Wave-Based Surveillance
Professor David H. Staelin
Massachusetts Institute of Technology
Lec18.5 - 1
2/6/01
A1
Antenna Aperture Transform Relations and Resolution
(
)
Define angular spectrum E ϕ x , ϕ y for
incoming monochromatic signals
x
y
ϕx
z
ϕy
E ( x, y ) ≅
[aperture]
1
E ( ϕx , ϕy ) ≅ 2
λ
Not to be confused with the radially
expanding and diminishing waves
characterized by E ( θ, ϕ,R )
∫
4π
(
−j
2π
xϕx + yϕy
λ
(
)
(
)
dΩ
π⎞
⎛
dxdy ⎜ For ϕx , ϕy << ⎟
2⎠
⎝
∆
Lec18.5 - 2
2/6/01
2π
xϕ x + yϕ y
λ
⎡⎣ vm−1ster −1 ⎤⎦
∫ E ( x, y ) e
A
)
E ϕx , ϕy e
+j
∆
Equivalently we let x / λ = x λ ; y/λ = y λ
A2
Antenna Aperture Transform Relations and Resolution
∆
∆
Equivalently we let x / λ = x λ ; y/λ = y λ
+ j2 π ( x λ ϕ x + y λ ϕ y )
E ( xλ, yλ ) ≅ ∫ E ϕx, ϕy e
dΩ
4π
(
)
(
)
E ϕ x , ϕ y ≅ ∫∫ E ( x, y ) e
(
− j2 π x λ ϕ x + y λ ϕ y
)
A
dx λ dy λ
Thus: E ( x λ , y λ ) ↔ E ( ϕx , ϕy )
↓
↓
2
RE τ λ
↔ E(ϕx , ϕy ) ∝ G ϕ ( transmitting)
where RE
Lec18.5 - 3
2/6/01
( )
(τ )
λ
∆
=
∞
∫ ∫ E (r ) E (r
( )
∗
λ
λ
)
− τλ dx λ dy λ
-∞
(Note : E is not stochastic )
A3
Single Aperture Resolution Limits
()
()
()
( )
( )
( )
Source image = TA ϕ = G ϕ ∗ TB ϕ
∆
fϕ =cycles per
radian (angle)
TA f ϕ = G f ϕ • TB f ϕ
( )
If G f ϕ = 0, there is no response
( )
in the image spectrum TA f ϕ
( )
( )
( )
Note : RE τλ ↔ G ϕ ↔ G f ϕ so RE ( τ ) ≅ G ( fϕ )
Lec18.5 - 4
2/6/01
A4
Single Aperture Resolution Limits
( )
( )
( )
Note : RE τλ ↔ G ϕ ↔ G f ϕ so RE ( τ ) ≅ G ( fϕ )
Example :
E ( xλ , yλ )
( )
Thus :
0
yλ
Dλ
( )
RE τ λ
G fϕ
D/λ
≅
τλ x
xλ
τλ y
fϕ x
fϕ y
fmax
D cycles
=
λ radian
Note : Zero response to source angular spectral components
with spatial frequencies beyond fm = D / λ cycles/radian
Lec18.5 - 5
2/6/01
A5
Antenna Responses for Stochastic Signals
(
)
{
Let E ( x λ , y λ , t ) vm −1 ster −1 = Re E ( t, x λ , y λ ) e jωt
}
↑
Slowly varying, narrowband random signal
Assume stochastic signals from different directions are
uncorrelated (so no systematic intensity variations in aperture).
