Lecture Notes 7 Fixed Pattern Noise Definition Sources of FPN Analysis of FPN in PPS and APS Total Noise Model Correlated Double Sampling EE 392B: Fixed Pattern Noise 7-1
Fixed Pattern Noise (FPN) FPN (also called nonuniformity) isthespatialvariationinpixeloutput values under uniform illumination due to device and interconnect parameter variations (mismatches) across the sensor It is fixed for a given sensor, but varies from sensor to sensor, so if v o is the nominal pixel output value (at unifrom illumination), and the output pixel values (excluding temporal noise) from the sensor are v ij for 1 i n and 1 j m, thenthefixedpatternnoiseisthesetofvalues v oij = v oij v o FPN consists of offset and gain components increases with illumination, but causes more degradation in image quality at low illumination FPN for CCD image sensors appears random CMOS (PPS and APS) sensors have higher FPN than CCDs and suffer from column FPN, which appears as stripes in the image and can result in significant image quality degradation EE 392B: Fixed Pattern Noise 7-2
FPN Images For CCD sensor For CMOS sensor EE 392B: Fixed Pattern Noise 7-3
Sources of FPN CCD image sensors only suffer from pixel FPN due to spatial variation in photodetector device parameters and dark current neither the CCDs nor the output amplifier (which is shared by all pixels) cause FPN (additional nonuniformity can result if more than one output amplifier is used, however) In CMOS image sensors pixel transistors cause additional pixel FPN and column amplifiers cause column FPN. As a result FPN is in general higher than in CCDs EE 392B: Fixed Pattern Noise 7-4
Main sources of FPN in PPS: i dc A D v T,C ol C f v o v REF + v op os Pixel FPN is mainly due to the variation in the photodetector parameters (e.g., area A D )anddarkcurrent Column FPN is due to the variation in the column amplifier parameters, e.g., offset voltage vos, op feedbackcapacitorvalue,reset transistor threshold voltage and overlap capacitance value C ol EE 392B: Fixed Pattern Noise 7-5
In APS v DD v DD v TR, C olr i dc v TF, W F L F A D C D v o i bias In addition to variation in the photodetector parameters and dark current, pixel FPN is caused also by variations in transistor parameters Column FPN is mainly due to variation i bias EE 392B: Fixed Pattern Noise 7-6
PPS and APS FPN APS suffers from higher pixel FPN than PPS but PPS generally suffers from higher column FPN PPS APS EE 392B: Fixed Pattern Noise 7-7
Quantifying FPN FPN is quantified by the standard deviation of the spatial variation in pixel outputs under uniform illumination (not including temporal noise). It is typically reported as a % of voltage swing (or well capacity) FPN standard deviation values of < 0.1% to > 4% of well capacity have been reported Experimentally, FPN is measured as follows: Set a constant uniform illumination level (including no illumination) Take many images For each pixel compute the average output value (to average out temporal noise) Estimate the standard deviation of the average pixel values Repeat the procedure for several uniform illumination levels EE 392B: Fixed Pattern Noise 7-8
Analysis of FPN Suppose we are given the standard deviation of each parameter that casues FPN, we now show how to compute its contribution to the total FPN Assume the parameter values to be random variables Z 1,Z 2,..., Z k expressed as Z i = z i + Z i, where z i is the mean of Z i (i.e., nominal value of the device parameter) and Z i is the variation of Z i from its mean, and has zero mean and standard deviation σ Zi Assuming sufficiently small device parameter variations, we can approximate the pixel output voltage (for a given illumination) as a function of the device parameters using the Taylor series expansion, as V o (Z 1,Z 2,...,Z k ) v o (z 1,z 2,...z k )+ k i=1 v o z i z1,z 2,...z k Z i EE 392B: Fixed Pattern Noise 7-9
where v o (z 1,z 2,...z k ) is the nominal output voltage and v o / z i is the sensitivity of v o w.r.t. the ith parameter (evaluated at the nominal parameter values) So the variation in V o can be represented by the random variable k v o z1 V o = Z i z i,z 2,...z k i=1 To quantify FPN, we find the standard deviation of the output voltage, σ Vo, i.e., the standard deviation of the r.v. V o Assuming that the Z i sareuncorrelated(maynotbeagoodassumption in general), we can write σ Vo = k ( vo ) 2 z1 σz 2 z i,z 2,...z i k i=1 EE 392B: Fixed Pattern Noise 7-10
