CMOS Image Sensor Design_h20_6_snr_model.pdf

in5350_h20_6_snr_model

1. INF5350 –CMOS image sensor design
Lecture 6 – Modelling S/N ratio
22-September-2020

2. Agenda
•Project milestone status
•Takeaways from previous lecture&exercises
•Modelling S/N ratio
21.09.2020 IN5350 2

3. Project schedule
Task/milestone Start Finish
Chose topic/scope 1-Sep 8-Sep
Create project plan (tasks, milestones, schedule) 8-Sep 15-Sep
MS1 –project plan approved by Johannes 15-Sep 22-Sep
Study literature on the topic 22-Sep 29-Sep
Design/simulation 29-Sep 13-Oct
Write up prelim report (increferences, design, results)13-Oct 20-Oct
MS2 –submit preliminary report to Johannes 20-Oct 20-Oct
Design/simulation 20-Oct 27-Oct
Write up final report (inclreferences, design, results)27-Oct 3-Nov
MS3 –presentation and discussion 3-Nov 3-Nov
MS4 –submit final report to Johannes 10-Nov 10-Nov
Exam 18-Nov 2020
✅
21.09.2020 IN5350 3
✅
✅

4. Key takeaways from previous lecture
•Temporal noise vs Fixed pattern noise
•Photon flux and electron flux are random Poisson
processes
–Mean value (μ) = Variance (σ
2
) = (RMS value, σ)
2
•
Independent noise sources can be summed
quadratically
σ
total= sqrt(σ
1
2+ σ
2
2+ σ
3
2+ ..)
21.09.2020 IN5350 4

5. SNR model based on σfor each block
•How to model σfor pixels, gain amplifiers, filters,
ADCs, etc?
21.09.2020 IN5350 5
CG
fd,
σ
kTC
G
sf,
σ
sf
G
adc,
σ
adc
S
in/σ
in S
out/σ
out
????????????
????????????????????????????????????/????????????
????????????????????????????????????=
????????????
????????????−????????????????????????
????????????????????????????????????
????????????????????????????????????
????????????????????????????????????
????????????
????????????−????????????????????????
????????????????????????
2
????????????
????????????????????????
2
????????????
????????????????????????????????????
2
+????????????
????????????????????????????????????
2
????????????
????????????????????????
2
????????????
????????????????????????????????????
2
+????????????
????????????????????????
2
????????????
????????????????????????????????????
2
+????????????
????????????????????????????????????
2
Floating
diffusion
Source
follower
A/D
converter

6. Modelling noise in frequency domain
•N
in(f): noise power density at input (V
2
/Hz)
•N
out(f): noise power density at output (V
2
/Hz)
•H(f): transfer function of a linear time- invariant system, e.g. pixel,
amplifier, filter, ADC, CDS, ..
21/09/2020 IN5350 6
dffHfNdffN
inoutout
2
00 )()()( ⋅==∫∫
∞∞
σ
H(f)N
in(f) N
out(f) (V
2
/Hz)
(V rms)

7. Noise spectral density at pixel SF output
•N
pix(f) = k
1+ k
2/f + k
3/f
2
(V
2
/ Hz)
21/09/2020 IN5350 7
Measured noise spectrum at the output of a CCD source follower
at -40C. CMOS source followers look similar. White noise (k
1)
dominates at high frequencies, typically above 1- 10MHz.
N
pix(f)

8. Digital CDS model
21/09/2020 IN5350 8
ℎ(????????????)=
2
????????????
????????????????????????????????????−1
????????????
????????????????????????????????????
{????????????
????????????–????????????????????????−????????????
????????????????????????????????????}
Low-pass filter
Sample&Hold
A/D
ALU
V
ref
Signal, V
pix
Noise, N
pix
Signal, ∆V
Noise, σ
ℎ????????????=1−????????????
−�
????????????
????????????????????????
δ(t) = delta-diracfunctionmodelstheADC input signal sampling
N
bit= number of ADC bits (typically 10b- 12b)
V
ref= ADC reference voltage (typically 1V at 1x gain)
T
CDS= time interval between sampling of reset level and signal level

