor a sphere of diameter
Dand in general depends on the target’s size and shape.
The translational motion of the target with respect to the
radar introduces aDoppler shiftνin the frequency of
the transmission. Both the time delay and the Doppler shift
of the echo can be predicted in advance from the target’s
ephemeris, which is calculated using the geodetic position
of the radar and the orbital elements of Earth and the target.
The predicted Doppler shift can be removed electronically
by continuously tuning the local oscillator used, for exam-
ple, for RF-to-IF frequency conversion (see Fig. 4). Some-
times it is convenient to “remove the Doppler on the uplink”
by modulating the transmission so that echoes return at a
fixed frequency. The predicted Doppler (i.e., the predicted
rate of change of the delay) must be accurate enough to
avoid smearing out the echo in delay, and this requirement
places stringent demands on the quality of the observing
ephemeris. Time and frequency measurements are critical
because the delay/Doppler distribution of echo power is
the source of fine spatial resolution and also can be used
to refine the target’s orbit. Reliable, precise time/frequency
measurements are made possible by high-speed data ac-
quisition systems and stable, accurate clocks and frequency
standards.
Because different parts of the rotating target will have
different velocities relative to the radar, the echo will be dis-
persed in Doppler frequency as well as in time delay. The
basic strategy of any radar experiment always involves mea-
surement of some characteristic(s) of the functionσ(τ,ν),
perhaps as a function of time and perhaps using more than
one combination of transmitted and received polarizations.
Ideally, one would like to obtainσ(τ,ν) with very fine
resolution, sampling that function within intervals whose
dimensionsτ×νare minute compared to the echo
dispersionsτTARGETandνTARGET. Figure 5 shows the
geometry of delay-resolution cells and Doppler-resolution
cells for a spherical target and sketches their relation toσ(τ)
andσ(ν).
2.4 Radar Waveforms
In the simplest radar experiment, the transmitted sig-
nal is a highly monochromatic, unmodulated, continuous
wave (cw) signal. Analysis of the received signal comprises
Fourier transformation of a series of time samples and yields
an estimate of the echo power spectrumσ(ν), but it contains
no information about the distance to the target orσ(τ). To
avoidaliasing,the sampling rate must be at least as large
as the bandwidth of the low-pass filter (see Fig. 4) and usu-
ally is comparable to or larger than the echo’s intrinsic dis-
persionνTARGETfrom Doppler broadening. Fast Fourier
transform (FFT) algorithms greatly speed the calculation
of discrete spectra from time series and are ubiquitous in
FIGURE 5 Time-delay and Doppler-frequency resolution of the
radar echo from a rotating sphere.radar astronomy. In a single FFT operation, a string ofN
time samples taken at intervals oftseconds is transformed
into a string ofNspectral elements with frequency resolu-
tionν= 1/(Nt).
To obtain delay resolution, one must apply some sort of
time modulation to the transmitted waveform. For exam-
ple, a short-duration pulse of cw signal lasting 1μs would
provide delay resolution of 150 m. However, the echo would
have to compete with the noise power in a bandwidth of or-
der 1 MHz (i.e., the reciprocal of 1μs), so the echo power
from many consecutive pulses would probably have to be
summed to yield a detection. One would not want these
pulses to be too close together, however, or there would
be more than one pulse incident on the target at once,
and interpretation of echoes would be insufferably ambigu-
ous. Thus, one arranges the pulse repetition periodtPRP
to exceed the target’s intrinsic delay dispersionτTARGET,
ensuring that the echo will consist of successive, nonover-
lapping “replicas” ofσ(τ) separated from each other by
tPRP. To generate this “pulsed cw” waveform, the transmit-
ter is switched on and off while the frequency synthesizer
(see Fig. 4) maintains phase coherence from pulse to pulse.
Then Fourier transformation of time samples taken at the
same position within each ofNsuccessive replicas ofσ(τ)
yields the power spectrum of echo from a certain delay res-
olution cell on the target. This spectrum has an unaliased
bandwidth of 1/tPRPand a frequency resolution of 1/(NtPRP).
Repeating this process for a different position within each