CN108768609B - Analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics - Google Patents

Analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics Download PDF

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CN108768609B
CN108768609B CN201810528220.XA CN201810528220A CN108768609B CN 108768609 B CN108768609 B CN 108768609B CN 201810528220 A CN201810528220 A CN 201810528220A CN 108768609 B CN108768609 B CN 108768609B
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李齐良
陈德望
包琪
胡淼
周雪芳
曾然
杨淑娜
唐向宏
杨国伟
毕美华
卢旸
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Abstract

The invention discloses a method for analyzing electro-optic phase chaotic dynamics similar to frequency-dependent time delay, which comprises the following steps of: the first step is as follows: establishing a dynamic mathematical model in the chaotic communication system; the second step is that: determining the relation between the time delay and the frequency; the third step: dividing time intervals and determining an initial value of an equation; the fourth step: calculating a delay signal at each moment in a certain period of time after delay through Fourier time-frequency transformation; the fifth step: converting the time delay differential equation into an ordinary differential equation; and a sixth step: and (5) carrying out numerical solution on the ordinary differential equation in the fifth step by using a Runge-Kutta method. The invention can accurately solve the differential-integral equation with the frequency-dependent time delay and accurately analyze the feedback chaotic dynamics and chaotic communication with the frequency-dependent time delay.

