Investigation of Doppler Frequency Shift Effect on the Performance of Four-Channel Space Chaotic Laser Communication

In this article, the wavelength division multiplexing (WDM) technology is applied to the space chaotic laser communication system. We deduce the bit error rate (BER) model of multi-channel space chaotic laser communication with Doppler frequency shift. Based on the BER model, the effect of Doppler frequency shift on the performance of four-channel space chaotic laser communication systems on the synchronous orbit(GEO)-low earth orbit(LEO) and GEO-mid earth orbit(MEO) has been discussed in detail. The numerical simulation results indicate that the effect of Doppler frequency shift on GEO-LEO system is much less than GEO-MEO system. Therefore, the Doppler frequency shift effect of GEO-LEO system can be overcome by redundant design, and the Doppler frequency shift effect of GEO-MEO system must be paid more attention to the optimum design. Besides, the relationship between the performance of communication system and frequency shift under different channels are similar. The result has an important implication for reducing the complexity of the system design. These results of the article are significant for the design of the four-channel space chaotic laser communication system.

a reference role in WDM combined with space chaotic laser communication.
As far as we know, the research on space chaotic laser communication is mainly in a single channel. While the research on multi-channel space chaotic laser communication has not been reported widely yet. The space chaotic laser communication causes the gain mismatch of the photoelectric oscillator and the power mismatch of the receiver due to the existence of pointing error. The main difference between multi-channel space chaotic laser communication and space chaotic laser communication is the introduction of WDM technology, which will have an impact on the performance of communication system. The influence is generated by the optical filter in the wavelength division multiplexer. On the one hand, the optical filter has an impact on the signal optical power, then affects the internal and external mismatch and the performance of the system. On the other hand, a small part of the signal light of other channels will enter one channel through the filter. The signal optical power of other channels has been defined as crosstalk, which have an impact on the performance of the multi-channel space chaotic laser communication system. And the optical filter is related to the frequency of the light. Therefore, the signal power and crosstalk are related to the frequency of the light. As we know, there is Doppler frequency shift between satellites, which will affect the signal power and crosstalk, and further affect the performance of multi-channel space chaotic laser communication system. Therefore, it's necessary to research the effect of Doppler frequency shift on the performance of the inter-satellite four-channel chaotic laser communication system.
In this article, we deduce the calculation formula of intersatellite Doppler frequency shift first. And considering the effect of Doppler frequency shift, the BER formula of multi-channel space chaotic laser communication is deduced. Due to the satellites in different orbits, the article will first discuss the effect of Doppler frequency shift on the mismatch and BER of the GEO-LEO satellite four-channel chaotic laser communication system in detail. Then, we will discuss the effect of Doppler frequency shift on the mismatch and BER of the GEO-MEO satellite four-channel space chaotic laser communication system. The numerical simulation results indicate that the BER deterioration of the GEO-MEO satellite chaotic communication system is greater than that of the GEO-LEO satellite chaotic communication system. Therefore, the Doppler frequency shift effect of GEO-MEO system must be paid more attention to the optimum design. And the Doppler frequency shift effect of GEO-LEO system can be overcome by redundant design. In addition, the characteristics of four channels are basically the same. The result shows that the design of a channel system has reference significance for other channels, which has a great significance for reducing the complexity of system design. These results of this article are significant for the design of the four-channel space chaotic laser communication system.

