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## 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
non-uniformity 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
non-uniformity DSNU, that is the response in
dark condition, and the photo response
non-uniformity 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 non-uniformity 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
sub-matrix 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 Q-function.

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 (dashed-dotted); 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) 228-231, 10.1016/j.nuclphysbps.2011.04.016 - D. Biagetti et
al.
*Beam test results for the RAPS03 non-epitaxial CMOS active pixel sensor*, Nucl. Instr and Meth A 628 (2011) 230–233

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