Column and Pixel FPN For a CMOS (PPS or APS) image sensor, let the column device parameters be Z 1, Z 2,..., Z l and the rest be the pixel device parameters, we can define the column variation as and the pixel variation as Y = X = l i=1 k i=l+1 v o z i z1,z 2,...z k Z i v o z i z1,z 2,...z k Z i We quantify column FPN by σ Y and pixel FPN by σ X (vary with illumination) Since (by assumption) X and Y are uncorrelated σ 2 V o = σ 2 Y + σ 2 X EE 392B: Fixed Pattern Noise 7-11
Offset and Gain FPN The pixel output voltage v o and FPN σ Vo vary with illumination The nominal output voltage from a pixel can be expressed in terms of the photocurrent density as v o = hj ph + v os where h is the pixel gain in V cm 2 /A (not to be confused with sensor conversion gain g) andv os is the pixel offset (which includes the dark signal as well as the offset voltages due to the amplifiers used, e.g., v op os for PPS) Assuming all photodetectors have the same QE, and thus under uniform illumination, they have the same photocurrent density, we can now write the pixel output voltage variation as V o = ( k i=1 h z i z1,z 2,...z k Z i = Hj ph + V os ) j ph + ( k i=1 v os z i z1,z 2,...z k Z i ) EE 392B: Fixed Pattern Noise 7-12
We quantify offset FPN by σ Vos and gain FPN by σ H j ph Offest FPN is reported as % of well capacity Gain FPN is referred to as Pixel Response Nonuniformity (PRNU) and is reported as % of gain factor variation, i.e., 100σ H /h Note that H and V os are not necessarily uncorrelated, since some device parameters can affect both offset and gain EE 392B: Fixed Pattern Noise 7-13
Analysis of FPN in PPS The figure shows the device parameters considered i dc A D vt,c ol C f v o v REF + v op os A D is the photodiode area, i dc is its dark current, v op os is the opamp offset voltage, C ol is the overlap capacitance, and v T is the threshold voltage EE 392B: Fixed Pattern Noise 7-14
The output voltage in steady state is given by 1 v o =(Q + C ol v T ) + v REF + v C os, op f where C ol v T is the feedthrough charge (when the reset transistor is turned off), and the charge Q accumulated on the photodiode capacitance Q =(j ph A D + i dc )t int The following table lists the absolute values of the parameter senitivities and effect on FPN v o z i Parameter Sensitivity Effect on FPN A D t int C j ph f pixel/gain pixel/offset i dc v op C f t int C f os 1 column/offset column/offset v T C ol i dc t int +C ol v T Cf 2 + A Dt int Cf 2 C ol C f v T C f j ph column/gain column/offset column/offset EE 392B: Fixed Pattern Noise 7-15
Offset FPN σ Vos = ( t int σ idc C f ) 2 + σ 2 v op os + (( ) ) 2 ( ) 2 ( ) 2 i dc t int + C ol v T vt Col Cf 2 σ Cf + σ Col + σ vt C f C f Gain FPN σ H j ph = j ( ) ( ) 2 2 t int A D t int ph σ AD + C f Cf 2 σ Cf Pixel FPN Column FPN σ X = (jph t int C f σ AD ) 2 + ( ) 2 tint σ idc C f (( ) ) 2 σ Y = σ 2 vos op + i dc t int + C ol v T + A D j ph t int Cf 2 σ Cf + ( ) 2 ( ) 2 vt Col σ Col + σ vt C f C f Note that the FPN variance σv 2 o = σx 2 + σ2 Y can be written as the sum of three terms, a term that is independent of the signal, a term that increases linearly with the signal, and a term that increases quadratically with the signal EE 392B: Fixed Pattern Noise 7-16
Example Assume the following device parameter means, standard deviations, and that t int =30ms Parameter Mean σ Sensitivity A D 50µm 2 0.4%A D 15 10 3 j ph V/µm 2 i dc 5fA 2%i dc 1.5mV/fA vos op 0V 2mV 1 C f 20fF 0.2%C f 11.6 10 11 V/F 37500j ph V/fF v TR 0.8V 0.2%v TR 0.02 C ol 0.4fF 0.4%C ol 0.04V/fF EE 392B: Fixed Pattern Noise 7-17
Offset FPN and Parameter Contribution to σ Vos i dc 0.15 mv vos op 2 mv C f 0.0464 mv v TR 0.032 mv 0.064 mv C ol σ Vos 2mV, which is basically equal to the opamp offset σ v op os value Gain FPN at j ph =2.64 10 6 A/cm 2 (high illumination) Parameter Contribution to σ H j ph A D 7.92 mv 3.96 mv C f and σ H j ph =8.85mV EE 392B: Fixed Pattern Noise 7-18
The following figure plots total FPN σ Vo, pixel FPN σ X, and column FPN σ Y,assumingmonochromaticilluminationF 0 photons/cm 2.s at quantum efficiency QE =0.3 10 9 Pixel FPN Column FPN Total FPN 8 7 6 FPN (mv) 5 4 3 2 1 0 10 12 10 13 10 14 illumination F o (photon/cm 2 s) EE 392B: Fixed Pattern Noise 7-19
Analysis of FPN in APS v DD v DD v TR, C olr i dc v TF, W F L F A D C D v o i bias In steady state and assuming soft reset, the output voltage is given by ( v o = v DD v TR Q ) 2LF v TF + i bias, C D k n W F where the charge accumulted on the photodiode is given by Q =(A D j ph + i dc )t int + C olr v DD The C olr v DD term is the feedthrough charge (when the reset transistor is turned off) EE 392B: Fixed Pattern Noise 7-20
