Transcript SAR System & Signals

SAR System and Signals Part 2
EE880 Synthetic Aperture Radar
M. A. Saville, PhD, PE
Summer, 2012
EE880 SAR System & Signals Part 2
Lesson Overview
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Imaging radar requirements
Array Basics
SAR signal modeling
Summary
EE880 SAR System & Signals Part 2
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Imaging Radar Requirements
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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
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Ideal 2D Radar Imaging Collection
• 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
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Realistic Down-range
Reconstruction
Ideal down-range
target profile
rect(𝑡) (infinite
bandwidth)
Time Domain
∆𝑡
Spectral Domain
-2000
-1500
-1000
Ideal receiver
filtering rect(𝑓)
(finite bandwidth)
Lost energy
-500
0
Time Domain
∆𝑡
500
1000
1500
2000
Profile distortion
& spreading
Reconstructed
down-range target
profile is IDFT of
windowed rect(𝑡)
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
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Down-range Digital Signal Processing
• Time/range domain
• Frequency domain
– finite signal bandwidth
B << W
– sampling period ΔT
– record length T
𝑇
Δ𝑇 =
2𝑊 = 𝑓𝑠
D
1
𝑓𝑠
𝑐Δ𝑇
𝑐
𝑐
Δ𝑅 =
=
=
2
2𝑓𝑠 4𝑊
– Unambiguous spectrum
𝑊 = fs/2
– spectral resolution Δf
D-1
𝑐𝑇
𝑐
𝑅=
=
2
2∆𝑓
Δ𝑓 =
𝑐
Δ𝑓 =
2𝑅
1
𝑇
1
2𝑊 =
∆𝑇
range results from scaling time
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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
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Cross-range Coordinates
End
synthetic
aperture
Θ
𝑅0
1. Collection
4. Scene center
reference
Start
synthetic
aperture
2. Coordinate
references
3. Synthetic
aperture reference
Ground
plane
𝑅0
Θ
EE880 SAR System & Signals Part 2
Slant
plane
𝑅𝑐𝑟 = 𝑅0 sin Θ
Cross range scene
extent is set by
beamwidth of
real aperture
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SAR Coordinate Reference
• SAR coordinates are different from detection and
tracking radar applications
• Coordinates are referenced to the scene center
• Synthetic aperture elements (spacing d and length L)
are referenced to scene center in angular
coordinates 𝛥𝜃 ← 𝛥𝜃 𝑑, 𝐿, 𝑅 , 𝛩 ← 𝛩 𝑑, 𝐿, 𝑅
• SAR is a receive array antenna
Angle
scene
Angle
scene
Range
radar
EE880 SAR System & Signals Part 2
Radar centric
Range
radar
Scene centric
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Cross-range Digital Signal Processing
• Array (angular) sampling: • Cross-range sampling
– array defined in linear
coordinates 𝑑, 𝐿
– array spacing 𝑑 ← Δ𝜃
– array length 𝐿 = 𝑄𝑑 ← Θ
– conceptually: spatial
samples
𝐿 = 𝑄𝑑
𝑑
Angles are scaled array
length and spacing
EE880 SAR System & Signals Part 2
– Unambiguous spectrum
Θ = 𝜃3dB
– cross-range extent 𝑌 ≈ RΘ
– cross-range resolution
Δ𝑌 = ℬ −1 Δ𝜃, Θ
𝑌
B
B-1
Δ𝑌
𝑌 is based on arc-length, but resolution
depends on the operator B and is
subject of course
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Antenna Array Basics
• Array - collection of
antenna elements
• Each element is a single
antenna
• Typically, elements have
identical radiation
patterns
• Isotropic elements used
in analysis for
convenience
EE880 SAR System & Signals Part 2
AN/SPY-1A
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Array Antenna (1/4)
Isotropic transmit
antenna
Received
P0
Power
level (dB) P0 - 6
P0 - 12
P0 - 18
R0
R0
2R0
4R0
8R0
Observation angle
ZL
Receive
antennas
Note: Antenna observation is defined in angle
coordinates because pattern is range-invariant
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Array Antenna (2/4)
Array of Q isotropic transmit elements
𝐄2
1
𝑃≈
=
𝑍𝐿
𝑍𝐿
Spherical
observation
surface
2
𝑄
𝐄𝑞
= 𝐺𝑃0
𝑞=1
ZL
𝐄 = 𝐞𝐸 𝑥, 𝑦, 𝑧, 𝑡
𝐸∈ℂ
Electric fields combine in a constructive or
deconstructive manner at different points on
the observation surface
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Array Antenna (3/4)
Radiation pattern of array
of isotropic elements
GP0
Received
GP0 - 6
Power
level (dB) GP0 - 12
GP0 - 18
G
R0
2R0
4R0
8R0
Observation angle
ZL
𝐹 𝑅 cos 𝜃 , 𝑅 sin 𝜃 = 𝑒
𝑄−1
−𝑗𝑘𝑅
𝑞=0
𝐼𝑞 𝑒 𝑗𝑞Δ𝜑
Δ𝜑 𝜃 = 𝑘𝑑 sin 𝜃
𝐹 𝜃 = 𝐼0 𝑒
𝑗
𝑄−1 Δ𝜑 𝜃
2
𝑄Δ𝜑 𝜃
2
Δ𝜑 𝜃
sin 2
sin
Null-to-null beamwidth 𝜃𝑁𝑁 ≈
𝜃
𝜃
𝑒 −𝑗𝑘𝑅
2𝜆
𝐿
Half-power beamwidth 𝜃3dB ≈
0.866𝜆
𝐿
Note: transmit array radiation pattern is the
same as the receive array pattern.
