Image Processing Reference
In-Depth Information
pure attenuated signal is distorted by some peak on the tail and a trigger condition has been
violated.
The plot in the 4th row and 3rd column on Fig. 3 shows a contribution of the DFT vs. the
absolute value of the 1st harmonic. For an exponential attenuated signal (with the attenuation
factor =
) the contribution of both real and imaginary coefficients decreases monotonically
with a significant value for all real coefficients. From the DFT definition we get:
β
N
1
n = 0 e β n e j 2 π N n
e N β
A 1
X k =
=
A
(8)
e β + j 2 N k
1
X k
e −β N
Re
(
)
(
1
+
)
ξ =
=
1
(9)
φ 2
X 1
e β cos
e 2 β sin 2
+
φ
k
N .
Calculating (8) for boundary factors
2
π
φ =
where
β
= (0.28, 0.42) (from the Auger database) and for k =
N/2 (as the lowest in a monotonically decreasing chain), we obtain forN=16:
= 24% and
28%, respectively. These values are too large to be use for triggering. Even an extension of the
DFT size does not help very much. For N = 32: we get still large values:
ξ
= 17% and 23%.
Almost vanishing higher DCT coefficients provide much natural trigger conditions. 32-point
FFT (roughly equivalent to 16-point DCT) does not offer better stability.
ξ
4. Genaral DCT algorithm
The DCT for real signal x n gives independent spectral coefficients for k = 0,1,...,N-1, changing
f k also from zero to
f sampl
2
but with f sampl
2 N grid. DCT vs. DFT gives twice better resolution.
Splitting the sum (1) and redefine the indices we get:
k
x n cos N
n
k
x N 1 n cos N
N
n
N
2
1
n = 0
1
2
1
2
X k = α k
+
+
(10)
Due to symmetry of the cosine function
n = 0 cos N n +
k
N
2
x n +(
k x N 1 n
1
1
2
X k = α k
×
1
)
(11)
We can introduce the new set of variables:
x n
0, ..., 2
+
x N 1 n
(
n
=
1
)
A n
=
(12)
N
2 , ..., N
(
=
)
x N 1 n
x n
n
1
DCT coefficients can be separated for even and odd indices respectively:
A n cos N
n
k
N
2 (
1
0
)
1
2
X
) = α k
+
(13)
even
odd
(
0
n
=(
)
N
1
Let us notice that (13) for even indices has the same structure as (1) with only shorter range of
indices. Recurrently we can introduce new sets of variables for the set of indices k = 2p, where
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