# BOC Signal Acquisition Algorithm Based on Similar Enfoldment

Authors: Qi Han; Kejia Zhu; Caibo Hu; He Zhao; Shuang Wu; Yu Fu
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

A BOC signal has been widely used in the Global Navigation Satellite System (GNSS). Although the BOC signal has many advantages, the autocorrelation function (ACF) of the BOC signal has lots of peaks, which makes signal acquisition difficult. This paper proposes a similar enfoldment acquisition (SEA) algorithm, which can be applied to sin-BOC and cos-BOC signals, with even and odd modulation orders. The SEA algorithm utilizes the similarity between the main peak and the secondary peak to construct a new ACF to eliminate the ambiguity of the BOC signal. This paper simulates the acquisition performance of the SEA algorithm, Martin algorithm, and SCPC algorithm, while the simulation result shows that the SEA algorithm is superior to the Martin algorithm and SCPC algorithm. Since the SEA algorithm does not need filters or auxiliary signals, its complexity is lower than that of the Martin algorithm and SCPC algorithm.


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## Body

## 1. Introduction

In recent years, the Global Navigation Satellite System (GNSS) has developed rapidly. Major countries or regions have established their own satellite navigation systems, such as the Global Positioning System (GPS) of the United States, the Galileo Satellite Navigation System of Europe (Galileo), the BeiDou Navigation Satellite System (BDS) of China, and the Global Navigation Satellite System (GLONASS) of Russia. Since the frequency bands of the major navigation systems are close to each other and the modulated signals become more and more, in order to save the frequency band resources, a binary offset carrier (BOC) signal appears [1].A BOCm,n signal is obtained by Pseudorandom Noise (PRN) codes multiplied with rectangular subcarriers, where the PRN code rate is fn=n∗f0, the subcarrier frequency is fm=m∗f0, the reference frequency is f0=1.023MHz, and the modulation order of the BOC signal is M=2m/n. According to whether the subcarrier phase is sinusoidal or cosine, the BOC signal can be divided into sin-BOC m,n and cos-BOC m,n.The BOC signal has many advantages: Firstly, owing to the spectral splitting characteristic of the BOC signal [2], BOC signal does not interfere with the original BPSK signal. Secondly, compared with the BPSK signal, the autocorrelation peak of the BOC signal is narrower, which leads to the BOC signal with higher tracking accuracy and better antimultipath performance [3]. However, due to the multipeak nature of BOC signal ACF, ambiguity is prone to occur in the phase of signal acquisition and tracking.In order to eliminate the ambiguity of BOC signal ACF, plenty of methods have been proposed by scholars all over the world. The main ideas are as follows: capturing the BOC signal with the acquisition method similar to the BPSK signal [2, 4–8], constructing auxiliary signals to eliminate the ambiguity of BOC signal ACF [3, 9–16], and constructing a filter to eliminate ambiguity of ACF [17]. Literature [5, 6] (hereinafter referred to as the Fishman method) filters the upper (lower) sideband of the received signal and the local BOC code, respectively, and after that processes the upper (lower) sideband separately. The authors of [7, 8] (hereinafter referred to as the Martin method) uses a filter, whose bandwidth includes the two main lobes of the received signal and all the secondary lobes between the main ones, to process the received signal. Generally speaking, a high complexity turns out due to applying filters to filter out the desired signals in References [5–8]. As demonstrated in [9], an autocorrelation side-peak cancellation technique (ASPeCT) algorithm is proposed to eliminate redundant edge peaks through a local auxiliary BPSK signal, whereas the ASPeCT method is only applicable to sin-BOC n,n signals. A general removing ambiguity via side-peak suppression (GRASS), which uses a local auxiliary signal and is suitable for sin-BOC kn,n signals, is proposed in [3]. The GRASS algorithm is improved by two auxiliary signals to make the algorithm suitable for cos-BOC m,n signals in literature [10]. Subcarrier phase cancellation (SCPC) technology [8, 11, 12] eliminates ambiguity of ACF by two auxiliary signals. In Reference [13], the ambiguity of ACF is eliminated by two symmetrical auxiliary signals, which is applicable to all BOC m,n signals. Other methods of constructing auxiliary signals can be referred to References [14–16]. In literature [17], a filter is constructed to process the ACF, and an unambiguous ACF is finally obtained. The acquisition for the GPS L1C signal, which is also an application of the BOC signal, is studied in [18]. A 2-D compressed correlator technique for the fast acquisition of BOC m,n signals is proposed in Reference [19]. Yang reconstructs [20, 21] the BOC signal ACF by the removing zeros of crosscorrelation (RZCC) algorithm, which greatly increases the peak value of the main peak. In addition, plenty of researchers focus on the tracking problem of the BOC signal [4, 9, 13, 22, 23]. Many scholars have studied the BOC signal multipath elimination technology [24–27], which can be mainly categorized into nonparametric and parameterized class [26]. The CCRW method is applied to the BOC 1,1 signal in Reference [24]. Wu and Dempster propose the “BOC-Gated-PRN” method [25]. Zhao et al. propose an improved CCRW method for BOC signals with odd modulation coefficient [26]. Another class of the multipath mitigation technique is the parameterization method such as that in [27], where the authors propose an averaging-FFT method, which achieves robustness for both low-order and high-order BOC signals well.Among the existing acquisition algorithms, some need to use filters, which are complex and easy to cause correlation loss, such as the Fishman method and Martin method. Some algorithms need to construct auxiliary signals, which increase the computational complexity, such as those in References [3, 10]. Some cannot be applied to all BOC m,n signals. For instance, literature [9] is suitable for the sin-BOC n,n signal, and literature [3] is applicable for the sin-BOC kn,n signal. Therefore, it is necessary to research on an algorithm which can be applied to all BOC signals with low complexity.


