me as theory
TRANSCRIPT
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THEORY OF MEASUREMENTSBrian Mason
Fifth NAIC-NRAO School on Single-DishRadio Astronomy
Arecibo, PRJuly 2009
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OUTLINE
Antenna-Sky Coupling
Noise
the Radiometer Equation
Minimum Tsys Performance measures
System Response
Gain & linearity
quantization
"#
G
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Antenna-Sky CouplingPower Received:
Source ^ ^ Antenna PatternSpecific intensity(surface brightness)
Effective Area ^
Factor !comes from one of two polarizations
Ae ="apertureAgeometrical
Pn(",#)$ d% =%Ant
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TAnt
=
1
"Ant
T(#,$)Pn(#,$)d"%
Compact, Isothermal BB source :
TAnt
="Src
"Ant
TSrc
Main-beam filling isothermal BB source:
TAnt
=
"MB
"Ant
TSrc
=#BTSrc
Very compact, non-thermal sources betterdescribed in terms of flux density:
TAnt =Aeff
2kSSrc
More collecting area is a big win for very compact sources.For Imaging sometimes you want a big dish, sometimes youWant a smaller dish.
Ideal value = 1Determined by optics& telescope illumination.
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The Antenna as a Spatial Filter
Imagine making a map by moving the telescope aroundand recording the antenna temperature at each(nyquist) point. In 1-D:
The Convolution Theorem gives
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The Antenna as a Spatial Filter
Imagine making a map by moving the telescope aroundand recording the antenna temperature at each(nyquist) point. In 1-D:
The Convolution Theorem gives which is also thesquare of FT ofAntenna illumination
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Antenna Pattern and its Fourier
Transform
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Observing Strategy & Spatial Filtering
The real world introduces contaminating signals whichmust be removed. This requirement often drives thechoice of observing strategy.
Beam Switching/Chopping differentialsky image.Emerson, Klein & Haslam (EKH) described how toaccount for this.
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reciprocal
f.t. ofreciprocal
TAnt(u)=
TSky(u)"B(u)
"D(u)
TAnt(x) =TSky(x)" (B(x)"D(x))
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Single-beam sky map
Tycho SNR w/Effelsburg 100m
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Nave Dual-beamBeamswitched sky map
Tycho SNR w/Effelsburg 100m
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Dual-beamBeamswitched sky mapAfter EKH
Tycho SNR w/Effelsburg 100m
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Variations, other Approaches
EKH-2 Use multiple chops to sample
missing spatial frequencies
Least Squares Map Making
Very general; suitable for
use with focal plane arrays Used by WMAP and other
CMB experiments E.g., Fixsen, Moseley &
Arendt (2000)
r
d =Ar
m
r
m =(ATA)
"1A
Tr
d
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Noise
("nrms
)2= n
2+ n
n ="
eh# /kT
$1
What is the intrinsic noise in the signal going into the telescope?Consider the case that were looking at a grey body.
Bose (wave noise) termDominates in long wavelength(RJ) regime.
At low count rates photon arrivaltimes are uncorrelated (Poisson /shot noise)
n: # photons/sec/Hz
!: emissivity
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Noise
("nrms
)2= n
2+ n
n ="
eh# /kT
$1
What is the intrinsic noise in the signal going into the telescope?Consider the case that were looking at a grey body.
n: # photons/sec/Hz
!: emissivity
Atmosphere: T~300K"=1 GHz, !=0.01 (radio) n~60
"=200 GHz, !=0.1 (mm) n~3
"=1 THz, !=0.05 (submm balloon) n~0.3
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Noise
("nrms
)2= n
2+ n
n ="
eh# /kT
$1
"T=T
"#$
What is the intrinsic noise in the signal going into the telescope?Consider the case that were looking at a grey body.
