Image Processing Reference
In-Depth Information
and
2
th
spectral line of
X
k
, k = 0,1,...,N-1 is lying on a symmetry axis: the real part is
symmetric, the imaginary part is asymmetric. The useful information is contained only in 1
st
2
+
1
spectral lines for k = 0,1,...,N/2 corresponding to frequencies
f
k
=
1
·
=
k
f
0
k
t
,
Δ
N
changing from zero to
f
smpl
2
with
f
smpl
N
grid.
3.1 Pedestal independence
The analog section of the FEB has been designed to have a pedestal of ca. 10 % of the full
FADC range in order to investigate undershoots. However, the pedestal is relatively sensitive
on the temperature. Daily variation of the pedestal may reach 5 ADC-counts. The trigger
pedestal-independent is very welcome. Let us consider signal with a constant pedestal:
N
1
n
=
0
(
x
n
+
ped
)
F
(
k
,
n
)=
−
N
1
n
=
0
F
(
k
,
n
)=
−
X
k
(
X
k
+
X
k
+
ped
)=
ped
ped
×
W
(3)
cos
k
N
(
1
2
)
F
(
k
,
n
)=
n
+
(4)
Due to symmetry and parity of the cosine, we get for odd and even indices respectively:
⎨
⎩
0
,
k
−
odd
cos
k
cos
N
n
k
N
2
−
1
n
=
0
2
1
2
k
2
N
2
−
W
=
2
+
−
=
1
n
=
0
F
(5)
2
(
k
,
n
)
,
k
−
even
By a recursion, repeating (5) we get finally
2
=
0,
2
. For
k
N
2
=
=
2 and
k
n
=
0
cos
2
n
+
1
1
2
=
0
(6)
In a consequence fork>0theDCTcoefficients are independent of the pedestal.
3.2 Scaling
The DCT algorithm has a significant advantage in comparison to the FFT one. The structure of
DCT coefficients is much simpler for interpretation and for a trigger implementation than the
structure of the FFT real and imaginary coefficients (compare 4th of the FFT data vs. 2nd row
for the DCT coefficients in Fig. 3). For the exponentially attenuated signals from the PMTs
higher DCT coefficients (scaled to the 1st harmonics)
X
k
X
1
ξ
k
=
(7)
are almost negligible, while both real and imaginary parts of the FFT (scaled to the module of
the 1st harmonics) give relatively significant contributions and are not relevant for triggering.
When a peak appears in the pure attenuated signal (last column in Fig. 3) the structure of the
DCT dramatically changes and trigger condition immediately expires, while modules of FFT
components almost do not change. The structure of FFT harmonics for the last graph in Fig. 3
would be more suitable for a trigger (almost negligible imaginary part for higher harmonics
and also relatively low real harmonics), however it corresponds just to situation, when the