## 2. Analysis of Existing Algorithms

The filter bandwidth of the Martin method includes two principal lobes and all the secondary lobes between the principal ones. The filtered signal is correlated with the local PRN code, which is shifted and kept at the same rate as the received signals. The flow chart of the Martin method is shown in Figure1, where parameter a of the shift factor is determined by Formula (1). Since the Martin method is less complex than the Fishman method, it is selected as the representative of the BPSK-like method for analysis in this paper.
(1)a=NBOC2.Figure 1
Flow chart of the Martin method.ASPeCT uses the PRN code as an auxiliary signal, and it can be applied to sin-BOCn,n. On the other hand, GRASS is suitable for sin-BOC kn,n while SCPC is applicable for BOC m,n. Therefore, we choose SCPC for analysis because of its wider applicability. SCPC shifts the phase of the BOC signal by π/2 to obtain the quadrature BOC signal (QBOC) and then uses QBOC as an auxiliary signal to eliminate the correlation peak ambiguity. The SCPC method in Reference [12] uses an ideal filter with a bandwidth of 2fm+fn to filter the received signal. The flow chart of the SCPC method is shown in Figure 2, and the final decision function is shown as
(2)RSCPCτ=RBOC/QBOC2τ+RBOC2τ.Figure 2
Flow chart of the SCPC method.The RZCC method captures the BOC signal by folding ACF, which can effectively improve the correlation peak value of ACF. The main idea of RZCC is as follows: the received signal is divided intoI and Q branches, which are correlated with the local PRN code and the local BOC code, respectively; then, RBOC and RBOC/PRN are obtained. For RBOC+RBOC/PRN, three peaks and two valleys are searched, respectively, and then, the final ACF is obtained by enfoldment. The flow chart of the RZCC method is shown in Figure 3.Figure 3
Flow chart of the RZCC method.As shown in Figure4, IAB‐BCD‐DEJ, where A, C, and E are the three peaks while B and D are the two valleys, is the initial ACF. In the first enfoldment, AFK‐KGE and ABC‐CDE are symmetrical about line 1. Hence, the coordinates of F can be expressed as
(3)xF=xB,yF=2yA−yB.Figure 4
Principle of enfoldment.
(a)
(b)
(c)
(d)The coordinates ofG and K can be determined with the same method. After the first enfoldment, ACF becomes IFK‐KGJ. In the second enfoldment, FHG and FKG are symmetrical about line 2. After the second enfoldment, ACF turns into IFHGJ, which is the final ACF, as shown in Figure 4(d).The RZCC algorithm is prone to frequency capture errors. It is because the RZCC algorithm folds theI branch and the Q branch separately and then calculates I2+Q2. If the peaks of the I and Q branches are not at the same point and the energy of the I and Q branches is similar (an example is given in Figures 5(a) and 5(b)), the peak points of the I and Q branches appear at 885 and 905, respectively. The result shows that the gain brought by enfoldment will not be very large. In this case, the peak after enfoldment of the adjacent frequency may exceed the correct frequency, which leads to a frequency capture error.Figure 5
Correlation results of the correct frequency.
(a)
Result ofI branch before enfoldment
(b)
Result ofQ branch before enfoldment
(c)
Results after enfoldment
(d)
Partial enlarged view of (c)For example, whenC/N0=40dB⋅Hz, the Doppler frequency shift is 0 Hz, the initial code phase is 905 chips, the IF is 50 MHz, and the sampling rate is 400 MHz; the sin-BOC (10, 5) signal is simulated. Figure 5 shows the results of the correct frequency. Figures 5(a) and 5(b) are the results of I and Q branches before enfoldment. Figure 5(c) is the correlation results after enfoldment, and Figure 5(d) is a partial enlarged view of Figure 5(c). The peak points of I and Q branches are at 885 and 905 with similar energy, respectively, but not at the same location. Figure 6 shows the results of adjacent frequency. The peak points of I and Q branches are at 905 and 361571, respectively. Although they are not at the same location, peak 361571 is much smaller than peak 905. Since the peak of adjacent frequency after enfoldment is 4.693e5 while the correct frequency is 4.097e5, the false acquisition of frequency will occur according to the RZCC capture criterion.Figure 6
Correlation results of the adjacent frequency.
(a)
Result ofI branch before enfoldment
(b)
Result ofQ branch before enfoldment
(c)
Results after enfoldment
(d)
Partial enlarged view of (c)


