IEEE Geoscience and Remote Sensing Magazine - March 2013 - 10

the received signals on-board. When referring to the time in
the range direction, it is often denoted to as fast time which
is an allusion to the velocity of the electromagnetic waves
travelling at the speed of light. The transmission and listen
procedure is repeated every PRI seconds, where the pulse
repetition interval (PRI) is the reciprocal of the pulse repetition frequency PRI = 1/PRF. Fig. 2 illustrates the typical
SAR geometry, where the platform moves in the azimuth or
along-track direction, whereas the slant range is the direction perpendicular to the radar's flight path. The swath width
gives the ground-range extent of the radar scene, while its
length depends on the data take duration, i.e., how long the
radar is turned on.
At any time t, the distance between the radar moving at
constant velocity v and a point on the ground, described by
its coordinates (x, y, z) = (x 0, 0, Th), is easily obtained applying Pythagoras' theorem

(vt) 2
r (t) = r 20 + (vt) 2 . r0 + 2r0

for vt/r0 % 1, (3)

where, without loss of generality t = t 0 = 0 is the time
of closest approach, when the distance is minimum and
r (t 0) = r0 = (H - Th) 2 + x 20 with the platform height
H. In general the distance r0 is much larger than vt during the illumination time Till a point on the ground is
observed; this allows expanding r (t) into a Taylor series
and neglecting all but the first two terms, which yields
the approximation on the right-hand side of (3). In the
above expression the time, given through the variable t, is
associated with the movement of the platform and therefore often denoted by slow time. The range variation of a
point target over time is directly related to the azimuth
phase by { (t) = - 4rr (t) /m, i.e., the phase variation has

c
eti
nth ture
y
r
S e
Ap

im
Az

Platform Height H

Sla

k
rac
ir T

Swath Width

e
r0

r (t )

x
Dh
Ground Range

FIGURE 2. Illustration of the SAR imaging geometry. r0 stands for
the shortest approach distance, H a for the azimuth beamdwidth
and v for the sensor velocity.

also a parabolic behavior (the factor 4r is due to the twoway range measurement of the SAR system). Note that
the quadratic approximation in (3) is done for the sake
of simplicity. Accurate SAR data processing takes into
account the complete phase history without any approximation [8], [9].
Being an imaging radar requires a two-dimensional resolution. The slant-range resolution d r is inversely proportional to the system bandwidth according to d r = c 0 /2B r,
where c 0 is the speed of light. The azimuth resolution d a
is provided by the construction of the synthetic aperture,
which is the path length during which the radar receives
echo signals from a point target. The beamwidth of an
antenna of length d a can be approximated by H a = m/d a .
From Fig. 2 it can be seen that the corresponding synthetic
aperture length is given through L sa = H a $ r0 = mr0 /d a . A
long synthetic aperture is favorable since it results in a
narrow virtual beamwidth H sa = m/2L sa (again, the factor
2 appears because of the two-way path from transmission
to reception) and a high azimuth resolution:

d
m
d a = r0 H sa = r0 2L = 2a . (4)
sa

The above equation suggests that a short antenna yields
a fine azimuth resolution. This appears surprising on the
first view. However, it becomes immediately clear if one
considers that a radar with a shorter antenna "sees" any
point on the ground for a longer time (the illumination
time can be approximated by Till . mr0 /vd a), which is equivalent to a longer virtual antenna length and thus a higher
azimuth resolution.
The received echo signal data form a two-dimensional
data matrix of complex samples, where each complex
sample is given by its real and imaginary part, thus representing an amplitude and phase value. The first dimension
corresponds to the range direction (or fast time); a range
line consists of the complex echo signal samples after being
amplified, down converted to base band, digitized and
stored in memory. The radar acquires a range line whenever
it travels a distance v $ PRI thus forming the second dimension of the data matrix, known as azimuth or slow-time.
The very nature of SAR is that the return echoes from the
illuminated scene are sampled both in fast time (range) and
slow time (azimuth).
Unlike optical sensors, visualizing raw SAR data does
not give any useful information on the scene. It is only
after signal processing that an image is obtained, as
shown in Fig. 3 which summarizes the basic SAR processing steps. In a very simplified way, the complete processing can be understood as two separate matched filter
operations along the range and azimuth dimensions. The
first step is to compress the transmitted chirp signals to
a short pulse. Instead of performing a convolution in the
time domain, a multiplication in the frequency domain
is adopted due to the much lower computational load.
Thus, each range line is multiplied in the frequency
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2013