Then: E ( x λ , y λ , t ) ↔ E ( ϕx , ϕy , t )
↓↓
↓↓
E ⎡RE τλ ⎤
⎣
⎦
(Double arrow implies
irreversibility for two reasons:
expectation and magnitude
operators used)
( )
ϕE
(
∆
=
2
⎡
τ xλ , τ yλ ↔ E E ϕ x , ϕ y , t ⎤
⎢⎣
⎥⎦
)
(
)
↑
−1 2
⎡⎣ vm ⎤⎦
Lec18.5 - 6
2/6/01
A6
Antenna Responses for Stochastic Signals
Then: E ( x λ , y λ , t ) ↔ E ( ϕx , ϕy , t )
(Double arrow implies
irreversibility for two reasons:
↓↓
↓↓
expectation and magnitude
E ⎡RE τλ ⎤
⎣
⎦
operators used)
∆
=
2
⎡
ϕE τ x λ , τ y λ ↔ E E ϕ x , ϕ y , t ⎤
⎢⎣
⎥⎦
↑ −1 2
⎡⎣ vm ⎤⎦
−2
−1
−1
( )
(
)
(
(
)
)
Can we deduce Ι ϕ x , ϕ y ⎡⎣ Wm ster Hz ⎤⎦ from E ( x λ , y λ , t ) ? (Yes)
(
⎡E ϕ , ϕ , f
E
φE τ xλ , τ yλ , f
x
y
⎢⎣
↔
2ηoB
2ηoB
(
)
↑ ( 377Ω )
(
)
−1 2
−1
vm
ohm Hz
= W m−2 Hz −1
Lec18.5 - 7
2/6/01
−1
(
v m−1 rad-1
ohm Hz
)
2
)
2
⎤
⎥⎦
= Ι ϕx , ϕy , f
(
)
W m−2 Hz −1 ster −1
[ster not a physical unit]
A7
Aperture Field Correlations for a Thermal Source
( )
square source @ TB K
say Ω s = ( 0.01 rad) ≅ 10 −4 ster
2
ϕx ϕy
aperture
y
2
⎡
x
E E ( ϕx , ϕy ) ⎤
⎢⎣
⎥⎦
kTB
2
• 2ηoB
=
Ι x ( ϕx , ϕy ,f ) = kTB / λ
2
λ
λ
↔
→
0
ϕx
(
φE τ xλ , τ yλ
)
Ωs
kTB
−4
10
•
η
•
2
B
o
λ2
ϕx
τ xλ
0.01 rad
τx
1
=
B is the signal
0.01 λ
bandwidth of interest
2
⎛ kTB
⎞
Note: φE ( 0, 0 ) = E E ( x, y, t ) = ⎜ 2 ΩsB ⎟ 2ηo as expected.
x
⎝ λ
⎠
0.01 rad
single polarization
{
Lec18.5 - 8
2/6/01
}
So ( watts )
A8
Time and Space Field Correlations (3D)
(
)
(
time /
frequency
)
(
)
(
φEx τλ x , τλ y , τ / 2ηo Wm−2 ↔ φEx τλ x , τλ y , f / 2ηo Wm−2Hz −1
(τ
( )
Ι x ϕ,f =
λ
↔ϕ
)
( ) Wm
kTB ϕ,f
−2
λ2
(one polarization)
Lec18.5 - 9
2/6/01
)
Hz −1 ster −1
A9
Aperture Synthesis
Assume field of size τλ xmax by τλ ymax within which two small
antennas can be moved independently
Note φE τ = φ∗E −τ
()
( )
(if stationary w.r.t.
τλy
τλy
r; i.e., true angular decorrelation
W ( τλ )
)
Therefore, in τ λ space, we need to
measure combinations in only two
B A
quadrants, e.g., A, B because the
τλx
∗
∗
follow.
