An Improved Phase Deviation Discriminator for Carrier Synchronization of APSK Signal in Satellite-to-Ground Communication Systems

Abstract

The conventional phase deviation discriminator used in the decision feedback loop for carrier synchronization of APSK signal requires symbol decision for every constellation symbol. When the number of constellation symbols used for coherent integration becomes larger, the complexity would increase greatly. To solve this problem, this work proposes an improved phase deviation discriminator based on symbol decisions. It firstly executes doubling frequency on the APSK signal to reduce the modulation order and eliminate the modulation phase of the received signal. Then, it rotates the constellation clockwise and selects the constellation symbols that on the X-axis to execute coherent integration and phase deviation extraction. Compared to the conventional discriminator in the decision feedback loop, the proposed discriminator can reduce the symbol decision times and steps. Therefore, when a large number of constellation symbols are used for coherent integration, it can reduce the computational complexity significantly. Moreover, with a large number of constellation symbols, the proposed discriminator achieves better BER and RMSE performance than other existing feedback loops.

1. Introduction

Satellite-to-ground communication technology, as a means of communication between space and earth, is widely used in disaster protection, navigation and positioning, geodesy, military reconnaissance and other fields [1,2]. With the continuous improvement of data transmission rate, the signal modulation mode develops toward higher-order modulation. Amplitude phase shift keying (APSK) is a higher-order modulation scheme designed for efficient transmission over satellite channels due to its intrinsic robustness against non-linear amplifier distortions, as well as spectral efficiency [3,4]. In the satellite-to-ground high-rate data communication system, the data should be modulated on a carrier at the transmitter and demodulated at the receiver. Under the space communication scenarios, relatively high-speed movement between the transmitter and receiver pair cause a Doppler shift on carrier frequency [5,6]. Carrier synchronization needs to recover a local carrier that has the same frequency and phase as the received carrier to adjust the local carrier phase. Carrier synchronization technology is of great research significance for ensuring the satellite-to-ground link to achieve high-quality communication [7].

Carrier synchronization can be implemented by estimating carrier phase deviation between the received carrier and the local carrier to adjust the local carrier phase [8]. The Carrier phase estimation method can be divided into a feed-forward-based method and a feedback-based method. Compared to the feed-forward-based method, the feedback-based method has the advantage of strong anti-interference ability and high estimation accuracy, which is more suitable for high-speed satellite-ground data transmission [9].

Feedback-based carrier phase estimation method is always designed based on the Phase-locked Loop (PLL) principle [10,11,12]. For an APSK signal which is a multi-level constellation signal, the phase deviation discriminator in the feedback loop needs to firstly eliminate the modulation phase, and then extract phase deviation. According to the different structures of the phase deviation discriminator, there are several kinds of PLL [13], such as the Costas loop, squaring loop, Mth-power loop, and decision-feedback loop. In the Costas loop, the discriminator eliminates the modulation phase by phase compensation and filtering [14,15,16,17]; its phase estimation accuracy is greatly affected by noise; moreover, the structure of the Costas loop would be more complex when the signal modulation order is high, thus it is not suitable for high-speed satellite-ground data transmission with limited resources. In the Mth-power loop, the discriminator eliminates the modulation phase by multiple frequency processing [18]; in order to meet the requirement of phase deviation extraction for a high-order modulation signal, multiple frequency processing should be performed more than once which may strengthen the noise term and reduce the sensitivity of the receiver. In the decision feedback loop, the discriminator eliminates the modulation phase by symbol decision and phase rotation [19,20]; its phase estimation accuracy is close to the maximum likelihood (ML) method; however, when the number of symbols in the constellation is large, the computation cost for symbol decision will increase greatly. To overcome this problem, and inspired by the principle of the phase deviation discriminator used in the decision feedback loop, this work aims to design an improved phase deviation discriminator based on symbol decision to reduce the computation cost while ensuring the high estimation accuracy.

The remainder of this paper is organized as follows. In Section 2, the system model of carrier synchronization based on a feedback loop is introduced and the problem is formulated. In Section 3, an improved phase deviation discriminator is proposed, and its design details are provided. In Section 4. Numerical simulation results are presented. The study is concluded in Section 5.

