### SAR System & Signals

```SAR System and Signals Part 2
M. A. Saville, PhD, PE
Summer, 2012
EE880 SAR System & Signals Part 2
Lesson Overview
•
•
•
•
Array Basics
SAR signal modeling
Summary
EE880 SAR System & Signals Part 2
2
•
•
•
•
•
Resolve scatterers in 1D,2D,3D
Construct geospatial image
Estimate reflectivity function
Estimate RCS of scene scatterers
Estimate cross-section coefficient
of clutter
• Image one uncompressed
range cell or voxel (3D case)
• Achieve specified resolution in
1, 2 or 3D
• Perform above within time and
computational constraints
EE880 SAR System & Signals Part 2
3
• Shown: ground plane imaging
• Down-range resolution set by
HRR waveform, i.e. bandwidth
• Cross-range resolution set
by narrow antenna beam
• Each echo resolves both dimensions
EE880 SAR System & Signals Part 2
4
Realistic Down-range
Reconstruction
Ideal down-range
target profile
rect() (infinite
bandwidth)
Time Domain
∆
Spectral Domain
-2000
-1500
-1000
filtering rect()
(finite bandwidth)
Lost energy
-500
0
Time Domain
∆
500
1000
1500
2000
Profile distortion
Reconstructed
down-range target
profile is IDFT of
windowed rect()
Note duality and reciprocity in Fourier Transforms. If we start with ideal S, transform to
s, window by applying a range-gate and inverse transform, we still observe spread in sw
EE880 SAR System & Signals Part 2
5
Down-range Digital Signal Processing
• Time/range domain
• Frequency domain
– finite signal bandwidth
B << W
– sampling period ΔT
– record length T

Δ =
2 =
D
1

Δ

Δ =
=
=
2
2 4
– Unambiguous spectrum
= fs/2
– spectral resolution Δf
D-1

=
=
2
2∆
Δ =

Δ =
2
1

1
2 =
∆
range results from scaling time
EE880 SAR System & Signals Part 2
6
Realistic Cross-range
Reconstruction
• Down-range resolved
• Cross-range not resolved
because of antenna beam
• Solution: apply
discrete-time
Fourier principles to
form narrow antenna beam
EE880 SAR System & Signals Part 2
7
Cross-range Coordinates
End
synthetic
aperture
Θ
0
1. Collection
4. Scene center
reference
Start
synthetic
aperture
2. Coordinate
references
3. Synthetic
aperture reference
Ground
plane
0
Θ
EE880 SAR System & Signals Part 2
Slant
plane
= 0 sin Θ
Cross range scene
extent is set by
beamwidth of
real aperture
8
SAR Coordinate Reference
• SAR coordinates are different from detection and
• Coordinates are referenced to the scene center
• Synthetic aperture elements (spacing d and length L)
are referenced to scene center in angular
coordinates  ←  , ,  ,  ←  , ,
• SAR is a receive array antenna
Angle
scene
Angle
scene
Range
EE880 SAR System & Signals Part 2
Range
Scene centric
9
Cross-range Digital Signal Processing
• Array (angular) sampling: • Cross-range sampling
– array defined in linear
coordinates ,
– array spacing  ← Δ
– array length  =  ← Θ
– conceptually: spatial
samples
=

Angles are scaled array
length and spacing
EE880 SAR System & Signals Part 2
– Unambiguous spectrum
Θ = 3dB
– cross-range extent  ≈ RΘ
– cross-range resolution
Δ = ℬ −1 Δ, Θ

B
B-1
Δ
is based on arc-length, but resolution
depends on the operator B and is
subject of course
10
Antenna Array Basics
• Array - collection of
antenna elements
• Each element is a single
antenna
• Typically, elements have
patterns
• Isotropic elements used
in analysis for
convenience
EE880 SAR System & Signals Part 2
AN/SPY-1A
11
Array Antenna (1/4)
Isotropic transmit
antenna
P0
Power
level (dB) P0 - 6
P0 - 12
P0 - 18
R0
R0
2R0
4R0
8R0
Observation angle
ZL
antennas
Note: Antenna observation is defined in angle
coordinates because pattern is range-invariant
EE880 SAR System & Signals Part 2
12
Array Antenna (2/4)
Array of Q isotropic transmit elements
2
1
≈
=

Spherical
observation
surface
2

= 0
=1
ZL
=  , , ,
∈ℂ
Electric fields combine in a constructive or
deconstructive manner at different points on
the observation surface
EE880 SAR System & Signals Part 2
13
Array Antenna (3/4)
of isotropic elements
GP0
GP0 - 6
Power
level (dB) GP0 - 12
GP0 - 18
G
R0
2R0
4R0
8R0
Observation angle
ZL
cos  ,  sin  =
−1
−
=0
Δ
Δ  =  sin
= 0

−1 Δ
2
Δ
2
Δ
sin 2
sin
Null-to-null beamwidth  ≈

−
2

Half-power beamwidth 3dB ≈
0.866

Note: transmit array radiation pattern is the
same as the receive array pattern.