## 3. Similar Enfoldment Acquisition Algorithm

The SEA algorithm is an improved algorithm for the RZCC algorithm. In this paper, two kinds of BOC signal acquisition algorithms are proposed, which are the similar enfoldment acquisition with the quadratic threshold (SEAQT) algorithm and similar enfoldment acquisition with the single threshold (SEAST) algorithm.The SEAQT flow chart is shown in Figure7, where the signal process is divided into I and Q channels and the local reference code is the BOC signal. After the received signal correlation with I and Q branches, I2+Q2 is calculated. The SEAQT method makes the first threshold decision for I2+Q2, and the decision threshold η1 equals to α∗η0, where η0 corresponds to the capture threshold for the ambiguous BOC method, 0≤α≤1. If the capture statistic exceeds threshold η1, it is folded to determine whether it exceeds threshold η2. If the capture statistic exceeds threshold η2, the signal is considered to be captured.Figure 7
Flow chart of the SEAQT method.Unlike RZCC, the SEAQT method does not require a PRN code as an auxiliary signal which applies two threshold decisions to determine whether signal exists or not.The flow chart of SEAST is shown in Figure8. SEAST searches for three peaks and two valleys of capture statistic and then folds them and makes a decision.Figure 8
Flow chart of the SEAST method.


### 3.1. Comparison between SEA and RZCC

There are two main differences between the SEA algorithm and RZCC: (1) SEA calculatesI2+Q2 first and folds later. (2) The SEA does not require a PRN code as an auxiliary signal.Still using the example of Figures5 and 6, I branch, Q branch, and I2+Q2 results before the enfoldment of the correct frequency are shown in Figure 9(a), and Figure 9(b) is a partial enlarged view of Figure 9(a). As can be seen in Figure 9(b), the peaks of I branch, Q branch, and I2+Q2 are all at 905 chips. After enfoldment, I2+Q2 is shown in Figure 10. It can be seen from the figure that the correlation peak corresponding to the correct frequency exceeds the left and right adjacent frequencies, which avoids false detection.Figure 9
Results of the correct frequency.
(a)
Results before enfoldment
(b)
Partial enlarged view of (a)Figure 10
I2+Q2 after enfoldment of correct and adjacent frequencies.
(a)
I2+Q2 of correct frequency
(b)
I2+Q2 of adjacent frequency 1
(c)
I2+Q2 of adjacent frequency 2


