A Carrier Synchronization Lock Detector Based on Weighted Detection Statistics for APSK Signals

Abstract

To solve the application limitations of conventional detectors caused by discrete phase distribution of high-order APSK signals, and the problem that the detection performance will degrade when the automatic control gain is unideal, a carrier synchronization lock detector based on weighted detection statistics is proposed for APSK signals. Based on the detection statistics of the Linn detector, the proposed detector calculates a weighted factor according to the amplitude difference of the signal on the APSK constellation to adjust the weight of detection statistics for different rings. The proposed detector solves the detection performance degradation problem of the Linn detector caused by uneven phase distribution. In order to further improve detection performance, the detection threshold and statistical signal length are reasonably designed. The expectation and variance properties are derived, and the lock detection probability is analyzed. The performance of the proposed detector is verified through simulations. Simulation results show that the proposed carrier synchronization lock detector has better performance than the Linn detector.

1. Introduction

Amplitude phase-shift keying (APSK) is a type of PSK modulation where the amplitude of the baseband pulse is not constant and takes values from another finite set [1]. APSK is a higher-order modulation scheme designed for efficient transmission over communication channels due to its intrinsic robustness against non-linear amplifier distortions, as well as its spectral efficiency [2,3]. Thus, it is widely used in digital communication, digital video broadcasting, satellite communication and other fields [4,5]. In a communication system, the transmitter modulates the baseband data to the radio frequency (RF) carrier and sends it out. The receiver estimates the carrier Doppler frequency and the phase information of the signal using signal synchronization technology, then peels off the carrier frequency, synchronizes the data clock, and completes the data demodulation [6]. Carrier synchronization as a key step of parameter estimation and compensation; its performance directly affects the correctness of demodulation data. The communication channel environment is changeable; the receiver usually needs to adjust the carrier synchronization algorithms according to the channel conditions. Synchronization lock detection can measure the synchronization performance of the current signal and guide the adjustment of the receiver’s demodulation scheme. Therefore, efficient and reliable synchronization lock detection technology has important research value.

The conventional method for carrier lock detection involves designing a lock detection statistic to extract the phase information of the received signals. The lock detection statistic is usually in the form of a cosine function of phase error. The detection statistics in the states of carrier lock and unlock are significantly different; thus, they can be used to detect the states of the carrier synchronization loop. A conventional carrier lock detector is the Mileant detector, which eliminates the effect of the modulation phase by quadrupling frequency and extracts the cosine value of the phase error as the detection statistic [7,8]. Based on this idea, a Fu detector, which uses cosine values of single and double phase errors to design the detection statistic, is presented [9]. The performance of the Fu detector is similar to the Mileant detector. In [10,11], a Lee detector, which uses absolute operation to replace multiplication operation, is introduced to save multiplier resources. In [12], a median filter is added in front of the Mileant detector to improve the performance of carrier synchronization lock detection. In [13], a maximum-likelihood-based carrier lock detector is derived for M-PSK signals, which has low implementation complexity.

The detectors analyzed above are all designed based on the assumption that the amplitude of the received signal is constant. However, the performances of these lock detectors are not ideal when their automatic gain controls (AGCs) are not ideal. To solve this problem, an improved Mileant detector is proposed in [14]; this detector simplifies the detection statistic and uses it in the signal-to-noise ratio (SNR) estimation to enhance detection performance. The Linn detector is another modified version of the Mileant detector [15,16,17,18,19]; the Linn detector improves lock detection performance by performing amplitude normalization. In the Linn detector, the order of detection statistics is the total number of discrete phases in the constellation. Therefore, for high-order APSK-modulated signals, the detector performance will be reduced because the detection order is too large. In addition, the Linn detector is only applicable to APSK signals with uniformly distributed discrete phases in their constellations; that is, the phase difference between any adjacent discrete phases is equal. For high-order APSK signals with uneven phase distribution, the detection performance of the Linn detector is seriously degraded and cannot be applied.