conjugates
A
,
B
A ∗ B∗
τ λ xmax
2L x
2τ λ y =
max
λ
We observe W τ λ • φE τ λ
max
( )
( )
() ()
and retrieve W ϕ ∗ E ϕ
Lec18.5 - 10
2/6/01
2
()
where E ϕ
2
is the desired image
C1
Maximum Antenna Separation Limits Resolution
( )
()
W τλ ↔ W ϕ =
0
ϕx
τλy
1
τλ x
max
τxλmax = L x / λ
ϕy
1
λ
=
2L x
Null at
2Ly
2τ λ xmax
λ
λ
Aperture synthesis: θ3dB ≅
,
2L x 2L y
λ λ
Re call, filled aperture: θ3DB ≅ , ( ηA = 1)
Lx Ly
Origin of difference:
Consider:
vs
L
y
Single uniform
Lx
Lx
aperture
Point -source
Good SNR
Vanishing SNR
response functions
Lec18.5 - 11
2/6/01
C2
Circuits for Interferometers
v1(t) ≅ k1 Re
{
v 2 (t ) ≅ k 2 Re
{
G1 ϕ
()
G2 ϕ
v1(t )
e + jγ
()
()
G1 ϕ • E ( x, y,t ) e jωt
(
}
)
v 2 (t )
G2 • E x − τ x , y − τ y ,t e jγ e jωt
↑ slowly varying
}
Narrowband
Case
Output = E ⎡⎣ v1 ( t ) v 2 ( t ) ⎤⎦ =
∗
k1k 2
⎡
Re G1G2 • E E ( x, y,t ) • E x − τ x , y − τ y ,t ⎤ e − jγ
⎣
⎦
2
{
(
=∆ G12
1
⎛
⎜⎝ Re call E ⎣⎡a ( t ) b ( t ) ⎦⎤ = Re
2
Lec18.5 - 12
2/6/01
)
}
( )
⎞
e
φ
=
φ
AB
,
Re
Im
} { }⎠⎟
{ } {
φE τλ ,f
∗
− jπ / 2
C3
Image from Discrete Antenna Array
E.G.
y
"T-array"
τλy
( )
(3,3) W τ λ
A sampled grid
τλx
(3,3)
∆τ x
Recall:
(
φ τ x λ , τ yλ ,f
2ηoB
x
A∗
) ↔ E ( ϕ,f )
( )
2ηoB
2
( )(
= Ι ϕ,f Wm−2ster −1Hz −1
( )
( )
)
()
Therefore: ⎡ φE τ λ ,f / 2ηoB⎤ • W τ λ ↔ Ι ϕ,f ∗ W ϕ
⎣
⎦
Lec18.5 - 13
2/6/01
C4
Image from Discrete Antenna Array
( )
( )
( )
()
⎡ φ τ λ ,f / 2ηoB⎤ • W τ λ ↔ Ι ϕ,f ∗ W ϕ
⎣ E
⎦
∆τ x
W ( τx ) =
τx =
2L x
•
∆τ x
W ( ϕx ) =
2L x
ϕx
λ / ∆τ x
λ / ∆τ x
1
0
∗
φx
λ / 2L λ
=
φx
Aliased images are confused if they overlap.
Lec18.5 - 14
2/6/01
C5
Image Aliasing in Synthesized Images
( φ ( τ ,f ) / 2η B) • W ( τ ) ↔ Φ ( ϕ,f ) ∗ W ( ϕ)
( )
E
λ
λ
0
ϕy
()
Ι ϕ,f ∗ W ϕ ⇒
ϕy
ϕx
ϕx
λ / ∆τ y
truth
λ / ∆τ x
truth ∗
0
∆ϕ source
λ / 2L x
To avoid image aliasing, let ∆τx < λ / ∆φ x ( source )
Note that objects in space often are isolated in empty fields,
so aliasing is not a problem. Objects imaged on the ground
have major aliasing problems, requiring Nyquist-sampled
τλ plane that puts all aliases in the weak sidelobes of G AB ϕ .
Lec18.5 - 15
2/6/01
( )
( )
()
()
Note: Ιˆ ϕ,f = ⎡Ι ϕ,f ∗ W ϕ ⎤ • G12 ϕ
⎣
⎦
()
C6
Aperture Synthesis Using Earth Rotation
Moveable on a track
Earth rotation moves effective τ x (t) τ
λy
ω
τ yλ
τλ
Earth
spins
τ xλ
1 day per curve
Gaps yield
sidelobes in
reconstructed
image
τλ x
∆τ λ
Radio astronomers Choose ∆τ λ sufficiently small
call τ λ x , τ λ y "u, v"
that no aliasing occurs.