Description

Analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics
Technical Field
The invention belongs to the technical field of optical information, and particularly relates to an analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics.
Background
In the current chaotic optical communication system, semiconductor lasers are often used to be mutually coupled to increase the degree of freedom thereof to generate chaotic signals. The chaos signal is used as the carrier of the signal, and the chaos has the characteristics of synchronization and robustness, so that the decoding of the transmitted information is realized. When transmitting signals, the chaotic carrier wave output by the transmitter serves as a carrier for transmitting signals. Since the amplitude of the signal is much smaller than the fluctuation of the chaotic carrier, it is difficult to separate the signal from the chaotic carrier. In particular, to decode a signal, the transmitted information needs to be coupled to a receiver that is highly similar to the transmitter. The signal can be recovered by monitoring the synchronization error between the output and the input of the receiver only under the condition that the chaotic carrier generated by the receiver is synchronous with the chaotic carrier generated by the transmitter. Therefore, chaotic synchronization is a key to realizing the entire chaotic transmission system. The physical parameters of the receiver and the transmitter are the same, and the receiver and the transmitter can realize synchronization. In order to prevent eavesdropping, the parameters of the transmitting terminal need to be hidden, and the chaotic dynamics of the transmitting terminal cannot be reconstructed if the parameters of the transmitter cannot be obtained, so that the communication safety is ensured. This is the chaos safety communication method of hiding the frequency-dependent delay time signature.
The invention provides an analysis method of similar frequency correlation delay electro-optic phase chaotic dynamics aiming at a similar frequency correlation delay electro-optic phase differential-integral equation.
Disclosure of Invention
Aiming at a general differential integral equation set with time delay and feedback, the invention researches the frequency-dependent delay time signature hiding encrypted by a digital key in the electro-optical feedback chaotic communication system, establishes a system model for the electro-optical feedback chaotic communication system, provides a differential integral equation set with frequency-dependent time delay and feedback, and analyzes the dynamic characteristic of frequency-dependent time delay electro-optical phase chaos. This also lays a further foundation for the system to achieve synchronization.
The invention adopts the following technical scheme:
an analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics is carried out according to the following steps:
the first step is as follows: establishing a dynamic mathematical model in the chaotic communication system;
the second step is that: determining the relation between the time delay and the frequency;
the third step: dividing time intervals and determining an initial value of a differential-integral equation describing electro-optic phase chaos;
the fourth step: solving a delayed signal by utilizing Fourier time-frequency transformation;
the fifth step: converting the time delay differential equation into an ordinary differential equation;
and a sixth step: and (4) carrying out numerical solution on the ordinary differential equation by using a Runge-Kutta method.
Further, the specific method of the first step is as follows:
modeling a transmitting end part in a chaotic communication system, wherein the system adopts a Mach-Zehnder interferometer for phase modulation, and the equation of the transmitting end is as follows:
Figure BDA0001676535070000021
Figure BDA0001676535070000022
where τ and θ are the response times for the high and low cut-off frequencies of the feedback loop, respectively, G is the electrical amplifier gain, Φ is the initial phase, and x, t represent the phase and time, respectively.
Further, the specific method of the second step is as follows:
the frequency dependent delay is generated by means of a ring resonator, the transmission equation being:
Figure BDA0001676535070000023
and is
A3＝A4ejβL (4)
Where r and k are the coupling coefficients of the ring resonator,
Figure BDA0001676535070000031
is the transmission constant, n is the refractive index, ω is the frequency of light, c is the speed of light, L is the cavity length, j is the complex unit, A1And A2The relationship of (1) is:
Figure BDA0001676535070000032
the transfer function is expressed as
Figure BDA0001676535070000033
Expressing H (ω) as amplitude and frequency, i.e.
H(ω)＝|H(ω)|eiφ (7)
Wherein phiHIs the phase of H (ω), and the frequency-dependent delay is expressed as
Figure BDA0001676535070000034
Further, the third step is specifically as follows:
dividing the total time T into N segments, each segment being long
Figure BDA0001676535070000035
Solving each time period according to the formulas (1) - (2), taking the value of the last time solved in the previous time period as the initial value of the next time period, and substituting the initial value into the next time period to solve, so that the operation is repeated until the equation is solved in the time T; for the first initial value of Δ T, the integral in differential equations (1) and (2) is made equal to 0, and then a stable solution of the equations is found, which is then taken as the first initial value of Δ T.
Further, the fourth step is specifically performed by:
as mentioned above, the present invention is directed to solving a differential equation with a frequency dependent delay, taking into account that this delay is frequency dependent, and then processing is performed in the frequency domain.
Equations 9 and 10 below represent [ x ], respectively1(t)+x2(t)]、[x2(t)]After being delayed, the inverse of the fourier transform.
Figure BDA0001676535070000036
x22(t)＝x2(t-T-τf)＝ifft{fft[x2(t)]·e-jωT} (10)
Wherein fft represents Fourier transform, ifft represents inverse Fourier transform, phiHIs the phase of the ring cavity transfer function H (ω).
Further, the concrete method of the fifth step is as follows:
the invention takes into account that during transmission, x is1And x2The delayed signal is already solved in the fourth step, and becomes an ordinary differential equation for the original equation, as follows
Figure BDA0001676535070000041
Figure BDA0001676535070000042
Further, the longge-kutta method is known equation derivative and initial value information, and when computer simulation is used, a process of solving a differential equation is omitted, and the process is specifically as follows:
for a general equation containing the initial problem:
y′i＝fi(t,y1y2·yi··yn),yi(t0)＝yi(0) and i is 1,2, n, n represents the number of equations.
Obtained by the fourth-order Longge Kutta method
Figure BDA0001676535070000043
h represents a time interval.
Wherein
k1＝fj(tj,yj)
Figure BDA0001676535070000044
Figure BDA0001676535070000045
Figure BDA0001676535070000046
Thus, the next value (y)i,j+1) From the present value (y)i,j) Plus the product of the time interval (h) and an estimated slope. This slope is a weighted average of the following slopes:
●k1is the slope at the beginning of the time period;
●k2is the slope of the midpoint of the time segment, and the slope k is adopted by the Euler method1To determine y at point tnA value of + h/2;
●k3is also the slope of the midpoint, but this time with slope k2Determining the value of y;
●k4is the slope of the end of the time period, the y value of which is k3And (6) determining.
Compared with the prior art, the invention has the following advantages:
1. the invention proposes a solution for processing signals with a frequency dependent delay.
2. The invention can also analyze the characteristics of the chaotic signal in the evolution along with time.
3. The invention has a great effect on analyzing the chaotic communication system with the hidden digital signature.
Drawings
FIG. 1 is an analytical flow chart of the present invention.
Fig. 2 is a schematic structural diagram of a communication system according to the present invention.
