SENSITIVITY
Outline
What is Sensitivity & Why Should You Care?
What Are Measures of Antenna Performance?
What is the Sensitivity of an Interferometer?
What is the Sensitivity of a Synthesis Image?
Summary
J.M. Wrobel - 19 June 2002
SENSITIVITY
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What is Sensitivity & Why Should You Care?
• Measure of weakest detectable radio emission
• Important throughout research program
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Technically sound observing proposal
Sensible error analysis in publication
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Expressed in units involving Janskys
•
– Unit for interferometer is Jansky (Jy)
– Unit for synthesis image is Jy beam-1
1 Jy = 10-26 W m-2 Hz-1 = 10-23 erg s-1 cm-2 Hz-1
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Common to use milliJy or microJy
J.M. Wrobel - 19 June 2002
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Measures of Antenna Performance
Source and System Temperatures
• What is received power P?
• Write P as equivalent temperature of matched
termination at receiver input
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–
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•
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Rayleigh-Jeans limit to Planck law P  k B  T  
Boltzmann constant kB
Observing bandwidth 
Amplify P by g2 where g is voltage gain
Separate powers from source, system noise
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–
2
Source antenna temperature Ta => source power Pa  g  k B  Ta  
System temperature Tsys => noise power PN  g 2  k B  Tsys  
J.M. Wrobel - 19 June 2002
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Measures of Antenna Performance
Gain
• Source power Pa  g 2  k B  Ta  
– Let Ta  K  S for source flux density S, constant K
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Then Pa  g 2  k B  K  S  
(1)
Pa  1  g 2 a  A  S  
2
Antenna area A, efficiency  a
(2)
But source power also
–
– Receiver accepts 1/2 radiation from unpolarized source
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Equate (1), (2) and solve for K
K  (a  A) /( 2  k B )  Ta / S
– K is antenna’s gain or “sensitivity”, unit degree Jy-1
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K measures antenna performance but no Tsys
J.M. Wrobel - 19 June 2002
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Measures of Antenna Performance
System Equivalent Flux Density
• Antenna temperature Ta  K  S
– Source power
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Express system temperature analogously
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Pa  g 2  k B  K  S  
Let Tsys  K  SEFD
SEFD is system equivalent flux density, unit Jy
System noise power PN  g 2  k B  K  SEFD  
SEFD measures overall antenna performance
SEFD  Tsys / K
– Depends on Tsys and K  (a  A) /( 2  k B )
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Examples in Table 9-1
J.M. Wrobel - 19 June 2002
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Interferometer Sensitivity
Real Correlator - 1
• Simple correlator with single real output that is
product of voltages from antennas j,i
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Interferometer built from these antennas has
– Accumulation time  acc , system efficiency  s
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SEFDi = Tsysi / Ki and SEFDj = Tsysj / Kj
Each antenna collects bandwidth 
Source, system noise powers imply sensitivity Sij
Weak source limit
– S << SEFDi
– S << SEFDj
Sij 
1
s

J.M. Wrobel - 19 June 2002
SENSITIVITY
SEFDi  SEFD j
2    acc
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Interferometer Sensitivity
Real Correlator - 2
• For SEFDi = SEFDj = SEFD drop subscripts
1
•
SEFD
S  
s
2    acc
–
Units Jy
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Accounts for electronics, digital losses
Eg: VLA continuum
Interferometer system efficiency  s
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Digitize in 3 levels, collect data 96.2% of time
Effective
s  0.81 0.962  0.79
J.M. Wrobel - 19 June 2002
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Interferometer Sensitivity
Complex Correlator
• Delivers two channels
– Real SR , sensitivity S
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Imaginary SI , sensitivity S
Eg: VLBA continuum
– Figure 9-1 at 8.4 GHz
– Observed scatter SR(t), SI(t)
– Predicted S = 69 milliJy
1
SEFD
S  
s
2    acc
–
Resembles observed scatter
J.M. Wrobel - 19 June 2002
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Interferometer Sensitivity
Measured Amplitude
• Measured visibility amplitude S m  S R2  S I2
– Standard deviation (sd) of SR or SI is S
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True visibility amplitude S
Probability Pr( Sm / S )
– Figure 9-2
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Behavior with true S / S
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High: Gaussian, sd S
Zero: Rayleigh, sd S 2  ( / 2)
Low: Rice. Sm gives biased estimate of S. Use unbias method.