II. PRINCIPLES AND THEORETICAL ANALYSIS
The configuration of four-channel space chaotic laser communication system is shown in Fig. 1(a). It is mainly composed of chaotic loop at transmitting terminal, wavelength division multiplexer (WDM), erbium-doped fiber amplifier (EDFA), and chaotic loop at receiving terminal. Chaotic signals of the transmitting terminal are combined into one signal by WDM and amplified by EDFA1. Then the mixed-signal reach the receiving terminal through the space channel. After the signal arrives at the receiving terminal, it will be amplified by EDFA2 and divided into four-channel through WDM. The specific configuration of the transmitter and receiver are shown in Fig. 1(b) and (c) separately. Chaotic carrier is generated by photoelectric feedback loop (Loop 1) composed of Mach-Zehnder modulator (MZ1), fiber delay line, avalanche photodiode (APD3), and RF amplifier (RFA1). Then, the signal emitted by the laser diode (LD3) will be coupled with the chaotic carrier through the 2×2 coupler at the transmitting terminal. At the receiver, the light is divided into two parts by a 1×2 coupler. More specifically, one part will be detected by APD1, and the other part will be used as the feedback signal of the photoelectric feedback loop (loop 2). The configuration and parameters of loop2 are the same as loop1. Chaotic carrier generated by photoelectric feedback loop (loop 2) will be detected by APD2. Considering that the receiver and transmitter have the same configuration and parameters, the chaotic carriers at the transmitting terminal are the same as receiver. Therefore, the message signal will be demodulated from the chaotic carrier by the power combiner.
It should be noted that four-channel space chaotic laser communication system is affected by the pointing error. The pointing error includes boresight error and jitter error, and the probability density function (PDF) of radius deviation can be obtained from Rice distribution [20].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where σ s = σ j L is boresight, A = A j L is jitter, L is the distance between two satellites, I 0 is the Zero-order Bessel function, r is the radial displacement of beam center and detector center.
The pointing error will lead to random misalignment between the beam center and the detector center, and the detector at the terminal will receive less power. Specifically, for Gaussian beams, the receiving power P T (r) is a function of the distance deviation r of the optical center, which can be given as [21] where P T = G EDFA1 P 1 α is the optical power without pointing error at receiving terminal. G EDFA1 represents the gain coefficient of EDFA1, P 1 is the power of LD1, geometric attenuation α = α loss D r 2 /W 2 . α loss is the coefficient of attenuation, D r is the aperture of receiver, W = W 0 +θL/2 is beam radius of receiver, W 0 is beam radius of transmitting terminal, θ is beam divergence angle.
In order to explain the chaotic signal with pointing error more concretely, the signals arriving at the receiving terminal APD1 and APD2 of the space chaotic laser communication system are [22] where G EDFA1 is gain coefficient of EDFA1, G EDFA2 is gain coefficient of EDFA2, V π 1 is dynamic half-wave voltage of MZ1, V π 2 is dynamic half-wave voltage of MZ2, d is mask ratio, P 2 is power of LD2, ϕ 1 is the fixed phase shift at the transmitting terminal, ϕ 2 is the fixed phase shift at the receiving terminal, x(t) is the feedback voltage of transmitting terminal at time t, y(t,r) is the feedback voltage of receiving terminal at time t, which is related to the radius deviation. For four-channel space chaotic laser communication system. One terminal is on the satellites in geosynchronous orbit, which is a satellite communication transfer station [23]. Other terminals are on the satellites in low Earth orbit or mid-Earth orbit. Four-channel space chaotic laser communication is different from space chaotic laser communication. The four-channel space chaotic laser communication system introduced WDM technology. WDM induces changes in the optical power of each channel. These changes are mainly caused by optical filter. The power through the optical filter can be obtained as [24] P where f 0 is the center frequency of filter, f FP is the bandwidth of filter. f LD is the bandwidth of laser. So when the bandwidth of laser is much smaller than the bandwidth of filter, the power through the optical filter P out can be modified as Since the optical filter has an impact on the optical power. Based on (6), the power of chaotic signal through the filter P(r) can be expressed as The optical filter also brings linear crosstalk to system. The power of the linear crosstalk P l (r) can be derived as where N is the number of channels, and P Ti (r) is the power of channel i signal through the filter. In addition, since the Doppler frequency shift has an impact on light frequency, and the power after the filter is related to the light frequency, the Doppler frequency shift will affect the performance of the system. Based on [25], the Doppler frequency shift between two orbits can be derived as where θ = wt is the angle between two satellites, R h is the distance between the GEO satellite and the earth's spherical center, R l is the distance between the other orbit satellite and the earth's spherical center, the relative angular velocity is w, t represents time and f is the initial frequency. Finally, at the receiving terminal, the power of PD1 and PD2 can be expressed as where K(r) is the amplitude of chaotic signal after APD1, K l (r) is the amplitude of crosstalk after APD1, e is elementary charge, η is quantum efficiency, h is Planck constant, v is laser light frequency. K' is the amplitude after APD2. Then two signals pass through the power combiner, and the signal m(t) is demodulated.