For APSK signals with uneven discrete phase distribution, the conventional carrier lock detection method involves using the symbols from the outmost ring of the constellation for detection and decision, which are then transformed into the problem of multiple phase-shift keying (MPSK) signal-locking detection. However, when the detection lock time is fixed, using only the outmost symbols will reduce the number of symbols for accumulation, thereby reducing detection performance. Based on the above analyses, this paper proposes a carrier synchronization lock detection method based on the weighted detection statistics for APSK signals. The main contributions of this article are summarized as follows:

(1)

Based on detection statistics used in the Linn detector, the proposed method weights the detection statistics according to the difference in signal amplitude on different rings of APSK constellation and generates a new detection statistic.

(2)

The expectation and variance of the detection statistics in the two states of carrier synchronization (lock and unlock) are analyzed theoretically.

(3)

The lock-detection probability and false-alarm probability of the proposed detection method are analyzed.

The work is organized as follows: after a brief introduction to the signal model in Section 2, Section 3 describes the proposed carrier synchronization lock detection method. Section 4 then gives the performance analyses of expectation, variance and lock detection probability. The numerical results are provided in Section 4. Section 5 concludes the paper.

2. Signal Model

The transmitted baseband APSK signal is [20]

$$m\left(t\right)={\displaystyle {\sum }_{-\infty }^{+\infty }{s}_n{\mathrm{e}}^{j{\varphi }_n}p\left(t-nT\right)}$$

(1)

where $s_n$ is the signal amplitude, which meets $s_n\in \mathsf{\Phi }$, $\mathsf{\Phi }={\left. [A_1,\dots ,A_i,\dots ,A_K\right]}^T$, $A_i$ is the amplitude of the $i$th ring, and $K$ is the number of rings on the constellation. Here, $p\left(t\right)$ is the baseband data pulse; ${\varphi }_n$ is the modulated phase, which meets ${\varphi }_n\in \mathsf{\Gamma }$; $\mathrm{\Gamma }=\left\{{\mathsf{\Gamma }}_1,\dots ,{\mathsf{\Gamma }}_i,\dots ,{\mathsf{\Gamma }}_K\right\}$, ${\mathsf{\Gamma }}_i$ is the phase set of the $i$th ring; and there are $h_i$ phases on the $i$th ring. Thus, ${\varphi }_n$ is written as

$${\varphi }_n=\frac{2\pi m}{h_i},m=0,1,2,\dots ,h_i-1$$

(2)

The transmission signal is corrupted by Gaussian noise at the receiver. Assume that the matched filter and data pulse filter conform to the Nyquist criterion for zero inter-symbol interference (ISI), and the sample of the matched filter output is at the ideal sampling time, which is indicated by the most open time on an eye diagram. The output signal $r\left(n\right)$ of the match filter can be presented as two orthogonal components

$$r\left(n\right)=I\left(n\right)+jQ\left(n\right)$$

(3)

Here, the $I\left(n\right)$ and $Q\left(n\right)$ branches are

$$I\left(n\right)=E_n\cos \left(\mathrm{\Delta }\omega nT+{\theta }_e+{\varphi }_n\right)+n_I\left(nT\right)$$

(4)

$$Q\left(n\right)=E_n\sin \left(\mathrm{\Delta }\omega nT+{\theta }_e+{\varphi }_n\right)+n_Q\left(nT\right)$$

(5)

where $\mathrm{\Delta }\omega$ represents the frequency offset, ${\theta }_e$ is the phase error, $T$ is the symbol period, $n_I\left(nT\right)$ and $n_Q\left(nT\right)$ are the Gaussian white noise, and $E_n$ is the signal power.

3. Methodology

The block diagram of the carrier synchronization lock detection method based on the weighted detection statistics for APSK signals is shown in Figure 1. After carrier synchronization and bit synchronization, $I\left(n\right)$ and $Q\left(n\right)$ APSK signals are stored and then sent to an APSK ring-judgment module and an APSK constellation-decision module. APSK ring judgment is performed according to the signal amplitude and outputs the results to the weighted-calculation module. The APSK constellation-decision module outputs decision results to a detection-statistic module. The detection statistic of each ring is calculated and weighted and then sent to the comparator. In the comparator, the weighted detection statistic is compared with the predefined threshold, and the lock state sign is output from the comparator.

In a carrier synchronization lock detector, the detection statistic is used to judge the carrier synchronization lock state. In general, the detection statistic is compared with a threshold. If the detection statistic is higher than the threshold, the carrier synchronization is judged as a lock state; otherwise, it is judged as an unlock state. Therefore, the key issue of the lock detector is the formulation of the detection statistic.