9. Fourier transfer of DCDS process
21/09/2020 IN5350 9
ℎ????????????=(????????????????????????−????????????(????????????−????????????
????????????????????????????????????))
????????????????????????????????????=(1−????????????
−????????????????????????????????????
????????????????????????????????????)(????????????=2????????????????????????)
????????????????????????????????????
2
=????????????????????????????????????�????????????−????????????????????????
????????????????????????????????????
2
=(1−????????????
−????????????????????????????????????
????????????????????????????????????)�(1−????????????
+????????????????????????????????????
????????????????????????????????????)
????????????????????????????????????
2
=(2−2�cos????????????????????????
????????????????????????????????????)=4�????????????????????????????????????
2
(
????????????????????????
????????????????????????????????????
2
)
(i.e. Vrst-Vsig)

10. DCDS transfer function example
21.09.2020 IN5350 10
????????????????????????????????????
2
=4�????????????????????????????????????
2
(
????????????????????????
????????????????????????????????????
2
)
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
1 10 100 1.000 10.000 100.0001.000.00010.000.000
|H(f)|^2
freq (Hz)
Tcds=5us
Low frequency
signals & noise
attenuated

11. DCDS with LP filter for noise limitation
21/09/2020 IN5350 11
ℎ(????????????)
=
2
????????????
????????????????????????????????????−1
????????????
????????????????????????????????????
{????????????(????????????)–????????????(????????????−????????????
????????????????????????????????????)}
Input
A/D and digital CDS
Outputℎ????????????=1−????????????
−�
????????????
????????????????????????
Low-pass filter
????????????(????????????)
2
=
1
1+????????????/????????????
0
2
????????????(????????????)
2
=4�
2
????????????
−1
????????????
????????????????????????????????????
�????????????????????????????????????
2
(
????????????????????????
????????????????????????????????????
2
)

12. DCDS w/LP filter transfer function example
21/09/2020 IN5350 12
Roll-off due to
low-pass filter
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
1 10 100 1.000 10.000 100.000 1.000.00010.000.000
|H(f)|^2
freq (Hz)
Tcds=5us
fc = 5MHz
????????????????????????????????????
2
=4�????????????????????????????????????
2
(
????????????????????????
????????????????????????????????????
2
)�
1
1+????????????/????????????
0
2

13. Output noise after digital CDS
21.09.2020 IN5350 13
????????????
2
=�
0
∞
????????????
????????????????????????????????????(????????????)
1+
????????????
????????????
????????????
2
�
????????????(????????????)
2
�????????????????????????
(LSB
2
)
????????????
2
=�
0
∞
????????????
????????????????????????????????????(????????????)
1+
????????????
????????????
????????????
2
�2
????????????
????????????????????????????????????−1
????????????
????????????????????????????????????
�4�????????????????????????????????????
2
(
????????????????????????
????????????????????????????????????
2
)�????????????????????????
????????????
2
=4�
2
????????????
????????????????????????????????????−1
????????????
????????????????????????????????????
��
0
∞
????????????
????????????????????????????????????(????????????)
1+
????????????
????????????
????????????
2
�????????????????????????????????????
2
(????????????????????????
????????????????????????????????????
2
)�????????????????????????

