826 SAR Basics-F15
826 SAR Basics-F15
826 SAR Basics-F15
Basics
1
Outline
Spatial resolution
Range resolution
• Short pulse system
• Pulse compression
• Chirp waveform
• Slant range vs. ground range
Azimuth resolution
• Unfocused SAR
• Focused SAR
Geometric distortion
Foreshortening
Layover
Shadow
Radiometric resolution
Fading
Radiometric calibration
2
Spatial discrimination
Spatial discrimination relates to the ability to resolve signals
from targets based on spatial position or velocity.
angle, range, velocity
Resolution is the measure of the ability to determine
whether only one or more than one different targets are
observed.
Range resolution, r, is related to signal bandwidth, B
3
Spatial discrimination
The ability to discriminate between targets is better when
the resolution distance is said to be finer (not greater)
Fine (and coarse) resolution are preferred to high (and low) resolution
Various combinations of resolution can be used to
discriminate targets
4
Range resolution
5
Range resolution
Short pulse radar
Tx T= 2R/c
The received echo, Pr(t) is
Pr t Pt t St
where Rx T
Pt(t) is the pulse shape point target echo
Example
r = 1 m 6.7 ns
r = 1 ft 2 ns
6
Range resolution
Clearly to obtain fine range resolution, a short pulse
duration is needed.
However the amount of energy (not power) illuminating the
target is a key radar performance parameter.
Energy, E, is related to the transmitted power, Pt by
E 0 Pt t dt
Pt
7
More Tx Power??
Why not just get a transmitter that outputs more power?
High-power transmitters present problems
Require high-voltage power supplies (kV)
Reliability problems
Safety issues (both from electrocution and irradiation)
Bigger, heavier, costlier, …
8
Simplified view of pulse compression
Energy content of long-duration, low-power pulse will be
comparable to that of the short-duration, high-power pulse
1 « 2 and P1 » P2
P1
1
Goal: P1 1 P2 2
2
P2
time
9
Pulse compression
Chirp waveforms represent one approach for pulse
compression.
Radar range resolution depends on the bandwidth of the
received signal.
c c c = speed of light, r = range resolution,
r = pulse duration, B = signal bandwidth
2 2B
The bandwidth of a time-gated sinusoid is inversely
proportional to the pulse duration.
So short pulses are better for range resolution
10
Pulse compression, the compromise
Transmit a long pulse that has a bandwidth corresponding
to a short pulse
Must modulate or code the transmitted pulse
to have sufficient bandwidth, B
can be processed to provide the desired range resolution, r
Example:
Desired resolution, r = 15 cm (~ 6”) Required bandwidth, B = 1 GHz (109 Hz)
Required pulse energy, E = 1 mJ E(J) = Pt(W)· (s)
Brute force approach
Raw pulse duration, = 1 ns (10-9 s) Required transmitter power, P = 1 MW !
Pulse compression approach
Pulse duration, = 0.1 ms (10-4 s) Required transmitter power, P = 10 W
11
FM-CW radar
Alternative radar schemes do not involve pulses, rather the
transmitter runs in “continuous-wave” mode, i.e., CW.
12
Linear FM sweep FM-CW radar
Bandwidth: B Repetition period: TR= 1/fm
Round-trip time to target: T = 2R/c
fR = Tx signal frequency – Rx signal frequency
B 4BR 4BR
f R T f m , Hz
TR 2 c TR c
2B
13
FM-CW radar
The FM-CW radar has the advantage of constantly illuminating the
target (complicating the radar design).
It maps range into frequency and therefore requires additional signal
processing to determine target range.
Targets moving relative to the radar will produce a Doppler frequency
shift further complicating the processing.
14
Blending the ideas of
Chirp radar
pulsed radar with linear
frequency modulation
results in a chirp (or linear
FM) radar.
Transmit a long-duration,
FM pulse.
Correlate the received
signal with a linear FM
waveform to produce
range dependent target
frequencies.
Signal processing (pulse
compression) converts
frequency into range.