Note: Simple targets that are characterized by the positions
or sizes of only a few key features or elements can be
deciphered using heavily aliased or burred images.
Lec18.5 - 16
2/6/01
C7
Interferometer Circuits
Professor David H. Staelin
Massachusetts Institute of Technology
Lec18.5 - 17
2/6/01
Basic Aperture Synthesis Equation
Recall:
( )
E r,t
( )
↓↓
( )
E ⎡RE τ λ ⎤ = φE τ λ
⎣
⎦
Lec18.5 - 18
2/6/01
↔
↔
( )
E ψ,t
{
↓↓
( )
E E Ψ,t
2
}
( ) ( )
∝ Ι Ψ ,TA Ψ
E1
Simple Adding Interferometers
( )
v o ψ = a2 + b2 + 2ab
G1
G2
(overbars mean
"time average" here)
Point
source
response
a(t)
+
( )2
Lec18.5 - 19
2/6/01
vo ( ψ x )
b(t)
θB
(single
antenna)
ψx
v o (t )
First null
λ /L
T
E2
Simple Adding Interferometers
ψ n u ll = sin − 1 ( λ / 2L )
≅ λ / 2L if λ < < L
S p re a d so u rce re sp o n se
v max
vo ( ψ x )
θB
λ /2
ψ n u ll
ψx
v min
DC: a2 + b2
AC: 2ab
L
=∆" T1 "
v max − v min j2πT1 T
∝ E ( τλ ) ↔ TA ( ψ )
Complex fringe visibility v =
e
v max + v min
∆
Lec18.5 - 20
2/6/01
E3
Interferometry as Fourier Analysis
( )
1
2
−D x 0
( )
( )
RE τλ
R e c a ll: φ A τ λ = R E τ λ
1
Dλ
τ λx
Example: 2 small
duplicate antennas
separated by Dλ in
the x direction
( )
∼
( )
TA ψ = G ψ
T A ( fψ
)
∼
= G ( fψ
∼
∗
∼
•
( )
TB ψ
∼
TB ( f ψ
)
1 + co s D λ ψ x
G sin ( ψ x ) ∝
0 1 + s in D ψ
λ
x
0
Lec18.5 - 21
2/6/01
∼
observed = antenna times signal
G co s ( ψ x ) ∝
2-element adding
interferometer
)
( )
φE τλ
•
ψx
ψx
E4
Interferometry as Fourier Analysis
1 + cos D λ ψ x
G co s ( ψ x ) ∝
2-element adding
interferometer
ψx
0 1 + s in D ψ
λ
x
G sin ( ψ x ) ∝
ψx
0
Interferometer directly measures Fourier components of source
Fringe patterns for all frequencies
Effect of finite bandwidth:
(colors) add in phase to create
"w h ite fr in g e "
ψx
fo + ∆ (H z )
Lec18.5 - 22
2/6/01
ψx
+
fo (H z )
∆ψ x ∝
ψx
⇒
+
fo − ∆ (H z )
1
B
B = 2∆
A delay line in one interferometer arm can redirect
the strong white fringe in other directions.
