noise diodes, calibration, baselines & nonlinearities ron maddalena nrao, green bank, wv shelly...
DESCRIPTION
Sig and Ref DefinitionsTRANSCRIPT
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Noise Diodes, Calibration, Baselines & Nonlinearities
Ron MaddalenaNRAO, Green Bank, WV
Shelly HynesLouisiana School for Math, Science and the Arts, Natchitoches, LA
Charles FiguraWartburg College, Waverly, IA
July 27, 2006
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Calibration Data June 18, 2006
C-Band – Off-On Observations Multiple calibration sources, same hardware, same
attenuator/filter settings Tcal calibration data – Various combinations of
polarization, high/low noise diodes Data consists of:
Sigon = On source, noise diode on Sigoff = On source, noise diode off Refon = Off source, noise diode on Refoff = Off source, noise diode off
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Sig and Ref Definitions
offonoffon
atmsrcoffonoffon
calrcvratmspillCMBon
rcvratmspillCMBoff
calsrcrcvratmspillCMBon
srcrcvratmspillCMBoff
srcsysoff
RefRefSigSigResids
ΔTνTRefRefSigSig
νTνTTTTRef
νTTTTRef
νTνTνTTTTSig
νTνTTTTSig
TT Sig
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Sig and Ref Definitions
2νTνTTTTT
2SigSigSig
2νTνTTTT
2RefRefRef
calrcvrsrcatmspillCMB
offon
calrcvratmspillCMB
offon
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Current Calibration Method
caloffon
A
caloffon
calsys
TνRefνRefνRef
νRefνRefνSigνT
TνRefνRefνRef
2TT
So Tsys loses all frequency information…
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Astronomical Tcal
= EfficiencyAp = Area
k = Boltzman’s Constant
=
ElevationOpacity
νS2k
ηAνT
e
SRefSigRefRefνS
calp
cal
sinθτ
offoff
offoncal
ηSA
2kTastrocalp
engcal If is unknown…
Source: Johnson et.al., 2002
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Linear Vector Tcal Expressions
AνSigBνTνTνT
AνSigBνTνT
AνRefBνTνT
AνRefBνT
onoAcalsys
offoAsys
onocalsys
offosys
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Now Tsys retains the frequency structure
νT
νRefνRefνRefνSig
νTνRefνRef
νRefνRef
νRefνSigνT
νTνRefνRef
νRef2
νTνT
caloffon
caloffon
A
caloffon
calsys
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Power Characterization
Power In0 5 10 15 20 25 30 35 40
Pow
er O
ut
0
20
40
60
80
100
Expected Linear/Nonlinear Power Response
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‘Resids’ for Spectral Processor
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Res
ids_
R
-4000
-3000
-2000
-1000
0
1000Epoch 3
1420
49
67
85
95
121172
178
1006 10121075
P_Out0.0 5.0 10.0 15.0 20.0 25.0 30.0
Res
ids_
L
-4000
-3000
-2000
-1000
0
1000
14
20
49
67
85
95
121
172
178
1006
10121075
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Resids vs. Power InC-Band
Power In0 1 2 3 4 5
Res
ids
0.00
0.01
0.02
0.03
0.04
0.05
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Nonlinear Theory Include a second-order correction for gain –
So the temperature equations become
2outout
'out P
BCPP
2ononsourcecalsys
2offoffsourcesys
2ononcalsys
2offoffsys
SigBCSigTTT
SigBCSigTT
RefBCRefTT
RefBCRefT
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Non-Linear SolutionsFor Calibration
src
onoffonoffononoffoff
2ononoffonoff
2off
2onoffcal
sys
srconoffonoffononoffoff
onoffonoff
srconoffonoffononoffoff
2on
2off
2on
2offcal
srconoffonoffononoffoffonoffonoffonoffonoffcal
TSigSigRefRefSigRefSigRefSigSigSigRefRefSigRefRefT
TSigSigRefRefSigRefSigRefSigSigRefRef C
TSigSigRefRefSigRefSigRefSigSigRefRef-B
TSigSigRefRefSigRefSigRefSigSigSigSigRefRefRefRefT
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Nonlinear Application Evaluate C, B, Tcal, using a known
calibration source. Because nonlinearity is encapsulated
within P’out, we can use the previous expressions from the linear case:
νT
νRefνRefνRefνSigνT
νTνRefνRef
νRef2
νTνT
caloffon
A
caloffon
calsys
calsysT
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Calibration Data June 18, 2006
C-Band – Off-On Observations Multiple calibration sources, same hardware, same
attenuator/filter settings Tcal calibration data – Various combinations of
polarization, high/low noise diodes July 15, 2006
