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; 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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