Fig. 3 is a diagram of the optical power time series of the transmitting end, a diagram of the receiving end, and a diagram of the power difference between the transmitting end and the receiving end. Indicating that the receiving end and the transmitting end are in strict synchronization.
Fig. 4 is a phase time sequence diagram of the transmitting end, an optical phase time sequence diagram of the receiving end and a phase difference between the transmitting end and the receiving end. It also indicates that the receiving end and the transmitting end are in strict synchronization.
Fig. 5 is a diagram of transmit-side optical power autocorrelation coefficients.
The upper diagram of fig. 6 shows the transmitted information and the lower diagram shows the demodulated signal.
Detailed Description
The following describes embodiments of the present invention in detail with reference to the accompanying drawings.
In this embodiment, an analysis method similar to the frequency-dependent time-delay electro-optic phase chaotic dynamics is performed according to the analysis flow shown in fig. 1. The analysis method is as follows:
the first step is as follows: and establishing a dynamic mathematical model in the chaotic communication system.
The invention models the transmitting end part in the chaotic communication system, the system adopts a Mach-Zehnder interferometer to carry out phase modulation, and the equation of the transmitting end can be written as
Figure BDA0001676535070000061
Figure BDA0001676535070000062
Where τ and θ are the response times for the high and low cutoff frequencies of the feedback loop, respectively, G is the electrical amplifier gain, and φ is the initial phase.
The second step is that: and determining the relation between the time delay and the frequency.
In the communication system, the frequency-dependent delay is generated by means of a ring resonator, and the transmission equation is
Figure BDA0001676535070000063
And is
A3＝A4ejβL (4)
Where r and k are the coupling coefficients of the ring resonator,
Figure BDA0001676535070000064
is the transmission constant, n is the refractive index, ω is the optical frequency, c is the optical velocity, L is the cavity length, A can be obtained1And A2The relationship (2) of (c).
Figure BDA0001676535070000065
Wherein the transfer function can be expressed as
Figure BDA0001676535070000066
If H (ω) is expressed in amplitude and frequency, i.e.
H(ω)＝|H(ω)|eiφ (7)
Wherein phiHIs the phase of H (ω), the frequency dependent delay can be expressed as
Figure BDA0001676535070000071
The third step: dividing time intervals and determining initial values of equations.
The invention equally divides the total time T into N sections, and each section has long time
Figure BDA0001676535070000072
The method solves each time period respectively, takes the value of the last time solved in the previous time period as the initial value of the next time period, and carries out the solution in the next time period, thus repeatedly carrying out the operation until the equation is solved in the time T. For the first initial value of Δ T, the integral in differential equations (1) and (2) may be made equal to 0, and then a stable solution of the equations is found, which is then taken as the first initial value of Δ T.
The fourth step: and solving the delayed signal by utilizing Fourier time-frequency transformation.
As mentioned above, the present invention is directed to solving a differential equation with a frequency dependent delay, taking into account that this delay is frequency dependent, and then processing is performed in the frequency domain.
Figure BDA0001676535070000073
x22(t)＝x2(t-T-τf)＝ifft{fft[x2(t)]·e-jωT}
(10)
WhereinFft represents the Fourier transform, ifft represents the inverse Fourier transform, phiHIs the phase of the ring cavity transfer function H (ω) mentioned above.
The fifth step: and converting the time delay differential equation into an ordinary differential equation.
The invention takes into account that during transmission, x is1And x2The delayed signal is already solved in the fourth step, and becomes an ordinary differential equation for the original equation, as follows
Figure BDA0001676535070000074
Figure BDA0001676535070000081
And a sixth step: the ordinary differential equations (11) and (12) are numerically solved using the longge-kutta method.
In this embodiment, as shown in fig. 2, after a laser at a transmitting end generates 10mW continuous light, information is phase-modulated by a phase modulator, the continuous light passes through a mach-zehnder interferometer with a loss coefficient of 0.082 and a bias voltage of 4.1V, then passes through a 1:1 optical splitter, one path is transmitted to a receiving end, the other path is transmitted through a cascade optical circulator with a delay frequency correlation and a maximum delay of 76ps, and then passes through a 2ns delay optical fiber, an optical signal is converted into an electrical signal by an electrical detector of light with a high cutoff frequency corresponding to 13.5ps and a low cutoff frequency corresponding to 5.5 μ s, the electrical signal is amplified by an electrical amplifier with a gain of 8 and fed back to the mach-zehnder interferometer for modulating a refractive index, and phase modulation of the optical signal is achieved. The output optical power chaotic time sequence and the chaotic phase sequence of the optical signal can be obtained by utilizing the calculation method. Through the autocorrelation operation of the signal, no obvious peak is found in the autocorrelation spectrum, see fig. 5, and there are no two clear peaks in the autocorrelation spectrum, which indicates that the delay time is hidden. Further illustrating the accuracy of the calculations in the summary of the invention above.
At the receiving end, the wave separator divides the received signal into two paths, one path passes through the photoelectric detector and the electric amplifier with the gain of 12. The other path drives a photoelectric phase oscillation ring with the structure and the parameters consistent with those of the transmitting end, the calculation shows that the generated optical power chaotic time sequence and the phase chaotic time sequence are completely synchronous with the transmitting end, and the calculation also shows that: the synchronous optical signal of the receiving end is detected by a photoelectric detector, the detected signal is amplified by an electric amplifier with the gain of 10, and the difference operation is carried out with the first path of detected signal, so that the transmitted information can be demodulated accurately. The calculation method has strong reliability.
The invention can accurately solve the differential-integral equation with the frequency-dependent time delay and accurately analyze the feedback chaotic dynamics and chaotic communication with the frequency-dependent time delay.
In the implementation process of this embodiment, the numerical calculation is performed by the above method to obtain: when no transmission information is added, the optical power time sequence diagrams of the transmitting end and the receiving end are shown in the front two diagrams of fig. 3, and the last diagram is two power differences which indicate that the signals are strictly in synchronization; and the upper diagram of fig. 4 is a phase time sequence diagram of the transmitting end, the middle diagram is an optical phase time sequence diagram of the receiving end, and the lower diagram is the phase difference between the transmitting end and the receiving end. It also indicates that the receiver and transmitter phases are in strict synchronization. The upper diagram of fig. 6 shows the information added by the transmitting end, and the lower diagram shows the demodulated information. The calculation of the method shows that due to the robustness of system synchronization, the two ends are strictly synchronized when no information is added, and the information is in a state of losing synchronization when the information is added, so that the information transmitted by the transmitting end can be recovered.
While the preferred embodiments and principles of this invention have been described in detail, it will be apparent to those skilled in the art that variations may be made in the embodiments based on the teachings of the invention and such variations are considered to be within the scope of the invention.