### 3.2. Enfoldment Gain of SEA

For the ambiguous BOC method, when the modulation order isM and the value of the main peak is normalized to 1, as a result, the second peak value is M−1/M2. Therefore, the main peak of capture statistic is 10lg1/M−1/M2=20lgM−1/MdB stronger than the second peak. For instance, when the modulation order M=2, the main peak is 6 dB stronger than the second peak. When the modulation order M=4, the main peak is 2.5 dB stronger than the second peak.The enfoldment process of the SEA method is shown in Figure11.Figure 11
Enfoldment process of the SEA method.The ordinates of pointsA and E are
(4)yA=yE=M−12M2.PointsB and B′ and D and D′ are symmetrical about line 1. According to the principle of symmetry, the ordinates of B′ and D′ are
(5)yB′=yD′=2yA−yB=2M−12M2.Similarly, the ordinate of pointC′ is
(6)yC′=2yA−yC=2M−12M2−1.PointC″ is the maximum correlation peak. Similarly, the ordinate of point C″ is
(7)yC″=2yB′−yC′=2M−12+M2M2.WhenM≥2, the third peak is M−2/M2. Therefore, the main peak is 10lg2M−12+M2/M−22dB stronger than the third peak.
(8)10lg2M−12+M2/M2M−2/M2=10lg2M−12+M2M−22dB.Compared with the ambiguous BOC method, the ratio of the main peak to the secondary peak increases:(9)10lg2M−12+M2M−22−20lgMM−1=10lg2M−12+M2M−22⋅M−12M2dB.When the modulation orderM=4, the main peak is 9.3 dB stronger than the secondary peak, which is 6.8 dB higher than that of the ambiguous BOC method.


## 3.1. Comparison between SEA and RZCC

There are two main differences between the SEA algorithm and RZCC: (1) SEA calculatesI2+Q2 first and folds later. (2) The SEA does not require a PRN code as an auxiliary signal.Still using the example of Figures5 and 6, I branch, Q branch, and I2+Q2 results before the enfoldment of the correct frequency are shown in Figure 9(a), and Figure 9(b) is a partial enlarged view of Figure 9(a). As can be seen in Figure 9(b), the peaks of I branch, Q branch, and I2+Q2 are all at 905 chips. After enfoldment, I2+Q2 is shown in Figure 10. It can be seen from the figure that the correlation peak corresponding to the correct frequency exceeds the left and right adjacent frequencies, which avoids false detection.Figure 9
Results of the correct frequency.
(a)
Results before enfoldment
(b)
Partial enlarged view of (a)Figure 10
I2+Q2 after enfoldment of correct and adjacent frequencies.
(a)
I2+Q2 of correct frequency
(b)
I2+Q2 of adjacent frequency 1
(c)
I2+Q2 of adjacent frequency 2


## 3.2. Enfoldment Gain of SEA

For the ambiguous BOC method, when the modulation order isM and the value of the main peak is normalized to 1, as a result, the second peak value is M−1/M2. Therefore, the main peak of capture statistic is 10lg1/M−1/M2=20lgM−1/MdB stronger than the second peak. For instance, when the modulation order M=2, the main peak is 6 dB stronger than the second peak. When the modulation order M=4, the main peak is 2.5 dB stronger than the second peak.The enfoldment process of the SEA method is shown in Figure11.Figure 11
Enfoldment process of the SEA method.The ordinates of pointsA and E are
(4)yA=yE=M−12M2.PointsB and B′ and D and D′ are symmetrical about line 1. According to the principle of symmetry, the ordinates of B′ and D′ are
(5)yB′=yD′=2yA−yB=2M−12M2.Similarly, the ordinate of pointC′ is
(6)yC′=2yA−yC=2M−12M2−1.PointC″ is the maximum correlation peak. Similarly, the ordinate of point C″ is
(7)yC″=2yB′−yC′=2M−12+M2M2.WhenM≥2, the third peak is M−2/M2. Therefore, the main peak is 10lg2M−12+M2/M−22dB stronger than the third peak.
(8)10lg2M−12+M2/M2M−2/M2=10lg2M−12+M2M−22dB.Compared with the ambiguous BOC method, the ratio of the main peak to the secondary peak increases:(9)10lg2M−12+M2M−22−20lgMM−1=10lg2M−12+M2M−22⋅M−12M2dB.When the modulation orderM=4, the main peak is 9.3 dB stronger than the secondary peak, which is 6.8 dB higher than that of the ambiguous BOC method.