EE880 SAR System & Signals Part 2
14
Array Antenna (4/4)
• Fields observed far from array
• Array pattern looks like I/DFT of rect
• Differential phase  on elements steers array
≫

+
= −1
=
−1
=0
0  Δ

Δ  =  sin  +
=
planar
wave fronts
−1

2
0
EE880 SAR System & Signals Part 2

2

sin 2
sin
Phase shift across dimension of array causes
angular shift (translation) to angle , i.e.
property of DFT.
15
Synthetic Array
• Synthetic aperture is a receive aperture
• Fields caused by scatterers (targets, clutter)
• Differential angle  causes differential phase
≫

+
= −1
=
−1
=0
0   Δ
+
Δ  = 2 sin
=
planar
wave fronts
−1

2
0
= Δ  +
EE880 SAR System & Signals Part 2

2

sin 2
sin
target
Synthetic array formed by correcting phases
caused by differential ranges. For linear array,
DFT along array dimension results in cross-range
compression, i.e. resolution.
16
Synthetic Aperture for Cross-range
Resolution
• SAR spatially samples along array dimension
Δ  = 2 sin
differential phase
shift across echoes
Incremental
path length
Point
target
2 sin  = DFT  Δ

=DFT{[]},  = 1, ⋯ ,
sin −Θ 2 ≤ sin  ≤ sin Θ 2
EE880 SAR System & Signals Part 2
Incremental Incremental
position
angle
Cross-range resolution
equals arc length ∆
17
SAR Signal Modeling Requirements
• N-D images require N-D signal representation
• Parameterize 2D signals (range,angle) with time
• Time has two scales (PRI- , and CPI- )
• System design must support stable collection
method and accurate coherent measurement
CPI (inter-pulse sampling)
0

slow time  [ms]
EE880 SAR System & Signals Part 2
PRI (intra-pulse sampling)

0
Δ

fast time  [s]
18
• SAR System differs from classic radar system
• Collection method (transmit and store), receiver
design to support imaging, signal processing
TX
Differences in CONOP
sTX(t)
s(t)
SAR Simple view
TX Ant
gc(t)
TX
Env
RT , σ
RG, σ0
RJ, sjam
RX
r(t)
yI(t)
yQ(t)
,
sRX(t)
Differences in RSP
RX
d[n]
DB
output
,
t, Tp, Fp, τ
EE880 SAR System & Signals Part 2
ℎ
RX Ant
RSP
SYNC
input
DM
ℎ−1
SAR is an inverse problem
19
Detailed SAR Modeling
• Signal development from signal processing
perspective
• Math development from inverse problem
perspective
• Algorithm processing from linear systems perspective
• Outline:
– Coordinate systems
– Transmit “signal”
– Scatterer response
– Operator representation
EE880 SAR System & Signals Part 2
20
Coordinate Systems (1/3)
• Lower case letters: global coordinates
• Primed lower case letters: local scene coordinates
• Upper case letters: local antenna coordinates

Antenna position
=  +  +

=  + +
′ = ′′ + ′’+′′
=  +  +
Scene center position
=  +  +

′
′

′

EE880 SAR System & Signals Part 2
Scene center position
relative to antenna
position
=   =  −
21
Coordinate Systems (2/3)
• Local coordinates show variation in position
Antenna position
Scene center position
+ ∆
+ ′
Scene center position
relative to antenna position
=  + ′ −  + ∆

∆

• Typically assume ∆ = 0
• Scene defined by ′
′
′

EE880 SAR System & Signals Part 2

′
′
=  + ′ −
=  0 + ′
• Position parameterized
with slow time
22
Coordinate System (3/3)
• Waveform definition in fast time coordinates
• Reference to scene center -- not antenna
• Signal has dependency on both  and
,  = ℝ      −
complex envelope
=   cos
=
EE880 SAR System & Signals Part 2
electromagnetic
wave behavior
+   sin
=  0  + ′

Can be phase, frequency, or
amplitude encoded
Assumes   ≪
Typically,  ′  = 0
23
Transmit Signal
• Wideband signal (LFM or
stepped frequency)
• Directional (line-of-sight to
scene)
,
= ℝ       −
∙
0
= 0  =
0
Cutaway view of a helix Traveling wave tube. (1)
Electron gun; (2) RF input; (3) Magnets; (4)
Attenuator; (5) Helix coil; (6) RF output; (7)
Vacuum tube; (8) Collector. [wikipedia.com]

∙   =  0  ∙   ≈ 0  + ∆
0
∆  ≈   ∙ ′
Differential path length for
arbitrary location in scene