T" Tobject +TRX +Tatmosphere +Tspillover # TSys
n: # photons/sec/Hz
!: emissivity
proportional to input signal (not Square Root) Other contributors to the signal add linearly to an overallSystem temperature. Typical Tsyss: few 10s of K 100 MHz BW reduce the noise by 10,000 in 1 sec
- few mK RMS in 1 sec
The Radiometer Equation:
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Minimum Tsysfor a Coherent Amplifier
Coherent Amplifier: phase-preserving
"E"t> h
"n"#>1
n1,"
1
n2=Gn
1
"2 = "1
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Minimum Tsysfor a Coherent Amplifier
Coherent Amplifier: phase-preserving
"E"t> h
"n"#>1
n1,"
1
n2=Gn
1
"2 = "1
"n1=
"n2
G!?!?
Problem is fixed by assuming the amplifier adds
~one photon per hz per second uncertaintyto the measurement of n1
TRX,min
=
h"
k
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1 GHz: 0.05 K > BG
If phase is not preserved- direct detection - this limit does notexist (Bolometers, optical CCD cameras, etc.)
Your measurement can in principle be limited by only the noise inthe input photon field [BLIP]
Note: even photon counting systems in the radio will not havepoisson statistics; they will obey the Radiometer Equation.
TRX,min
=
h"
k
Minimum Tsysfor a Coherent Amplifier
*these are theoreticalminimums real systems often noisier
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Performance Measures
From before:
An effective aperture of 2760 m2is requiredto give a sensitivity of 1.0 K/Jy.
TAnt =Aeff
2kSSrc "# $
TAnt
SSrc=
Aeff
2kGain
SNR =TAnt
"T# $
TSys
Units of 1/Janskys OR [meters2/Kelvin] (large is good)You sometimes see the System-Equivalent FluxDensity, SEFD (small is good)
Mapping Speed commonly defined as:
MappingSpeed=Area
(noise)2(time)
These are single-pixel measures. For mapping increase by Nfeeds.
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Gain Effects: deviations from linearity
Input Power
OutputPower Id
eal
Gain compression/saturation
TRx
Integrated power iswhat usually matters(RFI)
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Gain Effects: deviations from linearity
Input Power
OutputPower Id
eal
Gain compression/saturation
Be aware of the limitations of the instrumentYoure using & calibrate at a similar total powerLevel to what your science observations will see.
TRx
Integrated power iswhat usually matters(RFI)
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Gain Effects: fluctuations
Gain drifts over the course of an observingSession(s) easily removed with instrumentalCalibrators (e.g., noise diodes)
when gains or input power change,
attenuators often need to be changed &calibration is then needed also.
Short term gain fluctuations can be moreproblematic (see continuum lecture)
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Sampling/Quantization & Dynamic
Range: Postdetection
Diode or square law detector:Turns E-field into some outputProportional to power(E2)
A/D2Nlevels(N~14)
.
.
.
Common for continuum systems.
Robust & simple (large dynamic range)
"P =P
"#$
A/D sample time~1 level
" 2N#P
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Sampling/Quantization & Dynamic
Range: PredetectionSample the E-field itself.
Samplers must be much faster and sample much morecorasely (typically just a few levels)
Dynamic range limitations more important One usually requires variable attenuators to get it right (Balancing).
Small # of levels increases the noise level. (K-factor in Radiometer equation)
More levels -> greater dynamic range (RFI robustness), greater sensitivity
Monitor levels through your observation.
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THANKS
Don Campbell, Mike Davis Chris Salter
Editors of 1st SDSS proceedings
Further reading:Tools of Radio Astronomy (Rholfs & Wilson)Radio Astronomy (Krauss)Synthesis Imaging in Radio Astronomy II (Tayloer, Carilli & Pereley)
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Thanks To Don Campbell & Mike Davis; Chris Salter; and to theeditors of ASP Volume 278.
Further reading:Tools of Radio Astronomy (Rholfs & Wilson)Radio Astronomy (Krauss)Synthesis Imaging in Radio Astronomy II (Tayloer, Carilli & Pereley)