Astronomical Observational Techniques
and Instrumentation
RIT Course Number 1060-771
Professor Don Figer
Noise
1
Aims for this lecture
•
•
•
review how to calculate signal-to-noise ratio
describe processes that add noise to a detector signal
give examples of how to combat noise
2
Lecture Outline
•
•
signal-to-noise ratio
noise sources
–
shot noise
•
•
•
–
–
–
–
–
–
–
•
signal
dark current
background
thermal noise (Johnson)
kTC
1/f
read noise in the electronics (summation of many contributors)
electronic crosstalk
popcorn noise
pickup/interference
how to combat noise, improving SNR
3
Introduction
•
The ability to detect an object in an image is related to the
amplitude of the signal and noise (and by contrast in some
cases).
The signal can be increased by collecting more photons, e.g.
•
–
–
–
•
building a bigger lens,
using more efficient optics,
having a more efficient detector.
The noise can be reduced by making better electronics.
4
Signal-to-Noise Ratio
5
What is Signal? What is Noise?
6
SNR and Sensitivity Definitions
• SNR is the ratio of detected signal to uncertainty of the signal
measurement. Higher is better.
• Sensitivity is the flux level that corresponds to a given SNR
for a particular system and integration time. Lower is better.
7
SNR and Sensitivity
• Both SNR and sensitivity depend on
– signal
• brightness of source
• absorption of intervening material
– gas, dust
– particles in the atmosphere
– optics
• size of telescope
• efficiency of detector
• integration time
– noise
•
•
•
•
•
detector read noise
detector dark current
background (zodiacal light, sky, telescope, instrument)
shot noise from source
imperfect calibrations
8
Signal: definition
• Signal is that part of the measurement which is
contributed by the source.

S  total A
F tQE {e  }, where
h
A  area of aperture,
QE  quantum efficiency of detector,
  frequency bandwidth,
F  source flux,
 total  total transmiss ion, and
t  integratio n time.
9
Noise - definition
• Noise is that part of the measurement which is due to
sources other than the object of interest.
• In sensitivity calculations, the “noise” is usually equal
to the standard deviation.
• Random noise adds in quadrature.
N total 
N
2
i
.
i
10
Sensitivity: definition
• Sensitivity is the flux level that corresponds to a
given SNR.
Sensitivitydesired
SNR
 F ,desired
SNR
 ergs 
Noise

SNRdesired  2
, where


 cm s Hz 
total A
tQE
h
where, A  area of aperture,
QE  quantum efficiency of detector,
  frequency bandwidth,
F  source flux,
 total  total transmiss ion, and
t  integratio n time.
11
SNR Example
•
•
These images show synthetic noise added to a real image of a star cluster obtained
using Keck/LGSAO.
The maximum signal has been normalized to 100 and shot noise has been added.
SNR values are for brightest pixel.
SNR=10
read noise=0 e
SNR=9
read noise=5 e
SNR=4
read noise=20 e
SNR=7
read noise=10 e
SNR=2
read noise=50 e
12
Imaging at Night
• Signal is diminished as light level is reduced.
• Read noise is very important for object identification in this
case.
13
Read Noise vs. Telescope Size
Effective Telescope Size vs. Read Noise
Telescope Diameter (m)
80
70
60
50
40
30
20
0
1
2
3
4
5
6
Read Noise (electrons)
This plot shows a curve of constant sensitivity for a range
of telescope diameters and detector read noise values in
low-light applications. A 30 meter telescope and zero read
noise detector would deliver the same signal-to-noise ratio
as a 60 meter telescope with current detectors.
14
Noise and Performance
• These plots show system performance for NGST (now called
JWST).
• Note that lower noise gives better performance.
15
Tradeoff Example
• An improvement in SNR due to one detector property can
allow one to relax the performance in another property.
• One last step in making such a trade is to convert
improvements into dollars.
Joint Dark Energy Mission, Brown 2007, PhD Thesis
16
Noise Sources: Shot Noise in Signal
17
Shot Noise Described
• Photons arrive discretely, independently, and randomly. Their
arrival times are described by Poisson statistics.
• Poisson statistics tell us that the Root Mean square uncertainty
(RMS noise) in the number of photons per time interval is equal to
the square root of the mean photon fluence (average number of
photons detected per time interval). Said another way, the signal
equals the variance.
• For example, a star is imaged onto a pixel and it produces on
average 10 photo-electrons per second and we observe the star for 1
second, then the uncertainty of our measurement of its brightness
will be the square root of 10 i.e. 3.2 electrons. This value is the
‘Photon Noise’. Increasing exposure time to 10 seconds will
increase the photon noise to 10 electrons (the square root of 100) but
at the same time will increase the ‘Signal to Noise ratio’ (SNR).
• This noise is often called “Shot Noise” because it is analogous to
similar noise in electronics. The concept of shot noise was first
introduced in 1918 by Walter Schottky who studied fluctuations of
current in vacuum tubes.
• In the absence of other noise sources, the SNR will increase as the
square root of the exposure time due to shot noise of the signal.
18
Photon Shot Noise
• The uncertainty in the source charge count is simply
the square root of the collected charge.
N source