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Array Antenna (4/4)
• Fields observed far from array
• Array pattern looks like I/DFT of rect 𝜃
• Differential phase 𝛼 on elements steers array
𝑅≫𝐿
𝐿
𝜃+𝛼
𝐿 = 𝑄−1 𝑑
𝐹 𝜃 =
𝑄−1
𝑞=0
𝐼0 𝑒 𝑗𝑞Δ𝜑
𝜃
Δ𝜑 𝜃 = 𝑘𝑑 sin 𝜃 + 𝛼
𝐹 𝜃 =
planar
wave fronts
𝑄−1 𝜑 𝜃
𝑗
2
𝐼0 𝑒
EE880 SAR System & Signals Part 2
𝑄𝜑 𝜃
2
𝜑 𝜃
sin 2
sin
Phase shift across dimension of array causes
angular shift (translation) to angle 𝛼, i.e.
property of DFT.
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Synthetic Array
• Synthetic aperture is a receive aperture
• Fields caused by scatterers (targets, clutter)
• Differential angle 𝛼 causes differential phase
𝑅≫𝐿
𝐿
𝜃+𝛼
𝐿 = 𝑄−1 𝑑
𝐹 𝜃 =
𝑄−1
𝑞=0
𝐼0 𝑒 𝑗𝑞 Δ𝜑
𝜃 +𝛼
Δ𝜑 𝜃 = 2𝑘𝑑 sin 𝜃
𝐹 𝜃 =
planar
wave fronts
𝑄−1 𝜑 𝜃
𝑗
2
𝐼0 𝑒
𝜑 𝜃 = Δ𝜑 𝜃 + 𝛼
EE880 SAR System & Signals Part 2
𝑄𝜑 𝜃
2
𝜑 𝜃
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.
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Synthetic Aperture for Cross-range
Resolution
• SAR spatially samples along array dimension
Δ𝜑 𝜃 = 2𝑘𝑑 sin 𝜃
differential phase
shift across echoes
Incremental
path length
Point
target
𝑆 2𝑘 sin 𝜃 = DFT 𝑠 𝑞Δ𝜑 𝜃
𝑅
=DFT{𝑠[𝑞𝑑]}, 𝑞 = 1, ⋯ , 𝑄
sin −Θ 2 ≤ sin 𝜃 ≤ sin Θ 2
EE880 SAR System & Signals Part 2
Incremental Incremental
position
angle
Cross-range resolution
equals arc length 𝑅∆𝜃
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SAR Signal Modeling Requirements
• N-D images require N-D signal representation
• Parameterize 2D signals (range,angle) with time
• Time has two scales (PRI-𝑇𝑝 , and CPI-𝑀𝑇𝑝 )
• System design must support stable collection
method and accurate coherent measurement
CPI (inter-pulse sampling)
0
𝑇𝑝
slow time 𝜏 [ms]
EE880 SAR System & Signals Part 2
PRI (intra-pulse sampling)
𝑀𝑇𝑝
0
Δ𝑇
𝑇𝑝
fast time 𝑡 [𝜇s]
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SAR Radar System and Signals
• 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)
𝑠TX
Env
RT , σ
RG, σ0
RJ, sjam
Differences in receiver
RX
r(t)
yI(t)
yQ(t)
𝑅, 𝜎
sRX(t)
Differences in RSP
𝑠RX
d[n]
DB
output