## 4. Performance Analysis

### 4.1. Theoretical Analysis

The BOC signal broadcasted by the satellite can be expressed as(10)st=EbDtxPtxBtsin2πfct+θ.Among them,Eb is the signal energy, Dt is the data code broadcasted by the satellite, xPt is the PRN code, xBt is the BOC code, fc is the carrier frequency, θ is the initial phase of the signal.The signal after the downconverter and analog-to-digital converter (ADC) can be expressed as(11)sIFn=ADn−τxPn−τxBn−τsin2πfIF+fdn+θ+δn.Among them,A is the amplitude of the received signal, fIF is the intermediate frequency, δn is the Gauss white noise with zero mean and the variance σ2, Dn−τ is the data code of the received signal, xPn−τ is the PRN code of the received signal, and xBn−τ is the BOC code of the received signal.Detection statistic of the ambiguous BOC method is(12)V=1L∑n=0L−1I2n+Q2n.Among them,In and Qn are in-phase components and quadrature components obtained by correlation between local codes and the received signals, respectively, and L is the number of incoherent integrals. When the maximum correlation peak in detection V exceeds the threshold, the signal is detected. Vt, Pmd, and Pfa refer to the capture threshold, the probability of missed detection, and the probability of false alarm, respectively.The false alarm probabilityPfa corresponding to the capture threshold Vt is
(13)Pfa=∫Vt∞vσ2e−v2/2σ2dv=e−Vt2/2σ2.Thus, the capture thresholdVt is
(14)Vt=σ−2lnPfa.In this paper, the ambiguous BOC method, Martin method, SCPC method, and SEA method are selected for simulation verification. At the same time, the performance of RZCC and SEA is compared and analyzed. For the sin-BOC (10, 2) signal, the ACF of the four algorithms is shown in Figure12. The maximum correlation peak of the ambiguous BOC algorithm is normalized to 1. Because of filtering, the maximum peak of the Martin method and SCPC method is less than 1. Since the modulation order of the sin-BOC (10, 2) signal is 10, according to Formula (7), the maximum correlation peak of the SEA method is 2.62. As shown in Figure 12, the maximum correlation peak of SEA is 2.55, which is in accordance with the theoretical analysis results.Figure 12
ACF of four methods.


### 4.2. Applicability Analysis of the SEA Algorithm

The SEA algorithm applies the ACF similarity of the BOC signal. Whether it is the sin-BOC signal or cos-BOC signal, for the BOC signal of any modulation order, the adjacent two secondary peaks of the BOC signal ACF have similarities with the main peak. Therefore, the enfoldment method can be utilized. In other words, SEA is suitable for sin-BOC and cos-BOC signals of any order in theory.Four kinds of BOC signals, sin-BOC (10, 5), cos-BOC (1, 1), sin-BOC (15, 10), and cos-BOC (10, 4), are simulated with a sampling rate of 400 MHz, an IF of 30 MHz, a Doppler shift of 300 Hz, and an initial code phase of 1500 chips (after sampling). WhenC/N0=40dB⋅Hz, the crosscorrelation function (CCF) of the local BOC signal and the received signal is shown in Figure 13. The ACF of the four BOC signals with no noise is shown in Figure 14. As can be seen from Figures 13 and 14, the SEA method is suitable for sin-BOC and cos-BOC signals with even or odd modulation orders.Figure 13
CCF ofC/N0=40dB·Hz modulated signal.Figure 14
ACF of the four BOC signals.


