Statistical Modeling of SiPM Noise - GSI Indico …...S. Vinogradov, IEEE NSS/MIC 2009, TNS 2011,...

Statistical Modeling of SiPM Noise

Sergey Vinogradov

Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia

National Research Nuclear University «MEPhI», Moscow, Russia

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 1

Scope & outline

◙ Photon detection – a series of stochastic processes described by statistics

◙ SiPM response – a result of the stochastic processes – a random variable

◙ SiPM “noise” has a double meaning: as well as a “signal”

◙ Physics point of view – specific nuisance contributions to the response

◙ Dark counts - DCR

◙ Crosstalk - CT

◙ Afterpulsing – AP

◙ Statistics point of view – standard deviation of the response

◙ All above +…

◙ Multiplication – Gain

◙ Photo-conversion!!! – PDE

◙ Statistics of the response times

◙ Statistics of the response quantities

◙ Statistics of the response transients

◙

◙ Excess noise factor – ENF – as a Figure of Merit for all noise contributions


Definitions of statistics

◙ Statistics – wide sense – math related to random variables

◙ Random variable is fully described by its probability distribution Pr(X)

◙ Statistic – narrow sense – any function on probability distribution

Statistical moments nth order mn of random variable X:

― 𝑚𝑛(𝑋) = σ𝑖=0∞ 𝑖𝑛 Pr(𝑋 = 𝑖) X – discrete r.v.

― 𝑚𝑛(𝑋) = ∞−∞𝑥𝑛 Pr 𝑋 = 𝑥 𝑑𝑥 X – continuous r.v.

― first moment Mean (µ) = 𝑚1 ;

― second central moment – Variance (σ2) = 𝑚2 − 𝑚12

μ

σ

𝑷𝒓 𝑿 = 𝒙


Statistics of random times

◙ How probability distribution of signal and noise event times

are transformed by photon detection processes


Statistics of random times

https://doi.org/10.1016/j.nima.2015.07.009

Pr(∆t) or PDF– full characterization

Mean and Variance – not so obvious

because affected by recovery losses

and correlated events

~ Poisson affected by dead time

~ Poisson affected by afterpulsing

Pure Poisson – exponential distribution of ∆t

Poisson process:

No events in : Pr( | 0)

Interval between events : Pr( | 0) 1

Probability density function: ( )

Mean time 1/ Standard deviation ( ) 1/

DCR t

DCR t

DCR t

t t e

t e

PDF t DCR e

t DCR t t DCR

−

−

−

=

= −

=

= = =


Analysis based on probability density function (PDF)

= histogram of interarrival times

◙ Common-sense analysis of DCR and correlated events

◙ Contributions are fitted separately in specific time frames

P. Eckert et al, NIMA 2010

F. Acerbi et al, TNS 2016


Analysis based on cumulative distribution function (CDF)

= list of interarrival times

◙ Complimentary CDF approach: probabilities of independent “zero”

dark and “zero” correlated events in ∆t are multiplied:

( ) ( )

( )

.......

safterpulse andcrosstalk of separationfor

approach (C)CDF extendfurther can weMoreover,

shape itson sassumptionany without ),(or ),( analyse And

)exp(),,(11),(

: ),( events correlated ofon distributi pure

extract toallgot ve we'now So,

)exp(1)(),(1),(1),,(1

CDFary compliment1

APCTcorr

corrcorrcorrcorr

phtotalcorrcorr

corrcorr

dark

darkcorrcorrcorrtotal

darkcorrtotal

CCDFCCDFCCDF

PtfPtF

tDCRDCRNtFPtF

PtF

tDCRtFDCRtFPtFDCRPtF

CDFCCDFCCDFCCDFCCDF

=

−−=

−−=−−=−

−==


CCDF time distribution:

dark and correlated events

Pco

rr1

-P

co

rr

Experimental example – courtesy of E. Popova, D. Philippov… , NSS/MIC 2016

Correction on lost

dark counts at Tmax

should be applied:

)exp( max_ TDCRCCDF correctiondark −=


CCDF time distribution:

reconstruction of correlated event CDF

Pco

rr

Now we are ready to play with correlated

event CDF or PDF applying any models

and analysis.

