An approach for the S-curve fit I.Tsurin CERN 12.06.08 Page 1 ATLAS Pixel Calibration meeting 1.The new DSP algorithm to fit an error function 2.Goodness-of-fit.

An approach for the S-curve fit

I.Tsurin CERN 12.06.08

Page 1

ATLAS Pixel Calibration meeting

1. The new DSP algorithm to fit an error function

2. Goodness-of-fit test based on the likelihood ratio

3. Analysis of statistical and systematic errors

4. Matching precision with the DSP performance

5. Algorithm tests with the SCT threshold scans

A couple of reasons for improvements

Page 2

32 bits for 8-bit integers (unstable fits otherwise):

Unstable offline results:

Commissioning results:shown are the number ofmodules with more than 0.2% of pixels failing theonline (DSP) fits

(often mentioned)

ATLAS Pixel Threshold Scans

Page 3

Aim:

1. Threshold calibration (in terms of C-DAC settings)

2. Signal-to-noise ratio measurement

3. Finding the minimum detectable charge

Pixel parameters: signalgain, noise, time responseare measured using thethreshold scan techniques:

“Conventional” Threshold Scans

Aim:

1. Threshold calibration (in terms of T-DAC settings: Ti = K x Mean)

2. Signal-to-noise ratio measurement (plateau width)

3. Finding the lowest applicable threshold

Page 4

Allows for systematic effect studies: Baseline offset, TDAC non-linearity, etc.

S-curve parameterisation

Page 5

S-curve (bin statistics):

cumulative distribution function

(binomial Mean)

Gaussian probability densityfunction, could also be Laplaceor t-Student distribution

Actual S-curve Fit

Page 6

1. Find 3 points: X1, X2, X3 on the X-axis that correspond to 15%, 50% and 85%

of the number of generated pulses (compare the count readings with three

thresholds and take the nearest points that match this condition) . Drawback:

the 1st or the 3rd points could be missing for noisy / inefficient pixels.

2. Initial guess: Mean = X2, Sigma = (X1+X3) / sqrt(2)

3. Find the Mean with the minimum Chi2 in the range [X2-C ... X2+C] iteratively

4. Find the Sigma with the minimum Chi2 in the range [0.25*S ... 4*S] iteratively

Iterations

a.) keep M=const, take 3 Sigma values: S, 0.99S, 1.01S and return 0.99S

or 1.01S to approach minimum Chi2 for the entire data sample;

b.) keep S=const, take 3 Mean values: M, 0.99M, 1.01M and return 0.99M

or 1.01M to approach minimum Chi2 for the entire data sample;

c.) abort iterations when relative changes to Mean and Sigma are less

than some small value (currently 0.1%) at the same time.

5. Give up when the maximum number of iterations is reached :-(

An alternative method

Page 7

Derivative of the S-curve givesa Gaussian amplitude spectrum: Online

computation:

Skip negative F’ values and waitfor the next “good” Count reading !

Provides the minimum (analytically proven)when the parameterization is correct (no 1/ F noise)

(see page 5)

Pixel Calibration Data

Readout parameters:

Page 8

Offline computation:

Mean, RMS – characterize strips / pixels;Chi2, NDF – statistical significance of the measurement resultsData

reduction:

16 bytes / pixel – good trade off between saving the raw data (S-curves) and making DSP histograms which don’t characterize individual pixels

Goodness-of-fit test

Page 9

1. How confident are the and estimates

2. Find pixels / strips with pathologies

- test

Weighted Sum of residuals

Likelihood ratio test ( )

S-curve

Gaussian

Use -test as a reference method and develop the -test

S-curve

1. Weak against Type-I error

2. Expensive (in terms of DSP)

Binomial(expensive)

Multinomial (G-test)

Bin statistics = parameter

of the Likelihood function

Gaussian

Mean walk =

parameter of

the Likelihood

function

Page 10

Fit to the S-curve:

Fit to the Gaussian(S-curve derivative):

Chi-2 TestChi-2

statistics:

Be very careful with the approximation: (Pearson’s method)The Poisson approximation to the Binomial distribution works for

pi - cumulative distribution function

pi - probability density function

G-Test

Page 11

Neyman-Pearson lemma allows foran algebraic manipulation of the LRto see if there are key statistics in it.

The base of logarithm should provide the variable for the statistics

G-statistics (logarithm of the LR):

Multinomial distribution: consider the entire data sample distributed among k bins

Similarly to the binomial distribution,the probability density function:

to get N1, N2 ... Nk counts in bins 1, 2 ... ksimultaneously;To get the same bin counts

with the“true” (estimated) probabilities:

pi - probability in a singletrial in the i-th bin.

Likelihood Ratio ( ) Test in General Form

Mean = NDF, / NDF -> 1

Always show and NDF separately as theirratio is not enough to calculate the P-value

Probability to find the Mean Threshold on the right place :

Same for the “true” Mean:

Xi – threshold settings

For the S-curve derivative:

Page 12

Advantage over the G-test : the base oflogarithm is less dependent on the Ntrig Scale since there is no Nobs(i) multiplierfor each component of the sum

Page 13

Chi2 computation requiresa temporary data storagefor each pixel; this memorycould be shared betweendifferent scans (pipeline)

Use data from the previous scan tocalculate the Chi2-statistics (it hasto run faster than acquiring the newfrontend reading)

Receive the frontend readingand overwrite the memory cell;increment the RMS and Meanvariables

Towards the Hardware

Mean and StD results areinstantaneous (ready withthe last frontend reading)

Critical issues: 1. the total amount of memory needed for all pixels scanned in parallel; 2. complexity and performance of the new fit algorithm

What is the DSP ?