### 4.3. Capture Performance Analysis

Correct detection probabilityPcd and false detection probability Pfd are used to analyze capture performance. For BOC m,n signals, the phase difference between the main peak and the second peak code is fs/2fn, where fs is the signal sampling rate. The captured code phase and frequency are recorded as φc and fc, respectively; the actual code phase and frequency are recorded as φa and fa, respectively.The total number of Monte Carlo simulation isN. The number of correct detection N0 is defined as the number of times, while the φc−φa<fs/2fn and fc=fa are simultaneously satisfied in the simulation. Consequently, the correct detection probability Pcd is defined as
(15)Pcd=N0N.The false detection numberN1 is defined as the number of times φc−φa≥fs/2fn or fc≠fa in the simulation. Therefore, false detection probability Pfd is defined as
(16)Pfd=N1N.When the signal exists, if the capture statistic exceeds the capture threshold, the signal is considered to be detected, which corresponds to the detection probabilityPd. The opposite corresponds to the missed detection probability Pmd. Thus, the detection probability Pd and the missed detection probability Pmd satisfy the following relationship.
(17)Pd+Pmd=1.Considering that the correct detection probabilityPcd and the false detection probability Pfd in the paper are all defined under the condition that the signal is detected, the correct detection probability Pcd and the false detection probability Pfd satisfy the following relationship.
(18)Pcd+Pfd=Pd.The first set of simulation conditions is set as follows: for the sin-BOC (10, 2) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 30 MHz, the Doppler frequency shift is 300 Hz, the initial code phase is 905 chips, the C/N0 range is 34dB⋅Hz≤C/N0≤43dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.9 in the SEAQT method.As shown in Figure15, the correct detection probability Pcd of the SEAST method is the highest, followed by the SEAQT method. Taking 90% correct detection probability as an example, the SEAQT algorithm improves by about 1.2 dB compared with the SCPC algorithm. As C/N0≥37dB⋅Hz, the false detection probability Pfd of SEA is less than that of the Martin method or SCPC method. For the RZCC algorithm, both Pcd and Pfd are not performing well. In some cases, Pfd of RZCC exceeds 20%.Figure 15
Correct detection probability and false detection probability for sin-BOC (10, 2).The second set of simulation conditions is set as follows: for the sin-BOC (1, 1) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 40 MHz, the Doppler frequency shift is 580 Hz, the initial code phase is 1500 chips, the C/N0 range is 26dB⋅Hz≤C/N0≤35dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.7 in the SEAQT method.As shown in Figure16, the correct detection probability Pcd of SEAQT/SEAST is similar, which is slightly lower than that of the Martin method and SCPC, but the false detection probability Pfd performs better than that of the Martin method and SCPC method. The reason why SEA does not capture the BOC n,n signal well is that the second peak of the BOC n,n signal is only 1/4 of the main peak, which leads to the main peak after enfoldment is 1.5 times than the former one. In other words, the enfoldment signal may not be high enough to exceed the acquisition threshold, which results in a low detection probability. As for the RZCC algorithm, it performs even worse than SEA. When C/N0≥31dB⋅Hz, Pfd of RZCC reaches more than 25%, but Pcd is less than 60%.Figure 16
Correct detection probability and false detection probability for sin-BOC (1, 1).The third set of simulation conditions is set as follows: for the cos-BOC (15, 2.5) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 50 MHz, the Doppler frequency shift is -330 Hz, the initial code phase is 3220 chips, the C/N0 range is 36dB⋅Hz≤C/N0≤45dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.95 in the SEAQT method.As shown in Figure17, the correct detection probability Pcd of SEAST is the highest, followed by the SEAQT. Taking 90% correct detection probability as an example, the SEAQT algorithm improves by 2.1 dB compared with the Martin algorithm. When C/N0≥38dB⋅Hz, the false detection probability Pfd of SEA is less than that of the Martin method or SCPC method. As for the RZCC algorithm, Pfd of RZCC is smaller than SEA under certain circumstances. However, Pcd of RZCC performs worse than that of SEA. That is to say, RZCC sacrifices Pcd for the good performance of Pfd.Figure 17
Correct detection probability and false detection probability for cos-BOC (15, 2.5).The acquisition performances of the four methods are compared in Table1. The C/N0 corresponding to Pcd=90% and the average Pfd are compared. As can be seen from Table 1, when the modulation order is low (sin-BOC (1, 1)), the C/N0 required for Pcd=90% of the SEA algorithm is slightly higher than that of the Martin algorithm and SCPC algorithm, and the average Pfd is lower than that of the Martin algorithm and SCPC algorithm; when the modulation order is high, the C/N0 required for Pcd=90% and the average Pfd of the SEA algorithm are lower than those of the Martin and SCPC algorithm. By comparing comprehensively, the SEA algorithm is superior to the Martin algorithm and SCPC algorithm.Table 1
Comparison of capture performance for different algorithms.
BOC signal
Algorithm
C/N0 (Pcd=90%)
AveragePfd
sin-BOC (10, 2)
Martin
38.0 dB·Hz
3.48%
SCPC
38.0 dB·Hz
3.95%
SEAQT
36.8 dB·Hz
2.38%
SEAST
36.3 dB·Hz
3.14%
sin-BOC (1, 1)
Martin
29.4 dB·Hz
1.63%
SCPC
29.1 dB·Hz
1.82%
SEAQT
29.6 dB·Hz
1.16%
SEAST
29.6 dB·Hz
1.16%
cos-BOC (15, 2.5)
Martin
40.8 dB·Hz
6.39%
SCPC
41.0 dB·Hz
7.75%
SEAQT
38.7 dB·Hz
2.64%
SEAST
38.4 dB·Hz
3.91%