The constellation of the APSK signal presents a circular distribution. The symbols of the constellation are distributed on K rings. The lock metric of the detector is defined as $x={\left. [x_1,\dots ,x_i,\dots ,x_K\right]}^T$, where $x_i$ is the lock metric of the $i$th ring and $x_i$ is calculated as [18]

$$x_i=\frac{\mathrm{Re}\left. [{(I(n)+jQ(n))}^{p_i}\right]}{{\left(I^2(n)+Q^2(n)\right)}^{\frac{p_i}{2}}}$$

(6)

where $\mathrm{Re}[•]$ is the real part operation and $p_i$ is the order of APSK signal. The lock detection statistic is defined as $y={\left. [y_1,\dots ,y_i,\dots ,y_K\right]}^T$, where $y_i$ is the lock detection statistic of the $i$th ring, which is calculated as

$$y_i=\frac{1}{N_i}{\displaystyle \sum _{n=0}^{N_i-1}{x}_{i,n}}$$

(7)

where $N_i$ is the number of symbols belonging to the $i$th ring. The weighted lock detection statistic is expressed as

$$g=y^{T}w$$

(8)

where $w={\left. [w_1,\dots ,w_i,\dots ,w_K\right]}^T$ is the weight factor and $w_i$ is the weight of the $i$th ring. The weight factor can be calculated based on the following expressions

$$\left\{\begin{array}{l}\frac{E_1}{w_1}=\dots =\frac{E_i}{w_2}=\dots =\frac{E_K}{w_K}\\ w_1+\dots +w_i+\dots +w_K=1\end{array}\right.$$

(9)

where $E_i$ is the symbol power of the $i$th ring. $E_i$ can be calculated as $E_i=\sqrt{I^2(n)+Q^2(n)}$.

It can be seen that the weight of the detection statistic for each ring on the constellation is different. Thus, it is important to perform correct ring judgment for constellation symbols, i.e., to determine to which ring each symbol belongs. The ring judgment threshold is defined as $C={\left. [c_1,\dots ,c_i,\dots ,c_{K-1}\right]}^T$, where $c_i$ is the ring judgment threshold between the $i$th and $i+1$th ring. Based on the signal amplitude, the radius of each ring on the constellation is calculated as

$$\left\{\begin{array}{l}{\widehat{\rho }}_1=\sqrt{\frac{\left(1+{\displaystyle {\sum }_{i=2}\left(2i-1\right)}\right)E_i}{1+{\displaystyle {\sum }_{i=2}\left(2i-1\right)R_i{}^2}}}\\ {\widehat{\rho }}_i=R_i\times {\widehat{\rho }}_1\end{array}\right.$$

(10)

where ${\widehat{\rho }}_i$ is the $i$th radius and $R_i$ is the radius ratio of each ring to the first ring. $c_i$ is calculated as $c_i=\left({\widehat{\rho }}_i+{\widehat{\rho }}_{i+1}\right)/2$. Let $\rho$ denote the signal amplitude, and the ring judgment rules of $\rho$ are expressed as

$$\left\{\begin{array}{c}\hspace{1em}\rho <c_1 \to 1th ring\hfill \\ c_{i-1}\le \rho <c_i \to ith ring\hfill \\ \hspace{1em}\rho \ge c_{K-1} \to outmost ring\hfill \end{array}\right.$$

(11)

Taking 16APSK and 32APSK as examples, the diagrammatic sketches of ring judgment are shown in Figure 2 and Figure 3.

4. Performance Analysis

The performance of the lock detector is evaluated using the lock detection probability; it can be analyzed based on the statistic characteristics of the detection statistic, which is proposed in Section 3. The statistic characteristics of the detection statistic consist of expectation and variance. Thus, in this section, the expectation and variance of the proposed detection statistic are firstly derived under two conditions (lock and unlock). Then, the lock detection probability of the proposed detector is analyzed. Finally, the selection principle of the number of symbols used for accumulation is given.