 S  total A
F tQE {e }.
h
• Note that if this were the only noise source, then S/N
would scale as t1/2. (Also true whenever noise
dominated by a steady photon source.)
19
Noise Sources: Shot Noise from
Background
20
Noise - sources: Noise from Background
• Background photons come from everything but signal
from the object of interest!
• Note that the noise contribution is simply the
uncertainty in the background level due to shot noise.
N back


 Cback t   total A
Fback , tQE {e  }.
h
21
Atmospheric emission: Blackbody
Total power onto a detector:
P =  AW  esky B(Tsky)
:
transmission of all optics x Q.E.
esky: emissivity of sky
A:
telescope area
W:
solid angle subtended by focal plane
aperture
:
bandwidth
B(Tsky): Planck function
At 10 mm, typically:  ~ 0.2, e ~ 0.1, AW ~ 3x10-10 m2 Sr
 ~ 1.5 x 1013 Hz (10 mm filter), T ~ 270 K
P ~ 10-9 W or ~ 4 x 1010 g s-1
22
Atmospheric emission: A Note on Solid Angle
• Background from the sky can be calculated by multiplying the
blackbody equation by the area of the sky seen by the detector
and the solid angle subtended by the telescope aperture.
Flux  Asky Wtelescope  e sky B (Tsky ).
• The AW product is invariant in a system.
Asky Wtelescope  AtelescopeWsky .
1
Asky
0
Atelescope
detector
x
d
D
d 
A sky W telescope   x 
F 
2
F

2 2 
2
2
nˆ  dA  d  0 d 2 sin dd  d 
d2 2
d 2 2 ( D / 2) 2
 xD 
  x  2 (1  cos  0 ) ~ 2 x 0  2 x 

 .
 d 2   F x  0 0
d2
F
F
d2
F 
 2F 
x / 2    xD  .
 D  21d 
 
 D212  D2


2
d
F2
2
 2F 
2
A telescopeWsky
2
2
2
23
Noise Sources: Shot Noise from Dark
Current
24
Shot Noise of Dark Current
• Charge can also be generated in a pixel either by thermal
motion or by the absorption of photons.
• The two cases are indistinguishable.
• Dark current can be reduced by cooling.
N dark  C dark t {e  }.
Electrons per pixel per hour
10000
Dark Current vs. Temperature in Silicon
1000
100
10
1
-110
-100
-90
-80
-70
-60
-50
-40
Temperature Centigrade
25
Dark Current Mechanism
26
Dark Current in Infrared Materials
• Dark current is a function of temperature and cutoff
wavelength.
27
Noise Sources: Thermal Noise
28
Properties of Johnson Noise
• due to random thermal motion of electric charge in conductors
• independent of current flow
• noise is random at all frequencies (up to ~1/170 fs), i.e. it is
“white”
Vn ,thermal  4kTBR {Volts},
where k  Boltzmann’ s constant, 1.38(10-23 ) J/K,
T  temperature (K),
B  bandwidth of system (Hz), and
R  resistance (Ohms).
• For example, a 1 kΩ resistor at a temperature of 300 K has
Johnson noise of
29
Bandwidth
• The equivalent bandwidth of a circuit is described by the
illustration below.
• In the case of an RC circuit, the bandwidth is given by:
B  bandwidth 
f 3dB
2


1

.
2(2RC ) 4 RC
dB
 P1 
P1
Note that, dB  10 log  , so  10 10 , and 3dB is half - power point.
P2
 P2 
30
Noise Sources: kTC Noise
31
kTC Noise
• due to random thermal motion of electric charge in conductors,
just like Johnson noise in resistors
• act of “resetting” capacitor freezes in random fluctuation of
charge
• can be removed noiselessly through subtraction
Vn ,kTC  kT / C {Volts},
where k  Boltzmann’ s constant, 1.38(10-23 ) J/K,
T  temperature (K), and
C  capacitance.
Qn ,kTC 
CVn ,kTC
e
C kT / C
kTC


{# e}.
e
e
32
kTC Noise as Reset Noise
33
Removing Reset Noise with CDS
• The reset noise adds an upredictable offset voltage to the
signal.
• This offset can be removed by using correlated double
sampling.
34
Noise Sources: 1/f Noise
35
1/f noise
•
•
•
•
•
spectral density increases for lower frequencies
sometimes called “flicker noise”
caused by traps, often near surface interfaces
occurs in most devices
especially pronounced in FETs with small channels
2
1/ f
V