2. System Model and Problem Formulation

In this section, the system model of carrier synchronization based on a feedback loop is introduced first. Then, the phase deviation discriminator used in the decision feedback loop is analyzed and the problem is formulated.

2.1. System Model

The block diagram of carrier synchronization based on the feedback loop is shown in Figure 1. The sampled in-phase (I) and quadrature (Q) APSK signals to the digital mixer are expressed as:

$$\{\begin{array}{l}{s}_I\left(k\right)=\rho \left(k\right)\cos \left({\omega }_{d}k+{\phi }_0+\phi \left(k\right)\right)+N_I\left(k\right)\\ s_Q\left(k\right)=\rho \left(k\right)\sin \left({\omega }_{d}k+{\phi }_0+\phi \left(k\right)\right)+N_Q\left(k\right)\end{array}$$

(1)

where, $\rho \left(k\right)$ is the signal amplitude, $w_d$ is the Doppler frequency, ${\phi }_0$ is the unknown phase, $\phi \left(k\right)$ is the modulation phase, $N_I\left(k\right)$ and $N_Q\left(k\right)$ are the Gaussian white noise. $s_I\left(k\right)$ and $s_Q\left(k\right)$ are mixed with the local carrier signal which is generated by a voltage controlled oscillator (VCO). The output of the digital mixer can be written as:

$$\{\begin{array}{l}{x}_I\left(k\right)=\rho \left(k\right)\cos \left(\Delta {\omega }_{d}k+\Delta \phi +\phi \left(k\right)\right)+n_I\left(k\right)\\ x_Q\left(k\right)=\rho \left(k\right)\sin \left(\Delta {\omega }_{d}k+\Delta \phi +\phi \left(k\right)\right)+n_Q\left(k\right)\end{array}$$

(2)

where $\Delta {\omega }_d$ and $\Delta \phi$ are the carrier frequency offset and phase deviation, respectively. $n_I\left(k\right)$ and $n_Q\left(k\right)$ are the noise components. SNR estimation is performed on $s_I\left(k\right)$ and $s_Q\left(k\right)$ to generate a local constellation diagram which is expressed as:

$$\{\begin{array}{c}{X}_I\left(k\right)=\widehat{\rho }\left(k\right)\cos \left(\phi \left(k\right)\right)\\ X_Q\left(k\right)=\widehat{\rho }\left(k\right)\sin \left(\phi \left(k\right)\right)\end{array}$$

(3)

where, $\widehat{\rho }\left(k\right)$ is the estimated signal amplitude. The phase deviation $\Delta \phi$ between $x_I\left(k\right)$, $x_Q\left(k\right)$ and $X_I\left(k\right)$, $X_Q\left(k\right)$ is estimated by a phase deviation discriminator. According to the estimated phase deviation, the local carrier frequency is adjusted by a low-pass filter (LPF) and VCO. Then, this new local carrier frequency is mixed with the received APSK signals once again.

This work focuses on the research of phase deviation discriminators. The idea of this study is inspired by the phase discriminator used in the decision feedback loop, thus it is necessary to analyze the limitations of this discriminator, and then formulate the problem.

2.2. Problem Formulation

The phase discriminator in a decision-feedback loop estimates the phase deviation based on symbol decision. It performs symbol decision and phase rotation to eliminate the modulation phase. Then coherent integration and phase deviation extraction are performed to obtain the phase deviation. The block diagram of the carrier phase deviation discriminator based on symbol decision is shown in Figure 2. Take the 16-APSK signal as an example to analyze this method. In the second generation satellite digital video broadcasting (DVB-S2) standard [21], the 16-APSK constellation consists of two rings with an inner ring of four symbols and an outer ring of 12 symbols, as shown in Figure 3. The key component of this discriminator is the symbol decision which includes amplitude decision and phase decision. The symbol decision rules and phase rotation angle are shown in

Table 1. Symbol decision and phase rotation angle rules of phase discriminator used in decision feedback loop.