4.1. Expectation and Variance

The output of the lock detector can distinguish the locked and unlocked states of carrier synchronization so as to realize the decision of the carrier synchronization state. When $N$ is large enough, the output characteristics of the lock detector can be evaluated using the expectation and variance of $y_i$, denoted as $E_{y_i}$ and ${\sigma }_{y_i}^2$. It can be found in [17]:

$$\begin{array}{l}\mathrm{Re}\left. [{(I(n)+jQ(n))}^{p_i}\right]\\ ={\left(\sqrt{I^2(n)+Q^2(n)}\right)}^{p_i}\cdot \mathrm{Re}\left. [{\left(\cos {\phi }_n+j\sin {\phi }_n\right)}^{p_i}\right]\\ ={\left(I^2(n)+Q^2(n)\right)}^{\frac{p_i}{2}}\cdot \cos \left(M{\phi }_n\right)\end{array}$$

(12)

where ${\phi }_n$ is the signal phase with noise, ${\phi }_n=\mathrm{\Delta }\omega nT+{\theta }_e+{\varphi }_n+n$. Thus, the detection metric can be written as

$$x_i=\frac{{\left(I^2(n)+Q^2(n)\right)}^{\frac{p_i}{2}}\cdot \cos \left(p_i{\phi }_n\right)}{{\left(I^2(n)+Q^2(n)\right)}^{\frac{p_i}{2}}}=\cos \left(p_i{\phi }_n\right)$$

(13)

When carrier synchronization is locked, $\mathrm{\Delta }\omega =0$, ${\theta }_e=0$; thus, ${\phi }_n={\varphi }_n+n$. And there is

$$\begin{array}{cc}\hfill \cos \left(p_i\left({\phi }_n-{\varphi }_n\right)\right)& =\cos \left(p_i{\phi }_n\right)\cos \left(2\pi \left(m_i-k\right)\right)+\sin \left(p_i{\phi }_n\right)\sin \left(2\pi \left(m_i-k\right)\right)\hfill \\ & =\cos \left(p_i{\phi }_n\right)\hfill \end{array}$$

(14)

and

$$\cos \left(p_i{\phi }_n\right)=\cos \left(p_i\left({\phi }_n-{\varphi }_n\right)\right)=\cos \left(p_i\mathrm{\Delta }{\varphi }_n\right)$$

(15)

where $\mathrm{\Delta }{\varphi }_n$ is phase error caused by noise, and its probability density function is [21]

$$p\left(\mathrm{\Delta }\varphi |\chi \right)=\frac{1}{2\pi }{\mathrm{e}}^{-\chi }\left. [1+a{\mathrm{e}}^{\chi {\cos }^2\left(\mathrm{\Delta }\varphi \right)}{\displaystyle {\mathsf{\int }}_{-\infty }^{\cos \left(\mathrm{\Delta }\varphi \right)\sqrt{2\chi }}{\mathrm{e}}^{-\frac{x^2}{2}}dx}\right]$$

(16)

where $a=\sqrt{2\chi }\cos \left(\mathrm{\Delta }\varphi \right)$, $\chi =E_s/N_0$. Therefore, when carrier synchronization is locked, the expectation and variance of $y_i$ is

$$E_{y_i,lock}\cong {\displaystyle {\int }_{-\pi }^{\pi }{y}_i\left(n\right)p\left(\mathrm{\Delta }\varphi |{\chi }_i\right)d\left(\mathrm{\Delta }\varphi \right)}$$

(17)

and

$${\sigma }^2{}_{y_i,lock}={\displaystyle {\int }_{-\pi }^{\pi }{y}_i^2\left(n\right)p\left(\mathrm{\Delta }\varphi |{\chi }_i\right)d\left(\mathrm{\Delta }\varphi \right)}-E_{y,lock}^2$$

(18)

In order to analyze the expectation and variance performance under different SNRs, it is necessary to simplify the expression of $p\left(\mathrm{\Delta }\varphi |\chi \right)$. When the SNR is high, $p\left(\mathrm{\Delta }\varphi |\chi \right)$ is simplified as

$$\begin{array}{l}p\left(\mathrm{\Delta }\varphi |\chi \right)\\ \approx \frac{1}{2\pi }\exp \left(-\chi \right)\times \sqrt{2\chi }\exp \left(\chi \cdot {\cos }^2(\mathrm{\Delta }\varphi )\right)\cdot {\displaystyle {\int }_{-\infty }^{\infty }{e}^{-x^2/2}}dx\\ \approx \sqrt{\frac{\chi }{\pi }}\exp \left(-\chi {(\mathrm{\Delta }\varphi )}^2\right)\end{array}$$