Page 14

The second after the MCC intelligent device that processes the data

TMS320C6713B

No operating system (programis a loop with interrupt services)

L1 cache: 4 kb program memory,2x2kB (2-way) data memory,

L2 cache: 256 kb program anddata memory (re-configurable)

200 MHz maximum clock freq.

50 MHz maximum RAM accessfrequency

Strip & Pixel ROD (Readout Driver)

The DSP fit alone is ~50 kB of code + include files with the lookup data

DSP DOs and DON’Ts

Page 15

Recommended

Allowed

Processor-Specific

Arithmetical: +, -, *Logical: and, or, notRelations: > ,< , =

Logical: <<, >> (shift)

Arithmetical : 1/X, 1/sqrt(X)

Use integers whenever possible: operations on integers are 5x faster than on floating point values !

Many of the non-elementary functions are available throughlibraries or lookup tables but they are very expensive !

Solution for a serious computing task: use logarithms !

(4% error)

G-test and LR-test have an advantage over the Chi-2 test

“Quick-and-Dirty Logarithm”

Page 16

Real number:

N – shift > 22 and mask 0xFFY – mask 0x7FFFFF

Some useful Formulas:

Page 17

...and just for fun:

Gaussian Spectrum in the logarithmic scale is a parabola !

qdl2(X), pwr2(X) uncertainties

Page 18

Error cancellation:

qdl2(X) ~2% errorpwr2(X) ~6% error

DSP calculus vs. precise values

Chi2-tests: pwr2(qdl2(X))LogLR-tests: qdl2(X1) – qdl2(X2)

For both: the sum of oscillating terms in Chi-2 statistics or G-statistics shows smaller errors (smoothing effect):

qdl2(X) ~1% errorpwr2(X) ~4% errorThe latter has to be

takeninto account and corrected for

Maximum deviation 4e-7

Page 19

Statistical error : 1%Systematic error : 0.01%

Range:

Range, Sample size, Class intervals

7... 20

Class intervals:

From the critical values offor the 5% significance level:

to prevent Type-I errors

Bin statistics:

> 5Gaussian approximation

to the binomial distribution (Central Limit Theorem)

Page 20

Bandwidth and Memory Requirements

1 ROD -> 1 stave: 13 modules x 16 chips x 3000 pixels / 4 DSPs / 32 mask stages -> 5000 pixels

Analogue signal decay time ~3 us -> Maximum frequency of calibration pulses ~100 kHz (10 us intervals) -> 255 pulses for each threshold value -> 2.56 ms

Mean, RMS, G-stat variable increment ~100 clock cycles per pixel -> 500 ns without external RAM access (indeed should be faster due to 8x parallel DSP architecture)

200 e noise -> 200 e step (6 bins) -> 40 steps for 8000 e range -> 1 byte / reading-> 40 bytes / pixel (internal RAM is byte addressable ) -> 4 k-bytes / chip (32 maskstages) -> 64 k-bytes / module -> 800 k-bytes / stave -> 200 k-bytes / DSP

Conclusions: 1. The DSP is fast enough to scan 5000 pixels in parallel 2. The DSP has enough L2-cache memory (256 kb) for data (200 kb) and a small program (64 kb)

Page 21

Algorithm test with the SCT data

Mean amplitude and signal dispersion wereobtained from the SCT scans using the new fittechniques. Long and short strips with differentnoise levels are clearly visible from the plots.

This is due to over-estimated statistical errors(the SCT data could be averaged or smoothed).

The peak positionis OK but statisticsis poor

Page 22

SCT data analysis––––– original S-curve

. . . . . derivative function- - - fit curve

(+) accept fit (-) reject fit

Type - I errors :

Type - II errors:

Rejecting goodstrips / pixels

Accepting badStrips / pixels? How good is

“good”and how bad is “bad”

Summary

Recommended for scans of the silicon pixel, strip and pad detectors andany other systems with a binary readout.

Precise “on-the-fly” calculation of the population Mean and Variance

1. FAST (no iterations, no special functions, no memory access)2. ROBUST (no iterations and convergence requirements)3. Economic (a few DSP commands, no memory consumption)4. Elegant (analytic solution instead of numerical methods)

Fast and precise enough goodness-of-fit test (requires not much memory)

Last page

The algorithm has to be checked with the pixel data

float qdp2(float Dat){ int IQ; float Res; IQ = (int) ceil(Dat); Res = 1. + Dat - IQ; Res *= 1<<IQ; return 0.96 * Res;}

float qdl2(float Dat){ float Res; union{ int Tmp; float Val; } Arg; Arg.Val = Dat; Res = ((Arg.Tmp>>23) & 0x7F); Arg.Tmp &= 0x7FFFFF; Arg.Tmp += 0x3F800000; Res += Arg.Val; return Res;}

Base-2 Logarithm function

Power of 2 function

Code primitives