𝑅, 𝜎
t, Tp, Fp, τ
EE880 SAR System & Signals Part 2
ℎ
RX Ant
RSP
SYNC
input
DM
ℎ−1
SAR is an inverse problem
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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
– Received signal
– Operator representation
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Coordinate Systems (1/3)
• Lower case letters: global coordinates
• Primed lower case letters: local scene coordinates
• Upper case letters: local antenna coordinates
𝐙
Antenna position
𝐫𝑎 = 𝐱𝑥𝒂 + 𝐲𝑦𝒂 +𝐳𝑧𝒂
𝐘
𝐫 = 𝐱𝑥 + 𝐲𝑦+𝐳𝑧
𝐫′ = 𝐱′𝑥′ + 𝐲′𝑦’+𝐳′𝑧′
𝐑 = 𝐗𝑋 + 𝐘𝑌 + 𝐙𝑍
Scene center position
𝐫𝑔 = 𝐱𝑥𝒈 + 𝐲𝑦𝒈 +𝐳𝑧𝒈
𝐗
𝐫𝑎
𝐑
𝐳′
𝐲′
𝐳
𝐲
𝐫𝑔
𝐱′
𝐱
EE880 SAR System & Signals Part 2
Scene center position
relative to antenna
position
𝐑 = 𝐑 𝟎 = 𝐫𝑔 − 𝐫𝑎
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Coordinate Systems (2/3)
• Local coordinates show variation in position
Antenna position
Scene center position
𝐫𝑎 + ∆𝐑
𝐫𝑔 + 𝐫′
Scene center position
relative to antenna position
𝐑 = 𝐫𝑔 + 𝐫′ − 𝐫𝑎 + ∆𝐑
𝐙
𝐘
∆𝐑
𝐫𝑎
𝐗
𝐑
• Typically assume ∆𝐑 = 0
• Scene defined by 𝐫′
𝐳′
𝐲′
𝐳
𝐲
𝐱
EE880 SAR System & Signals Part 2
𝐫𝑔
𝐫′
𝐱′
𝐑 = 𝐫𝑔 + 𝐫′ − 𝐫𝑎
𝐑 = 𝐑 0 + 𝐫′
• Position parameterized
with slow time 𝜏
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Coordinate System (3/3)
• Waveform definition in fast time coordinates 𝑡
• Reference to scene center -- not antenna
• Signal has dependency on both 𝑡 and 𝜏
𝑠 𝑡, 𝜏 = ℝ𝕖 𝑝 𝑡 𝑒 𝑗𝜔𝑡 𝑒 −𝑗𝑘𝑅
complex envelope
𝑝 𝑡 = 𝑎 𝑡 cos 𝜓 𝑡
𝑅 𝜏 = 𝐑 𝜏
EE880 SAR System & Signals Part 2
electromagnetic
wave behavior
+ 𝑗𝑎 𝑡 sin 𝜓 𝑡
= 𝐑 0 𝜏 + 𝐫′ 𝜏
𝜏
Can be phase, frequency, or
amplitude encoded
Assumes 𝐑 𝜏 ≪ 𝑐
Typically, 𝐫 ′ 𝜏 = 0
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Transmit Signal
• Wideband signal (LFM or
stepped frequency)
• Directional (line-of-sight to
scene)
𝑠 𝑡, 𝐤 𝜏
= ℝ𝕖 𝑝 𝑡 𝑒 𝑗𝜔𝑐 𝑡 𝑒 −𝑗𝑘
𝐤∙𝐑 𝜏
𝐑0 𝜏
𝐤 𝜏 = 𝐑0 𝜏 =
𝑅0 𝜏
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]
𝐙
𝐘
𝐗
𝐤 ∙ 𝐑 𝜏 = 𝐑 0 𝜏 ∙ 𝐑 𝜏 ≈ 𝑅0 𝜏 + ∆𝑅 𝜏
𝐑0
∆𝑅 𝜏 ≈ 𝐤 𝜏 ∙ 𝐫′
Differential path length for
arbitrary location in scene
EE880 SAR System & Signals Part 2