(19)

Thus, the expectation and variance of $x_i$ are

$$\begin{array}{l}E\left. [x_i\right]={\displaystyle {\int }_{-\pi }^{\pi }{x}_i}p(\mathrm{\Delta }\varphi |{\chi }_i)\cdot d\mathrm{\Delta }\varphi \\ \approx \sqrt{\frac{{\chi }_i}{\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\cos }(p_i\mathrm{\Delta }\varphi )\cdot \exp \left(-{\chi }_i{(\mathrm{\Delta }\varphi )}^2\right)d\mathrm{\Delta }\varphi \\ =\exp \left(\frac{-p_i{}^2}{4{\chi }_i}\right)\end{array}$$

(20)

and

$$\begin{array}{l}{\sigma }^2\left. [x_i\right]=\frac{1}{N_i}{\displaystyle {\int }_{-\pi }^{\pi }{\cos }^2(p_i\mathrm{\Delta }\varphi )}p(\mathrm{\Delta }\varphi |\chi )\cdot d\mathrm{\Delta }\varphi -E^2\left. [x_i\right]\\ \approx \sqrt{\frac{\chi }{\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\left(\frac{1+\cos (2p_i\mathrm{\Delta }\varphi )}{2}\right)}\exp \left(-\chi {(\mathrm{\Delta }\varphi )}^2\right)d\mathrm{\Delta }\varphi -E^2\left. [x_i\right]\\ =\frac{1}{2}+\frac{1}{2}\exp \left(\frac{-p_i{}^2}{\chi }\right)-\exp {\left(\frac{-p_i{}^2}{4\chi }\right)}^2\end{array}$$

(21)

According to (7), $y_i$ follows the distribution $y_i\sim N\left(E\left. [x_i\right],{\sigma }^2\left. [x_i\right]/N_i\right)$. Therefore, the expectation and variance of $g={\mathbf{y}}^T\mathbf{w}$ are

$$E[g]={\displaystyle \sum _{i=1}^{K}E\left. [x_i\right]w_i}$$

(22)

and

$${\sigma }^2[g]={\displaystyle \sum _{i=1}^K\frac{{\sigma }^2\left. [x_i\right]}{N_i}{w}_i^2}$$

(23)

The statistical characteristics of the detection metric in the state when the carrier synchronization is unlocked are analyzed as follows. According to [10], when the carrier frequency offset is larger than 0.4 times the traction frequency range of the phase locked loop, the loop needs a long time to track the carrier. At this time, it can be considered that the carrier is in the state of losing lock. When the loop losses lock, the phase noise $\mathrm{\Delta }\varphi$ is a random process, and is independent of amplitude random process. Therefore, $\mathrm{\Delta }\varphi$ approximately obeys uniform distribution in the range [−π/4, π/4] [8]. When the carrier synchronization is unlocked, the expectation is

$$E{\left. [y_{i,unlock}\right]}_{N_i\to \infty }=\frac{1}{N_i}{\displaystyle \sum _{n=0}^{N_i-1}{x}_{i,n}}=0$$

(24)

In addition, when the carrier synchronization is unlocked, the variance of $x_i$ is

$${\sigma }^2\left. [x_i\right]=E\left. [x_i^2\right]-E^2[x_i]\le {\left(\sup \left|x_i\right|\right)}^2=1$$

(25)

Thus, $y_{i,unlock}\sim N\left(0,1/N_i\right)$, and $g\sim N\left(0,{\displaystyle \sum _{i=1}^K\frac{1}{N_i}{w}_i^2}\right)$.

4.2. Lock Detection Probability

The general method for carrier lock detection involves comparing detector outputs with a threshold, denoted by $V_h$. If the output is higher (lower) than $V_h$, the carrier is considered as locked (unlocked). The performance of the lock detector is measured using lock detection probability and false alarm probability, denoted by $P_d$ and $P_{fa}$.

