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Aerospace Radar- Lesson 4: SYNTHETIC APERTURE RADAR
Hon.-Prof. Dr.-Ing. Joachim Ender
Head of
Fraunhoferinstitut für Hochfrequenzphysik and Radartechnik FHR
Neuenahrer Str. 20, 53343 Wachtberg
J.H.G. Ender: Radarverfahren- 2 -
SYNTHETIC APERTURE RADAR
Imaging radar (SAR, ISAR) is based on ..
Measurement of range
(pulse compression)
Measurement of range change via phase
(-> cross range resolution)
Relative aspect change of the
scene, the object necessary
SAR: via motion of the
platform
ISAR: via motion of the target
J.H.G. Ender: Radarverfahren- 3 -
Spatial Fourier transformation
One dimensional spatial Fourier
transformation
Let a(x) be a complex function of the spatial variable x
Kx is a spatial frequency [rad/m] which represents the phase variation along the space.
• d-dimensional spatial Fourier transformation
• Inverse spatial Fourier transformation
Examples
Most rapid phase change in direction of r0
1)
2)
rK
rtj
dea 0
2
1)(
Plane wave with
J.H.G. Ender: Radarverfahren- 4 -
2D Fourier correspondence
r0
J.H.G. Ender: Radarverfahren- 5 -
Restriction to a K-set
Back transformation over a K-set
yields an unsharp reconstruction of a(r)
• Different choices of the K-set effect different (in phase and speckle) unsharp reconstructions.
• In the SAR-context the generation of multiple images via different K-sets is called 'Multilook'
• Incoherent summation of the unsharp reconstructions reduces the speckle.
J.H.G. Ender: Radarverfahren- 6 -
Restriction to a K-set
Fourier
transformation
Back
Transf.
Restriction to K-sets
Kx
Ky
J.H.G. Ender: Radarverfahren- 7 -
Radar imaging with turn table
Radar
Turn table
Beam
360°
T
rRange histories of isolated scatterers
J.H.G. Ender: Radarverfahren- 8 -
Geometry for the turn table experiment
sincos0 yxR
y
xr
Far field approximation!
J.H.G. Ender: Radarverfahren- 9 -
Radar signals from the turntable experiment
Signal from a single scatterer at (x,y) after range pre-processing
• Superposed signals, reflectivity a(x,y)
• Define:
• Variable substitution (Re-formatting)
J.H.G. Ender: Radarverfahren- 10 -
Polar Reformatting
To apply a DFT, the measurements on a polar grid have to be interpolated to a rectangular grid
J.H.G. Ender: Radarverfahren- 11 -
Point spread function
Measured K-set:
• The point spread function is given by the Fourier transform of the Indicator function of the K-set
J.H.G. Ender: Radarverfahren- 12 -
Resolution
Extension of the K-set in x- and y-direction
(for small angles):
• Resolution:
Ky
Kx
KKy
Kx
FFT columns, x S*(f)
r
T
IFFT colums2D FFT
(Amplitude)
y
x
2D IFFT
(Amplitude)
ky
kx
Polar Reformatting
J.H.G. Ender: Radarverfahren- 14 -
Turntable ISAR-imaging at 94 Ghz
J.H.G. Ender: Radarverfahren- 15 -
Turntable ISAR-imaging at 220 Ghz (8 GHz bandwidth)
J.H.G. Ender: Radarverfahren- 16 -
ISAR film of rotating TIRA antenna
J.H.G. Ender: Radarverfahren- 17 -
SYNTHETIC APERTURE RADAR
Applications
Geology, Cartography: Chartering of areas which are permanently under dense cloud cover, surveying of geological structures, worldwide surveying of human settlings and cultivated land.
Oceanography: Analysis of ocean waves, conclusions on wind fields, research on tsunamis.
Hydrology: Surveying of floods and soil moisture.
Glaciology: Monitoring of ice in the arctic regions, icebergs, and snow coverage.
Nautical shipping and economy: Recording of the sea states, monitoring of shipping lanes, chartering of seaweed appearances.
Agriculture and Forestry: Identidentification of cultivation types, control of fallowing schemes, monitoring of planting states, observation of grasshopper swarms.
Emergency management: Recognition of surface changes, e.g. cost erosion, detection of earth- and seaquakes, search and rescue of missing people.
Environmental monitoring: In addition to the applications mentioned above: Recognition and imaging of pollution, e.g. oil films on water surfaces, chartering of damage to forests.
J.H.G. Ender: Radarverfahren- 18 -
SYNTHETIC APERTURE RADAR
Applications: Analysis of floods
SAR-image
of the Elbe
river near
Dömitz at
times of a
disastrous
flooding.
