ECEN 5254 Remote Sensing Signals and Systems Professor Albin J. Gasiewski Lecture #19 – March 20, 2012 ECOT 246 303-492-9688 (O)
[email protected] Classes: ECCS 1B14 TR 9:30-10:45 AM ECEN 5254 Remote Sensing Signals & Systems
Spring 2012
University of Colorado at Boulder
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Administration • Reading: – RSSS slides – Thomson, Moran, Swenson - Ch 2 – Additional references on D2L
• Term paper abstracts due March 22
ECEN 5254 Remote Sensing Signals & Systems
Spring 2012
University of Colorado at Boulder
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Last Lecture
• Photon (shot) noise • Gain fluctuations • Sensitivity and CNR
ECEN 5254 Remote Sensing Signals & Systems
Spring 2012
University of Colorado at Boulder
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Today’s Lecture
• • • • •
Coherent detection Quantum limit Interferometry Van Cittert-Zernike Theorem Interferometric imaging
ECEN 5254 Remote Sensing Signals & Systems
Spring 2012
University of Colorado at Boulder
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Optical Detector Output Statistics Consider AC and DC components after low pass filtering: Gv (ideal) +
is(t) ↑ id(t) ↑ iT(t) ↑ iA(t) ↑ R
vo(t)
C 1
DC:
1/(2πRC)
f
AC: 5
Optical Detector Relative Sensitivity Long integration time limit: RC » pulse duration
Decreases as 1/√RC ~ 1/√integration time (as expected) Gain reduces thermal and amplifier noise Gain fluctuation noise factor reduces sensitivity If photon (rather than thermal) limited then relative sensitivity is inversely proportional to square root of count rate. This behavior is characteristic of shot noise.
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Minimum Detectable Power Variation Long integration time limit: RC » pulse duration
Np = Expected # photons received during integration time interval (shot noise limit) 7
Coherent Optical Detection
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γ
ES(t) fS
-
optical diplexer
RL
coherent source (laser)
+
Optical Heterodyne Receiver
ELO(t) fLO
Gv +
vd(t)
B
vIF(t)
BPF
τi
vo(t)
Coherent signal ES(t) of optical bandwidth B in USB at fS Use any quantum optical detector (VP, PMT, PIN, APD…)
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γ
ES(t) fS
-
optical diplexer
RL
coherent source (laser)
+
Optical Heterodyne Receiver
Gv +
vd(t)
B
vIF(t)
BPF
τi
ELO(t) fLO
vo(t)
Consider detected signal vd(t) : |ES|2 |ELO|2
2ESELO 10
γ
ES(t) fS
-
optical diplexer
RL
coherent source (laser)
+
Optical Heterodyne Receiver
ELO(t) fLO
Gv +
B
vd(t)
vIF(t)
BPF
τi
vo(t)
Can increase LO power to make » , , thermal and amplifier noise. Provided that (fS - fLO) « 1/τpulse we have:
(e.g., τpulse ~ 1 nsec, fS - fLO ~ 0.1-10 MHz)
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Optical Heterodyne Sensitivity If LO power large, IF frequency small compared to 1/τpulse :
Relative sensitivity:
Compare to:
(incoherent case, large )
Compared to incoherent case, coherent downconversion has removed effects of : 1) dark counts and 2) thermal & amplifier noise. Quantum efficiency and gain fluctuations are still important. LO power fluctuations may also be important, but can be mitigated by balanced detection – per previous lectures. 12
Quantum Limit
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Radio Frequency Limit of OHR Consider thermal input signal at radio frequencies:
Ultimate radiometer sensitivity is TQ = hfS/k (cannot exceed!)
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Basis for Interferometry
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Recall: Radiation from Aperture
x
z
y 16
Aperture Inverse Problem: Point Source Consider contribution to field in aperture caused by plane wave from a distance point source of solid angle dΩ located in ^: direction -k i
x
z
y
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Aperture Inverse Problem: Extended Sources Now consider contribution to field in aperture caused by plane waves from many distant point sources each located in ^ . Evaluating in direction -k i the aperture plane (z = 0) yields:
x
z
y
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Aperture Inverse Problem: Extended Sources Continuing…
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Aperture Inverse Problem: Extended Sources …where the Jacobian of the transformation from (θ,ϕ) to (u,v) is used:
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Fourier Transform Relationships
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Fourier Transform Relationships
Coordinate reversal property of Fourier transform…
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Fourier Transform Relationships
Wiener Khinchine theorem for pulse functions…
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Fourier Transform Relationships
Assuming stochastic plane wave fields - apply expectation operator…
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Fourier Transform Relationships
Relation between autocorrelation function and coherency matrix (or, equivalently, Stokes parameters)…
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Fourier Transform Relationships Assuming that aperture fields are statistically homogeneous (i.e., spatially stationary) and ergodic (ensemble average=time average):
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Van Cittert-Zernike Theorem
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Van Cittert-Zernike Theorem W/(m2-st-Hz)
Spatial form of Wiener-Khinchine theorem for propagating plane waves
V2/m2 Aside from multiplicative factors, the intensity distribution as a function of angle is the Fourier transform of the (complex) correlation function in the aperture plane.