E5
Broad-Bandwidth Effects in Interferometers
E ( x, y,t )( the "slowly varying" part )
may vary rapidly enough that the offset
time L sin ϕx c is significant
ϕx
v2 ( t)
L
L sin φx
seconds
c
x
v1 ( t )
{
L sin ϕ x ⎞
⎡
⎛
E [ v1 ( t ) v 2 ( t ) ] =
G12 ( φ x ) E R e E ⎜ x, y, t +
⎟
⎢
2
2c ⎠
⎝
⎣
L sin ϕ x ⎞
jωL sin ϕ x / c − jγ ⎤
∗⎛
E ⎜ x − τx , y − τy , t −
e
e
•
⎟
⎥⎦
2c ⎠
⎝
⎡ e jω( t + L sin ϕx / 2c )e − jω( t −L sin ϕx / 2c ) = e jωL sin ϕx / c ⎤
⎣
⎦
k1k 2
E [ v1 ( t ) v 2 ( t ) ] =
k1k 2
2
( )
G12 ϕs R e
cross-gain
Lec18.5 - 23
2/6/01
{
}
}
⎛ L sin ϕ x ⎞
j2πL sin ϕx λ
e e
• φE( t ) ⎜
⎟
c
⎝
⎠
monochromatic
fringe envelope
fringe
−j
γ
E6
Bandwidth-Limited Angular Response
φ ( ) ( τ)
E ⎡⎣ v ( t ) v ( t ) ⎤⎦
E(f)
2
E t
E.G.
−B 0
null at
1/2B sec
↔
B
f
0 sinφ
x
Null @ L
c
τ sec.
null
( )
G12 ϕ s
1
sec
=
2B
1
2
white fringe
ϕx
Fringe envelope
φE( t ) (L sin ϕ x / c )
In broadband optical interferometer all colors contribute to
central "white" fringe; sidelobe fringes appear colored.
Therefore ϕxnull = sin−1 ( c / 2LB ) ≈ c / 2LB for ϕx ≅ 1
3 × 108
e.g. ϕ xnull ≅
= 1.5 radians for L = 10 m, B = 10 MHz
7
2 × 10 × 10
↑ 10 MHz
10-m
If B = 1 GHz, and L = 100 m, then ϕ xnull = 1.5 mrad ≅ 5 arc min
B = 3 × 1014 Hz and L = 100 m, then ϕ xnull = 10 −3 arc sec.
Lec18.5 - 24
2/6/01
E7
Dicke Adding Interferometer
y ( ψ ) point source response
+1 Dicke
−1
a2 + b2 + 2ab
ψ
(Also called "lobe-switching"
interferometer)
"Adding" interferometer
Dicke switch
oscillator
±1
∫
τ
×
×
a
b
+
⇒
( )2
y ( ϕ ) = ( ±a + b )
This circuit cancels D.C.
term, leaving only 4ab
as source traverses
beam ⇒
ψ
Lec18.5 - 25
2/6/01
2
⎡a2 + b2 + 2ab⎤ − ⎡a2 − 2ab + b2 ⎤ = 4ab
⎣
⎦ ⎣
⎦
Can add second adder and
square-law device operating
on a and –jb to yield sine
terms in Fourier expansion
G1
Dicke Adding Interferometer
±1
Yields φ ( τ λ ) for
even a stationary
source
"Multiplying"
interferometer
×
×
±1
90
×
×
Note: If no Dicke
switch, have gain
fluctuation
vulnerability
Lec18.5 - 26
2/6/01
∫
×
±1
τ
{ ( )}
Re φ τ λ
∫
τ
{
Ιm φ ( τ λ )
}
G2
“Lobe-Scanning” Interferometer
Lobes are scanned at
ωm , demodulated, and
averaged to yield φ ( τ λ )
Because all bias and
large-scale sources yield
no fine-scale response,
can integrate long times
seeking fine structure,
e.g. 10-3 Jansky point
sources like stars (can
measure stellar diameters
at ~10-3 arc sec)
τλ
ωs
L.O.