### 4.4.α Value Impact Analysis

Theα value of the SEAQT algorithm is used for the first detection, which is a screening of the capture statistic to reduce the probability of false detection. When the α value approaches zero, the SEAQT will degenerate into SEAST. Considering that a detection error is more likely to occur when the modulation order is high, a larger α value can be used to minimize Pfd in this case. As the enfoldment gain of the low-order BOC signal is low, the capture statistic is difficult to exceed the threshold, which makes the false detection possibility low, and Pfd is not prone to occur. Therefore, the value of α can be smaller than the high-order BOC.Taking the sin-BOC (10, 2) signal as an example, whenPfa=10−4, C/N0=36dB⋅Hz, the range of α is between 0,1 with step 0.05, and 1000 simulations are carried out. The relationship curves between Pcd and Pfd versus α are shown in Figure 18. As can be seen from the figure, in order to obtain higher Pcd and lower Pfd, α=0.65 should be taken; in order to ensure lower Pfd, α=0.95 should be taken. With the increase in the α value especially when α>0.7, Pcd gradually decreases while the size of the α value has a little effect on Pfd.Figure 18
The relationship curves betweenPcd and Pfd versus α.


### 4.5. Complexity Analysis of Algorithms

When the local code is correlated with the received signal in the frequency domain, the frequency band number of the captured signal is recorded asnf and the coherence integration length is recorded as N0. The computational complexity of a FFT operation is counted as NFFT, the computational complexity of filtering is marked as Nfilter, the computational complexity of complex multiplication is marked as Nm, and the computational complexity of real addition is marked as Na.In addition to FFT operation, the Martin algorithm needs one filtering for theI/Q branch separately and one complex multiplication for upper and lower sidebands. Then, the complexity of the Martin algorithm is
(19)NAndia=nf4NFFT+2Nfilter+2Nm+3Na.Among them,4NFFT is the sum of the FFT computational complexity of the upper/lower sideband I/Q branch, 2Nfilter is the sum of the computational complexity of the I/Q branch filter, 2Nm is the sum of the computational complexity of the upper and lower sideband complex multiplication, 3Na is the sum of the computational complexity of the upper/lower side bands of I2 plus Q2 and the upper I2+Q2 plus lower I2+Q2 side band.Similarly, the complexity of the SCPC algorithm is(20)NSCPC=nf4NFFT+2Nfilter+3Na.In the SEAQT algorithm, the computational complexity of the first threshold decision is counted asNdetection, the computational complexity of searching three peaks and two valleys is counted as Nfind, and the computational complexity of enfoldment is counted as Nfold. Therefore, the complexity of the SEAQT algorithm is as follows:
(21)NSEAQT=nf2NFFT+Na+Ndetection+Nfind+Nfold.The complexity of the SEAST algorithm is as follows:(22)NSEAST=nf2NFFT+Na+Nfind+Nfold.Since the SEA algorithm is not computationally intensive for searching the three peaks and two valleys and the enfoldment calculation is a simple multiplication and addition, the complexity of the SEA algorithm is lower compared with that of the Martin algorithm and SCPC algorithm.


