In the range measurement, the ranging signal is used to inform the signal arrival and can be regarded as a trigger signal for the receiver. Thus, the form of the uplink signal can be diverse, not necessarily the pure ranging signal. To conduct a range measurement, the ranging information is imbedded in the ranging signal and transmitted between objectives. The ranging information could also be integrated into the communication signal and would have little interference on the communication data transmission. Thus, there is a possibility that the communication and the range measurement can be performed simultaneously.

Figure  1(a) presents the time scheme of the simultaneous communication and ranging method and figure  1(b) indicates the designed signal symbol structure. It should be noted that the uplink and downlink signals share a similar symbol structure. For the uplink signal, there is an uplink symbol identifier at the head of the symbol. Following the identifier is the communication data. The symbol identifier contains two parts: the synchronization sequence and the sequence number (SN). The synchronization sequence is generally a pseudo random code with a fine autocorrelation property, used for the acquisition of the symbol. The SN is the unique identifier of the symbol that is used for the time synchronization of the range measurement. The SN is transmitted to the slave station and sent back to the master station. By pairing up the received SN with the local SN at the master station, we could obtain the arrival time of the specific SN. Knowing the emitting time of the specific SN, we could acquire the two-way light travel time τ . Subsequently, the range can be calculated by

where c is the speed of light.

Fig.  1.

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	Fig.  1. Simultaneous communication and ranging method: (a) time scheme of the proposed method, (b) signal symbol structure.

The flow charts of the signal processing procedures in the master station and the slave station are presented in Figs.  2 and 3, respectively. It can be seen that the master station and the slave station share some similarities in the signal processing procedure. In the following sections, we discuss the system structure by analyzing the key points of the proposed method.

Fig.  2.

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	Fig.  2. Flow chart of signal processing in the master station.

Fig.  3.

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	Fig.  3. Flow chart of signal processing in the slave station.

2.2.1. Modulation scheme

The proposed method utilizes the circular polarization modulation to modulate the x-ray signal. Polarization is the unbalanced vibration of the light vector on the plane perpendicular to the light propagation direction.^[18] A polarization state shows fine stability in space propagation, ^{[19, 20]} which makes it a proper choice for space communication.^{[21, 22]} The circular polarization modulation is to convert binary codes into circular polarization states. To describe the circularly polarized signal, the Stokes vector is adopted

where A_x and A_y are the amplitudes of the light components, and δ is the phase error between the two light components, i.e., δ = θ _x − θ _y, with θ _x and θ _y being the phases of the light components indicated in Fig.  4. The δ determines the type of polarization state. Then, we can express the circularly polarized signal as

where M is the length of the signal sequence,

and s(t) is the signal wave defined as

where T is the slot duration, g(t) is the gate signal, and {d_i} is the signal in binary form.

Fig.  4.

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	Fig.  4. Light vector components of polarized light.

The detection of the circularly polarized signal is realized with a series of optical devices and the differential detection is adopted to mitigate the noise interference. The details in detecting the circularly polarized signal will be presented in the sections of performance analysis.

2.2.2. Signal synchronization

Signal synchronization aims at separating one signal symbol from another and setting the boundaries of the symbols. The synchronization is conducted by correlating the synchronization sequence of the symbol with the local sequence. The property of the correlation process determines the quality of the signal acquisition. Thus, choosing a proper synchronization sequence becomes an issue of great importance.

The m-sequence and Gold sequence are widely used synchronization sequences in various areas, such as spread spectrum communication, navigation, signal synchronization, etc. The Gold sequence is generated by the modulo-2 addition of two m-sequences.^[23] As to the correlation property, the m-sequence has a better auto-correlation property than the Gold sequence while the Gold sequence shows better performance in cross-correlation, as shown in Fig.  5. For the proposed method, we are concerned more about the autocorrelation property of the sequence. Thus, the m-sequence is chosen as the synchronization sequence for the proposed method.

Fig.  5.

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Fig.  5. Normalized autocorrelation property of the m-sequence and the Gold sequence. The generation of the m-sequence is based on a simple sequence register generator (SSRG) with an order of 6, and the Gold sequence is generated with two m-sequences with the order of 6 using the parallel generation structure.[23, 24] For the m-sequence, the ratio of the correlation peak value to the second peak value, termed as the peak ratio, is 10.5, while for the Gold sequence it is only 3.7. It means that the m-sequence has a better autocorrelation property for detection than the Gold sequence under the same condition.