Left: 29
August 2002,
Right: 22
October
2003.
FHR
J.H.G. Ender: Radarverfahren- 19 -
Antenna
footprint
Flight path
Synthetic array
Nadir
L
x
Azimuth
Slan
t ran
ge
Ground range
Strip w
idth
La
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Stripmap mode
J.H.G. Ender: Radarverfahren- 20 -
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Focused SAR as synthetic antenna
Phase along the synthetic aperture
Distance of a point scatterer at rto an aperture coordinate x
Azimuth resolution after phase compensation and integration along synthetic aperture Lx
Maximum length of a synthetic array
J.H.G. Ender: Radarverfahren- 21 -
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Focused SAR as synthetic antenna
Length of the antenna footprint
(ua denotes the beamwidth of the real
antenna):
For fixed antenna look direction the
distance the platform is flying from a
scatterer entering into the mainbeam
until its disappearance is La, too. So for
this mode, the length of the synthetic
array Lx is bounded by
Consequently, the achievable azimuth
resolution for stripmap SAR is
For a real antenna we have
It follows
J.H.G. Ender: Radarverfahren- 22 -
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Focused SAR as synthetic antenna
There are some differences compared to a real array:
The phase terms always are related to two-ways traveling. Compared to a real array always the double phase has to be inserted. As a consequence, for the same aperture the half beamwidth is obtained. On the other hand, the minimum spatial sampling interval is reduced by the factor 2.
Omnidirectional antennas would have to be spaced at the raster l/4. The phase centers of directed antennas with the real aperture lx are not allowed to move more than lx/2 from pulse to pulse.
The pattern of the synthetic antenna shows only to a one-way characteristics, since the way forward and backward had been regarded in the phase terms. Unfortunately, this leads to a higher side lobe level compared to the two-way pattern of a real array!
Because of the long synthetic aperture a far field focusing is no longer possible.
The quadratic term of the phase along the synthetic aperture is related to a distinct range. Like for an optical lens a high quality focusing is obtained only in a certain range interval, the depth of focus.
J.H.G. Ender: Radarverfahren- 23 -
SYNTHETIC APERTURE RADAR
Depth of focus
J.H.G. Ender: Radarverfahren- 24 -
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Cylinder coordinates
J.H.G. Ender: Radarverfahren- 25 -
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Cylinder coordinates - azimuth/slant range coordinates
Since the range from the antenna to a scatterer depends only on
the azimuth coordinate and the radius r, it is convenient to
introduce a cylindric coordinate system (x, r, J ), where the
third parameter, the elevation angle, does not influence the range
function.
It merely modulates the signal amplitude by the antenna
characteristics.
So we can reduce the geometry to two dimensions (x; r).
J.H.G. Ender: Radarverfahren- 26 -
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Two-dimensional coordinates
J.H.G. Ender: Radarverfahren- 27 -
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The azimuth signal
Range and direction histories
We regard a scatterer at x = 0. Representative histories:
J.H.G. Ender: Radarverfahren- 28 -
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The azimuth signal
Derivatives with respect to the platform position:
Values at x = 0:
Taylor approximations: Valid for small angular
deviations from the normal
direction, where the range
hyperbola can sufficiently
be approximated by the
range parabola.
J.H.G. Ender: Radarverfahren- 29 -
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The azimuth signal
J.H.G. Ender: Radarverfahren- 30 -
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The azimuth chirp
Due to the range and direction histories we get the following model signal in the slow time domain:
Quadratic approximation of range history:
J.H.G. Ender: Radarverfahren- 31 -
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The azimuth chirp - Phase and Doppler history
Quadratic approximation:
Phase and Doppler history:
J.H.G. Ender: Radarverfahren- 32 -
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The azimuth chirp
Time histories of the
range (phase), the angle,
the Doppler frequency
and the azimuth chirp
J.H.G. Ender: Radarverfahren- 33 -
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Azimuth chirp and synthetic aperture
J.H.G. Ender: Radarverfahren- 34 -
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Azimuth point spread function by summing up over a synth. aperture
without phase compensation ("unfocused SAR")
Increasing
length of
synthetic
aperture
J.H.G. Ender: Radarverfahren- 35 -
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Summary of the interdependencies between the variables
J.H.G. Ender: Radarverfahren- 36 -
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The azimuth chirp - important parameters
Maximum length of the synthetic aperture
Doppler modulation rate
Duration of illumination For small beamwidth
At time T=0:
J.H.G. Ender: Radarverfahren- 37 -
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The azimuth chirp - important parameters
Doppler bandwidth
Time-bandwidth product
for an antenna with length lx
The time-bandwidth product (compression rate) is equal to the ratio between the
synthetic and the half real aperture
J.H.G. Ender: Radarverfahren- 38 -
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The azimuth chirp - important parameters
Numerical example
J.H.G. Ender: Radarverfahren- 39 -
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The flying radar as a (slow-)time-invariant linear filter
Measured signal for azimuth reflectivity normalized to slow time
NoiseModel signal
The flying radar acts on the azimuth reflectivity signal like a time-invariant
linear filter, whose pulse response is s0(T).
