Me as Theory

8/12/2019 Me as Theory

1/28

THEORY OF MEASUREMENTSBrian Mason

Fifth NAIC-NRAO School on Single-DishRadio Astronomy

Arecibo, PRJuly 2009


2/28

OUTLINE

Antenna-Sky Coupling

Noise

the Radiometer Equation

Minimum Tsys Performance measures

System Response

Gain & linearity

quantization

"#

G


3/28

Antenna-Sky CouplingPower Received:

Source ^ ^ Antenna PatternSpecific intensity(surface brightness)

Effective Area ^

Factor !comes from one of two polarizations

Ae ="apertureAgeometrical

Pn(",#)$ d% =%Ant


4/28


5/28

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.


6/28

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


7/28

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


8/28

Antenna Pattern and its Fourier

Transform


9/28

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.


10/28

reciprocal

f.t. ofreciprocal

TAnt(u)=

TSky(u)"B(u)

"D(u)

TAnt(x) =TSky(x)" (B(x)"D(x))


11/28

Single-beam sky map

Tycho SNR w/Effelsburg 100m


12/28

Nave Dual-beamBeamswitched sky map



13/28

Dual-beamBeamswitched sky mapAfter EKH



14/28

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


15/28

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


16/28

Noise

("nrms

)2= n

2+ n

n ="

eh# /kT

$1


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


17/28

Noise

("nrms

)2= n

2+ n

n ="

eh# /kT

$1

"T=T

"#$


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:


18/28

Minimum Tsysfor a Coherent Amplifier

Coherent Amplifier: phase-preserving

"E"t> h

"n"#>1

n1,"

1

n2=Gn

1

"2 = "1


19/28


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


20/28

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


*these are theoreticalminimums real systems often noisier


21/28

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.


22/28

Gain Effects: deviations from linearity

Input Power

OutputPower Id

eal

Gain compression/saturation

TRx

Integrated power iswhat usually matters(RFI)


23/28

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)


24/28

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)


25/28

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


26/28

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.


27/28

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)


28/28

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)

Me as Theory

Documents

Transcript of Me as Theory