Key parameters:
B, chirp bandwidth
, Tx pulse duration
15
Chirp radar
Linear frequency modulation (chirp) waveform
s( t ) A cos 2 f C t 0.5 k t 2 C
for 0 t
fC is the starting frequency (Hz)
k is the chirp rate (Hz/s)
C is the starting phase (rad)
B is the chirp bandwidth, B = k
16
Stretch chirp processing
17
Challenges with stretch processing
Low-pass A/D
Received signal Digitized signal
filter converter
(analog)
To dechirp the signal from extended targets, a
Reference local oscillator (LO) chirp with a much greater
chirp bandwidth is required. Performing analog
dechirp operation relaxes requirement on A/D
Echoes from targets at various converter.
ranges have different start
times with constant pulse
duration. Makes signal
processing
LO
more difficult.
near
Tx
frequency
frequency
B Rx near
time
far far
time
18
Pulse compression example
Key system parameters
Pt = 10 W, = 100 s, B = 1 GHz, E = 1 mJ , r = 15 cm
19
Pulse compression example (cont.)
With stretch processing a reduced video signal bandwidth
is output from the analog portion of the radar receiver.
video bandwidth, Bvid = k Tp (where Tp = 2 Wr /c is the swath’s slant width)
for Wr = 3 km, Tp = 20 s Bvid = 200 MHz
This relaxes the requirements on the data acquisition
system (i.e., analog-to-digital (A/D) converter and
associated memory systems).
20
Correlation processing of chirp signals
Avoids problems associated with stretch processing
Takes advantage of fact that convolution in time domain
equivalent to multiplication in frequency domain
Convert received signal to freq domain (FFT)
Multiply with freq domain version of reference chirp function
Convert product back to time domain (IFFT)
Freq-domain
reference chirp
21
Signal correlation examples
Input waveform #1
High-SNR gated sinusoid, no delay
Input waveform #2
High-SNR gated sinusoid, ~800 count delay
22
Signal correlation examples
Input waveform #1
High-SNR gated sinusoid, no delay
Input waveform #2
Low-SNR gated sinusoid, ~800 count delay
23
Signal correlation examples
Input waveform #1
High-SNR gated chirp, no delay
Input waveform #2
High-SNR gated chirp, ~800 count delay
24
Signal correlation examples
Input waveform #1
High-SNR gated chirp, no delay
Input waveform #2
Low-SNR gated chirp, ~800 count delay
25
Chirp pulse compression and time sidelobes
r = r
range resolution
27
Why time sidelobes are a problem
Sidelobes from large-RCS targets with can obscure signals
from nearby smaller-RCS targets.
Time sidelobes are related to pulse duration, .
fb
fb = 2 k R/c
28
Window functions and their effects
Time sidelobes are a side
effect of pulse compression.
Windowing the signal prior to
frequency analysis helps
reduce the effect.
Some common weighting
functions and key
characteristics
29
Window functions
Basic function:
30
Window functions
31
Detailed example of chirp pulse compression
received signal
s( t ) a cos 2 f C t 0.5 k t 2 C , 0 t
dechirp analysis
s( t ) s( t T) a cos 2 f C t 0.5 k t 2 C a cos 2 f C ( t T) 0.5 k ( t T) 2 C
which simplifies to sinusoidal term
a2 cos (2 f C T 2 k t T k T 2 )
s( t ) s( t T )
2 2 2
cos 2 k t 2 f C t k t 0.5 k T f C T 2 C
chirp-squared
term
quadratic linear
frequency frequency phase terms
dependence dependence sinusoidal
term
after lowpass filtering to reject harmonics
q( t )
a2
2
cos 2 f CT k T t 0.5 k T 2
32
Pulse compression effects on SNR and blind range
SNR improvement due to pulse compression: B
Pt G t G r l2s
SNRcompress B
4 R k T B F
3 4
33
Pulse compression
Pulse compression allows us to use a reduced transmitter
power and still achieve the desired range resolution.
The costs of applying pulse compression include:
added transmitter and receiver complexity
must contend with time sidelobes
increased blind range
The advantages generally outweigh the disadvantages so
pulse compression is used widely.
34
Slant range vs. ground range
Cross-track resolution in
the ground plane (x) is the
projection of the range
resolution from the slant plane
onto the ground plane.