Kf
wLC ox
f
{V 2 }, where
f
K f is strongly dependent on technology, and
is typically ~ 3(10-24 ) V 2 F,
f  bandwidth of FET,
Cox  capacitance of FET,
w  width of FET, and
L  length of FET.
36
Noise in Time Domain
white noise
1/f noise
37
Noise in a MOSFET
• Noise is a combination of thermal and 1/f noise, with the latter
dominating at low frequencies.
2
2
Vn ,1/ f
Vn , Johnson
Vn


f
f
f
2
4kT
K 1


.
g m wlC ox f
38
Noise Sources: Popcorn Noise
39
Popcorn Noise
• This is the minimum noise you can hear in a movie theatre during a tense
scene.
• This is noise produced by traps in FET channels that temporarily change
the properties of the channel.
• The summation of traps is thought to be a potential source of 1/f noise.
40
Read Noise
41
Read Noise
• Read noise is produced by all the electronics in a detector
system, e.g.
–
–
–
–
Johnson noise,
1/f noise,
electronic current shot noise,
unstable power supplies
• It is usually measured as the standard deviation in a sample of
multiple reads taken with minimum exposure time and under
dark conditions.
42
Read Noise and SNR Example
Fig. 13 - Text superposed on a tonal gradient, with superposed read noise. The contrast
difference between text and background rises linearly across the image,
while the noise stays constant; thus the S/N ratio rises linearly from zero to eight going from left
to right.
43
Noise Sources: Electronic Crosstalk
44
Electronic Crosstalk
signal
Φ1
ΦR1
Φ2
ΦR2
Φ3
ΦR3
45
Clocking Feedthrough (Crosstalk)
• read time, 10.18 us
• wait time for convert, 8.7 us
• bandwidth ~ 160 kHz
46
Noise Sources: Pickup or Interference
47
Pickup or Interference
• This noise is produced by ambient electromagnetic fields from
nearby radiators.
• Interference doesn’t have to be periodic, but it often is.
• As an example, consider the image below which shows
~20kHz interference pattern in a small section of a full 2Kx2K
detector readout.
48
Improving SNR
49
SNR
• SNR can be improved by maximizing the numerator and/or
minimizing the denominator of the full SNR equation.
• The choice of what to optimize often comes down to money.
That is, some things are expensive to improve and some are
not.
• For instance, the background flux can be reduced in
astrophysics applications by launching the system into space
(for the cost of billions of dollars….).

F tQE
S
h
SNR  
.
N



 

2

A
F
tQE


A
F
tQE

i
t

 inst



N
read


inst
back ,

dark
h
h

 

inst A
50
Improving SNR
• Optical effects
– Throughput: bigger aperture, anti-reflection coatings
– Background: low scatter materials, cooling
• Detector effects
– Dark current: high purity material, low surface leakage
– Read Noise: multiple sampling, in-pixel digitization, photon-counting
– QE: thickness optimization, anti-reflection coatings, depleted
• Atmospheric effects
–
–
–
–
Atmospheric absorption: higher altitude
OH emission: OH suppression instruments
Turbulence: adaptive optics
Ultimate “fix” is to go to space!
51
Increasing Integration Time/Coadds
• Signal increases with exposure time. Noise can also increase,
but not by as much.
• “Coadding” is summing individual exposures – similar to
increasing exposure time.

F tQE
S
h
SNR  
.
N



 


A
F
tQE


A
Fback , tQE   idark t  N 2read
 inst

   inst
h
h

 

inst A

F tQE  t.
h

inst A
F tQE
noise
h
SNR read


 t.
N read
noise
SNR shot

 inst A
52
Read noise vs. Shot Noise Limited Case
LOG(S/N)
4
3
slope=1/2 (flux dominated)
2
slope=1 (read noise limited)
1
0
1
2
LOG(time)
3
4
53
Fowler Sampling
• Fowler sampling uses the averages of groups of nondestructive reads at the beginning and end of exposure.
• This sampling mode generally reduces random noise by the
square root of the number of reads.
http://www.stsci.edu/hst/wfc3/documents/ISRs/WFC3-2007-12.pdf
54
Improving SNR: multiple sampling
55
Up-the-ramp Sampling
• In up-the-ramp sampling, the signal is non-destructively read
out many times during an exposure.
• This read mode generally reduces random noise by the square
root of the number of reads.
• The math can be more difficult than for Fowler sampling.
• This mode is potentially good for removing cosmic rays.
56
Read Noise Limited Case
• In the read noise limited case, SNR can be improved through
multiple non-desctructive reads.
• However, some amount of exposure time is lost by performing
many reads, so there is a tradeoff that depends on how long it
takes to do a read.
57
Background Limited Case
• In the background limited case, it does not help to obtain
multiple samples. It just wastes time and reduces exposure
time.
58
Sampling Mode Summary: CDS
59
Sampling Mode Summary: Up-the-ramp
60
Sampling Mode Summary: Fowler
61
Other Electronic Techniques
• bandwidth limiting filters
• increasing gain (before noise is injected)
• reducing the unit cell capacitance (thereby increasing in-pixel
gain).
62