EE880 SAR System & Signals Part 2
Flight path
′
′

′
Scene
′
24
Scattered Signal
• Clutter & targets, atmospheric and space loss L
– In SAR, heterogeneous clutter = “target”
– Approximate target signal model is simple sum of
isotropic point scatterers:
amplitude scaled, time, frequency/phase shifted
− 20
,  =
−20
−2 0
+∆

• EM physics (with typical approximations)
,  =   − 20
−20
−20

′ =   ′ −
′  −2
∙ ′
′

SAR approximates scene’s reflectivity function
EE880 SAR System & Signals Part 2
25
• Signal comprises all echoes during synthetic aperture
′ ,  ;  = 1, ⋯ ,
• Inertial navigation system provides motion
compensation timing, i.e., compensates for aperture
deviation from flight path compensation
,  =  ′ ,   2 ∙∆
=   =  0  =  0,
Flight path

∆
• Slow-time recorded in
angle coordinates

0, ′
′

,  =   ,
EE880 SAR System & Signals Part 2
Scene
′
26
• Fast-time signals sampled according to signal
bandwidth
,  ;  = 1, ⋯ ,
• Signals recorded either with absolute time or relative
to initial or middle pulse in collection with respect to
scene center
• LFM signal recovered using deramp and deskew
receiver -- relates sample time to instantaneous
frequency
EE880 SAR System & Signals Part 2
27
SAR Signal Processing Overview
• Signal model after A/D
,  =
−  − 2 0,
rect

=0
−1
×  Φ
,
′  −2
∙ ′
′

• LFM transmit phase profile
Φ ,  = 2  +   −
2
Chirp  [Hz/sec]
• LFM receive (deramp) phase profile
Φ  ,  = −
4
20,
+  −  −

EE880 SAR System & Signals Part 2
− 0, +
4
− 0,
2

2
28
• Recall LFM waveform with chirp  Hz/sec [Sullivan, 7.2]:
transmit
−   2 +
= 0 rect

,
,
−
−  − 2
=  rect

x
2 −2  + − −2
2
2
Reference to Scene Center
(motion compensation point)
,  =
EE880 SAR System & Signals Part 2
2 −20  + − −20

2
29
• Mix reference signal with echo
,
X
=  −
,
fast time within PRI
conj  ,
intermediate frequency
,
− 2   −4
=  rect

20
+
−
−0

2
4
2  −0

Received pulse train from q-th target
,  =
Φ ,  = −
EE880 SAR System & Signals Part 2
−1
=0
rect
− 2   Φ

4
20
+−

,
− 0 +
4
− 0
2

2
30
• Signal phase
Φ ,  = −
4
20
+−

− 0 +
4
− 0
2

linear phase,
easily compensated
2
not easily corrected,
often dismissed as
phase error term
• For a fixed target range, the instantaneous received
frequency is
,
1 Φ
2
=
=−
− 0
2

constant range-dependent frequency is dechirped or deramped
EE880 SAR System & Signals Part 2
31
frequency

time
Near
Scene

2
Scene
Center

Far
Scene
before
deramp
frequency
after
Targets at different ranges
have different frequencies
<
time
Deramping also reduces A/D
sampling speeds
EE880 SAR System & Signals Part 2
32
• Each echo contains multiple tones from scatterers at
different ranges in the scene that occur at different
times
2  − 0

=

=−

• SAR processing requires one-to-one mapping of
frequency to sample time, i.e. no time-delay
• Correct as
Φ ← Φ  ,
2
−

• IFT each echo to
recover frequencies
EE880 SAR System & Signals Part 2
frequency
Deramped and deskewed
<
time
33
Operator Modeling
ℒ

TX
ENV

ℳ1 ℒ
RX
RSP
MF
ℱ −1
ℳ2 ℒ
PFA,
CBP

represents antenna radiation of signal from transmitter
ℒ represents scattering from scatterer
ℳ represents receiver front end (mixing, matched filtering, etc…)
= ℳ1 ℒ
These operations can be
approximated as a forward
Fourier transform
EE880 SAR System & Signals Part 2
≈ ℱℴ
The approximation depends on simple
linear superposition of scatterers and far
field reception
34
Summary of
SAR Systems & Signals Part 2
•
•
•
•
Imaging requirements
Antenna array
SAR signal modeling
Operator modeling
EE880 SAR System & Signals Part 2
35
Lesson References
• [Levanon] N. Levanon, Radar Signals, Wiley-IEEE Press, 2004.
• [Stimson] G. Stimson, Introduction to Airborne Radar, SciTech Publishing
Inc., 1998.
• [Sullivan] R. Sullivan, Foundations for Imaging and Advanced Concepts,
SciTech Publishing Inc., 2004.
EE880 SAR System & Signals Part 2
36
```