Transcript cal_meet_1206

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An approach for the S-curve fit
ATLAS Pixel Calibration meeting
1.
2.
3.
4.
5.
I.Tsurin
The new DSP algorithm to fit an error function
Goodness-of-fit test based on the likelihood ratio
Analysis of statistical and systematic errors
Matching precision with the DSP performance
Algorithm tests with the SCT threshold scans
CERN
12.06.08
A couple of reasons for improvements
32 bits for 8-bit integers (unstable fits otherwise):
Commissioning results:
shown are the number of
modules with more than
0.2% of pixels failing the
online (DSP) fits
Unstable offline results:
(often mentioned)
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ATLAS Pixel Threshold Scans
Pixel parameters: signal
gain, noise, time response
are measured using the
threshold scan techniques:
Aim:
1. Threshold calibration
(in terms of C-DAC settings)
2. Signal-to-noise ratio
measurement
3. Finding the minimum
detectable charge
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“Conventional” Threshold Scans
Allows for systematic effect
studies: Baseline offset,
TDAC non-linearity, etc.
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
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S-curve parameterisation
S-curve (bin statistics):
(binomial Mean)
cumulative distribution function
Gaussian probability density
function, could also be Laplace
or t-Student distribution
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Actual S-curve Fit
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 :-(
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An alternative method
Derivative of the S-curve gives
a Gaussian amplitude spectrum:
(see page 5)
Online computation:
Skip negative F’ values and wait
for the next “good” Count reading !
Provides the minimum
(analytically proven)
when the parameterization is correct (no 1/ F noise)
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Pixel Calibration Data
Readout parameters:
Data reduction:
Offline computation:
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Mean, RMS – characterize strips / pixels;
Chi2, NDF – statistical significance of
the measurement results
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
1. How confident are the
and
estimates
2. Find pixels / strips with pathologies
Likelihood ratio test (
Weighted Sum of residuals
S-curve
- test
S-curve
Bin statistics = parameter
of the Likelihood function
Gaussian
1. Weak against Type-I error
2. Expensive (in terms of DSP)
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Use
Binomial
(expensive)
)
Gaussian
Mean walk =
parameter of
the Likelihood
function
Multinomial
(G-test)
-test as a reference method and develop the
-test
Chi-2 Test
Chi-2 statistics:
Fit to the S-curve:
pi - cumulative
distribution function
Fit to the Gaussian
(S-curve derivative):
pi - probability
density function
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Be very careful with the approximation:
(Pearson’s method)
The Poisson approximation to the Binomial distribution works for
G-Test
Multinomial distribution: consider the entire data sample distributed among k bins
Similarly to the binomial distribution,
the probability density function:
To get the same bin counts with the
“true” (estimated) probabilities:
G-statistics (logarithm of the LR):
Neyman-Pearson lemma allows for
an algebraic manipulation of the LR
to see if there are key statistics in it.
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The base of logarithm should provide
the variable for the
statistics
to get N1, N2 ... Nk
counts in bins 1, 2 ... k
simultaneously;
pi - probability in a single
trial in the i-th bin.
Likelihood Ratio (
Mean = NDF,
) Test in General Form
/ NDF -> 1
Probability to find the Mean
Threshold on the right place :
Same for the “true” Mean:
Xi – threshold settings
For the S-curve derivative:
Always show
and NDF separately as their
ratio is not enough to calculate the P-value
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Advantage over the G-test : the base of
logarithm is less dependent on the Ntrig
Scale since there is no Nobs(i) multiplier
for each component of the sum
Towards the Hardware
Mean and StD results are
instantaneous (ready with
the last frontend reading)
Chi2 computation requires
a temporary data storage
for each pixel; this memory
could be shared between
different scans (pipeline)
Critical issues:
1. the total amount of memory
needed for all pixels scanned
in parallel;
2. complexity and performance
of the new fit algorithm
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Use data from the previous scan to
calculate the Chi2-statistics (it has
to run faster than acquiring the new
frontend reading)
Receive the frontend reading
and overwrite the memory cell;
increment the RMS and Mean
variables
What is the DSP ?
The second after the MCC intelligent device that processes the data
TMS320C6713B
Strip & Pixel ROD (Readout Driver)
No operating system (program
is a loop with interrupt services)
L1 cache: 4 kb program memory,
2x2kB (2-way) data memory,
L2 cache: 256 kb program and
data memory (re-configurable)
200 MHz maximum clock freq.
50 MHz maximum RAM access
frequency
The DSP fit alone is ~50 kB of code + include files with the lookup data
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DSP DOs and DON’Ts
Recommended
Arithmetical: +, -, *
Logical: and, or, not
Relations: > ,< , =
Allowed
Logical:
<<, >> (shift)
Processor-Specific
Arithmetical :
1/X, 1/sqrt(X)
(4% error)
Use integers whenever possible: operations on integers are
5x faster than on floating point values !
Many of the non-elementary functions are available through
libraries or lookup tables but they are very expensive !
Solution for a serious computing task: use logarithms !
G-test and LR-test have an advantage over the Chi-2 test
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“Quick-and-Dirty Logarithm”
Real number:
N – shift > 22 and mask 0xFF
Y – mask 0x7FFFFF
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Some useful Formulas:
Gaussian Spectrum in the logarithmic scale is a parabola !
...and just for fun:
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qdl2(X), pwr2(X) uncertainties
DSP calculus vs. precise values
qdl2(X)
pwr2(X)
~2% error
~6% error
Error cancellation:
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)
pwr2(X)
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~1% error
~4% error
The latter has to be taken
into account and corrected for
Maximum deviation 4e-7
Range, Sample size, Class intervals
Range:
Class intervals:
7... 20
From the critical values of
for the 5% significance level:
to prevent Type-I errors
Bin statistics:
>5
Gaussian approximation
to the binomial distribution
(Central Limit Theorem)
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Statistical error :
1%
Systematic error : 0.01%
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 mask
stages) -> 64 k-bytes / module -> 800 k-bytes / stave -> 200 k-bytes / DSP
Conclusions:
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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)
Algorithm test with the SCT data
Mean amplitude and signal dispersion were
obtained from the SCT scans using the new fit
techniques. Long and short strips with different
noise levels are clearly visible from the plots.
The peak position
is OK but statistics
is poor
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This is due to over-estimated statistical errors
(the SCT data could be averaged or smoothed).
SCT data analysis
––––– original S-curve
. . . . . derivative function
- - - fit curve
(+) accept fit
(-) reject fit
Type - I errors :
Rejecting good
strips / pixels
Type - II errors:
Accepting bad
Strips / pixels
? How good is “good”
and how bad is “bad”
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Summary
Recommended for scans of the silicon pixel, strip and pad detectors and
any 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)
The algorithm has to be checked with the pixel data
Last page
Code primitives
Base-2 Logarithm function
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;
}
Power of 2 function
float qdp2(float Dat){
int IQ;
float Res;
IQ = (int) ceil(Dat);
Res = 1. + Dat - IQ;
Res *= 1<<IQ;
return 0.96 * Res;
}