Flight path
𝐳′
𝐲′
𝐑
𝐫′
Scene
𝐱′
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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
𝐴𝑞 𝑝 𝑡 − 2𝑇0 𝑒 𝑗𝜔𝑐
𝑠 𝑡, 𝜏 =
𝑡−2𝑇0
𝑒 −𝑗𝑘2 𝑅0
𝜏 +∆𝑅𝑞 𝜏
𝑞
• EM physics (with typical approximations)
𝑠 𝑡, 𝜏 = 𝐿𝑝 𝑡 − 2𝑇0 𝑒 𝑗𝜔𝑐
𝑡−2𝑇0
𝑒 −𝑗2𝑘𝑅0
𝜏
𝜌 𝐫 ′ = 𝛿 𝐫 ′ − 𝐫𝑞
𝜌 𝐫 ′ 𝑒 −𝑗2𝑘𝐤
𝜏 ∙𝐫 ′
𝑑𝐫 ′
𝑆𝑐𝑒𝑛𝑒
SAR approximates scene’s reflectivity function
EE880 SAR System & Signals Part 2
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Received Signal (1/2)
• Signal comprises all echoes during synthetic aperture
𝑠 ′ 𝑡, 𝜏𝑚 ; 𝑚 = 1, ⋯ , 𝑀
• Inertial navigation system provides motion
compensation timing, i.e., compensates for aperture
deviation from flight path compensation
𝑠 𝑡, 𝜏𝑚 = 𝑠 ′ 𝑡, 𝜏𝑚 𝑒 𝑗2𝑘𝐤𝑚 ∙∆𝐑𝑚
𝐤 𝑚 = 𝐤 𝜏𝒎 = 𝐑 0 𝜏𝒎 = 𝐑 0,𝑚
Flight path
𝐙
𝐘
∆𝐑 𝑚
• Slow-time recorded in
angle coordinates
𝐗
𝐑 0,𝑚 𝐳′
𝐲′
𝐑𝑚
𝜙𝑚 , 𝜃𝑚 = 𝜙 𝜏𝒎 , 𝜃 𝜏𝒎
EE880 SAR System & Signals Part 2
Scene
𝐱′
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Received Signal (2/2)
• Fast-time signals sampled according to signal
bandwidth
𝑠 𝑡𝑛 , 𝜏𝑚 ; 𝑛 = 1, ⋯ , 𝑁
• 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
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SAR Signal Processing Overview
• Signal model after A/D
𝑆𝐼𝐹 𝑡𝑛 , 𝜏𝑚 =
𝑡𝑛 − 𝑚𝑇𝑝𝑟𝑖 − 2 𝑅0,𝑚 𝑐
rect
𝑇𝑝𝑢𝑙𝑠𝑒
𝑚=0
𝑀−1
× 𝑒 𝑗Φ
𝑡𝑛 ,𝜏𝑚
𝜌 𝐫 ′ 𝑒 −𝑗2𝑘𝐤
𝜏 ∙𝐫 ′
𝑑𝐫 ′
𝑆𝑐𝑒𝑛𝑒
• LFM transmit phase profile
Φ 𝑡, 𝜏𝑚 = 2𝜋𝑓𝑐 𝑡 + 𝜋𝛾 𝑡 − 𝑚𝑇𝑝𝑟𝑖
2
Chirp 𝛾 [Hz/sec]
• LFM receive (deramp) phase profile
Φ 𝑡𝑛 , 𝜏𝑚 = −𝑗
4𝜋𝛾 𝑓𝑐
2𝑅0,𝑚
+ 𝑡𝑛 − 𝑚𝑇𝑝𝑟𝑖 −
𝑐 𝛾
𝑐
EE880 SAR System & Signals Part 2
𝑅𝑞 − 𝑅0,𝑚 +
4𝜋𝛾
𝑅𝑞 − 𝑅0,𝑚
2
𝑐
2
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Deramp & Deskew Receiver (1/5)
• Recall LFM waveform with chirp 𝛾 Hz/sec [Sullivan, 7.2]:
transmit
𝑡 − 𝑚𝑇𝑝𝑟𝑖 𝑗 2𝜋𝑓𝑐 𝑡+𝜋𝛾
= 𝐴0 rect
𝑒
𝑇𝑝𝑢𝑙𝑠𝑒
𝑆𝑇𝑥 𝑡, 𝜏𝑚
receive
𝑆𝑅𝑥 𝑡, 𝜏𝑚
𝑡−𝑚𝑇𝑝𝑟𝑖
𝑡 − 𝑚𝑇𝑝𝑟𝑖 − 2 𝑅𝑞 𝑐
= 𝑎𝑞 rect
𝑇𝑝𝑢𝑙𝑠𝑒
x𝑒
𝑗 2𝜋𝑓𝑐 𝑡−2𝑅𝑞 𝑐 +𝜋𝛾 𝑡−𝑚𝑇𝑝𝑟𝑖 −2𝑅𝑞 𝑐
2
2
Reference to Scene Center
(motion compensation point)
𝑆𝑅𝑒𝑓 𝑡, 𝜏𝑚 =
EE880 SAR System & Signals Part 2
𝑗 2𝜋𝑓𝑐 𝑡−2𝑅0 𝑐 +𝜋𝛾 𝑡−𝑚𝑇𝑝𝑟𝑖 −2𝑅0 𝑐
𝑒
2
