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

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

Sig and Ref Definitions

offonoffon

atmsrcoffonoffon

calrcvratmspillCMBon

rcvratmspillCMBoff

calsrcrcvratmspillCMBon

srcrcvratmspillCMBoff

srcsysoff

RefRefSigSigResids

ΔTνTRefRefSigSig

νTνTTTTRef

νTTTTRef

νTνTνTTTTSig

νTνTTTTSig

TT Sig

Sig and Ref Definitions

2νTνTTTTT

2SigSigSig

2νTνTTTT

2RefRefRef

calrcvrsrcatmspillCMB

offon

calrcvratmspillCMB

offon

Current Calibration Method

caloffon

A

caloffon

calsys

TνRefνRefνRef

νRefνRefνSigνT

TνRefνRefνRef

2TT

So Tsys loses all frequency information…

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

Linear Vector Tcal Expressions

AνSigBνTνTνT

AνSigBνTνT

AνRefBνTνT

AνRefBνT

onoAcalsys

offoAsys

onocalsys

offosys

Now Tsys retains the frequency structure

νT

νRefνRefνRefνSig

νTνRefνRef

νRefνRef

νRefνSigνT

νTνRefνRef

νRef2

νTνT

caloffon

caloffon

A

caloffon

calsys

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

‘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

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

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

Non-Linear SolutionsFor Calibration

src

onoffonoffononoffoff

2ononoffonoff

2off

2onoffcal

sys

srconoffonoffononoffoff

onoffonoff

srconoffonoffononoffoff

2on

2off

2on

2offcal

srconoffonoffononoffoffonoffonoffonoffonoffcal

TSigSigRefRefSigRefSigRefSigSigSigRefRefSigRefRefT

TSigSigRefRefSigRefSigRefSigSigRefRef C

TSigSigRefRefSigRefSigRefSigSigRefRef-B

TSigSigRefRefSigRefSigRefSigSigSigSigRefRefRefRefT

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

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

Tsys Comparison

NonlinLin

Tcal Comparison

NonlinLin

Baseline Improvement

NonLin

Linear

Cat

Traditional

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

Source of ‘Baselines’ – Traditional Non-Linear

2νT))(T)TTT(21(νT

))(T)(T2(TTT[Smooth]ν

cal2rcvr

RefAtmSpillCMBrcvr

calrcvrRefatm

Sigatmsrc

fccSmooth

ffcT

BaselinesScalarsys

Time Dependence

S13C11x July 15, 2006, t = 0 hours

NonLin

Linear

Cat

Time Dependence

S34C11x July 15, 2006, t = 2 hours

NonLin

Linear

Cat

TA Comparisons

t = 2 hoursRed – EarlyBlue - Late

Time Dependence

S69C11x July 15, 2006, t = 4 hours

NonLin

Linear

Cat

TA Comparisons

t = 4 hoursRed – EarlyBlack - Late

Time Dependence

NonLin S34C11x

NonLin S377C11x

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

System Determination

-3dB (IF)-6dB (IFR)-6dB (CR)-6dB (IF)-10dB (IF)

System Determination

+3dB (IF)+6dB (CR)+6dB (IF)+10dB (IFR)+10dB (IF)

System Determination-3dB IFRack

NonLin

Linear

Cat

System Determination-6dB IFRack

NonLin

Linear

Cat

System Restoration

NonLin

Linear

Cat

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.

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.

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

Catalog Calibration

νS2k

ηAνT

νlogcνlogbaνSlog

catpcat

A

2cat

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)