Noise analysis of particle sensors and pixel detectors.

A fundamental
quantity for a detector is
the noise. The noise, which is
usually an additive term, gives
a limit to the sensitivity of
detector since is impossible or,
anyway, very difficult to
extract the information from a
signal where the informative
component has the same or a
lower strength than the noise.
In the following will be
described the behavior of the
sensors in absence of any
stimuli.
Fixed
pattern noise FPN
Even if a
matrix of pixels is in dark
condition, the pixels are not at
the same level. Due to the
inevitable differences of a
pixel to any other in the
matrix, caused by unavoidable
nonuniformity in the
realization processes, each
pixel has a different level in
dark.
The fixed
pattern noise FPN represents
the response of the pixels
matrix in case of dark or with a
uniform illumination. Strictly
speaking, the FPN includes two
different components: the dark
signal nonuniformity DSNU, that
is the response in dark
condition, and the photo
response nonuniformity PRNU,
that represents the different
manner with which the pixels
react to a uniform irradiation.
However, as first approximation,
we have assumed that PRNU is
negligible and in the following
with the term FPN we only refer
to the DSNU.
The dark
signal nonuniformity represents
the offset (often called pedestal)
to subtract to all the frames in
order to retrieve the signal
produced by the radiation. This
operation is always performed
when we are interested to detect
the incident radiation. During
our experiments, in fact, the
sensors are always shielded by
the ambient light and, in order
to evaluate the signal produced
by the ionizing particles,
before each sets of
measurements, the
FPN is evaluated, stored in the
DAQ system and used to calculate
the differences unless
the sensor does not perform this
operation automatically.
In Figure 1
the FPN of
one pixel matrix was
evaluated collecting 500
consecutive frames and making
their average. is shown a
representation of this
elaboration for the pixels.
Fig
1 Response of pixel matrix in
dark condition (a), with its
linear regression to a plane (b)
and the relative residues (c).
As can be
seen there is a component of the
signal that rises linearly with
the column number: using a
multiple regression algorithm
can be extracted the plane shown
in Figure 1(b). Subtracting this
plane to the measured FPN the
values are brought back all
around zero (Figure 1c).
The equation of plane is:
Y=
rα_{r} +
cα_{c} +
k
where r is
the row number and c is the
column, α_{r } and α_{c} are
the inclination of the plane
respectively along columns and
along rows and k is a constant
term.
Subtracting
the interpolating plane the two
components discussed above are
cancelled and the resulting
distribution is centered on
zero. In Figure 2a it is shown
the distribution of that
residuals for the pixel matrix.
The distribution has a Gaussian
shape with a standard deviation
of few tens of counts.
Increasing the integration time
the shape starts to be slightly
different from the Gaussian, in
Figure 2b this behavior is
emphasized using a logarithmic
scale to show the differences at
the left of the Gaussian lobe.
Fig. 2 (a)
FPN distribution after
subtraction of the interpolating
plane of the pxiel matrix
(frames collected with 33ms of
integration time), and (b) the
comparison between this last one
and with the same distribution
calculated at 262ms of
integration time.
Increasing
the integration time the pixels
voltage must decrease because of
the dark current, but the pixels
of a matrix don’t behave in a
coherent manner: some pixels go
towards the discharge faster
than others; this means that the
dark current is different for
each pixel as it should be.
Single
Pixel Noise
Each pixel
fluctuate around its pedestal
from an acquisition to another.
The main cause of this
fluctuation at the pixel level
is the so called kT/C
noise but
there are some other
contributions which increase the
level of measured noise as the
noise of readout buffer, the
quantization noise, external
electromagnetic interference,
bonding, cables,
PCB board, etc.
The Figure 3a
is the distribution of the
signal measured during 500
consecutive frames on the same
pixel: the signal has a Gaussian
distribution centered on the
pixel’s pedestal. The standard
deviation of that distribution
differs from pixel to pixel and
with the integration time; the
Figure 3b shows the distribution
of that values for the pixel
matrix at different integration
times. Increasing the
integration time the mean noise
rise almost linearly in the