Through the above analysis, it is found that the main factor affecting the quality of four-channel space chaotic laser communication is the power mismatch caused by pointing error and optical filter.
The mismatch is an important parameter that affects space chaotic laser communication system. The reference [26] has a comprehensive analysis for the mismatch, which can be given by where <n 2 > represent the chaotic mismatch noise, Δϕ is phase deviation, ΔK(r) is external mismatch caused by filter and pointing error, the synchronization error of the system <ε 2 (r)> is given by [27].
Meη hv (21) where β 1 is the gain of chaotic loop at transmitting terminal, G 1 is the gain coefficient of RFA1. Δβ(r) is internal mismatch, β 2 is the gain of chaotic loop at receiving terminal, G 2 is the coefficient of the receiving terminal, β l is gain of crosstalk in chaotic loop at receiving terminal, ΔT is the mismatch of time delay, τ is the high cutoff response time, and Δτ is the mismatch of high cutoff response time.
Based on [26], the signal-to-noise ratio (SNR) of the system can be modified as Then, the BER(r) can be modified as below Since the existence of the pointing error, BER is related to the radial displacement r between the center of the beam and the center of the detector, as shown in (23). In light of the pointing error varying much more slowly than the transmitted digital signals, the ensemble average BER can be expressed as the integral of BER(r) and the probability density function (PDF) of radius deviation ρ(r) based on [22].

III. NUMERICAL SIMULATION RESULT
Based on the BER model in the second section, firstly we will analyze the effect of Doppler frequency shift on the external and internal mismatch on the four-channel space chaotic laser communication between GEO and LEO satellite. Then, the effect of Doppler frequency shift on external and internal mismatch on the four-channel space chaotic laser communication between GEO and MEO satellite will be analyzed. Finally, the effect of the frequency shift on system performance will be discussed further in detail.
The numerical simulation are based on these parameters as follow: the center wavelengths of the four channels are 1549.2 nm, 1550 nm, 1550.8 nm, and 1551.6 nm respectively, filter bandwidth is 0.4 nm, both the jitter and boresight are 1 μrad, power of the emitter P 1 = 4 mW, power of the receiver P 2 = 2 mW, loss efficiency α loss = 1, receiving aperture D r = 1 m, divergence angle θ = 20 μrad, APD gain factor M = 80, quantum efficiency η = 0.75, the gain factor of RFA1 and RFA2 G 1 = G 2 = 10, time delay mismatch ΔT = 1 ps, high cutoff response time τ = 50 ps, high cutoff response time mismatch Δτ = 1 ps, offset phase mismatch Δϕ = 0.02.