36
Along-track resolution
Consider an airborne radar system flying at a constant speed
along a straight and level trajectory as it views the terrain.
For a point on the ground the range to the radar and the radial
velocity component can be determined as a function of time.
Radar position = (0, vt, h), Target position = (xo, yo, 0), Range to target, R(t)
R t 0 x o 2 v t yo 2 h 02
R 0 x o2 yo2 h 2
v v t yo
R t
dR
dt x o v t yo h 2
2 2
yo v
R 0
x o2 y o2 h 2
f D R 0
2 2 yo v
l l x o2 y o2 h 2
37
Along-track resolution
38
Along-track resolution
Example
Airborne SAR
Altitude: 10,000 m
Velocity: 75 m/s
Five targets on ground
All cross-track offsets = 5 km
Along-track offsets of -1000, -500, 0, 500, and 1000 m
39
Along-track resolution
20000
Alt = 10 km
19000
18000
17000
16000
Range (m)
15000
14000
13000
1000 m
500 m
12000
0m
11000 -500 m
-1000 m
10000
-15000 -10000 -5000 0 5000 10000 15000
Along-track position (m)
40
Along-track resolution
1500
Alt = 10 km, Vel = 75 m/s, l = 10 cm 1000 m
500 m
1000 0m
-500 m
-1000 m
Doppler frequency (Hz)
500
0
300
200
-500 100
0
c
-100
-1000 -200
-300
-1000 -500 0 500 1000
-1500
-15000 -10000 -5000 0 5000 10000 15000
Along-track position (m)
41
Along-track resolution
42
Along-track resolution
Example
Airborne SAR
Altitude: 10,000 m
Velocity: 75 m/s
Five targets on ground
All along-track offsets = 0 m
Cross-track offsets of 5, 7.5, 10, 12.5, and 15 km
43
Along-track resolution
20000
Alt = 10 km
19000
18000
17000
16000
Range (m)
15000
14000
13000
15 km
12.5 km
12000
10 km
11000 7.5 km
5 km
10000
-15000 -10000 -5000 0 5000 10000 15000
Along-track position (m)
44
Along-track resolution
1500
Alt = 10 km, Vel = 75 m/s, l = 10 cm 15 km
12.5 km
1000 10 km
7.5 km
5 km
Doppler frequency (Hz)
500
0
100
75
50
-500
25
0
c
-25
-1000 -50
-75
-100
-750 0 750
-1500
-15000 -10000 -5000 0 5000 10000 15000
Along-track position (m)
45
Along-track resolution
46
Along-track resolution
Now solve for R and fD for all target locations and plot lines
of constant range (isorange) and lines of constant Doppler
shift (isodops) on the surface.
47
Along-track resolution
Isorange and isodoppler lines for aircraft flying north at 10 m/s at a 1500-m altitude.
r = 2 m, V = 0.002 m/s, fD = 0.13 Hz @ f = 10 GHz, l = 3 cm 48
Along-track resolution
Without the spatial filtering of the antenna, the azimuth chirp waveform
covers a wide bandwidth.
49
Along-track resolution
Samples of phase variations due to a changing range
throughout the aperture are provided with each pulse in the
slow-time domain.
Note that the range chirp has been frequency shifted to baseband.
50
Along-track resolution
Squint-mode operation (or moving targets) will skew the
Doppler spectrum. This skew can be detected and
accommodated.
51
Strip-map SAR signal example (no squint)
52
Strip-map SAR signal example (no squint)
Time domain characteristics of single point target.
Magnitude of phase history mapped in azimuth and range.
• Constant amplitude in range
axis indicates uniform pulse
amplitude (no windowing).
• Variation in azimuth
represents antenna beam
pattern in azimuth plane.
53
Strip-map SAR signal example (no squint)
Time domain characteristics of single point target.
Real part of phase history mapped in azimuth and range.