Transfer function:
J.H.G. Ender: Radarverfahren- 40 -
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Azimuth compression
Analogue to the pulse compression in fast time, the point spread function in
azimuth is the response of the compression filter to the azimuth model signal:
To compress the azimuth signal, we use a filter with pulse response h(T)
The SNR-optimum compression filter is the matched filter, given by
The point spread function in azimuth is the Fourier back-transform of
J.H.G. Ender: Radarverfahren- 41 -
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Azimuth compression - inverse filter
Inverse filter over a Doppler-frequency set F with S0(F) not equal to 0.
J.H.G. Ender: Radarverfahren- 42 -
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Evaluation of the azimuth chirp - spectrum
Rough estimate of the spectral power:
Direction related to Doppler F, power from this direction
So we get approximately
Again, the Doppler power spectrum is a scaled version of the two-ways antenna
pattern, as similar stated for the clutter spectrum before.
The clutter spectrum is generated by many ground based scatterers at different
azimuth positions, observed over a short time.
Here, we assumed a single scatterer observed for the whole time of illumination.
J.H.G. Ender: Radarverfahren- 43 -
The integral of a chirp (see 'Anatomy of a
chirp'):
For a smooth signal s(t) we get:
Transferred to t=t0:
More general:
(Chirp selection theorem)
Main contribution to the integral
SYNTHETIC APERTURE RADAR
Chirp selection theorem
J.H.G. Ender: Radarverfahren- 44 -
We regard an integral of the form
Let s(t) be slowly varying and F(t) two
times differentiable and running through
many 2 intervals.
Let be a unique t0 with
We can expand the phase around t=t0:
Then:
For more than one point of stationary phase, the
contributions sum up.t0 is called 'point of stationary phase'.
SYNTHETIC APERTURE RADAR
Principle of stationary phase
J.H.G. Ender: Radarverfahren- 45 -
We regard a signal of the form
and compute the Fourier transform:
with slowly varying s(t) and a phase F(t)
with the above mentioned properties,
Remark, that we spare an integral with this approximation!
with t0 determined by
and
SYNTHETIC APERTURE RADAR
Application to the Fourier transform
J.H.G. Ender: Radarverfahren- 46 -
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Fourier transform of the azimuth chirp
The phase:
J.H.G. Ender: Radarverfahren- 47 -
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Fourier transform of the azimuth chirp
The value of the stationary phase:
F2
J.H.G. Ender: Radarverfahren- 48 -
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Fourier transform of the azimuth chirp
Second derivative of the phase:
J.H.G. Ender: Radarverfahren- 49 -
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Fourier transform of the azimuth chirp
Result:
Essence The power spectrum of the azimuth chirp is a scaled version of the magnitude-
square of the two-way antenna characteristics.
The complex azimuth point target reference spectrum S0(F) can be approximated by the
principle of the stationary phase by above equations. This approximation can be used
to perform the azimuth compression in the Doppler domain.
J.H.G. Ender: Radarverfahren- 50 -
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Compression of the azimuth chirp - compression filters
The point spread function is the inverse Fourier transform of
The SNR-optimum azimuth compression filter is the matched filter
Window in the slow-time domain, or in the Doppler domain:
Inverse filter over a set F of frequencies with non-vanishing signal spectrum:
J.H.G. Ender: Radarverfahren- 51 -
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Compression of the azimuth chirp - compression filters
Fourier transform of the point spread function for inverse filtering:
Fourier transform of the compressed scene i
Fourier transform of the azimuth reflectivity scaled to slow-time:
J.H.G. Ender: Radarverfahren- 52 -
SYNTHETIC APERTURE RADAR
Compression of the azimuth chirp - Spotlight and sliding mode
For the stripmap mode the antenna must not be longer then twice the desired azimuth
resolution. That means that a very short antenna has to be used to obtain a fine
resolution. But this is a problem in several aspects:
The antenna gain will be low leading to a poor SNR.
The Nyquist condition for azimuth sampling (at least one pulse per movement of
the half antenna length) requires a high PRF leading to a high data rate and perhaps
range ambiguities.
