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
Lec20.4-1
2/26/01
( )
∝ 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
Lec20.4-2
2/26/01
↔ 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.
Lec20.4-3
2/26/01
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
Lec20.4-4
2/26/01
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 ⎟
4 πR
4 πR
⎝ 4πR ⎠ 4π
Wm−2 at target
2
Prec =
σ "scattering cross-section" is equivalent capture
cross-section for a target scattering isotropically
Lec20.4-5
2/26/01
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
Lec20.4-6
2/26/01
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)
Lec20.4-7
2/26/01
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
Lec20.4-8
2/26/01
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
Lec20.4-9
2/26/01
f0
f0 + ∆f
f
N5
Range-Doppler Response for a CW Pulse
σ ( t, f )
Range band
ω
E
North
C
A
D B
E
C
Moon
e.g. Moon
B
D
f
Doppler Band
E
C
t (range)
A
Note north-south ambiguity
Lec20.4-10
2/26/01
N6
Radar Range and
Doppler Ambiguity
Professor David H. Staelin
Massachusetts Institute of Technology
Lec20.4-11
2/26/01
Optimum CW Pulse Radar Receiver
P ( t)
Given:
f ≅ fo
CW transmitted pulse at fo ,
power = P(t)
T
Receives for point source:
r(t) (volts) = k σ P(t) cos ω o t + m(t)
↑
↑
Usually Gaussian
∝ G2 etc.
white noise
A bank of matched filters could test every possible delay,
but the output envelope of a single filter matched to the
transmitted waveform is equivalent.
Lec20.4-12
2/26/01
P1
Optimum CW Pulse Radar Receiver
P ( t)
Given:
f ≅ fo CW transmitted pulse at fo , power = P(t)
T
r ( t)
Optimum receiver is matched filter
h(t )
y ( t)
0
⎡ P ( −t ) cos ωo t ⎤
⎥
y ( t ) ∝ ⎡ σ P ( t ) cos ωo t + m ( t ) ⎤ ∗ ⎢
⎣
⎦ ⎢
⎥
P (0)
⎦
↑
h ( t) → ⎣
Scattering
P ( t − τ)
∆ ∞
R P ( t ) = ∫ P ( τ)
dτ
−∞
P (0)
Signal h ( t )
t =
∗
2T
T
( )2
Lec20.4-13
2/26/01
g(t )
Envelope D et ector
z ( t ) ≅ k 2 σ oR 2
P
( t) + n ( t)
P2
Optimum CW Pulse Radar Receiver
r ( t)
( )2
h(t )
g(t )
z ( t ) ≅ k 2σoR 2P ( t ) + n ( t )
Envelope D et ector
R2 P ( τ ) is "ambiguity function"
in delay (blur function)
z ( t ) ∝ R2 r ( τ )
n ( t ) , noise floor
−T
0
Ambiguity function
↓
2
E ⎡⎣ z ( t ) ⎤⎦ = k σ ( t ) ∗ R2 P ( t ) + n ( t )
↑
Scattering cross-section
Lec20.4-14
2/26/01
T
Note: The matched filter can
operate on the RF signal
(as here) or on its
detected envelope.
P3
Range-Doppler Matched-Filter Receiver
Pulsed CW (Continuous Wave)
transmitted signal (e.g.):
E(t )
0
P(t )
T
2T
r(t )
h ( t, f0 + ∆f1 )
y1(t )
t
h ( t, f0 + ∆f2 )
y 2 (t )
Peak detector
t estimates delay
h ( t, f0 + ∆fN )
y 3 (t )
0
Lec20.4-15
2/26/01
Peak
"D" and doppler
t
D
("∆f2 " Here)
Best
Estimate :
D, ∆f2
t (delay)
P4
Range-Doppler Matched-Filter Receiver
Alternative range-doppler matched filter receiver
y(t )
h ( t, f0 + ∆f1 )
r(t ) +
n(t )
h ( t, f0 + ∆f2 )
h ( t, f0 + ∆fN )
Filter Bank
z(t )
2
()
h2 (t )
z(t )
( )2
h2 (t )
z(t )
( )2
h2 (t )
Envelope
detector
h ( t,f0 + ∆f1 ) = p ( − t ) cos ( − ( ω + ∆ω1 ) t )
Lec20.4-16
2/26/01
Matched
Filter for
Envelope
t Estimates:
Delay "D"
and
t
Doppler
"∆f2 " Here
t
D, ∆f2
or:
Peak
detector
D, ∆f2
P5
Range-Doppler Ambiguity Function
T
CW Pulse:
Original E(t)
Doppler-Shifted E(t)
First null in y(t, ∆f):
t
0
t
Special case:
180 out of phase
yields y(t) null response
0
∆f 1/ 2 cycle
=
f
fT cycles
Therefore ∆f = 1 2 T Hz for first Doppler null
Shifted
Doppler ambiguity function for y(t):
y(τ=0, ∆f )
0
Doppler resolution ≅ 1 2 T Hz
in confusion limit
(better if point-source reflector)
∆f(Hz )
null at 1 2T Hz
Lec20.4-17
2/26/01
Q1
Range-Doppler Ambiguity Function
Range-doppler ambiguity function for y(t);
Represents point-source response:
y(t) peak
τ (range offset, seconds)
0
1/ 2T Hz
∆f(Hz ) Doppler Offset
T
Half-power width in
range ≅ T seconds;
width in Doppler ≅ 1/ 2T Hz
Heisenberg uncertainty principle:
∆t∆f ≅ 1 for
Lec20.4-18
2/26/01
or: BT ≅ 1 = time-bandwidth product,
where B ≅ 1/ T Hz here
Q2
CW Pulse Radar Response Ζ ≅ σ( τ, ∆f ) ∗ R( τ,∆f )
Ambiguity function R(τ, ∆f) = z(τ, ∆f) for point target =
“impulse response” for radar.
Therefore
z(τ, ∆f ) = R( τ, ∆f ) ∗ σs ( τ, ∆f )
where σs ( τ, ∆f ) = target response function
That is: Radar response = (ambiguity function) ∗ (target response)
Note: For good image reconstruction of complex
images we are limited largely by T and 1/2T
resolution in delay, Doppler
Lec20.4-19
2/26/01
Q3
CW Pulse Radar Response – Simple Targets
For a point target, delay and Doppler resolution can achieve
~0.01T and 1/200T, or better, if the SNR is sufficiently high;
almost the same is possible for 2 point targets
~0.01T
T
Single point target
range delay
Lec20.4-20
2/26/01
τ
T
Pair of point targets
Q4
Improved Resolution for Signals with BT>>1
Example:
E(t)
T
t
0
BT >> 1 for white noise,
PRN (pseudo-random
noise), binary data with
~BT bits per block
Noise, B Hz
Binary Code Example:
s(t)
0
1 0 11010 0 …
T
1
∼ sec
B
t
n bits
T = n(1/ B), so BT = n >> 1
Lec20.4-21
2/26/01
Q5
Improved Resolution for BT>>1 Signals
Ambiguity function z( t, ∆f ) for BT
1:
1
∆f ≅ Hz
T
∆τ ≅ 1/ B sec
Sidelobe
ambiguity function
1
∆τ∆f ≅
BT
τ
1
T
B
f
Design of s(t) cos ωt to yield minimum ambiguity sidelobes
is difficult; trial and error is common design technique.
Lec20.4-22
2/26/01
Q6