Can be shown that contour of constant phase follows:
K t 2 K a 2
Where
K is the pulse chirp rate
Ka is the azimuth chirp rate
t is the fast time index
is the slow time index
is a constant
• Positive range chirp (K > 0),
negative azimuth chirp (Ka < 0)
• Contours of constant phase
map as hyperbolae
54
Strip-map SAR signal example (no squint)
55
Strip-map SAR signal example (no squint)
Time domain characteristics of single point target.
Real part of phase history mapped in azimuth and range.
Can be shown that contour of constant phase follows:
K t 2 K a 2
Where
K is the pulse chirp rate
Ka is the azimuth chirp rate
t is the fast time index
is the slow time index
is a constant
• Negative range chirp (K < 0),
negative azimuth chirp (Ka < 0)
• Contours of constant phase
map as ellipses
56
Unfocused SAR
Processing SAR phase data to achieve a fine-resolution
image requires elaborate signal processing.
In some cases trading off resolution for processing
complexity is acceptable.
In these cases a simplified unfocused SAR processing is
used wherein only a portion of the azimuth phase history is
used resulting in a coarser azimuth resolution.
In unfocused SAR processing consecutive azimuth
samples are added together (in the slow-time domain).
Since addition a is simple operation for digital signal
processors, the image formation processing is much easier
(less time consuming) than fully-focused SAR processing.
57
Unfocused SAR
Summing consecutive samples,
also known as a coherent
integration or boxcar filtering, is
useful so long as the signal’s
phase is relatively constant over
the integration interval.
Example
For a 20-sample interval the
central portion of the chirp
waveform (zero Doppler) is
relatively constant.
For the outer portions of the chirp
the phase varies significantly and
integrating produces a reduced
output.
58
Unfocused SAR
Example (cont.)
Over a 38-sample interval phase
variations within the central portion
of the chirp waveform results in a
reduced output (0.8 peak vs. 1).
The magnitude of the first sidelobe
is also larger (0.4 vs. 0.3).
The width of the main lobe is
narrower.
59
Unfocused SAR
The resolution improves with increased integration length up to a point
when oscillations in the signal are included in the integral.
60
Focused SAR
To realize the full potential of SAR and achieve fine along-
track (azimuth) resolution requires matched filtering of the
azimuth chirp signal.
Stretch chirp processing, correlation processing, tracking
Doppler filters, as well as other techniques can be used in
a matched filter process.
However the range processing is not entirely separable
from the azimuth processing as an intricate interaction
between range and azimuth domains exists which must
also be dealt with to achieve the desired image quality.
61
Focused SAR
Consider the phenomenon known as range walk or range-
cell migration.
Variations in range to a
target over the synthetic
aperture not only introduce
a quadratic phase change
(resulting in the azimuth
chirp) but may also
displace echo in the
range (fast-time) domain.
62
Focused SAR
If the range to the target
varies by an amount
greater than the range
resolution then the range-
cell migration must be
compensated during the
image formation
processing.
64
Radiometric resolution -- signal fading
For extended targets (and targets composed of multiple
scattering centers within a resolution cell) the return signal
(the echo) is composed of many independent complex
signals.
The overall signal is the vector sum of these signals.
Consequently the received voltage will
fluctuate as the scatterers’ relative
magnitudes and phases vary spatially.
Consider the simple case of only two
scatters with equal RCSs separated
by a distance d observed at a range Ro.
65
Signal fading
As the observation point moves along the x direction, the
observation angle will change the interference of the
signals from the two targets.
The received voltage, V, at the radar receiver is
V V0 e j 2 k R a V0 e j 2 k R b
where d d
Ra R0 sin , R b R 0 sin
2 2
2d
V 2 V0 cos sin
l
The measured voltage varies
from 0 to 2, power from 0 to 4.
Single measurement will not
provide a good estimate of the
scatterer’s s.
Note: Same analysis used
for antenna arrays.
66
Fading statistics
Consider the case of Ns independent scatterers (Ns is large)
where the voltage due to each scatterer is Vi e j
i
i 1
where Ve and are the envelope voltage
and phase.
It is assumed that each voltage term,
Vi and i are independent random variables and that i is
uniformly distributed.
The magnitude component Vi can be decomposed into
orthogonal components, Vx and Vy
Vx Vi cos i and Vy Vi sin i
where Vx and Vy are normally distributed.