In the spotlight mode the antenna beam is steered to a fixed point at the earth. In this
way the synthetic aperture can be elongated nearly arbitrarily to achieve fine
resolution. The instantaneous Doppler bandwidth is still given by B = 2V/lx and the
PRF has to be adopted to this bandwidth, while the center Doppler frequency changes
according to the varying look direction. A disadvantage of the spotlight mode is the
restriction of the azimuth length of the scene to the width of the antenna footprint.
J.H.G. Ender: Radarverfahren- 53 -
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Compression of the azimuth chirp - Spotlight and sliding mode
A continuous intermediate state between stripmap and spotlight mode can be obtained
by the sliding mode.
Here the center point of the antenna footprint moves along the earth surface with a
velocity which is normally between zero and the platform velocity, but sometimes it is
also desirable to let the footprint slide over the earth with a higher velocity or even
opposite to the flight direction. In all these cases, a feasible balance between azimuth
extension of the scene and azimuth resolution can be achieved.
J.H.G. Ender: Radarverfahren- 54 -
SYNTHETIC APERTURE RADAR
The signal in the x- r domain
We go back to the original raw data before pre-processing to the normal form. If the
transmit signal s(r) - written as function of the spatial variable r - is used, the
following raw data signal in the (r,x)-plane from a point scatterer at the position (x,r)
is got:
Now we consider a chirp as transmit signal, written in the spatial variable as
J.H.G. Ender: Radarverfahren- 55 -
SYNTHETIC APERTURE RADAR
The signal in the x- r domain
Signal if the wave form is a chirp:
The phase of this expression (the approximation is valid in the vicinity of (r, x))
J.H.G. Ender: Radarverfahren- 56 -
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The signal in the x- r domain
Since the exponents, which are both approximately quadratic in their respective
variables add, the lines of equal phases are approximately ellipses, if the signs of the
quadratic terms coincide, otherwise they are hyperbolas. Since the azimuth chirp is
always a down-chirp, ellipses appear if the transmit signal is also a down chirp, and
hyperbolas, otherwise.
Around a fixed point targets we observe in the first case a sequence of concentric
ellipses, which have the same sequence of radius as the fringes of a Fresnel zone
plate which are the cuts of concentric sphere shells with equidistantly growing radius.
These interference fringes are known from holography. Since such Fresnel
interference patterns can be focused using coherent light, can SAR raw data, exposed
on a photographic film, be focused to an image using optical methods, as it was done
in the early period of SAR processing.
J.H.G. Ender: Radarverfahren- 57 -
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Signal in the azimuth-range plane (down-chirp)
J.H.G. Ender: Radarverfahren- 58 -
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Real raw data in the azimuth-range domain
J.H.G. Ender: Radarverfahren- 59 -
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The signal in the x - kr - domain
After the application of a Fourier transform from the r- to the kr-dimension followed
by an inverse filtering, the signal assumes the form
The phase of the azimuth chirp is scaled with kr !
J.H.G. Ender: Radarverfahren- 60 -
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Signal in the x - kr planeR
F b
andw
idth
J.H.G. Ender: Radarverfahren- 61 -
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The signal in the kx - kr domain
A spatial Fourier transform along x yields
Again, the principle of stationary phase can be applied. The complete phase is given
by
J.H.G. Ender: Radarverfahren- 62 -
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The signal in the kx - kr domain
The right side of this equation is the phase gradient along the synthetic aperture.
Obviously, that position x0 is searched for which this phase gradient is equal to kx.
For each kx, a position x0 has to be determined where the derivation of the phase
history is zero, resulting in
J.H.G. Ender: Radarverfahren- 63 -
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The signal in the kx - kr domain
Since the absolute value of the phase gradient is always lower or equal to 2kr, we can
characterize kx as the abscissa component of a vector with length 2kr in the k-domain
(kx; kr).
In this plane, kx = 2kr cos b, where b is the angle from the kx-axis to the vector.
J.H.G. Ender: Radarverfahren- 64 -
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The signal in the kx - kr domain - geometrical interpretation
The phase can be reduced to a length of a way.
J.H.G. Ender: Radarverfahren- 65 -
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The signal in the kx - kr domain - geometrical interpretation
J.H.G. Ender: Radarverfahren- 66 -
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The signal in the kx - kr domain - geometrical interpretation
Computation of the minimum way g(x0).
J.H.G. Ender: Radarverfahren- 67 -
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The resulting phase - interpreted as the Thales half circle
J.H.G. Ender: Radarverfahren- 68 -
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The signal in the kx - kr domain
Doppler bandwidth
RF
ban
dw
idth
J.H.G. Ender: Radarverfahren- 69 -
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The signal in the kx - kr domain
Computation of the second derivative of
the phase.