interval 33ms  262ms and the
spread of the values increase.
Due to the asymmetric shape of
the distribution the mean value
of the measured noise, for a
given matrix, differs from the
most probable value.
Fig 3 (a)
Single pixel noise. (b)
Distribution of the single pixel
noise for all the pixels of an
pixel matrix measured for
different integration times.
In Figure 4
are drawn both quantities. As
expected the noise is smaller
for the matrices with the large
photodiode because of its larger
capacitance, (indeed the term C
occurs at denominator in the
expression of kT/C noise power).
Another
aspect that can be noted is the
dependence of the noise on the
integration time: the kT/C noise
is, by definition, independent
by the time, but this is not
true for other contributions
such as the leakage current.
Indeed, the signal is
proportional to the electric
charge accumulated into the
photodiode and this charge is
subjected to a continuous loss
due to the leakage current.
Fig 4 (a)
Most probable value of the
single pixel noise for four
submatrix of a pixel matrix
chip, for different integration
times. (b) Standard deviation of
the noise values for all the
pixels of two pixel matrices,
evaluated at different
integration times.
The charge is
lose proportional to the time
with the relation: Q = I × t: if
we assume I as a stochastic
process with a given variance, Q
is a stochastic process in turn
with a variance t times higher
than the variance of the leakage
current.
During
acquisition operations, usually,
the signal of each pixel,
obtained by the difference with
the pedestal, is compared with a
certain threshold, if the
threshold is crossed it is
assumed that a particle has
crossed the detector and the
corresponding frame is
collected. Fake crossings,
caused by noise, can occur.
Usually the threshold must be as
lower as possible in order to
detect weak signals but the
noise puts a lower limit to this
parameter. Each frame is
composed by N_{p} pixels,
assuming the same Gaussian noise
for each pixel with a variance σ
the probability P_{FH} to
have a fake hit is given by:
where p is
the probability of not crossing
the threshold T on a pixel and
g(x) is the Gaussian function;
using the normalized threshold τ
= T/σ the equation becomes:
where Q(x) is
the so called Qfunction.
The P_{FH} can
be estimated with the ratio
between the number of frames
which have crossed a certain
threshold on the total frames
during an acquisition in dark
condition. The measured P_{FH} is
reported in Figure 5 with a
dotted line. As can be seen the
measured quantity differs
drastically from the theoretical
one. However, the results of the
analysis show that this method
is not enough effective to
reduce the fake hits. An
explanation of this behavior is
that there are some pixels
(called bad pixel) which have
not a Gaussian noise, as shown
in Figure 6 where it is reported
the evolution of the signal (the
graph on the left) and its
cumulative distribution (on the
right) measured on a particular
pixel of the pixel matrix along
about 500 consecutive frames.
Fig
5. Comparative between the
theoretical probability of fake
hit (P_{FH} ) in
function of the threshold (in
units of σ) and the measured
one. The bold curves represent
the measured probability
including into the trigger
control: all the pixel (dotted
line); only the pixels with
noise less than the most
probable one (dasheddotted);
excluding the Bad Pixels
(solid).
Fig 6. Signal measured on a pixel of the matrix during about 500 consecutive frames (the graph on left) and its cumulative distribution (on right).
If we measure the standard deviation of this pixel, for example, between the 300^{th} and the 400^{th} frame of the figure we will find a value compatible with the most probable noise value of the frame and we would not exclude that pixel from the trigger check, but the leaps clearly visible in the figure could generate fake hits.
References

D. Passeri et al., Characterization
of CMOS Active Pixel Sensors for particle detection: beam test of the four
sensors RAPS03 stacked system, Nucl.
Instr. and Meth. A 617 (2010) 573–575

D.Passeri,et al. Tilted
CMOS Active Pixel Sensors for Particle Track Reconstruction, IEEE
Nucl. Sci. Symp. Conf. Rec. NSS09 (2009) 1678. July 2006.

L. Servoli et al. . Use
of a standard CMOS imager as position detector for charged particles ,
Nucl. Instr. and Meth. A 215 (2011) 228231, 10.1016/j.nuclphysbps.2011.04.016

D. Biagetti et al. Beam
test results for the RAPS03 nonepitaxial CMOS active pixel sensor,
Nucl. Instr and Meth A 628 (2011) 230–233
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