67
Fading statistics
The fluctuation of the envelope voltage, Ve, is due to fading
although it is similar to that of noise.
The models for fading and noise are
essentially the same.
Two common envelope detection schemes are considered,
linear detection (where the magnitude of the envelope voltage is
output) and square-law detection (where the output is the square
of the envelope magnitute).
Linear detection, VOUT = |VIN| = Ve
It can be shown that Ve follows a Rayleigh distribution
Ve Ve2 2 s2
, Ve 0 where s2 is the variance
pVe 2 s 2
e
Ve 0 of the input signal
0,
68
Fading statistics (linear detection)
70
Fading statistics (linear detection)
The mean value, VL, is unaffected by the averaging process
VL s 2 Ve
71
Fading statistics (square-law detection)
Square-law detection, Vs = Ve2
The output voltage is related to the power in the envelope.
It can be shown that Vs follows an exponential distribution
1 Vs 2 s2
, Vs 0 where s2 is the variance
pVs 2 s 2
e
Vs 0 of the input signal
0 ,
73
Fading statistics (square-law detection)
The mean value, VL, is unaffected by the averaging process
VL 2 s2
However the magnitude of the fluctuations are reduced
Vs
2 2
Vsac N
And the effective SNR due
to fading improves as 1/N.
As more exponential
distributed samples are
averaged the distribution
begins to resemble a 2(2N)
distribution.
For large N, (N > 10), the
distribution becomes Gaussian.
74
Independent samples
Fading is not a noise phenomenon, therefore multiple
observations from a fixed radar position observing the
same target geometry will not reduce the fading effects.
Two approaches exist for obtaining independent samples
change the observation geometry
change the observation frequency (more bandwidth)
Both methods produce a change in which yields an
independent sample.
Estimating the number of independent samples depends
on the system parameters, the illuminated scene size, and
on how the data are processed.
75
Independent samples
In the range dimension, the number of independent
samples (NS) is the ratio of the range of the illuminated
scene (Wr) to the range resolution (r)
N S R r
76
Independent samples
When relative motion exists between the target and the
radar, the frequency shift due to Doppler can be used to
obtain independent samples.
The number of independent samples due to the Doppler
shift, ND, is the product of the Doppler bandwidth, fD, and
the observation time, T
ND f D T
N NS N D
N=1 N = 10
N = 50 N = 250
78
Radiometric calibration
Translating the received signal power into a target’s radar
characteristics (cross section or attenuation) requires
radiometric accuracy.
From the radar range equation for an extended target
l2 Pt G 2 s A
Pr
4 3
R 4
79
Radiometric calibration
Transmit power, Pt
Addition of an RF coupler or power splitter at the transmitter output
permits continuous monitoring of the transmitted signal power.
Antenna gain, G
The antenna’s radiation pattern must be well characterized. In
many cases the antenna must be characterized on the platform
(aircraft or spacecraft) as it’s immediate environment may affect the
radiation characteristics. Furthermore the characterization may
need to be performed in flight.
Target range, R
Radar’s inherent ability to measure range accurately minimizes any
contribution to radiometric uncertainty.
Resolution cell area, A
Difficult to measure directly, requires measurement data from
extended target with known s.
80
Radiometric calibration
Calibration targets
Radiometric calibration of the entire radar system may
require external reference targets such as spheres,
dihedrals, trihedrals, Luneberg lens, or active calibrators.
81
Radiometric calibration
Flat plate
82
Radiometric calibration
Dihedral and trihedral corner reflectors
83
Radiometric calibration
Dihedral and trihedral corner reflectors
84
Radiometric calibration
Luneberg lens
85
Radiometric calibration
Luneberg lens
86
Radiometric calibration
87
Radiometric calibration
Active radar calibrator [Brunfeldt and Ulaby, IEEE Trans. Geosci. Rem. Sens., 22(2),
pp. 165-169, 1984.]
88
Radiometric calibration
Active radar calibrator
89
Radiometric calibration
90
Radiometric calibration
RCS of some common shapes
91
Radiometric calibration
92
Radiometric calibration
93