3C147 C-Band, low-diode, linear polarization
Various attenuator settings at various places S-band Scal calibration data – Various combinations
of polarization, high/low noise diodes
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Tsys Comparison
NonlinLin
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Tcal Comparison
NonlinLin
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Baseline Improvement
NonLin
Linear
Cat
Traditional
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Source of ‘Baselines’ – Traditional
Assuming Linear
2νTνTTTTT
2SigSigSig
2νTνTTTT
2RefRefRef
calrcvrsrc
SigatmspillCMB
offon
calrcvr
RefatmspillCMB
offon
ScalarsysTνRef
νRefνSigνTA
2νTνTTTT
TTTν
calrcvr
RefatmspillCMB
Refatm
Sigatmsrc
ScalarsysT
Baselines
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Source of ‘Baselines’ – Traditional Non-Linear
2νT))(T)TTT(21(νT
))(T)(T2(TTT[Smooth]ν
cal2rcvr
RefAtmSpillCMBrcvr
calrcvrRefatm
Sigatmsrc
fccSmooth
ffcT
BaselinesScalarsys
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Time Dependence
S13C11x July 15, 2006, t = 0 hours
NonLin
Linear
Cat
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Time Dependence
S34C11x July 15, 2006, t = 2 hours
NonLin
Linear
Cat
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TA Comparisons
t = 2 hoursRed – EarlyBlue - Late
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Time Dependence
S69C11x July 15, 2006, t = 4 hours
NonLin
Linear
Cat
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TA Comparisons
t = 4 hoursRed – EarlyBlack - Late
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Time Dependence
NonLin S34C11x
NonLin S377C11x
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Noise Estimates – Vector Tcal
sigref
sigT tChanWidthtNchan
t
11TT2T)TT(
νTνRefνRef
νRefνSigνT
2
cal
src2sys
2srcsys
2
caloffon
A
Assume Tsys = 10*Tcal, NChan = 1, tref = tsig
Tsrc = 0 --- σ2=2*Tsys/(ChanWidth * tsig)
Tsrc = Tcal --- σ2=4.2*Tsys/(ChanWidth * tsig)
Tsrc = 2Tcal --- σ2=10.4*Tsys/(ChanWidth * tsig)
Tsrc = Tsys --- σ2=205*Tsys/(ChanWidth * tsig)
Assuming Tcal << Tsys
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System Determination
-3dB (IF)-6dB (IFR)-6dB (CR)-6dB (IF)-10dB (IF)
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System Determination
+3dB (IF)+6dB (CR)+6dB (IF)+10dB (IFR)+10dB (IF)
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System Determination-3dB IFRack
NonLin
Linear
Cat
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System Determination-6dB IFRack
NonLin
Linear
Cat
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System Restoration
NonLin
Linear
Cat
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Summary Using a vector form of Tcal for baselines is better than
traditional, regardless of linear or non-linear assumptions.
Baselines are slightly improved by the quadratic approximation,
Cannot achieve good noise, good baselines, and good calibration simultaneously – Compromise!!
System rebalancing restores original nonlinearity. Data taken with major ‘distortions’ to power levels can
be recovered. ‘C’ remains fairly constant with time.
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Conclusions
Extended sources should not be used to determine linearities, Scals, etc.
Polarized sources must be corrected for.
Very bright sources cannot be handled by the 2nd order nonlinear approximation.
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Recommendations
At a minimum, use the vector form of Tcal. Use compact sources for calibrations. For many observations, the linear
approximation is sufficient. Balancing often isn’t necessary and actually
may be detrimental since C will change. Don’t skimp on channels – more channels,
less compromises between noise and baseline
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Catalog Calibration
νS2k
ηAνT
νlogcνlogbaνSlog
catpcat
A
2cat
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Non-Linear Derivation
Prefoff = A + BPout + CPout2 (Eq1)
Prefon = A + B (Pout+Pcal) + C (Pout+Pcal)2 (Eq2)
Psourceoff = A + B (Pout+Psource) + C (Pout+Psource)2 (Eq3)
Psourceon = A + B (Pout+Pcal+Psource) + C (Pout+Pcal+Psource)2
(Eq4)