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CALCULATION OF COMPLEXITY OF A PULSE
TRANSFORMATION IN TIME-VARYING MEDIUM
Ruzhytska N.N., Nerukh A.G.
Kharkiv National University of RadioElectronics,
14 Lenin Avenue, Kharkov, 61166, Ukraine. E-mail: [email protected]
Nerukh D.A.
Unilever Centre for Molecular Informatics, Department of Chemistry,
University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK. E-mail:
[email protected]
Abstract: - A complexity of electromagnetic signals transformed by a short sequence of
cycles of medium parameters time changing is investigated. Dependence of the
complexity on the modulation parameters is considered and a difference between
forward and backward waves that are inevitable result of the medium time changing is
shown.
INTRODUCTION
Interactions between electromagnetic pulses and semiconductor active media in
waveguides are of significant importance in optical communication technology and
time-domain techniques for solving such electromagnetic problems have received
increased attention in the literature recently. Parametric phenomena in active media are
especially important in optoelectronic systems as they allow controlling of
electromagnetic signals by the temporal adjustment of the medium parameters. It is
known that a single abrupt change of medium parameters leads to changing of a pulse
shape [1-3]. The complexity of the phenomenon arises when only a few cycles of the
modulation are considered that happens in ultrafast optics. In this paper a change of the
complexity of the initial electromagnetic pulse E0 (t , x) with a number of modulation
cycles is considered. In the modulation cycles the permittivity receives constant
1
magnitude
on
the
disturbance
intervals
of
the
cycles
(n  1)T  t  T1  (n  1)T , n  1,..., N and the permittivity has constant magnitude 
on the quiescence intervals of the cycles T1  (n  1)T  t  nT , n  1,..., N . Here, T is
the duration of the cycle of the parameters change, T1 is the duration of the disturbance
interval of each cycle.
TRANSFORMATION OF THE PULSE AND ITS COMPLEXITY
Investigation of the pulse transformation is based on the Volterra integral equation
method, which allows considering problems with arbitrary primary signal [4]. If the
initial field has the form E0 (t , x)  f ( x  vt ) then the transformed field is described by
the following expressions: on the disturbance interval of the n-th cycle, n  1 En (t , x)  En (  ) (t  x / v1 )  En (  ) (t  x / v1 ) , where
En (  ) (t  x / v1 ) 

t  x / v1
t  x / v1
a

(a  1) Fn(1) (
 (a  1)(n  1)T )  ( a  1) Fn(1) ( 
 ( a  1)( n  1)T ) 

2
a
a

(1)
En (  ) (t  x / v1 ) 
t  x / v1
t  x / v1
a

(a  1) Fn(1) (
 (a  1)(n  1)T )  ( a  1) Fn(1) (
 ( a  1)( n  1)T ) 

2
a
a

()
()
on the quiescent interval of the n-th cycle Fn (t , x)  Fn (t  x / v)  Fn (t  x / v) ,
where
Fn (  ) (t  x / v)  F0 (t  x / v) 


1
2a 2
n 1


 (a  1)  E
()
k 1
k 0
(
t  x / v a 1
t  x / v a 1


(kT  T1 ))  Ek( 1) (

kT )  
a
a
a
a

t  x / v a 1
t  x / v a 1


(a  1)  Ek( 1) (

(kT  T1 ))  Ek( 1) (

kT )  ,
a
a
a
a


(2)
Fn (  ) (t  x / v) 

1
2a 2
n 1


 (a  1)  E
()
k 1
k 0
(
t  x / v a 1
t  x / v a 1


(kT  T1 ))  Ek( 1) (

kT )  
a
a
a
a

t  x / v a 1
t  x / v a 1


(a  1)  Ek( 1) (

(kT  T1 ))  Ek( 1) (

kT ) 
a
a
a
a


In order to estimate how complex the transformed signals are we calculated the
‘finite statistical complexity’ measure of the signals [5, 6]. This approach of estimating
the complexity of dynamical process rests on such well-known theories as KolmogorovChaitin algorithmic complexity and Shannon entropy. This measure of complexity
5
0.8
0.6
E, backward quiescence
Complexity
E, forward quiescence
Complexity
0.3
5
1.0
4
3
0.4
2
0.2
0
10
4
3
0.2
2
1
0.1
0
5
10
15
20
25
partition
0.0
20
partition
0.0
-0.1
-0.2
-5
0
5
10
15
20
25
-20
t-x/v
a=1.5 N=5 NT=8 T1=0.5T quiescence
0
-10
0
10
20
t+x/v
1
2
3
4
5
a=1.5 N=5 NT=8 T1=0.5T quiescence
1
2
3
4
5
Fig. 1. Transformation of the pulse on the quiescent interval when the medium change
beginning coincides with the beginning of the pulse, a   / 1 .
shows how much information is stored in the signal. It also indicates how much
information is needed to predict the next value of the signal if we know all the values up
to some moment in time. The algorithm of computing the finite statistical complexity
follows the method described in [7].
Dependence a signal shape and its complexity on the number of modulation
cycled is shown in Fig. 1, 2. The complexity increases with the number of partitions and
asymptotical behaviour at infinite number of partitions has a binary logarithm character
[8].
6
6
4
3
4
2
E backward
Complexity
20
E forward
1
0
10
complexity
6
5
0
5
10
15
20
partition
4
2
0
0
2
2
4
6
8
10
partition
0
0
-2
-10
0
0.0
4.0
8.0
0
1
a=1.5 T=8 T1=0.5 back
2
3
40
60
80
t+x/v2
t-x/v2
a=1.5 T=8 T1=0.5T
20
12.0
4
1
2
3
4
5
5
Fig. 2. Transformation of the pulse on the disturbed interval when the medium change
beginning coincides with the pulse maximum.
CONCLUSION
Investigations show a strong transformation of the electromagnetic signal by only a
short sequence of modulation cycles. Overall the complexity of the signals increases
with the number of cycles. However, in those cases when the signals get narrower
keeping their shape the complexity may decrease. The backward signals can be either
more complex than the forward ones or less complex.
In many cases the calculated complexity allows to quantitatively estimate the
relative informational contents of the signals. This is especially true in the situations
when it is impossible to judge by eye either when the signals have simple and very
similar shape or they are too complex to compare.
REFERENCES
1. Morgenthaler F.R., IRE Transactions MTT-6, p. 167, 1958
2. Ostrovsky L. A. and N.S. Stepanov, Radiophysics Quantum Electronics (English
transl.), 14, pp. 489-529, 1971.
3. Harfoush F.A. and A.Taflove, IEEE Trans. On Antennas and Propag., 39, pp. 898906, 1991.
4. Nerukh A.G., J. of Physics D: Applied Physics, 32, pp. 2006-2013, 1999.
5. Crutchfield J. P. and K. Young, Phys. Rev. Lett., 63, 105-108,. 1989.
6. Crutchfield J. P., Physica D, 75, 11, 1994.
7. Perry N. and P.-M. Binder, Phys. Rev. E, 60, 459, 1999.
8. Cover T. M. and J.A. Thomas, Elements of information theory, John Wiley & Sons,
Inc., 1991.
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