Based on above analyses, $x_i$ is independent and identically distributed. When $N_i$ is large enough, $y_i$ approximately obeys Gaussian distribution. Thus, the expectation and variance of $y_i$ can be expressed as

$$E_{y_i}=E_{x_i}, {\sigma }_{y_i}^2=\frac{{\sigma }_{x_i}^2}{N_i}$$

(26)

where $N_i$ is the number of symbols used for accumulation on the $i$th ring. The lock detection probability $P_d$, the false alarm probability $P_{fa}$ and the threshold $V_h$ are calculated as follows:

$$P_d\left({\chi }_i,N\right)=P\left(y_i>V_h|lock\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{V_h-E_{y_i,lock}}{{\sigma }_{y_i,lock}/\sqrt{N_i}}\right),$$

(27)

$$P_{fa}\left({\chi }_i,N\right)=P\left(y_i>V_h|unlock\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{V_h-E_{y_i,unlock}}{{\sigma }_{y_i,unlock}/\sqrt{N_i}}\right),$$

(28)

$$V_h=\frac{{\sigma }_{y_i,unlock}{\mathrm{erfc}}^{-1}\left(2P_{fa}\right)}{\sqrt{N_{}}}+E_{y_i,unlock}$$

(29)

where, $\mathrm{erfc}(•)$ denotes the complementary error function. $N$ is calculated as $N={\displaystyle {\sum }_{i=1}^K{N}_i}$. From (27) and (28), $P_d$ and $P_{fa}$ are the function of $N_i$ and SNR. In order to accurately distinguish carrier synchronization state, the selection of $N_i$ needs to meet the following expression:

$$N_i={\left(\frac{{\mathrm{erfc}}^{-1}\left(2P_d\right)\cdot {\sigma }_{y_i,lock}-{\mathrm{erfc}}^{-1}\left(2P_{fa}\right)\cdot {\sigma }_{y_i,unlock}}{E_{y_i,unlock}-E_{y_i,lock}}\right)}^2$$

(30)

It can be noted that with the same $P_{fa}$ and SNR, the higher the value of $N_i$, the higher the lock detection probability that the detector achieves; but it will increase the implementation resources. In addition, the higher the value of $N_i$, the stronger the anti-interference ability of the detector; but it may lead to slow down the update frequency of the lock detector, resulting in untimely state update. Therefore, $N_i$ should be designed in accordance with actual needs.

5. Numerical Results

Taking 16APSK and 32APSK signals as examples, the lock detection performance of the proposed detector was evaluated and compared with the Linn detector. Because the conventional carrier synchronization lock detector usually uses the outmost constellation symbols to obtain the detection statistics for the APSK signal with uneven phase distribution, the simulation of the proposed detector was carried out under two conditions: one was to select all constellation symbols; the other was to select the outmost constellation symbols. To avoid confusion, in the simulation results, the detector that selected all constellation symbols was called “The proposed”, and the detector that selected the outmost constellation points was called “The outmost”. In the simulation, the number of symbols used for accumulation was set to 1024. For each case, 2000 Monte Carlo simulations were performed. Main simulation settings are summarized in

Table 1. The coding principle and dictionary update of the CS-based LZW algorithm.

Parameters	16APSK	32APSK
$K$	2	3
$h_i$	[4,12]	[4,12,16]
$p_i$	[4,12]	[4,12,16]
${\rho }_i$	[1.41,3]	[1.41,3,5]
$c_i$	2.21	[2.21,4]
$w_i$	[0.18,0.82]	[0.06,0.25,0.69]
$\mathrm{Symbol} \mathrm{Rate} R_s$(sps)	1000	1000
$\mathrm{Sample} \mathrm{Rate} f_s$(Hz)	16,000	16,000
Roll-off factor	0.25	0.25
$E_s/N_0$(dB)	0~30	0~30
Simulation time(s)	0.0639	0.0639

.

5.1. Expectation and Variace

In the simulation results, let E(z) and Sig(z) denote the expectation and variance of the detection statistic of the lock detector, where ‘z’ denotes the detection statistic. Figure 4a,b show the simulation results of expectation and variance versus the SNR under the condition that carrier synchronization is locked for the 16APSK signal. It can be seen from Figure 4a that the theoretical value of expectation of the proposed detector is basically consistent with the simulation value. With an increase in the SNR, the expectation of all detectors shows an upward trend. The expectation of the proposed detector is similar to that of the outmost detector and is greater than that of the Linn detector. It can be seen from Figure 4b that the theoretical value of variance is only consistent with the simulation value when the SNR is greater than 10 dB, but separation occurs when the SNR is low. This is because there will be errors in ring judgment according to signal amplitude in the simulation at low SNRs, and the error of ring judgment is not considered in the theoretical analyses. When the SNR is lower than 10 dB, the variance of the proposed detector is larger than that of the Linn detector, but smaller than that of the outmost detector. When the SNR is larger than 15 dB, the variance of the proposed detector is smaller than those of the other two detectors.