14. Correlated Multi-Sampling (CMS)
•Sample Vrstand Vsigtwice and take average to
reduce noise (need twice as fast ADCs)
21.09.2020 IN5350 14
Low-pass filter
A/D
ALU
ℎ????????????=1−????????????
−�
????????????
????????????????????????
ℎ(????????????)=
2
????????????
−1
????????????
????????????????????????????????????
{????????????????????????+????????????(????????????−????????????)–????????????(????????????−????????????
????????????????????????????????????)–????????????(????????????−????????????−????????????
????????????????????????????????????)}/2
CDS

15. CMS S/N model
21.09.2020 IN5350 15
ℎ(????????????)=
2
????????????
−1
????????????
????????????????????????????????????
{????????????????????????+????????????(????????????−????????????)–????????????(????????????−????????????
????????????????????????????????????)–????????????(????????????−????????????−????????????
????????????????????????????????????)}/2
????????????????????????????????????=
2
????????????
−1
2�????????????
????????????????????????????????????
1+????????????
−????????????????????????????????????
−????????????
−????????????????????????????????????
????????????????????????????????????−????????????
−????????????????????????(????????????+????????????
????????????????????????????????????)
????????????????????????????????????
2
=
2
????????????
−1
2�????????????
????????????????????????????????????
{1+cos(????????????????????????)−cos(????????????????????????
????????????????????????????????????)−
1
2
�????????????????????????????????????
????????????????????????
????????????????????????????????????−????????????−
1
2
�????????????????????????????????????
????????????????????????
????????????????????????????????????+????????????}
????????????
2
=�
0
∞
????????????
????????????????????????????????????(????????????)
1+
????????????
????????????
????????????
2
�
????????????(????????????)
2
�????????????????????????
(LSB
2
)

16. Example of Poisson distributed random
values (e.g. photons and dark current)
21.09.2020 IN5350 16
Mean(A) = 2 => sigma(A) = 1.4
Mean(B) = 10 => sigma(B) = 3.2
Mean(A+B) = 12 => sigma(A+B) = 3.5

17. Noise from color crosstalk
•R’=a
1R+a
2G+a
3B e.g. 1.35R-0.2G-0.15B
•G’=b
1R+b
2G+b
3B e.g. -0.05R+1.1G-0.05B
•B’=c
1R+c
2G+c
3B e.g. -0.05R-0.15G+1.2B
21.09.2020 IN5350 17
Matrix ‘A’ = Color correction matrix (CCM)
3x3 matrix with fixed values stored in non-
volatile memory
Ref to green
crosstalk, b
1
Blue to green
crosstalk, b
3

18. Noise from color crosstalk (cont.)
•Green signal, G’ = b
1R + b
2G + b
3B
•Green noise, σ
G’
2= b
1
2σ
R
2 + b
2
2σ
G
2 + b
3
2σ
B
2
•Assuming R=G=B=S
e->>RN (photon noise limited)
•The larger the CCM values, the lower the SNR.
(same for blue and red pixels.)
•NB! Above assumes b
1+b
2+b
3=1 (unchanged
brightness)
21.09.2020 IN5350 18
)()(
)(
2
3
2
2
2
1
2
3
2
2
2
1
321
bbb
S
bbbS
bbbS
SNR
e
e
e
++
=
++
++
=
−
−
−

19. Photoresponsenon-uniformity
•Fixed pattern noise
•Several potential root causes
–Quantum efficiency variation due to manufacturing spread in the optical stack and/or
in the PD implants
–Conversion gain (CG) variation due to parasitic capacitance variation
–Source follower gain variation due to Vth variation in active device
•PRNU (in e-rms, V rms, or LSB rms) is proportional to the signal level
•Therefore, it is usually quoted in terms of % rmsof signal level
•Let’s assume a signal level of 500e
-
and 1% rmsPRNU. Then,
21/09/2020 IN5350 19
????????????
????????????????????????????????????????????????=500????????????
−
�0.01=5????????????
−
????????????????????????????????????