As shown in Fig.  2, an SN generator is used to generate SN for the signal symbol. The function of SN is for the time synchronization in the system. It should be noted that SN increases monotonically when generated. The repetition rate of the SN generator is defined as f_{rep_SN} = 1/2^l_SN, where l_SN is the length of SN. It should be noted that the repetition rate is not directly related to time, but to the length of SN. On the other hand, the ranging repetition rate is related to the length of the symbol, defined as f_{rep_range} = 1/MT, where M is the length of the signal symbol in units of slots and T is the slot duration. To guarantee a proper ranging repetition rate, the length of the symbol is limited.

As shown in Fig.  1, the data region of the signal symbol has a length of M − l_SS − l_SN, where l_SS is the length of the synchronization sequence. As the signal is modulated by the circular polarization modulation method, the data transmitted by one symbol is M − l_SS − l_SN bits. Thus, the data rate (bit/s) is defined as

From Eq.  (6), we know that the data communication rate is related to the symbol structure and the slot duration. Besides, the data communication rate conflicts with the ranging repetition rate. Therefore, there should be a balance between the data communication rate and the ranging repetition rate.

As discussed in the previous section, the length of SN determines the repetition rate of SN, f_{rep_SN}. Further, f_{rep_SN} influences the maximum unambiguous range (R_max). The maximum number of symbols marked by SN is N_SN = 2^l_SN. For the proposed method, the maximum unambiguous range is defined as half of the maximum range that the symbol travels before another symbol with the same SN is emitted. Then, the maximum unambiguous range L_m can be expressed as

Based on Eq.  (7), with a larger l_SN, a longer R_max could be obtained. It can also be concluded that R_max is in conflict with the range repetition rate. To obtain a long R_max, the ranging repetition rate is limited. Besides, it should be noted that SN only contributes to the ranging process and has little impact on the communication data transmission.

For the proposed method, an m-sequence with a length of l_SS is adopted as the synchronization sequence, which is used for the signal symbol acquisition. We define the synchronization sequence as

where {m_i,   i = 0, … , l_SS − 1} is the m-sequence code and g(t) is the gate function defined as

Passing through the deep space channel, s_SS (t) is influenced by noise and becomes

where n(t) is a zero-mean additive Gaussian white noise with a two-sided power spectral density of N₀/2. To acquire the signal symbol, the received synchronization sequence is correlated with the local synchronization sequence. The correlation between s_SS(t) and can be written as

where T_corr is the correlation time and ξ _SS is the influence of noise on the correlation. We have

The ξ _SS(k) is a Gaussian variable with zero mean value. As n(t) has a two-sided power spectral density of N₀/2, the variance of ξ _SS(k) is N₀T_corr/2.

Fig.  6.

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	Fig.  6. Impact of noise on synchronization: (a) average correlation peak values, (b) peak ratio of the correlation results under different SNR.

A simulation is performed to show the impact of noise on the synchronization. For the simulation, m-sequences with different orders (n = 7, 8, 9, 10) are used and additive Gaussian noises with different power levels are added to the m-sequences. Then, the noise-interfered m-sequence is correlated with the local sequence without noise. The slot duration is set to T = 10  ns. Figure  6(a) shows the average correlation peak values. For each SNR in the figure, the peak values of the correlation results are averaged over 100 trials. It can be observed in Fig.  6(a) that the correlation peak is high and fluctuant at lower SNR. The high and fluctuant correlation at low SNR is caused by the high noise power. When SNR is large, the peak values become steady, which indicates the weak influence of noise. Figure  6(b) is the peak ratio of the correlation results under different SNR. Here, we define the peak ratio as p_r = p_{in− phase}/p_max, where p_{in− phase} is the peak value of the correlation result when the two sequences are in-phase, termed as the in-phase peak, and p_max is the peak value of the correlation results excluding the in-phase peak. Similar to Fig.  6(a), for each SNR, the correlation process is performed 100 times and the peak ratios are averaged over the 100 trials. It can be seen that as the SNR increases, the peak ratio increases accordingly, which means with high SNR, the peak value can be clearly separated. At low SNR, the peak ratio is small because of the influence of noise. When the SNR is extremely low, the peak ratio is under 1. The reason is that the in-phase peak is overwhelmed by noise and then the in-phase peak value cannot be correctly detected.