For example, typical assumption on

exponential time distribution, but take care

about multiple exponents (see backup):

moreandeP

Ptf

moreandePPtF

corr

corr

t

corr

corrcorrcorrcorr

t

corrcorrcorrcorr

...),,(

...1),,(

−

−

=

−=

( ) )exp(),,(11),( tDCRDCRNtFPtF phtotalcorrcorr −−=

Experimental example – courtesy of E. Popova, D. Philippov , NSS/MIC 2016

More details – next talk by Eugen Engelmann @ ICASiPM


Empirical CDF (list mode) vs PDF (histogram):highest precision due to full information from all data points

◙

[1] Y. Cao, L. Petzold, “Accuracy limitations and the measurement of errors in the stochastic simulation…”, J. Comput. Phys. 212, 2006


◙ How probability distributions of signal and noise quantities

are transformed from input to output

Statistics of random quantities


◙ Pr(Nout|Nin) – full characterization

◙ Mean and Variance – partial characterization

Statistics of random quantities

Input photon distribution Photodetection processes Output charge distribution

Pr(Nin)

Pr(Nout)

Process i

Process j

)( inN

inN outN

)( outN

Photons (Poisson) photoconversion (Bernulli) photoelectrons (Poisson)

Dark+photoelectrons (Poisson) triggering (Bernulli) primary avalanches (Poisson)

Primary avalanches (Poisson) , (TBD) secondarCT AP

y avalanches (TBD)

All avalanches(TBD) multiplication (TBD) output electrons (TBD)


Correlated stochastic processes of CT & AP

Branching

Poisson

Process

Geometric

Chain Process

Poisson number of primaries <N>=μ

e.g. SiPM photoelectron spectrum

Single primary event N≡1

e.g. SiPM dark electron spectrum

Process

Models

Non-random (Dark) event

Primary

1st event

2nd event

No event

Ran

dom

AP

even

ts…

Ra

nd

om

CT

ev

ent

s

Random Photo eventsRandom primary (Photo) events

Ran

dom

AP

even

ts

…

…

…

Non-random (Dark) event

Ran

dom

CT

even

ts

…

… …

…

…

…

Random primary (Photo) events

Ran

dom

CT

even

ts

…

…

λµ

S. Vinogradov, IEEE NSS/MIC 2009, TNS 2011, NDIP 2011


Models Geometric chain process Branching Poisson process

Primary event

distribution

Non-random

single (N≡1)

Poisson (μ) Non-random single

(N≡1)

Poisson (μ)

Total event

distribution Geometric (p)

Compound

Poisson (μ, p) Borel (λ) Generalized Poisson (μ, λ)

P(X=k) ( )ppk −− 11

−

−

sp

s

eL 1

1~

( ) ( )!

exp1

k

kkk

−−

( ) ( )

!

exp1

k

kkk

−−+−

E[X] p−1

1

p−1

−1

1

−1

Var[X] 2)1( p

p

−

2)1(

)1(

p

p

−

+

( )31

−

( )31

−

ENF p+1 )...(2

31

)1ln(1

1

1

1 32 pppp

+++−+

=−

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Threshould [fired pixels]

Dark

Co

un

t R

ate

[H

z]

Experiment

Borel

Geometric

Analytical results for CT & AP statistics

Pct=40%

Pct=10%

S. Vinogradov, NDIP 2011 (experiment – R. Mirzoyan, 2008 (MEPhI SiPM)) S. Vinogradov, NDIP 2011 (experiment – LPI, Hamamatsu MPPC)

0%

10%

20%

30%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

k, number of events

Pro

bab

ilit

y o

f k e

ven

ts

p=0 L=3

p=0.2 L=3

p=0.5 L=3

Crosstalk


Statistics of transient signals - stochastic functions

◙ How random quantities are developed in time


Statistics of filtered marked correlated point processes

Signal IntensityDark noise,

Background

Dark eventCorrelated eventDark event

Single Electron Response (IRF)

Random Gain (amplitude)

False signal detectionDiscriminator

True signal detection

Signal events


Mean and Variance of

filtered marked Poisson point process

dt

thdN

V

VthENFN

dt

tVoutd

tVVar

VthENFVtVVar

tPDENtRGainq

VthVtVoutE

specificSIPM

tXVartttXCOVREMARK

thA

AttIRFCOVXCOVtYVar

tdtNthAtIRFEtXEtYE

NiiidrvAthAtIRFttIRFtY

NiiidrvtPoissonianNtttX

sersptrph

pe

ser

noisesersptrphgainpe

out

t

noisegainserout

sptrphph

fall

loadserser

ttinin

A

ttinout

t

inout

ii

N

i

iout

i

N

i

iin

)(

)(

)(

)]([

)()]([

)()()()]([

)]([)()(][:

)(1)]([][)]([

)()()]([)]([)]([

)...1()()()()(

)...1()()(

2

22

222

2

2

22

0

1

1

+

==

+=

=

=

→−=

+=−=

===

=−=−=

=−−−=

→

=

=

=

/ /

2

Example: baseline fluctuation due to DCR

( ) ( )

Var =V2

fall riset t

fall riseserout

t DCR const h t e e

V ENF DCR

− −= = = −

+


Figure of Merit for all noise contributions

◙ How to compare noises and optimize SiPM operations:

Excess Noise Factor – ENF – approach


ENF: two definitions

◙ ENF in multiplying photodetectors (APD, PMT):

◙ ENF in amplifying electronics:

Noise factor (F) measures degradation of the signal-to-noise ratio (SNR), caused by components in a signal chain.

