2. SAR Signal Specifications:
The LFM signal equation has same form in spatial and frequency domain.
Range frequency in LFM has direct relation with range frequency:
f Kr . t
If t change from 0 to T, f change from 0 to BW.
f = BW
f = 0 Hz
t = Tt = 0
3. SAR Signals:
Strip-map mode
Moving direction (Nadir line)
Delay time in receive signal
The plane in moving during send and receive.
Doppler effect and plane’s moving makes
SAR system complex.
Range Cell Migration is to find exact place of
the surface where the receive signal hit it and
reflect.
4. SAR Signals and RCM:
Complexity hierarchical
1. Line beam, no movement
2. Line beam, with movement
3. Antenna beam, no movement (-)
4. Antenna beam, with movement(+)
(-): Pulse-signal SAR (Doppler effect)
stop-and-go situation
(+): LFM-CW SAR
6. SAR Processing Algorithm
Send and Received Signals:
Signal Bandwidth is defined by pulse duration and Kr parameter.
The echo signal multiply by reference signal and digitized by ADC.
*
Redc Transmit ceiveS S S= ×
rBandWidth K
T
τ
= ×
7. De-chirp Signal Bandwidth:
Frequency Domain:
Bandwidth shift by center of image, using Lowpass filter
Range bandwidth:
Plane height = 1000 m
Plane speed = 50 m/s
Plane angle = 50 degree
Squint angle = 0
Range variations from 1300 to 2000 meter
De-chirp Signal Bandwidth is determined by SAR range of detection.
The echo signal multiply by reference signal and digitized by ADC.
2
0 0 0 0 0( , , , ) ( , ) exp( 2 )dc r rS t r s r s j f j K j K tτ σ π τ π τ π τ= × ∆ − ∆ + ∆
2
0 0 0( ) ( , ) exp( 2 ) exp( 2 )dc r rF S r s j f j K j K t j ft dtσ π τ π τ π τ π= × ∆ − ∆ + ∆ × −∫
8. De-chirp Signal Bandwidth:
Range frequency:
Bandwidth = 1 MHz
Range variations from 1300 to 2000 meter
Delay is dictated by slant range:
Range frequency spectrum:
10. Stationary-Phase Principle:
Phase-Stationary to get a mathematical view to Fourier transform of De-chirp signal
For example:
2
0 0 0( , ) exp( 2 2 ) exp( 2 ).dc r r sSf r s j f j K j K t j f t dtσ π τ π τ π τ π= × ∆ − ∆ + ∆ × −∫
2
02 2 2fourier r r s sf K K t f tθ π τ π τ π τ π= ∆ − ∆ + ∆ −
2 2 2
022 sr v tR
c c
τ
+
= =
' 0
d
dt
θ
θ = =
Phase-Stationary is a place where amplitude variation is ignorable against phase
variation.
2
0 0 0 0 0( , , , ) ( , ) exp( 2 )dc r rS t r s r s j f j K j K tτ σ π τ π τ π τ= × ∆ − ∆ + ∆
11. Taylor approximation used to linear azimuth and range in formula
Taylor approximations:
Taylor approximations can be used
12. Taylor approximations:
Considering phase of Fourier transform, it composed of three nonlinearity need to be
solved before proceeding.
The first term is modulation of azimuth , the second is RCM which is linear
to range frequency, and the last is secondary range compression. Each part must be
compensated to derive a phase in two in-depended dimension.
These nonlinearities and dependencies can be solved in frequency domain or by
changing the variables and domains. The second is defined as w-k algorithm. In this
method the range and azimuth frequency domain change to w-k domain which in the
new domain the domains are orthogonal too.
In w-k algorithm the Reference Function Multiply is used for decoupling, and created
in w-k domain.
13. Reference Function Multiply
Reference function multiply In w-k algorithm
Rc is the slant range from plane to center of the target scene
Rc is the center of Bandwidth in range’s Fourier transform
Reference function multiply change the phase of signal by distance from Rc
14. Focusing in LFM-CW Omega-K :
Returning from w-k domain by inverse Fourier transform:
The position of the target appear as a spike function (Delta) in range-azimuth domain.
The Phase Stationary algorithm can locate targets by approximation.
15. Simulation’s parameters and values:
The simulations is done in lower sampling frequency, for fast results.
Here are the aspects of the simulations:
h = 300 m , plane altitude
Squint angle from 0 degree to 5 degree
v = 50 m/s, speed of the plane
fs = 2 Msps , sampling frequency after de-chirping
xc = 357 m, angular center of image
PRF = 1000 Hz
Pulse Duration = 1 ms
Carrier frequency = 9.6 GHz
Chirp Bandwidth = 200 MHz
Xnear and Xfar are 252 and 519 meters respectively.
Rnear and Rfar are 391 and 600 meters respectively.
Azimuth and range resolutions are 0.2 and 0.75 meters respectively.
The signals are generated in Matlab and C, for time saving.