J.M. Wrobel - 19 June 2002
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Interferometer Sensitivity
Measured Phase
• Measured visibility phase
m  arctan( S I / S R )
• True visibility phase 
• Probability Pr(  m )
– Figure 9-2
– Behavior with true S / S
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High: Gaussian
Zero: Uniform
Seek weak detection in phase, not amplitude
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Single Polarization
• Simplest weighting case where visibility samples
1
SEFD
– Have same interferometer sensitivities S  
s
2    acc
– Have same signal-to-noise ratios w
– Combined with natural weight (W=1), no taper (T=1)
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Image sensitivity is sd of mean of L samples, each
with sd S , ie, I m  S / L
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1
 N  ( N  1)
2
No. of interferometers
No. of accumulation times tint /  acc
SEFD
So I  1 
m
s
N  ( N  1)    tint
J.M. Wrobel - 19 June 2002
SENSITIVITY
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L   N  ( N  1)  (tint /  acc )
2
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Image Sensitivity
Dual Polarizations - 1
• Single-polarization image sensitivity I m
• Dual-polarization data => image Stokes I,Q,U,V
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Polarized flux density
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Gaussian noise in each image
Mean zero, I  Q  U  V  I m / 2
P  Q2  U 2
Rayleigh noise, sd Q  2  ( / 2)  U  2  ( / 2)
Cf. visibility amplitude, Figure 9-2
Polarization position angle
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1
2
   arctan( U / Q )
Uniform noise between   / 2
Cf. visibility phase, Figure 9-2,  
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Dual Polarizations – 2
• Eg: VLBA continuum
– Figure 9-3 at 8.4 GHz
– Observed
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T: Stokes I, simplest weighting
B: Gaussian noise I = 90 microJy beam-1
– Predicted
I  I m / 2  S / 2  L
1
 N  ( N  1)  (tint /  acc )
2
Previous eg S
Plus here L = 77,200
So I = 88 microJy beam-1
L
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•
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Dual Polarizations – 3
• Eg: VLBA continuum
– Figure 9-3 at 8.4 GHz
– Observed
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T: Ipeak = 2 milliJy beam-1
B: Gaussian noise I = 90 microJy beam-1
– Position error from sensitivity?
1
I
   HPBW 
2
I peak
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Gaussian beam  HBPW = 1.5 milliarcsec
Then  = 34 microarcsec
Other position errors dominate
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Dual Polarizations – 4
• Eg: VLA continuum
– Figure 9-4 at 1.4 GHz
– Observed
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Q, U images, simplest weighting
Gaussian Q  U = 17 microJy beam-1
– Predicted
Q  U  I m / 2  S / 2  L
S 
L
1
s

SEFD
2    acc
1
 N  ( N  1)  (tint /  acc )
2
• Q  U = 16 microJy beam-1
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Dual Polarizations – 5
• Eg: VLA continuum
– Figure 9-4 at 1.4 GHz
– Observed
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Q, U images, simplest weighting
Q  U = 17 microJy beam-1
Form image of P  Q 2  U 2
Rayleigh noise in P
Sd 11 microJy beam-1
Predicted
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Sd Q  2  ( / 2)  U  2  ( / 2)
Sd 11 microJy beam-1
J.M. Wrobel - 19 June 2002
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Image Sensitivity
Dual Polarizations – 6
• Eg: VLA continuum
– Figure 9-4 at 1.4 GHz
– Observed
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•
I, Q, U images, simplest weighting
Gaussian noise Q  U  I
– I, Q, U will have same sd if each is limited by sensitivity
• Recall I  Q  U  V  I m / 2
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Other factors can increase I
Suspect dynamic range as Ipeak = 10,000 I
Lesson: Use sensitivity as tool to diagnose problems
J.M. Wrobel - 19 June 2002
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Sensitivity
Summary – 1
• One antenna
– System temperature Tsys
– Gain K
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Overall antenna performance is measured by system
equivalent flux density SEFD
SEFD  Tsys / K
–
Units Jy
J.M. Wrobel - 19 June 2002
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Sensitivity
Summary - 2
• Connect two antennas to form interferometer
– Antennas have same SEFD, observing bandwidth 
– Interferometer system efficiency  s
– Interferometer accumulation time  acc
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Sensitivity of interferometer
1
SEFD
S  
s
2    acc
– Units Jy
J.M. Wrobel - 19 June 2002
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Sensitivity
Summary - 3
• Connect N antennas to form array
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Antennas have same SEFD, observing bandwidth 
Array has system efficiency  s
Array integrates for time tint
Form synthesis image of single polarization
Sensitivity of synthesis image
1
SEFD
I m  
s
N  ( N  1)    tint
–
Units Jy beam-1
J.M. Wrobel - 19 June 2002
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