2

2

22

1

( )1

( )

Mm MENF

m M M

= = +

2 22 2

2 2

in outout in

in out

ENF Res Res

= =


https://en.wikipedia.org/wiki/Signal-to-noise_ratio

https://en.wikipedia.org/wiki/Signal_chain_(signal_processing_chain)

ENF: two definitions are equal for Poisson input


2 2

2 2

1( 1) 1 ( ) 1

( )

Input quantity N Poisson ( ) ( 1)

M Min in

in in

ENF X vs ENF X NFano NM M

ENF X N ENF X

= + = = +

= =

Resolution at output is a product of Resolution at input and √ENF(Xin≡1)

Total ENF for a sequence of specific processes is approximately

ENF as a measure of SNR (resolution) degradation

)1()(

1)()(

1

)1(

)1(1)()(

2

2

=

=

+= inin

in

inout

outinout XENFXRES

XFanoXFanoY

XRESYRES

...)()()( 21 processprocessintotalinout ENFENFXRESENFXRESYRES ==

Input photon distribution Photodetection processes Output charge distribution

RES(Xin)

RES(Yout)= RES(Xin)√ENF

ENFi

ENFj


ENF related to Gain, PDE and DCR

◙ Multiplication (single electron)

◙ Photon detection (single photon)

◙ Dark counts (Poisson dark+pe)

pedcr N

tDCRENF += 1

PDEPDE

PDEPDEENFpde

1)1(1

2=

−+=

SiPM ofmost for 05.1...01.1

12

2

+=

gain

gain

gain

ENF

GainENF

Xin=1

PDF(Yout) ?

PDF(Xin)=δ(x)

<Yout>=Gain

Photoelectrons

RES(Yout)

Dark counts

Xin=1

PDF(Yout) = BernoulliPDF(Xin)=δ(x)

<Yout>=PDE

<Yout>=Npe+Nd


Pr(0)

Methods of ENF measurements

1

10

100

1000

10000

100000

1000000

10000000

0 1 2 3 4 5 6 7 8 9 10

Amplitude [Single Electron Response]

Eve

nts

Crosstalk amplitude histogram

N=1

N=1 non-random primary

_________________________________

N ϵ Poisson random primaries

Nph could also be measured by reference PD

22 2

2 2 2( )

ln(Pr(0))

outin outin

in out out

ENF X N N

N

= = =

= −

2

2( 1) 1 out

in

out

ENF X

= +

Remark: ENF measurements in electronics: noise generator + noise power meter


σ/µ

σ/µ

Key SiPM parameters in ENF metrics

Noise source Key parameter Typical values for

SiPM ENF expression

SiPM

ENF

PMT

ENF

Fluctuation of

multiplication

Mean gain and standard

deviation of gain

μ(gain) ~ 106

σ(gain) ~ 105 )(

)(1

2

2

gain

gainFm

+= 1.01 1.2

Crosstalk Probability of crosstalk

event Pct 5% – 40%

1

1 ln(1 )Fct

Pct=

+ − 1.05 – 2 1

Afterpulsing Probability of afterpulse

Pap 5% – 20% 1Fap Pap= + 1.05 – 1.2 1.01

Shot noise of dark

counts

Dark count rate DCR

Signal events in Tpulse 105 – 106 cps 1

DCR TpulseFdcr

Nph PDE

= +

1.01 1

Noise / losses of

photon detections

Photon detection

efficiency PDE 20% – 50%

1Fpde

PDE= 2 – 5 3 – 5

Total noise in linear

dynamic range Total ENF_linear _ENF linear Fm Fct Fap Fdcr Fpde= 3 – 12 4 – 6

Binomial

nonlinearity

Mean number of possible

single pixel firings n

exp( ) 1; ; binomial distribution

Nph PDE nn ENFpix

Npix n

−= =

Dead time

nonlinearity

Mean rate of possible

single pixel firings λ ; 1 ; nonparalizible model

nENFrec Trec

Tpulse = = +

total

nlapctdcrpdemtotaltotal

ph

ph

out

out

ENFDQEFFFFFFENFENF

N

N

N

NPNR

1

)(

)(

)(

)(

*

* ===


The end

Thank you for your attention!

Questions?

Objections?

Opinions?

…

[email protected]


FBK results on Generalized Poisson model

A. Gola, A. Ferri, A. Tarolli, N. Zorzi, and C. Piemonte, “SiPM optical crosstalk amplification due to scintillator crystal: effects on timing

performance,” Phys. Med. Biol., vol. 59, no. 13, p. 3615, 2014.

Sergey Vinogradov LIGHT – 2017 19 October 2017 26

DESY results on Generalized Poisson

◙ E. Garutti group @ DESY: V. Chmill et al., NIMA 2017

Sergey Vinogradov LIGHT – 2017 19 October 2017 27

Statistical Modeling of SiPM Noise - GSI Indico …...S. Vinogradov, IEEE NSS/MIC 2009, TNS 2011,...

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