Besides the correlation results, the acquisition probability, denoted by P_acq, is utilized to evaluate the acquisition performance. The acquisition probability is defined as the probability of successfully acquiring the peak value of the correlation results. Then, we have the following expression:

Though ξ _SS(0) and ξ _SS(k) are both Gaussian distributions with zero mean value, it should be noted that ξ _SS(0) and ξ _SS(k) are not independent. This is because s_SS(t − kT) and s_SS(t) are coupled by the time delay. Since the length of the synchronization sequence is limited and the influence of the coupling on the integration can be ignored, ξ _SS(0) and ξ _SS(k) can be considered to be approximately independent. Therefore, ξ _SS(k) − ξ _SS(0) follows a Gaussian distribution with a zero mean value and a variance of N₀T_corr, and

Then equation (13) can be rewritten as

Figure  7 shows the acquisition probability under different SNR. In the simulation, m-sequences of different orders (n = 5, 6, 7, 8) are used and Gaussian white noises of different power levels are added to the sequences. Then the acquisition probability is calculated. As shown in the figure, under low SNR, the synchronization is influenced by the noise and the acquisition probability is small. With increasing SNR, the acquisition probability rises accordingly because of the weak noise influence. It can also be concluded that a longer synchronization sequence shows better performance in acquisition. The reason is that the long synchronization sequence has better correlation performance and anti-noise property.

Fig.  7.

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	Fig.  7. Acquisition probability under different SNR.

Because of the noise in the communication channel, the signal is distorted during the propagation. To evaluate the distortion, the bit error rate (BER) is commonly used. As for BER, it varies with different choices of the modulation methods. Besides, passing through the deep space channel, x-ray scintillation occurs. The x-ray scintillation will influence the intensity of the light and consequently have an impact on the BER.

For the scintillation, there are several models available.^[25] The most widely used models are the Lognormal distribution model, the K-distribution model, and the Gamma– Gamma distribution model. The Lognormal distribution model is a simple model used to analyze the light scintillation, which is defined as

where I is the irradiance of the light, Ī is the mean of I, and σ _s denotes the scintillation index. The K-distribution is defined as

where Γ is the Gamma function, K_t is the modified Bessel function of the second kind, ^{[26, 27]} and α is a parameter concerning the effective number of discrete scatters, satisfying α > 0. Here α determines the order of the Bessel function. In Eq.  (17), we assume that the light irradiance is proportional to the detector output current. The K_t can be defined as

where I_t(x) is the modified Bessel function of the first kind. Finally, the Gamma– Gamma distribution is

It can be seen in Eqs.  (17) and (19) that these two models share some similarities. Actually, when β = 1, the Gamma– Gamma distribution becomes the K-distribution. The Lognormal distribution generally applies to the conditions of weak fluctauation. When the turbulence turns stronger and different scattering effects occur, the K-distribution finds its application. The Gamma– Gamma distribution is a widely used model in various cases, which connects its paramaters with the propagation channel conditions.

Fig.  8.

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	Fig.  8. Differential detection of circularly polarized signal.

To calculate the BER, we should know the detection of the circularly polarized signal besides the scintillation model. For the circularly polarized signal detection, the differential detection method is adopted, which is shown in Fig.  8. The circularly polarized signal at the receiver side has the form

where r(t) is the received signal in the binary format and τ ′ is the one-way time delay after the space propagation. Being through a quarter wave plate (QWP), S becomes

where

and J_QWP is the Mueller matrix of QWP.^[28] Apparently, L is a linearly polarized signal. Then, L passes through a polarization beam splitter (PBS). The PBS aims at separating two orthogonal light components into two independent channels, which are denoted as channels l₁ and l₂, respectively. For channel l₁, we have

Similarly, for channel l₂, we have

Thus, with the left-hand polarization state received, in channel l₁ and in channel l₂. On the other hand, with the right-hand polarization state transmitted, there is in channel l₁ and in channel l₂. At the same time, the output current of the photoelectric detector is proportional to the light intensity. Thus, by calculating the current difference between the two photoelectric detectors, we can demodulate the polarized signal. Take the left-hand circular polarization state as an example. When the left-hand polarization state is received, the output current is

where i_s is the current caused by the input light, and i_n1 and i_n2 are the detector noises which follow a Gaussian distribution with a zero mean value and a variance of . The differential current Δ i_l can be defined as

By assuming that i_n1 and i_n2 are independent, i_n1 − i_n2 follows a Gaussian distribution of . Similarly, for the right-hand circular polarization state, we have

Then, the probability density function of the differential current caused by the left-hand and the right-hand polarization states can be written as

For the left-hand polarization state, the BER is

where erfc is the complementary error function. Similarly, for the right-hand polarization state, there is

Taking the scintillation effect into account, we can define the average bit error rate as

where SNR is the signal-to-noise ratio defined as , and ī _s is the average output current of the photoelectric detector. As indicated by Eq.  (34), P_BER is influenced by the SNR. This can also be illustrated by the simulation results, which is shown in Fig.  9.