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Interferometric Imaging Principle: Measure the complex field correlation function REa(-ρx, -ρy,0) in an aperture plane, then apply a 2-D spatial Fourier transform to obtain the angular distribution of radiation intensity. Practical issues include: • Sampling (density, range, angular sensitivity) • Integration noise and bandwidth (fringe washing) • Absolute calibration (magnitude and phase) • Data correlation techniques
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Two-Element Interferometer
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z
Two-Element Interferometer y
x
ejγ v1(t) B
F = complex effective length of antenna:
v2(t) B
× γ = phase adjustment
τi vo(t) 31
Two-Element Interferometer Let:
(e.g., x-polarized)
*
Adjust phase γ to measure complex xx component of REa :
Similar for yy, xy components…
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Two-Element Complex Correlator z
x-polarized : y x
B
Requires two real correlation channels
B
× ×
τi
τi
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Complex Polarimetric Correlator z
x
Assumed dual (orthogonal-linear) polarized antennas, all complex y elements of REa can be obtained using six real correlation channels:
B
B
× ×
τi
τi
Two each real channels required for each of xx, yy, and xy terms 34
z
Adding Interferometer y
x
Need to remove constant DC terms (may also not be constant due to fluctuations)
B B
+ +
(v)2
τi
(v)2
τi
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Very Long Baseline Interferometry z
y
v1(t)
v2(t)
x
×
× ~ × ~
B
B
Ts A/D
τi
v2(t) v1(t)
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Correlation Function Sampling For time-invariant radiation fields may move around one element to sample correlation function. Time-varying fields require many element pairs, or baselines. However, correlation function symmetry can be exploited: ρy Calculate these pairs w/o measurement ("no cost")
ρx
Noise Measure these baseline pairs
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Interferometric Imaging Systems
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Radio Image (M51)
Very Large Array (VLA) in Socorro, New Mexico 27 antennas, Y-shaped array Each antenna 25 meters dia “A” array: 36 km maximum baseline Resolution: 40 mas at 43 GHz (highest frequency) (“golf ball at 100 miles”) Optical Image (M51)
(Photos: NRAO/AUI)
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VLA Image of Saturn at 15 GHz Cold rings mask bright radiation from planet (de Pater and Dickel, 1982, NRAO/AUI) 40
Possible "Einstein Ring" observed at 15 GHz using the VLA (5 arcsec FOV) Source: 4C 05.51, a low-surface brightness ring with two diametrically opposedcompact sources. The structure strongly suggests that it is due to gravitational lensing by a massive foreground object. This image may be evidence for a symmetric case of gravitational lensing as proposed by Einstein in 1936 (from J.N. Hewitt and E.L. Turner, NRAO/AUI)
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Very Long Baseline Array (VLBA)
Ten radio antennas, each 25 meters in diameter. Maximum baseline ~8,000 km. (“like reading the New York Times from Los Angeles”)
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Central galaxy of Virgo cluster Distance: ~6x107 ly Size: ~7’
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Cygnus A
Cygnus A radio galaxy observed at 6 cm wavelength using VLA. The distance between the outer radio lobes of Cygnus A is ~5x the size of the Milky Way galaxy. 44
Resolution ~0.15 mas using VLBI at 15 GHz (1 mas = 1/3600 deg) (Photo from NRAO/AUI)
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Rapidly evolving galactic nucleus with T~6E13 K (AO 0235+164) observed with ~3 orders of magnitude higher resolution than VLA image at same frequency (S. Frey et al., 2001)
VSOP: VLBI Space Observatory Program (Japan) Launched February, 1997 8 meter diameter radio telescope in elliptical orbit with perigee/apogee heights of 560/21,000 km Frequencies at 1.6, 5 GHz, Resolution ~0.45 mas
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SMOS – Soil Moisture and Ocean Salinity ESA Project: L-band, Polar low-Earth orbit, Launched November 2, 2009
Instantaneous (non-aliased) FOV :
L-band: 1400-1427 MHz 69 total elements in Y-array (21 elements per arm X three arms) 6.75-m maximum baseline Dual polarimetric (Tx,Ty) Surface resolution: ~50 km at 775 km altitude
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SMOS – Soil Moisture and Ocean Salinity L-band, Low-Earth orbit, Launched November 2, 2009
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SMOS Imagery over Scandinavia
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SMOS Imagery – Dec 2009
Courtesy UPC / DEIMOS
SMOS Imagery – Dec 2009
Courtesy UPC / DEIMOS
February 13, 2008
GeoSTAR Concept 2-D Geostationary Sounder/Imager
GeoSTAR spatial response pattern for 298 elements with 2.8lspacing • ~50 km spatial resolution • Full disk image every one hour • No moving parts • ~2.5m maximum baseline • NASA/JPL concept
Y-Array of ~300-600 receiver elements and many tens of thousands of one-bit correlators in AMSU A/B bands of 50-56 and 183 GHz
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Optical Interferometry Cambridge Optical Aperture Synthesis Telescope (COAST) Three 0.4 m diameter telescopes at λ~1 μm with maximum baseline 6 m apart Resolution ~10 mas – better than HST or best ground-based adaptive optics system Illustrated evolution of split binary star Capella (separated by ~50 mas)
~0.6 AU or ~5 lmin 42 ly distance from Earth
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Next Lecture
• • • •
Sampling, resolution, fringe washing Noise Aperture thinning Intensity correlation (Hanbury-BrownTwiss Experiment)
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