Amplitude Decision Results	Phase Decision Results	Symbol Decision Results	Phase Rotation Angle
inner-circle	$0~\pi /2$	s12	$\pi /4$
inner-circle	$\pi /2~\pi$	s14	$3\pi /4$
inner-circle	$\pi ~3\pi /2$	s15	$5\pi /4$
inner-circle	$3\pi /2~2\pi$	s13	$7\pi /4$
outer-circle	$0~\pi /6$	s4	$\pi /12$
outer-circle	$\pi /6~\pi /3$	s0	$\pi /4$
outer-circle	$\pi /3~\pi /2$	s8	$5\pi /12$
outer-circle	$\pi /2~2\pi /3$	s10	$7\pi /12$
outer-circle	$2\pi /3~5\pi /6$	s2	$3\pi /4$
outer-circle	$5\pi /6~\pi$	s6	$11\pi /12$
outer-circle	$\pi ~7\pi /6$	s7	$13\pi /12$
outer-circle	$7\pi /6~4\pi /3$	s3	$5\pi /4$
outer-circle	$4\pi /3~3\pi /2$	s11	$17\pi /12$
outer-circle	$3\pi /2~5\pi /3$	s9	$19\pi /12$
outer-circle	$5\pi /3~11\pi /6$	s1	$7\pi /4$
outer-circle	$11\pi /6~2\pi$	s5	$23\pi /12$

. For example, if a received signal sample is judged as on the inner ring according to amplitude decision, and the phase decision result is 0~$\pi /2$, then the symbol decision result is s12 and the phase rotation angle is $\pi /4$.

Generally, the phase deviation discriminator based on symbol decision in the decision-feedback loop needs to execute all steps (as shown in Figure 2) for every constellation symbol. For each symbol, the computational cost of carrier phase estimation contains four parts: symbol decision, phase rotation, coherent integration and phase deviation extraction. When the number of constellation symbols becomes larger, the computational cost will increase greatly. In addition, the phase deviation estimation accuracy is determined by the result of symbol decision. Symbol decision error would reduce the phase estimation accuracy, resulting in the decline of loop performance and an increase in bit error rate. Based on these analyses, this study considers designing an improved phase deviation discriminator which can reduce the symbol decision times and steps.

3. Methodology

Take the 16-APSK signal as an example to describe the proposed phase deviation discriminator. The diagram of the proposed discriminator is shown in Figure 4. It firstly executes frequency doubling on $x_I\left(k\right)$ and $x_Q\left(k\right)$. Then, the constellation is rotated clockwise by $\pi /2$. After constellation rotation, an amplitude decision is performed to judge whether the symbol is on the inner ring. If the symbol is on the inner ring, coherent integration and phase deviation extraction are performed to obtain phase deviation; otherwise, the symbol needs to perform a phase decision to judge whether it is on the X-axis. If the symbol is on the X-axis, it is selected to perform coherent integration and phase deviation extraction.

Frequency doubling on $x_I\left(k\right)$ and $x_Q\left(k\right)$ is performed to reduce the modulation order and eliminate the modulation phase. Take an inner ring symbol as an example, after frequency doubling, the signals can be expressed as:

$$\{\begin{array}{ll}{x}_{I\_2}\left(k\right)& ={\rho }_{}^2\left(k\right)\cos \left(2\Delta {\omega }_{d}k+2\Delta \phi \pm \frac{\pi }{2}\right)+n_{I\_2}\left(k\right)\\ & =\mp {\rho }_{}^2\left(k\right)\sin \left(2\Delta {\omega }_{d}k+2\Delta \phi \right)+n_{I\_2}\left(k\right)\\ x_{Q\_2}\left(k\right)& ={\rho }_{}^2\left(k\right)\sin \left(2\Delta {\omega }_{d}k+2\Delta \phi \pm \frac{\pi }{2}\right)+n_{Q\_2}\left(k\right)\\ & =\pm {\rho }_{}^2\left(k\right)\cos \left(2\Delta {\omega }_{d}k+2\Delta \phi \right)+n_{Q\_2}\left(k\right)\end{array}$$