16. Test Conditions:
The test condition change by:
Number of targets
Position of targets
Squint angle
Test conditions :
1.One-Target in Middle of image, Squint angle = 0 degree
•Two-Target in Xnear and Xfar, Squint angle = 0 degree
• one-Target in Corner, Squint angle change from 4.5 to 7.5 degree
• Four Targets in corners, (Xnear, Xfar, strip’s width= 10m)
• Multi-Target, Squint angle = 0 degree
• Multi-Target, Squint angle = 5.3 degree
17. Test Conditions 1:
Target’s location is [Xc,0],
Squint angle = 0
PSLR = -13.41 dB
Distance from 1th-peak to 2th-peak in
azimuth direction is about 4 pixel.
Range distinction ability is 1 pixel.
Drop in range direction is about 56 dB 460 480 500 520 540
-10
0
10
20 X: 501
Y: 19.33
DetectioninAzimuth-Direction
SignalStrength[dB]
X: 497
Y: 5.917
X: 505
Y: 5.917
0 500 1000 1500 2000
-100
-50
0
50
DetectioninRange-Direction
SignalStrength[dB]
995 1000 1005 1010
-40
-20
0
20
20. Test Conditions 2:
Two target at [Xnear,0] , [Xfar,0]
Plane’s Height = 300 m
Xnear and Xfar (from 40 to 60 degree from Nadir line to Target) equals 252 and 519
meters respectively, so, Rnear and Rfar equals 391 and 600 meters.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-100
-80
-60
-40
-20
0
20
DetectioninRange-Direction
SignalStrength[dB]
900 950 1000 1050 1100 1150 1200
-20
0
20
X: 1179
Y: 16.85
SignalStrength[dB]
X: 901
Y: 19.81
DetectioninRange-Direction
As show in range detection image, the peaks happened at range indexes 901, 1179.
The Range Resolution equals 0.75 meter, and Rc is 466 meter, so, the differences is
-75 and 133 for Rnear and Rfar. (75/0.75 = 100, 133/0.75 = 177)
21. Test Conditions 2:
Two target at [Xnear,0] , [Xfar,0].
Azimuth detection:
0 200 400 600 800 1000
-60
-40
-20
0
20
DetectioninAzimuth-Direction
SignalStrength[dB]
450 500 550
-10
0
10
20
DetectioninAzimuth-Direction
SignalStrength[dB]
480 490 500 510 520 530
-20
-10
0
10
20
DetectioninAzimuth-Direction
PSLR = -14.9 dB
The second peak happened at 4th
pixel away
the center.
24. OutputImage
995 1000 1005 1010
1150
1200
1250
Test Conditions 3 (squint = 7.5):
One target at [Xc,0] , squint angle equals to 7.5 degree (half of the data is valid).
Range’s resolution is 1 pixel, in azimuth direction PSLR=-12.78dB (in 9 pixel).
1190 1200 1210 1220 1230 1240
-30
-20
-10
0
10
DetectioninAzimuth-Direction(squint=7.5degree)
0 500 1000 1500 2000
-80
-60
-40
-20
0
20
DetectioninRange-Direction(squint=7.5degree)
SignalStrength[dB]
1000.5 1001 1001.5
1205
1210
1215
1220
25. OutputImage
500 1000 1500 2000
200
400
600
800
1000
1200
Test Conditions 3 (squint = 5.3):
One target at [Xnear,0] , squint angle equals to 5.3 degree. Range’s resolution is 1
pixel, in azimuth direction PSLR=-13.84dB. Maximum acceptable squint is 5.3 degree.
870 880 890 900 910 920 930 940
980
990
1000
1010
1020
0 500 1000 1500 2000
-80
-60
-40
-20
0
20
DetectioninRange-Direction(squint=5.3degree)
SignalStrength[dB]
950 1000 1050 1100
-20
-10
0
10
DetectioninAzimuth-Direction(squint=5.3degree)
SignalStrength[dB]
26. Test Conditions 4:
Four-Target in corners, (Xnear, Xfar, strip’s width=10m)
The peaks happened at 900 and
1179 in range, 401 and 601 in
azimuth direction.
Rnear = 391 m
Rfar = 600 m
Rc = 466 m
Range’s resolution = 0.75 m
(466 - 391)/0.75 = 100
(600-466)/0.75 = 177
Image boundary is detected.
0 500 1000 1500 2000
-80
-60
-40
-20
0
20
DetectioninRange-Direction
SignalStrength[dB]
0 500 1000 1500
-40
-20
0
20
DetectioninAzimuth-Direction
SignalStrength[dB]
27. Test Conditions 4:
Four Targets in corners, (Xnear, Xfar, strip’s width=10m)
(Image’s corners detection)
OutputImage
900 950 1000 1050 1100 1150
400
450
500
550
600
28. Test Conditions 5:
Multi-Target, squint angle = 0 degree
White pixels of the image separated by 1 black pixel in azimuth and 2 black pixel
in range direction. The white pixels have values.
Each pixel width equals 1
meter.
Image size is 8 * 28 meters,
Image starts from Xc to Xc+28
Range Direction
OutputImage
1155 1160 1165 1170 1175 1180 1185 1190
550
600
650