J.H.G. Ender: Radarverfahren- 70 -
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The signal in the kx - kr domain
Resulting formula for the signal in this domain
J.H.G. Ender: Radarverfahren- 71 -
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K-set and point spread function
Platform position
Reference point at the surface
Point in the neighborhood of the
reference point
Local approximation of
the range history
withpointing to the
reference point
J.H.G. Ender: Radarverfahren- 72 -
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K-set and point spread function
Signal in the neighborhood of the reference point
(The two-way characteristics is written as function of the double k-vector)
Measurements
where a describes the re°ectivity in the vicinity of the reference point.
J.H.G. Ender: Radarverfahren- 73 -
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K-set and point spread function
Phase compensation with respect to the reference point
SAR processing: correlation with the model signal given above, normalization
J.H.G. Ender: Radarverfahren- 74 -
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K-set and point spread function
By variation of kr in the interval and the variation of the platform
position within a set of wave number vectors is generated:
The k-set is a circular ring segment (see Figure).
The generated SAR image can be written as reconstruction over this k-set:
J.H.G. Ender: Radarverfahren- 75 -
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K-set and point spread function
For rectangular main beam the circular ring segment can be restricted to the main
beam, leading to the k-set
We get the simplified reconstruction
Obviously the reconstruction is obtained by the Fourier transformation of the motion
compensated data over the k-set.
J.H.G. Ender: Radarverfahren- 76 -
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K-set and point spread function
k-setpoint spread
function
Fourier transf.
J.H.G. Ender: Radarverfahren- 77 -
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K-set and point spread function
If for zmc the motion compensated signal of a point scatterer at the reference point is
inserted, we get the point spread function
J.H.G. Ender: Radarverfahren- 78 -
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Generic SAR processor
The measured signal z(x; r) consists of the superposition of the echoes of many point
scatterers with the reflectivity at a(x,r):
After transformation into the (x, kr)-domain
J.H.G. Ender: Radarverfahren- 79 -
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Generic SAR processor
Principally, the raw data could be processed to an image optimally, if for each possible
position of a point scatterer the two-dimensional normalized matched filter for the
model signal would be applied:
Fast time domain:
Range frequency domain:
The computational effort would be much to high. The first approach is to apply
separated processing, i.e. range compression and azimuth compression are applied
one after the other.
J.H.G. Ender: Radarverfahren- 80 -
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Principal processing
J.H.G. Ender: Radarverfahren- 81 -
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The range-Doppler processor
The range-Doppler processor starts with the classical range compression as a rule as
'fast correlation' performed in the range frequency domain.
Now the effect that the signal runs along a hyperbolic curve through the range cells -
the range curvature problem - has to be solved.
There is a possibility to reduce the numerical effort of the described interpolation
along the range hyperbolas:
First the data are transformed by an azimuth FFT into the kx domain. The signals of
scatterers with the same r but varying x-coordinates are shifted in slow time; but by
the Fourier transform the pathes in the kx-domain are overlayed1. For this reason, the
range curvature correction can be performed in the kx-domain simultaneously.
J.H.G. Ender: Radarverfahren- 82 -
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Range curvature correction
J.H.G. Ender: Radarverfahren- 83 -
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The range-Doppler processor
J.H.G. Ender: Radarverfahren- 84 -
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The range-migration processor (Omega-k processor)
The main steps in the range-migration processor are done in the double-frequency
domain.
Remember:
The measured data are the superposition of these signals with the azimuth
reflectivity:
J.H.G. Ender: Radarverfahren- 85 -
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The range-migration processor (Omega-k processor)
Variable substitution:
We get in the new (kx,k_r)-domain:
with A(kx, kr) being the two-dimensional Fourier transformation of the reflectivity.
This substitution has changed the data in the (kx, kr)-domain into the 2D Fourier
transformation of the reflectivity distribution, modulated by the antenna
characteristics.
The grid in (kx, kr) has to be transferred into a grid in the (kx, kr)-domain. For this
the Stolt-interpolation is applied. At the end only a two-dimensional inverse Fourier
transformation has to be performed to get the focused SAR image
J.H.G. Ender: Radarverfahren- 86 -
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The range-migration processor (Omega-k processor)
J.H.G. Ender: Radarverfahren- 87 -
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The back-projection processor
J.H.G. Ender: Radarverfahren- 88 -
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Effects in SAR imaging
Top of tower appears here
(Foreshortening)
Radar shadow
Layover