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

2

νTνTTTTT

2

SigSigSig

2

νTνTTTT

2

RefRefRef

calrcvrsrcatmspillCMB

offon

calrcvratmspillCMB

offon

Current Calibration Method

caloffon

A

caloffon

calsys

TνRefνRef

νRefνRef

νRefνSigνT

TνRefνRef

νRef2

TT

So Tsys loses all frequency information…

Astronomical Tcal

= Efficiency

Ap = Area

k = Boltzman’s Constant

=

ElevationOpacity

νS2k

ηAνT

e

S

RefSig

RefRefνS

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

νRefνRef

νRefνSig

νTνRefνRef

νRef

νRef

νRefνSigνT

νTνRefνRef

νRef

2

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_Out

0.0 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

B

CPP

2ononsourcecalsys

2offoffsourcesys

2ononcalsys

2offoffsys

SigB

CSigTTT

SigB

CSigTT

RefB

CRefTT

RefB

CRefT

Non-Linear SolutionsFor Calibration

src

onoffonoffononoffoff

2ononoffonoff

2off

2onoffcal

sys

srconoffonoffononoffoff

onoffonoff

srconoffonoffononoffoff

2on

2off

2on

2offcal

srconoffonoffononoffoff

onoffonoffonoffonoffcal

TSigSigRefRefSigRefSigRefSigSigSigRefRefSigRefRef

T

TSigSigRefRefSigRefSigRef

SigSigRefRef C

TSigSigRefRefSigRefSigRef

SigSigRefRef-B

TSigSigRefRefSigRefSigRefSigSigSigSigRefRefRefRef

T

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

νRefνRef

νRefνSigνT

νTνRefνRef

νRef

2

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

Nonlin

Lin

Tcal Comparison

Nonlin

Lin

Baseline Improvement

NonLin

Linear

Cat

Traditional

Source of ‘Baselines’ – Traditional

Assuming Linear

2

νTνTTTTT

2

SigSigSig

2

νTνTTTT

2

RefRefRef

calrcvrsrc

SigatmspillCMB

offon

calrcvr

RefatmspillCMB

offon

ScalarsysTνRef

νRefνSigνTA

2

νTνTTTT

TTTν

calrcvr

RefatmspillCMB

Refatm

Sigatmsrc

ScalarsysT

Baselines

Source of ‘Baselines’ – Traditional Non-Linear

2

νT))(T)TTT(21(νT

))(T)(T2(TTT[Smooth]ν

cal2rcvr

RefAtmSpillCMBrcvr

calrcvrRefatm

Sigatmsrc

fccSmooth

ffc

T

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 hours

Red – Early

Blue - Late

Time Dependence

S69C11x July 15, 2006, t = 4 hours

NonLin

Linear

Cat

TA Comparisons

t = 4 hours

Red – Early

Black - Late

Time Dependence

NonLin S34C11x

NonLin S377C11x

Noise Estimates – Vector Tcal

sigref

sigT tChanWidthtNchan

t

11

T

T2T)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)