×
×
×
USB
Filter
×
ωm
Filter
ωm
90
×
×
∫
∫
τ
{
∝ G12 Re φ ( τ λ )
Lec18.5 - 27
2/6/01
ωs
τ
}
{
∝ G12Ιm φ ( τ λ )
}
G3
Cross-Correlation Interferometer Spectrometer
Let a ( t ) =∆ a1
a ( t − τ ) =∆ a2
×
a
b
+
Delay line or shift register
±1
±1
(
)
×
∫
∫
×
×
c + = a1a2 + b1b2 + a1b2 + a2b1
c − = −a1a2 − b1b2 + a1b2 + a2b1
φ τλ , f
×
T
T
(
φˆ τ λ , τ
) ∫
T
Computer
FFT
( )
if a ⊥ b, φ ( τ , f ) → 0
Note: if a = b, φ τ λ , f → φ ( 0, f ) = S ( f )
λ
Lec18.5 - 28
2/6/01
G4
Alternate Cross-Correlation Interferometer Spectrometer
L.O.
a
b
L.O.
Delay line
×
×
×
∫
∫
∫
Computer
(
φ τλ , f
)
Note: Figures omit down-converters and bandlimiting filters
Lec18.5 - 29
2/6/01
G5
Mechanical Long Distance Phase Synchronization
For L >> λ, L.O. synchronization can be degraded by random
phase variations in path length L between two (or more) sites
(due principally to thermal and acoustic variations)
One standard solution:
Line Stretcher
∆φ
Lec18.5 - 30
2/6/01
Feedback
"Line stretcher" varies path so that ∆φ ≅ 0
Communications and telemetry systems can
similarly be phase synchronized
J1
Synchronizing with Remote Atomic Clocks
If distance L is too great to synchronize L.O.'s, then we can use
a remote clock, e.g. "very- long baseline" interferometry, "VLBI"
×
Hydrogen maser
Cesium beam
Global Positioning System (GPS)
Loran − C
Temperature-stabilized crystal oscillators
Crystal oscillators
Tape
⇒
⇒
⇒
⇒
⇒
⇒
∼ 10 −14 , 10 −15
∼ 10 −12
∼ 10 −8 sec
∼ 10 −6 sec
∼ 10 −5 − 10 −8
∼ 10 −4 − 10 −5
If clocks perfect: cross-correlate to find time offset, correct for it, then
correlate the signals, albeit with an unknown fixed phase offset φ0
[unless reference source (in the sky) or phase is available].
If clocks imperfect and delays each way are identical: at site A measure
delay between clock B and A; do the same at B, and subtract results to
yield twice the clock offsets. Use this offset to align A and B data streams.
Lec18.5 - 31
2/6/01
J2
Synchronizing with Remote Atomic Clocks
Alternative approaches if clocks and transmissions are imperfect:
a) Track and correct phase shifts:
then average φˆ ( τ λ , f )
+π
Phase β
t
0
−π
Example: A 10 −12 cesium clock drifts 2π in ~ 100 sec, so φˆ ( τλ , f ) might be
computed for 5 − 10 sec blocks before averaging; then only φˆ is known.
b) Same, but set phase using strong resonant line point source,
ψy
Φ(f)
monochromatic
Strong point source
point source
ψx
f
c) or separable point source in space, modulation, etc.
To source
To reference
Lec18.5 - 32
2/6/01
J3
Multiband Synchronization of Clocks
Use multiple frequencies for wideband sources:
Which fringe F is source on?
ψ
0
Switch across all f's within coherence time of clock.
2πF
Source 1
Source 2
Observed bands
(collectively adequate)
0
Lec18.5 - 33
2/6/01
f
J4
Phaseless Interferometry
Hanbury-Brown and Twiss
Visible interferometer at
Narrabri, Australia
TA
Phototube
TA
a
B
TR
⎡
⎤
v o rms
TR2
≅ 2
⎢For TR >> TA ,
⎥
v
τ
T
2W
⎢⎣
⎥⎦
o
A
b
f
B
f
W ≅ 1 GHz
( )2
( )2
aa
LPF
×
bb
0W
LPF
aabb
∫
τ
Lec18.5 - 34
2/6/01
( )
∝ φE τλ
2
f
One might (wrongly) think
photodetectors would lose
all phase information and
ability to measure source
structure at λ/D resolution.