Figure 5a,b show the simulation results of expectation and variance versus the SNR under the condition that carrier synchronization is unlocked for the 16APSK signal. The expectation of the proposed detector is larger than that of the other two detectors. The variance of the proposed detector is similar to the Linn detector at lower SNRs, and is lower than the other two detectors at higher SNRs.

Figure 6a,b show the simulation results of expectation and variance versus the SNR under the condition that carrier synchronization is locked for the 32APSK signal. It can be seen from Figure 6a that the theoretical value of expectation of the proposed detector is basically consistent with the simulation value. With an increase in the SNR, the expectation of all detectors shows an upward trend. The expectation of the proposed detector is similar to that of the outmost detector and is greater than that of the Linn detector. It can be seen from Figure 6b that the theoretical value of variance is only consistent with the simulation value when the SNR is greater than 14 dB, but separation occurs at lower SNRs. The reason has been analyzed above. When the SNR is lower than 14 dB, the variance of the proposed detector is larger than Linn detector, but smaller than the outmost detector. When the SNR is larger than 15 dB, the variance of the proposed detector is smaller than that of the other two detectors.

Figure 7a,b show the simulation results of expectation and variance versus the SNR under the condition that carrier synchronization is unlocked for the 32APSK signal. When the SNR is larger than 25 dB, the expectation of the proposed detector is larger than the outmost detector and smaller than the Linn detector. When the SNR is smaller than 25 dB, the expectation of the proposed detector is larger than the other two detectors. When the SNR is larger than 18 dB, the proposed detector has the smallest variance compared to other detectors.

5.2. Lock Detection Probability

Figure 8 shows ROC curves of the 16APSK signal carrier synchronization lock detector at two different SNRs ($E_s/N_0=6.2706\mathrm{dB}$ and $E_s/N_0=11.2706\mathrm{dB}$). It can be seen that the performance of the three detectors is similar at lower SNRs, and the proposed detector has the best performance over the other two detectors at higher SNRs. Figure 9 shows ROC curves of the 32APSK signal carrier synchronization lock detector at two different SNRs ($E_s/N_0=7.2397\mathrm{dB}$ and $E_s/N_0=12.2397\mathrm{dB}$). It can be seen that the performance of the three detectors is similar at lower SNRs. However, when the SNR is higher, the performance of the proposed detector is similar to that of the outmost detector and higher than Linn detector.

Figure 10 and Figure 11 show the lock detection probabilities of 16APSK and 32APSK signals, respectively. In the simulation, set $P_{fa}={10}^{-3}$. For the 16APSK signal, when $E_s/N_0<7\mathrm{dB}$ and $E_s/N_0>11\mathrm{dB}$, the three detectors have similar performance. When $7\mathrm{dB}\le E_s/N_0\le 11\mathrm{dB}$, the proposed detector has the best performance compared with the other two detectors. For the 32APSK signal, when $E_s/N_0<10\mathrm{dB}$ and $E_s/N_0>14\mathrm{dB}$, the three detectors have a similar performance. When $10\mathrm{dB}<E_s/N_0<14\mathrm{dB}$, the proposed detector shows a similar performance to the outmost detector and a better performance than the Linn detector.

6. Conclusions

Aiming to solve the application limitations of the Linn detector, a carrier synchronization lock detector based on weighted detection statistics is proposed for APSK signals. According to the amplitude difference of the signal on the APSK constellation, a weighted factor is generated to adjust the weight of detection statistic on different rings, and a new detection statistic is obtained. The expectation and variance properties are derived and the lock detection probability is analyzed. The performance of the proposed detector is verified through simulations. In the simulations, 16APSK and 32APSK signals are chosen as the received signals. The simulation results indicate that the proposed detector has similar performance in term of expectation and better performance in term of variance compared to other detectors. Moreover, the proposed detector has better performance in terms of lock detection probability than the Linn detector and the outmost detector. Therefore, the proposed detector has better application values.