Figure  9 shows the BER calculated under different scintillation models. Apparently, the BER is influenced by both the parameters of the scintillation models and the SNR. For the Lognormal distribution in Fig.  9(a), its performance is influenced by the scintillation index σ _s. A smaller σ _s leads to a lower BER curve. For the K-distribution in Fig.  9(b), it is influenced by the order of the Bessel function. However, the difference caused by the order is small. For the Gamma– Gamma distribution, there are two key parameters α and β . Different combinations of α and β lead to different BER curves. The Gamma– Gamma model is actually an extension of the K-distribution. When β = 1, the Gamma– Gamma distribution becomes the K-distribution, which can be known from Figs.  9(b) and 9(c). It can be seen from all the figures that a high SNR leads to a low BER, indicating the effects of noise on the BER. It can also be observed that the BER curves share a similar tendency under the same SNR conditions and that there is only a tiny difference between the BER curves of different scintillation models. Because of the complex and changeable deep space circumstances, no model can perfectly describe the scintillation effects. As described in Ref.  [25], these models have varying degrees of success. Thus, the scintillation model is generally chosen based on the condition of the propagation channel.

Fig.  9.

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	Fig.  9. BER under different scintillation models: (a) the Lognormal distribution, (b) the K-distribution, (c) the Gamma-Gamma distribution.

As discussed in Subsection 2.1, the ranging process is conducted by matching the SN from the downlink with the local SN. Thus, the synchronization process will have an impact on the ranging jitter because the inaccurate synchronization will lead to the time synchronization error for the range measurement.

The synchronization sequence is defined in Eq.  (8) and the signal passing through the propagation channel is expressed by Eq. (10). For the synchronization process, the aim is to find k with the maximum value of R_corr(k), i.e.,

As ξ _SS(k) is a random variable, thus κ can be seen as a random variable. Apparently, κ is influenced by the noise of the synchronization sequence. Then, the variance of κ T, , contributes to the ranging jitter. The ranging jitter, denoted as σ _R, can be presented as

Because we cannot have an analytical form for Eq.  (35), we use the Monte Carlo simulation to analyze the variance of κ and give the relation between the ranging jitter and the SNR.

In the simulation, the Monte Carlo method is used to analyze the impact of the synchronization error on the ranging jitter under different SNR levels, and m-sequences of different orders are used. For each SNR, 1000 correlation trials expressed by Eq.  (35) are conducted and the variance of κ T is calculated. Based on Eq.  (36), the ranging jitter is calculated and presented in Fig.  10. From the figure, we can see that when the SNR is low, the synchronization error causes a large ranging jitter, indicating the impact of the noise on the range measurement. As the SNR increases, the ranging jitter goes down. We can also know that an m-sequence of high order has a better ranging jitter performance. The reason is that the high order m-sequence has a better synchronization performance, which has been demonstrated in the previous sections. It should be noted that under lower SNR, the ranging jitter of the high order m-sequence is larger than that of the lower order one. The reason is that at lower SNR, the noise contributes more to the correlation process of m-sequences with high order than those with the low orders, which influences the in-phase judgment. This is also the reason for the high correlation peak at low SNR in Fig.  6(a).

Fig.  10.

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	Fig.  10. Ranging jitter.

For the proposed method, the ranging signal and the communication signal are integrated into a single signal symbol structure. Thus, the impact of the ranging signal on the communication signal mainly lies in the macroscopic aspects, such as the power consumption, the data rate, etc. In the following, we analyze the interference mainly from these aspects.

The power consumption depends on the signal symbol structure. Because of the existence of the synchronization sequence and the SN sequence, the power is not entirely assigned to communication. We define the power efficiency η as the ratio of the power assigned to communication to the total signal power. Based on the signal symbol structure, η can be expressed as

Apparently, large l_SN and l_SS will decrease the communication power efficiency when the symbol length is fixed. Small l_SN will improve the communication power efficiency, but it will in turn decrease the ranging repetition rate and lower the ranging jitter performance.

When it comes to the data rate, we have given an analysis in Subsection 3.1. As indicated in Eq.  (6), the SN for the range measurement occupies part of the symbol and the increasing length of SN will decrease the data rate. The length of SN also influences the maximum unambiguous ranging distance. Shorter SN leads to a smaller R_max, which is indicated in Eq.  (7). The interactive influence between different parameters indicates that the signal symbol structure design contributes to the system performance and that the balance of different parameters should be considered.