In order to analyze the relationship between Doppler frequency shift and the performance of four-channel space chaotic laser communication system in detail, the change of Doppler frequency shift on the GEO-LEO and GEO-MEO satellite optical communication system will be obtained firstly. Base on (9), the angle between satellites is one of the most important factors that affect frequency shift. When the angle changes from 45°to 45°, the Doppler frequency shift between GEO and LEO satellite, and the Doppler frequency shift between GEO and MEO satellite are shown in Fig. 2. For the GEO-LEO satellite system, the Doppler frequency shift is 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz respectively when angle is 0°, ±15°, ±30°, and ±45°. And the Doppler frequency shift between GEO and MEO satellite is 0 GHz, 9.9 GHz, 14.7 GHz, and 16.3 GHz respectively. The Doppler frequency shift between GEO and MEO satellite system is greater than that of GEO-LEO satellite system. In the  In addition, the purpose of introducing chaos technology is to improve the security of the system from the physical layer. The BER of authorized receiver (Bob) and the BER of eavesdropper (Eve) in GEO-LEO satellite four-channel chaotic laser communication are shown in Fig. 3. The numerical simulation results indicate that BER of Bob is less than 10 −5 in all channels. While, compared with the BER of Bob, the BER of Eve is always larger than 10 −2 in all channels. Therefore, it is difficult for the eavesdropper to get the effective message. The situation of GEO-MEO satellite four-channel chaotic laser communication  is the same as the case of GEO-LEO satellite chaotic laser communication. Here we do not give the detail again. As we know, the mismatch is an important parameter of a chaotic communication system. So, we will observe the effect of Doppler frequency shift on mismatch of channel 1. According to (16), the external mismatch is affected by the amplitude of chaotic signal after APD1 K(r), the amplitude of crosstalk after APD1 K l (r), and the amplitude after APD2 K'. As shown in Table I, the value of K is always less than K'. Therefore, the (16) can be written as ΔK = K'+K l (r)-K(r). When frequency shift is set 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz, the value of K'+ K l is 149.96 mA, 149.94 mA, 149.93 mA, and 149.93 mA separately. Obviously, the value of K'+ K l is almost unchanged under different frequency shift. So, the value of K'+ K l (r) can be assumed as a constant K s . For internal mismatch, (18) should be expressed as Δβ = β 1 +β l (r)-β 2 (r) and the value of β 1 +β l should be assumed as a constant β s . Specifically, when frequency shift is set 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz, the value of β s is 523.49 mA, 523.47 mA, 523.46 mA, and 523.45 mA separately. The value of β 1 +β l is also almost unchanged under different frequency shift.
Based on the above conclusion, we will analyze the mismatch in detail. As shown in Fig. 4(a), when the frequency increases, the external mismatch will become larger. Specifically, when Δf is 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz, the external mismatch rate is 23%, 25.1%, 30.1%, and 36% respectively. The reason is that the power after the filter becomes smaller with the increase of frequency shift. It causes the optical power of APD1 smaller. Fig. 4(b) shows the probability distribution of current amplitude K under different frequency shifts. K s represents the value of K'+K l (r). When the frequency shift is 0 GHz (0°), 3.5 GHz frequency shift (15°), 6.2 GHz (30°), and 8.1 GHz (45°), the average value of K(r) is K 1 , K 2 , K 3 , and K 4 separately. We can observe from the Fig. 4(b) that K 1 >K 2 >K 3 >K 4 . PDF curve shifts left when frequency shift increases. Therefore, the external mismatch rate ΔK/K = (K s -K)/K will decrease with the increase of frequency shift. The red line depicts the relationship between external mismatch rate ΔK/K and K, and the red line is not a straight line. The reason is that when the external mismatch rate ΔK/K = (K s -K)/K is modified to ΔK/K = K s /K-1, it can be easy to find that the external mismatch rate ΔK/K and K is an inverse proportional function.
Then, we analyze the relationship between frequency shift and internal mismatch. As shown in Fig. 4(c), when Δf increase from 0 GHz to 8.1 GHz, the internal mismatch rate will deteriorate 7.8%. The reason is similar to the external mismatch. As shown in Fig. 4(d), β s represent β 1 +β l (r). The average gain of loop2 β 2i (i = 1, 2, 3, 4) will become smaller when Δf increases. Therefore, the value of Δβ/β 1 will be increased when frequency shift increases. The red straight line depicts the relationship between β and internal mismatch rate. In addition, we can observe that when Doppler frequency shift is 0 GHz, both external mismatch and internal mismatch rate are not 0. The main reason is that when Doppler frequency shift is 0 GHz, the pointing error will still introduce the internal mismatch and external mismatch into four-channel space chaotic laser communication system.