Example Consider the following parameter means and standard deviations parameter mean σ effect on FPN i dc 5fA 2%i dc pixel/offset A D 50µm 2 0.4%A D pixel/gain C D 20fF 0.4%C D pixel/offset,gain v TR 1.1V 0.2%v TR pixel/offset C olr 0.4fF 0.4%C olr pixel/pffset v TF 0.9V 0.2%v TF pixel/offset W F L F 4 2 0.2% W F L F pixel/offset i bias 1.88µA 1%i bias column/offset You will compute the FPN component values in the homework EE 392B: Fixed Pattern Noise 7-21
Image Sensor Total Noise Model Combining temporal noise and FPN, we can express the total input referred noise charge as where Q n = Q shot + Q reset + Q readout + Q fpn, Q shot is the r.v. representing the noise charge due to photodetector photo and dark current shot noise and is Gaussian with zero mean and variance 1 q (i ph + i dc )t int electrons 2 Q reset is the r.v. representing the reset noise and is basically independent of the signal Q readout is the r.v. representing the readout circuit noise (possibly including quantization) and is basically independent of the signal EE 392B: Fixed Pattern Noise 7-22
Q fpn is the r.v. representing FPN (in electrons), and can be represented either as a sum of pixel and column components Q fpn = 1 g (X + Y ) where g is the sensor conversion gain in V/electron, or offset and gain components Q fpn = 1 g ( Hj ph + V os ) Thus it has one component that is independent of signal and one that grows with the signal The noise components are assumed independent Thus the total average noise power is the sum of three components: One that does not depend on the signal (due to reset and readout noise and offset FPN) One that increases linearly with the signal (i ph or j ph )(duetoshot noise and gain FPN) One that increases quadratically with the signal (due to gain FPN) EE 392B: Fixed Pattern Noise 7-23
Noise as Function of Photocurrent 10 3 Average Noise power (V 2 ) 10 4 10 5 10 6 10 7 10 16 10 15 10 14 10 13 10 12 10 11 i ph (A) EE 392B: Fixed Pattern Noise 7-24
Correlated Double Sampling (CDS) CDS is a multiple sampling technique commonly used in image sensors to reduce FPN, and reset and 1/f noise You sample the output twice; once right after reset and a second time with the signal present. The output signal is the difference between the two samples CDS only reduces offset FPN (does not reduce gain FPN) CDS does not cancel offset FPN due to dark current variation In CCDs, PPS, photogate and pinned diode APS, CDS cancels reset noise. In photodiode APS it increases it EE 392B: Fixed Pattern Noise 7-25
CDS in PPS Word Reset SS SR C os C or Reset SR Word, SS EE 392B: Fixed Pattern Noise 7-26
Cancells FPN due to vos, op v T,andC ol Temporal noise due to reset (terms Vo2 2 and V o3 2 Readout noise due to op-amp 1/f noise Does not cancel Adds in our analysis) Offset FPN due to i dc variation. This is called Dark Signal Non-uniformity (DSNU) Gain FPN (or PRNU) Other temporal noise components Opamp noise due to reset read (Vo4 2 term) noise due to SS and SR transistors KT C EE 392B: Fixed Pattern Noise 7-27
To summarize, the total noise charge for the two samples are given by: Q n1 = Q reset + Q read1 + Q fpn1 Q n2 = Q shot + Q reset + Q readout2 + Q fpn2 Note that Q fpn1 is simply an offset FPN whereas Q fpn2 is the sum of offset and gain FPN (PRNU). However, Q fpn1 does not include the offset FPN due to dark current variation (DSNU), whereas the offset part of Q fpn1 includes it The difference between the two samples is thus: Q n2 Q n1 = Q shot +(Q readout2 Q readout1 )+Q prnu + Q dsnu EE 392B: Fixed Pattern Noise 7-28
PPS FPN With and Without CDS The following figure plots PPS FPN with and without CDS (assuming that v op os, v T,andC ol are eliminated) 10 9 CDS w/o CDS 8 7 6 FPN (mv) 5 4 3 2 1 0 10 12 10 13 10 14 illumination F o (photon/cm 2 s) EE 392B: Fixed Pattern Noise 7-29
PPS Offset FPN With and Without CDS 10 10 20 20 30 30 40 40 50 50 60 60 10 20 30 40 50 60 without CDS 10 20 30 40 50 60 with CDS EE 392B: Fixed Pattern Noise 7-30
CDS in 3T APS Reset SS Word SR C os C or Word SS Reset SR EE 392B: Fixed Pattern Noise 7-31
Cancells All offset FPN terms involving v TR, v TF, C olr, W F L F,andi bias Does not cancel Adds DSNU Reset noise PRNU Readout noise Reset noise kt 2C D (reset noise component during reset readout independent of that during signal readout) Readout noise during reset readout kt C due to SS and SR transistors EE 392B: Fixed Pattern Noise 7-32
To summarize, the total noise charge for the two samples are given by: The difference is: Q n1 = Q shot + Q reset1 + Q readout1 + Q fpn1 Q n2 = Q reset2 + Q readout2 + Q fpn2 Q n1 Q n2 = Q shot +(Q reset1 Q reset2 )+(Q read1 Q read2 )+Q prnu + Q dsnu An important advantage of photogate and pinned diode APS is that reset noise is eliminated using CDS instead of doubled EE 392B: Fixed Pattern Noise 7-33