(4)

where $n_{I\_2}$ and $n_{Q\_2}$ are the noise components. The constellation after frequency doubling is shown in Figure 5b. It can be seen that the inner ring symbols are equivalent to BPSK modulation, and the outer ring symbols are equivalent to 6-PSK modulation. In order to further decrease the computational cost, only the constellation symbols on a straight line are used to estimate the phase deviation. However, the symbols on the X-axis are easy to perform phase deviation extraction. Therefore, it is necessary to rotate the constellation clockwise by $\pi /2$. After constellation rotation, $x_{I\_2}\left(k\right)$ and $x_{Q\_2}\left(k\right)$ become:

$$\{\begin{array}{ll}{x}_{I\_2}^{\prime }\left(k\right)& =\mp {\rho }_{}^2\left(k\right)\sin \left(2\Delta {\omega }_{d}k+2\Delta \phi -\frac{\pi }{2}\right)+n_{I\_2}\left(k\right)\\ & =\pm {\rho }_{}^2\left(k\right)\cos \left(2\Delta {\omega }_{d}k+2\Delta \phi \right)+n_{I\_2}\left(k\right)=x_{Q\_2}\left(k\right)\\ x_{Q\_2}^{\prime }\left(k\right)& =\pm {\rho }_{}^2\left(k\right)\cos \left(2\Delta {\omega }_{d}k+2\Delta \phi -\frac{\pi }{2}\right)+n_{Q\_2}\left(k\right)\\ & =\pm {\rho }_{}^2\left(k\right)\sin \left(2\Delta {\omega }_{d}k+2\Delta \phi \right)+n_{Q\_2}\left(k\right)=-x_{I\_2}\left(k\right)\end{array}$$

(5)

It can be seen that constellation rotation is equivalent to the exchange of the I and Q signals without additional calculation cost. The constellation after rotation is shown in Figure 5c.

As described above, after constellation rotation, only the symbols on the X-axis are selected for phase deviation estimation. Therefore, it is important to judge whether a received symbol is on the X-axis or not. As shown in Figure 5c, if a symbol is on the inner ring, it must be on the X-axis. Thus, an inner ring symbol only needs to perform amplitude decisions. However, if a symbol is on the outer ring, it may be on the X-axis or far away from X-axis. Thus, an outer ring symbol needs to perform both amplitude decision and phase decision. Based on these analyses, the symbol decision of the proposed discriminator consists of two steps:

Step 1: Amplitude decision for every received symbol.

Let $D_1$ denote the amplitude decision threshold, and it can be calculated as:

$$D_1=\left({\widehat{\rho }}_1+{\widehat{\rho }}_2\right)/2$$

(6)

where ${\widehat{\rho }}_1$ and ${\widehat{\rho }}_2$ are the radii of the inner ring and outer ring, respectively. If the amplitude of a symbol meets $\rho <D_1$, judge it as the inner ring symbol and it is selected for coherent integration and phase deviation extraction; otherwise, judge the symbol as the outer ring symbol, and it needs to perform phase decision.

Step 2: Phase decision for outer symbols.

If a symbol is judged as an outer ring symbol according to the rule described in step 1, it needs a phase decision to judge whether it is on the X-axis. According to the distribution characteristic of the symbol on the constellation which is shown in Figure 5c, using the value of $\left|Q\right|/\left|I\right|$ could judge whether a symbol is on the X-axis. If $\left|Q\right|/\left|I\right|<\sqrt{3}/3$, the symbol is judged as on the X-axis, then coherent integration and phase deviation extraction are performed for it; otherwise, it is judged as far away from X-axis and discarded.

Compared to the conventional phase deviation discriminator used in the decision-feedback loop, the proposed discriminator does not need to execute symbol decision for all constellation symbols, thus it could reduce the computational cost obviously. Moreover, the symbol decision of the conventional discriminator needs to perform both amplitude decision and phase decision for all symbols. However, the proposed discriminator executes amplitude decision and phase decision for outer ring symbols and executes only amplitude decision for inner ring symbols. Thus, it could further reduce the computational cost. For 16-APSK signal, conventional discriminator requires amplitude decision 16 N times and phase decision 16 N times, respectively. By contrast, the proposed discriminator only requires amplitude decision 4 N times and phase decision 2 N times, respectively. The number of complex multiplication used in the conventional discriminator for phase rotation is N. Meanwhile, the number of complex multiplication used in the proposed discriminator for doubling frequency is N. For simplicity, only the above different components in the conventional discriminator used in the decision feedback loop and the proposed discriminator are chosen for complexity comparison, which is shown in

Table 2. Complexity comparison of the two discriminators for 16-APSK signal.