After observing the relationship between mismatch and Doppler shift of channel 1, the effect of frequency shift on internal and external mismatch for other channels will be discussed. From Fig. 5, it can be found that the trends of the four channels are basically the same, with only minor differences. Specifically, when frequency shift is 0 GHz and 8.1 GHz, the external mismatch of all channels are about 23% and 36% respectively. And the internal mismatch of all channels are about 18.5% and 26.3% respectively. The external mismatch of all channels has deteriorated by about 13% with the increase of Doppler frequency shift. And the internal mismatch rate increases about 7.8%.
Then, the reason for this phenomenon will be analyzed. From (14) and (19), we can know that the value of K' and β 1 of all  II  PARAMETERS THAT AFFECT INTERNAL AND EXTERNAL MISMATCH OF  ALL CHANNELS ON THE GEO-LEO SATELLITE FOUR-CHANNEL SPACE  CHAOTIC LASER COMMUNICATION Table II, the average current amplitude K of all channels is equal at a specific frequency shift, and the average loop gain β 2 of all channels is equal at a specific frequency shift. Specifically, when frequency shift is set 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz, the average current amplitude K is 122 mA, 120 mA, 115 mA, and 111 mA respectively, and the average loop gain β 2 is 427 mA, 418 mA, 402 mA, and 386 mA respectively. In addition, both the average gain of crosstalk in chaotic loop β l and the average current amplitude of crosstalk K l are much smaller than other parameters. This leads to little change in the mismatch. So, the trends of the four channels are basically the same.
To better discuss the effect of Doppler frequency shift on the performance of four-channel space chaotic laser communication system between GEO and LEO satellite, the relationship between frequency shift and BER is depicted in Fig. 6. The x-axis represents the channel. The -log10(BER) will deteriorate at 0.87 when the frequency shift changes from 0 GHz (0°) to 8.1 GHz (45°). And the BER of all channels is almost the same at a specific frequency shift. When the frequency shift is 0 GHz, the -log10 (BER) of all channels is about 6. The -log10 (BER) of all channels is about 5.13 when the frequency shift is 8.1 GHz.
From (23), BER is related to SNR directly. To explain the relationship between frequency shift and BER, we further calculate the average SNR. The data is shown in Table III. The average SNR of all channels is about 24.9, 22.3, 18.2, and 15.2 when Δf is set 0 GHz, 3.5 GHz, 6.2 GHz, and 8.1 GHz separately. The average SNR of all channels is almost the same at a specific  frequency shift. It is obvious that the SNR trends of all channels are almost the same. That means Doppler frequency shift has almost the same effect on all channel. Therefore, the designer only needs to focus on one channel. The result simplifies the design of a four-channel chaotic laser communication system. In addition, the value of SNR will deteriorate about 9.7 when the frequency shift becomes larger. The above results show that in the design process of the four-channel chaotic laser communication system between GEO and LEO satellites, the influence of the Doppler frequency shift is limited and can be overcome by redundant design. The effect of Doppler frequency shift on the performance of four-channel space chaotic laser communication between GEO and LEO satellite is discussed above, the GEO-MEO satellite communication system will be discussed likewise as below. Fig. 7(a) and (c) depict the external mismatch and the internal mismatch separately of channel 1. When the frequency shift becomes larger, both external mismatch and internal mismatch deteriorate. Specifically, when frequency shift is 0 GHz (0°), 9.9 GHz (15°), 14.7 GHz (30°), and 16.3 GHz (45°), the external mismatch rate is 21.3%, 39.8%, 62.9%, and 72.9% respectively, and the internal mismatch rate is 17.3%, 28.5%, 38.5%, and 42% respectively. The reason is the same as above. As shown in Fig. 7(b) and (d), the average current amplitude K and the average loop gain β 2 become smaller when Δf increases, both K s (equal to K'+K l (r)) and β s (equal to β 1 +β l (r)) are assumed as constant. Therefore, the external mismatch ΔK/K and internal mismatch Δβ/β 1 will deteriorate when the frequency shift becomes larger.  