20. SNR model w/FPN sources
21.09.2020 IN5350 20
G
asc,
σ
asc
S
in/σ
in S
out/σ
out
????????????
????????????????????????????????????/????????????
????????????????????????????????????=
????????????
????????????????????????
????????????????????????????????????
????????????
????????????????????????
????????????????????????????????????
2
+????????????
????????????
2
????????????
????????????????????????????????????????????????
2
????????????
????????????????????????????????????
2
+????????????
????????????????????????????????????????????????
2
????????????
????????????????????????????????????
2
+????????????
????????????????????????????????????????????????
2
S
e: number of photo- electrons generated by photodiode
S
out: ADC output signal (DN or LSBs)
σ
PRNU: photoresponsenon-uniformity (% rms )
σ
DSNU: dark signal non- uniformity (e- rms)
σ
vfpn: rmsnoise from vertical column FPN (LSB rms )
G
asc: conversion factor of analog signal chain, (LSB/e- )
CG
fd: floating diffusion conversion gain (V/e- )
If S
e>>1 => SNR=1/σ
PRNU

21. Example noise curve
21.09.2020 IN5350 21
1
10
100
1000
1 10 100 1.000 10.000100.000
Noise (e -rms)
Light level (e-)
noise
(e- rms)
Photon shot noise
dominated
PRNU dominated
RN=3e-

22. Example SNR plot
21.09.2020 IN5350 22
-20
-10
0
10
20
30
40
50
1 10 100 1.000 10.000 100.000
SNR (dB)
Light level (e-)
SNR (dB)
Photon shot
noise dominated
PRNU dominated

23. PRNU dominates over photon shot noise at high
signal levels
•PRNU for most sensors is <1% rms
•Negligible at small signal levels (small FW) but becomes
dominant at high signal levels
21.09.2020 IN5350 23
0,00
1,00
2,00
3,00
4,00
5,00
6,00
0 0,2 0,4 0,6 0,8 1
Noise ratio (no unit)
PRNU (% rms)
PRNU/PN @1K
PRNU/PN @10K
PRNU/PN @100K
PRNU/PN @1M
Linear (PRNU/PN @1K)
Linear (PRNU/PN @10K)
Linear (PRNU/PN @100K)
Linear (PRNU/PN @1M)

24. Example of picture with random noise
Possible noise
sources in this
picture:
•Photon noise in
‘grey’ medium
bright areas
•Dark current (DC)
in dark areas
•Electronic circuit
noise (read noise)
in dark areas
•Photoresponse-
non-uniformity
(PRNU) in bright
areas
21.09.2020 IN5350 24

25. Vertical fixed pattern noise (VFPN)
21.09.2020 IN5350 25
Example of column FPN (vertical FPN)
22222222
VFPNRNGGCGNGGCGS
adcsfpixdcadcsfpixeADCout +++=
−
σ
When VFPN is randomly distributed across columns, it can be
modelled similarly to temporal noise:

26. Signal, Noise and Dynamic Range
IN535021.09.2020 26
Source: Nakamura et.al.

27. Key takeaways
•Noise sources added together in power domain or
variance (ie V
2
, I
2
, or σ
2
)
•CDS eliminates in-pixel kTC-noise and strongly
attenuates low-frequency (1/f) noise
•The larger the color correction matrix (CCM)
values, the more shot noise is added to the pixel
•FPN is calculated after removing (averaging)
temporal noise. Thus, FPN relates to the image,
not a single pixel.
21.09.2020 IN5350 27

28. APPENDIX
21.09.2020 IN5350 28

29. Random variables
•Discrete random variables modelled as probability mass
function
–x is integer (example: number of photons or electrons in a Poisson
process)
•A continuous random variable (x real, e.g. analogue
voltage) modelled as probability density function
–x is real (example: analogue voltage in a Gaussian process)
21.09.2020 IN5350 29
�
????????????
????????????????????????=1????????????????????????≥0
????????????x≥0,�
−∞
+∞
????????????
????????????????????????????????????=1

30. Gaussian random variable
21.09.2020 IN5350 30
????????????????????????=
1
2????????????????????????
2
????????????
−
(????????????−????????????)
2
2????????????
2
f(x)
σ