Discriminator	Amplitude Decision	Phase Decision	Multiplication
Conventional discriminator	16 N	16 N	N
Proposed discriminator	4 N	2 N	N

. It can be seen that the proposed discriminator has lower complexity than the conventional discriminator used in the decision feedback loop.

4. Simulation Results

The phase deviation estimation performance of the decision feedback loop based on the proposed phase deviation discriminator was verified by MATLAB simulation. Other conventional three types of feedback loops, including the Mth-power loop, conventional decision-feedback loop and squaring decision-feedback loop, were compared to the proposed feedback loop. The input signal was a 16-APSK signal with a symbol rate of 200 Mbps and a bit rate of 800 Mbps. The performance of the bit error rate (BER) and Root Mean Square Error (RMSE) was simulated in this part.

4.1. BER Simulation

BER is a performance index to measure the correctness of data transmission in a specified time. Figure 6 shows the BER performance versus $T_{coh}$ for different feedback loops. $T_{coh}$ varied from 0.0025 ms to 0.05 ms, and the corresponding data symbols for coherent integration are from 500 to 10,000. SNR ($E_s/N_0$) is set as 15 dB. When $T_{coh}<0.005$, the performance of the conventional decision-feedback loop and squaring loop are better than the Mth-power loop and the proposed decision feedback loop. This is because the proposed discriminator only selects the data symbols on the X-axis for phase discrimination. When $T_{coh}$ is small, it has fewer data symbols that participate in coherent integration leading to degradation of BER performance. However, when $T_{coh}$ is large, more data symbols are selected for phase discrimination, the BER performance of the proposed discriminator is improved. Therefore, when $T_{coh}>0.005$, the proposed feedback loop has better performance than the other three feedback loops. Figure 7 shows the BER performance versus SNR for different feedback loops. SNR ($E_s/N_0$) varied from 10 dB to 22 dB in 1 dB intervals and $T_{coh}$ is set as 0.02 ms. By comparison, the proposed feedback loop is superior in performance to other feedback loops, provided that the curve is closer to the theoretical curve.

4.2. RMSE Simulation

RMSE is another index to verify the performance of carrier phase estimation. In this part, 10,000 Monte Carlo simulations are conducted. Figure 8 shows the RMSE performance versus $T_{coh}$ for different feedback loops. $T_{coh}$ varied from 0.0025 ms to 0.05 ms, and the corresponding data symbols for coherent integration are from 500 to 10,000. SNR ($E_s/N_0$) is set as 15 dB. When $T_{coh}<0.005$, the proposed feedback loop has better performance than the Mth-power loop, but worse performance than the other two feedback loops. This is because when $T_{coh}$ is small, it has fewer symbols participate in coherent integration leading to degradation of RMSE performance. When $T_{coh}>0.005$, the proposed feedback loop is superior to the other three feedback loops. Figure 9 shows the RMSE performance versus SNR for different feedback loops. The Mth-power loop has the worst performance and the proposed feedback loop has the best performance compared to the other three feedback loops.

5. Conclusions

This work presents an improved phase deviation discriminator for carrier synchronization based on a feedback loop. The proposed discriminator executes frequency doubling and constellation diagram rotation, then selects the symbols that are on the X-axis of the constellation for phase discrimination. This article takes the 16-APSK signal as an example to describe the principle of the proposed phase deviation discriminator. Simulation results show that the phase deviation estimation of the feedback loop based on the proposed phase deviation discriminator has advantages in aspects of BER and RMSE performance compared with other existing feedback loops. Moreover, the proposed discriminator could greatly reduce the computational cost of carrier phase estimation compared to other phase deviation discriminators. The proposed phase deviation discriminator can be also applied for carrier phase estimation of higher order APSK signals, such as 32-APSK and 64-APSK signals. The proposed phase discriminator would be applied in the satellite-to-ground communication systems in the further, due to its advantages of estimation performance and computational complexity. Therefore, the research content of this article is of great significance to the development of aerospace technology.