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Deramp & Deskew Receiver (2/5)
• Mix reference signal with echo
𝑆𝑅𝑥 𝑡, 𝜏𝑚
X
𝑡 = 𝑡 − 𝑚𝑇𝑝𝑟𝑖
𝑆𝐼𝐹 𝑡, 𝜏𝑚
fast time within PRI
conj 𝑆𝑅𝑒𝑓 𝑡, 𝜏𝑚
intermediate frequency
𝑆𝐼𝐹 𝑡, 𝜏𝑚
𝑡 − 2 𝑅𝑞 𝑐 −𝑗4𝜋𝛾
= 𝑎𝑞 rect
𝑒 𝑐
𝑇𝑝𝑢𝑙𝑠𝑒
𝑓𝑐
2𝑅0
+
𝑡−
𝑅𝑞 −𝑅0
𝛾
𝑐
2
4𝜋𝛾
𝑗 2 𝑅𝑞 −𝑅0
𝑒 𝑐
Received pulse train from q-th target
𝑆𝐼𝐹 𝑡, 𝜏𝑚 = 𝑎𝑞
Φ 𝑡, 𝜏𝑚 = −𝑗
EE880 SAR System & Signals Part 2
𝑀−1
𝑚=0
rect
𝑡 − 2 𝑅𝑞 𝑐 𝑗Φ
𝑒
𝑇𝑝𝑢𝑙𝑠𝑒
4𝜋𝛾 𝑓𝑐
2𝑅0
+𝑡−
𝑐 𝛾
𝑐
𝑡,𝜏𝑚
𝑅𝑞 − 𝑅0 +
4𝜋𝛾
𝑅𝑞 − 𝑅0
2
𝑐
2
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Deramp & Deskew Receiver (3/5)
• Signal phase
Φ 𝑡, 𝜏𝑚 = −𝑗
4𝜋𝛾 𝑓𝑐
2𝑅0
+𝑡−
𝑐 𝛾
𝑐
𝑅𝑞 − 𝑅0 +
4𝜋𝛾
𝑅𝑞 − 𝑅0
2
𝑐
linear phase,
easily compensated
2
quadratic phase,
not easily corrected,
often dismissed as
phase error term
• For a fixed target range, the instantaneous received
frequency is
𝑓 𝑡, 𝜏𝑚
1 𝑑Φ
2𝛾
=
=−
𝑅𝑞 − 𝑅0
2𝜋 𝑑𝑡
𝑐
constant range-dependent frequency is dechirped or deramped
EE880 SAR System & Signals Part 2
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Deramp & Deskew Receiver (4/5)
Adapted from [SUL,7.2]
frequency
𝑐𝑇𝑝𝑢𝑙𝑠𝑒
𝑇𝑝𝑢𝑙𝑠𝑒
𝑓𝑐
𝐵
time
Near
Scene
𝑇𝑝𝑢𝑙𝑠𝑒
2𝑆 𝑐
Scene
Center
𝑆
Far
Scene
before
deramp
frequency
after
Targets at different ranges
have different frequencies
𝐵𝐼𝐹 < 𝐵
time
Deramping also reduces A/D
sampling speeds
EE880 SAR System & Signals Part 2
32
Deramp & Deskew Receiver (5/5)
• Each echo contains multiple tones from scatterers at
different ranges in the scene that occur at different
times
2 𝑅𝑞 − 𝑅0
𝑓𝑞
𝑡𝑞 =
𝑐
=−
𝛾
• SAR processing requires one-to-one mapping of
frequency to sample time, i.e. no time-delay
• Correct as
Φ ← Φ 𝑡𝑞 , 𝜏
𝜋𝑓𝑞 2
−𝑗 𝛾
𝑒
• IFT each echo to
recover frequencies
EE880 SAR System & Signals Part 2
frequency
Deramped and deskewed
𝐵𝐼𝐹 < 𝐵
time
33
Operator Modeling
ℒ 𝜎𝒦 𝑝𝑇𝑥
𝒦 𝑝𝑇𝑥
TX
ENV
𝑝𝑇𝑥
𝜎
ℳ1 ℒ 𝜎𝒦 𝑝𝑇𝑥
RX
RSP
MF
ℱ −1 𝑆𝑅𝑥
ℳ2 ℒ 𝜎𝒦 𝑝𝑇𝑥
PFA,
CBP
𝜎
𝒦 represents antenna radiation of signal from transmitter
ℒ represents scattering from scatterer
ℳ represents receiver front end (mixing, matched filtering, etc…)
𝑆𝑅𝑥 = ℳ1 ℒ 𝜎𝒦 𝑝𝑇𝑥
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