2
Recall: E ⎡⎣aabb⎤⎦ = a2 b2 + 2ab ,
( )
where ab is φE τy here.
⇒ source size, etc.
L1
Phaseless Recovery of Source Structure
( )
Recall: E [aabb] = a b + 2ab , where ab is φE τ y here.
2
2
2
( )
Recall: E ( x, y ) ↔ E ψ
Purely real if source
is even function of
position, allowing
perfect source
reconstruction
↓
↓
( )
( )
φE τλ
↓
( )
φE τλ
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↔ E ψ
2
( )
⇒Ι ψ
↓
2
↔
R
( )
E ψ
2
( ∆ψ )
L2
Phaseless Interferometer Interpretation: Independent Radiators
D
g
Source, independent
thermal radiators g and h
h
R
a
θ
b
a,b are uncorrelated if ∆φa − ∆φb > 2π
g
[∆φa is ∆φ at "a" for rays g,h]; or if
φs
= φ > ( λ 2) L.
2
Thus a,b decorrelated if φs > λ L.
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D
h
φ=
R
λ 2 a
L
b
φs
2
L3
Phaseless Interferometer Diffraction-Limited Source
D
R
Source "Aperture"
θB ≅ λ / D
θ
L
a
b θ ≅ L /R
If θ > θB ≅ λ D, then a and b are ~uncorrelated
Therefore decorrelated if
Lec18.5 - 37
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Dθ ≥ λ
or if
DL R ≥ λ
since θ ≅ L R
or if
φs ≥ λ L
since φs ≅ D R
L4
Radar Equation
Target v
ω
Issues: Signal design
Propagation, absorption, Scattering
Processor design refraction, scintillation,
scattering, multipath
Antenna
Wm−2 at transmitter
Pt
σ
⎛ Gλ ⎞ σ
•
A
•
•
G
Watts
=
P
t
t
t⎜
2
2
2 ⎟
4πR
4πR
⎝ 4πR ⎠ 4π
Wm−2 at target
2
Prec =
σ "scattering cross-section" is equivalent capture
cross-section for a target scattering isotropically
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N1
Radar Scattering Cross-Section
σ "scattering cross-section" is equivalent capture
cross-section for a target scattering isotropically
Note: Corner reflector can have σ >> size of target
Biastatic radars:
R1
If target is unresolved, Prec ∝ 1/ R12R 22
If target is resolved by the transmitter,
Prec ∝ 1/ R22
R2
Note resolution enhancement:
Prec ∝ R G where G ( θ ) has
a narrower beam than G ( θ )
−4
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2
G ( θ)
2
θ
G2 ( θ )
θB
N2
Target Scattering Laws
1) Specular
3) Faceted
θ
θ
2) Scintillating
P ( θ)
0
4) Lambertian
θ
δ << λ ⇒∼ nulls
cos θ ∝ power scattered
(geometric projection only)
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N3
Target Scattering Laws
5) Random Bragg Scattering (frequency selective)
At Bragg angles ∆Path1,2 = n • 2π
n = 0, ± 1, ± 2,...
Path 2 between
phase fronts A and B
B
A
Path 1
A, B
nπ
Quasi-periodic surface
6) Sub-Surface Inhomogeneities
0
z
0
ε ( z)
∼ random Bragg
scattering
x, y
e.g. rocks
z
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N4
Target Range-Doppler Response
Narrowband Pulsed Radar
σ ( t, f0 )
e.g.
Impulse response
function for a deep target
t
Moon
Deep Target
CW Radar
Moon
"Spins"
σ(t, f )
Return Doppler,
shifted to f0 + ∆f
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f0
f0 + ∆f
f
N5
Range-Doppler Response for a CW Pulse
σ ( t, f )
Range band
ω
E
North
C
A
B
D
E
C
Moon
e.g. Moon
B
D
f
Doppler Band
E
C
t (range)
A
Note north-south ambiguity
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N6