Claims (1)

1. An analysis method of similar frequency-dependent time-delay electro-optic phase chaotic dynamics is characterized by comprising the following steps:
the first step is as follows: establishing a dynamic mathematical model in the chaotic communication system;
the second step is that: determining the relation between the time delay and the frequency;
the third step: dividing time intervals and determining an initial value of a differential-integral equation describing electro-optic phase chaos;
the fourth step: solving a delayed signal by utilizing Fourier time-frequency transformation;
the fifth step: converting the time delay differential equation into an ordinary differential equation;
and a sixth step: carrying out numerical solution on the ordinary differential equation by using a Longge-Kutta method;
the specific method of the first step is as follows:
modeling a transmitting end part in a chaotic communication system, wherein the system adopts a Mach-Zehnder interferometer for phase modulation, and the equation of the transmitting end is as follows:
Figure FDA0002743224660000011
Figure FDA0002743224660000012
wherein τ and θ are response times corresponding to a high cut-off frequency and a low cut-off frequency of the feedback loop, respectively, G is an electrical amplifier gain, Φ is an initial phase, and x, t represent phase and time, respectively;
the specific method of the second step is as follows:
the frequency dependent delay is generated by means of a ring resonator, the transmission equation being:
Figure FDA0002743224660000013
and is
A3＝A4ejβL (4)
Where r and k are the coupling coefficients of the ring resonator,
Figure FDA0002743224660000014
is the transmission constant, n is the refractive index, ω is the frequency of light, c is the speed of light, L is the cavity length, j is the imaginary unit, A1And A2The relationship of (1) is:
Figure FDA0002743224660000015
the transfer function is expressed as
Figure FDA0002743224660000021
Expressing H (ω) as amplitude and frequency, i.e.
H(ω)＝|H(ω)|eiφ (7)
Wherein phiHIs the phase of H (ω), and the frequency-dependent delay is expressed as
Figure FDA0002743224660000022
The third step is specifically as follows:
dividing the total time T into N segments, each segment being long
Figure FDA0002743224660000023
Solving each time period according to the formulas (1) - (2), taking the value of the last time solved in the previous time period as the initial value of the next time period, and substituting the initial value into the next time period to solve, so that the operation is repeated until the equation is solved in the time T; for the first initial value of Δ T, the integral in differential equations (1) and (2) is made equal to 0, then the stable solution of the equations is solved, and this solution is taken as the initial value of the first Δ T;
the fourth step comprises the following specific steps:
[x1(t)+x2(t)]、[x2(t)]after being delayed, Fourier transformedThe inverse transforms are respectively represented as:
Figure FDA0002743224660000024
x22(t)＝x2(t-T-τf)＝ifft{fft[x2(t)]·e-jωT} (10)
wherein fft represents Fourier transform, ifft represents inverse Fourier transform, phiHIs the phase of the ring cavity transfer function H (ω);
the concrete method of the fifth step is as follows:
x1and x2The delayed signal has been solved in a fourth step and becomes an ordinary differential equation for the original equation, as follows
Figure FDA0002743224660000031
Figure FDA0002743224660000032
The Runge-Kutta method is known equation derivative and initial value information, and a process of solving a differential equation is omitted when computer simulation is used;
the process of the Longge-Kuta method is as follows:
for a general equation containing the initial problem:
y′i＝fi(t,y1y2·yi··yn),yi(t0)＝yi(0) i is 1,2, …, n, n represents the number of equations;
obtained by the fourth-order Longge Kutta method
Figure FDA0002743224660000033
h represents a time interval;
wherein
k1＝fj(tj,yj)
Figure FDA0002743224660000034
Figure FDA0002743224660000035
Figure FDA0002743224660000036
The next value (y)i,j+1) From the present value (y)i,j) Adding the product of the time interval (h) and an estimated slope, said slope being a weighted average of: k is a radical of1Is the slope at the beginning of the time period; k is a radical of2Is the slope of the midpoint of the time segment, and the slope k is adopted by the Euler method1To determine y at point tnA value of + h/2; k is a radical of3Also the slope of the midpoint, slope k2Determining the value of y; k is a radical of4Is the slope of the end of the time period, the y value of which is k3And (6) determining.
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