In addition, the variation range of external mismatch and internal mismatch is 51.6% and 24.7% separately. The deterioration is larger than the GEO-LEO satellite communication system. The reason is that the frequency shift between GEO and MEO satellite is greater than the frequency shift between GEO and LEO satellite at the same angle. Then, we discuss the relationship between frequency shift and mismatch of all channels. As shown in Fig. 8, when the angle between satellites changes from 0 without frequency shift to 45°with 16.3 GHz frequency shift, the external mismatch and internal mismatch deteriorate 51.6% and 24.7% separately. We observe the curves of each channel are very similar from Fig. 8. The reason for this phenomenon is similar to the GEO-LEO satellite four-channel space chaotic communication system. The average current amplitude of crosstalk K l and the average gain loop of crosstalk β l are obtained from Table IV. Just like K l and β l of channel 1, when Doppler frequency shift is 0 GHz, 9.9 GHz, 14.7 GHz, and 16.3 GHz, the value of K l is 0.11, 0.119, 0.131, and 0.136 respectively, and the value of β l is 0.385, 0.415, 0.457, and 0.475 respectively. The value of K l and β l are much smaller than K and β 2 respectively.
To better discuss the effect of Doppler frequency shift on the performance of four-channel space chaotic laser communication between GEO and MEO satellite, the relationship between the frequency shift and BER is depicted in Fig. 9. The -log10(BER) will deteriorate 3 when the frequency shift changes from 0 GHz to 16.3 GHz. And the BER of all channels is almost the same at a specific frequency shift. When the frequency shift is 0, the -log10(BER) of all channels is about 6.1. The -log10(BER) of all channels is about 3.1 when frequency shift is 16.3 GHz. For the GEO-MEO satellite four-channel space chaotic laser communication system, the Doppler frequency shift has a great impact on communication performance. The relationship between Doppler frequency shift and BER will be explained in detail below.   Table III, we can discover the attenuation of SNR in the GEO-MEO satellite communication system is greater than that of the GEO-LEO satellite system. That leads to the deterioration of BER on the GEO-MEO satellite chaotic communication system is greater than that of the GEO-LEO satellite chaotic laser communication system. Therefore, for the design of the GEO-MEO satellite four-channel space chaotic laser communication system, compensation for the Doppler frequency shift is necessary.

IV. CONCLUSION
In summary, the relationship between the performance of the inter-satellite four-channel chaotic laser communication and Doppler frequency shift is analyzed in detail. The Doppler frequency shift will affect the power through the optical filter then bring the mismatches of amplitude current and optoelectronic oscillator gain to chaotic synchronous demodulation process. The BER analysis with mismatch caused by Doppler frequency shift on four-channel space chaotic laser communication system is conducted. To better explain the relationship between the performance of four-channel space chaotic laser communication and frequency shift, we analyzed the effect of the frequency shift on the GEO-LEO and GEO-MEO satellite communication systems respectively. Specifically, The GEO-MEO satellite chaotic communication system external mismatch and internal mismatch deteriorate 51.6% and 24.7% separately. The -log10(BER) of the communication system deteriorates about 3. And the GEO-LEO satellite chaotic communication system external mismatch and internal mismatch deteriorate 13% and 7.8% separately. The -log10(BER) of the GEO-MEO communication system only deteriorates about 0.87. The numerical simulation results indicate that the effect of Doppler frequency shift on GEO-LEO system is much less than GEO-MEO system. Therefore, the Doppler frequency shift effect of GEO-MEO system must be paid more attention to the optimum design. While the GEO-LEO system can be overcome by redundant design. In addition, we have observed the relationship between the performance of communication system and frequency shift under different channels are similar. The result shows that the design of a channel system has reference significance for other channels, which has a great significance for reducing the complexity of system design. These results are significant for the design of the four-channel space chaotic laser communication system.