## 4.1. Theoretical Analysis

The BOC signal broadcasted by the satellite can be expressed as(10)st=EbDtxPtxBtsin2πfct+θ.Among them,Eb is the signal energy, Dt is the data code broadcasted by the satellite, xPt is the PRN code, xBt is the BOC code, fc is the carrier frequency, θ is the initial phase of the signal.The signal after the downconverter and analog-to-digital converter (ADC) can be expressed as(11)sIFn=ADn−τxPn−τxBn−τsin2πfIF+fdn+θ+δn.Among them,A is the amplitude of the received signal, fIF is the intermediate frequency, δn is the Gauss white noise with zero mean and the variance σ2, Dn−τ is the data code of the received signal, xPn−τ is the PRN code of the received signal, and xBn−τ is the BOC code of the received signal.Detection statistic of the ambiguous BOC method is(12)V=1L∑n=0L−1I2n+Q2n.Among them,In and Qn are in-phase components and quadrature components obtained by correlation between local codes and the received signals, respectively, and L is the number of incoherent integrals. When the maximum correlation peak in detection V exceeds the threshold, the signal is detected. Vt, Pmd, and Pfa refer to the capture threshold, the probability of missed detection, and the probability of false alarm, respectively.The false alarm probabilityPfa corresponding to the capture threshold Vt is
(13)Pfa=∫Vt∞vσ2e−v2/2σ2dv=e−Vt2/2σ2.Thus, the capture thresholdVt is
(14)Vt=σ−2lnPfa.In this paper, the ambiguous BOC method, Martin method, SCPC method, and SEA method are selected for simulation verification. At the same time, the performance of RZCC and SEA is compared and analyzed. For the sin-BOC (10, 2) signal, the ACF of the four algorithms is shown in Figure12. The maximum correlation peak of the ambiguous BOC algorithm is normalized to 1. Because of filtering, the maximum peak of the Martin method and SCPC method is less than 1. Since the modulation order of the sin-BOC (10, 2) signal is 10, according to Formula (7), the maximum correlation peak of the SEA method is 2.62. As shown in Figure 12, the maximum correlation peak of SEA is 2.55, which is in accordance with the theoretical analysis results.Figure 12
ACF of four methods.


## 4.2. Applicability Analysis of the SEA Algorithm

The SEA algorithm applies the ACF similarity of the BOC signal. Whether it is the sin-BOC signal or cos-BOC signal, for the BOC signal of any modulation order, the adjacent two secondary peaks of the BOC signal ACF have similarities with the main peak. Therefore, the enfoldment method can be utilized. In other words, SEA is suitable for sin-BOC and cos-BOC signals of any order in theory.Four kinds of BOC signals, sin-BOC (10, 5), cos-BOC (1, 1), sin-BOC (15, 10), and cos-BOC (10, 4), are simulated with a sampling rate of 400 MHz, an IF of 30 MHz, a Doppler shift of 300 Hz, and an initial code phase of 1500 chips (after sampling). WhenC/N0=40dB⋅Hz, the crosscorrelation function (CCF) of the local BOC signal and the received signal is shown in Figure 13. The ACF of the four BOC signals with no noise is shown in Figure 14. As can be seen from Figures 13 and 14, the SEA method is suitable for sin-BOC and cos-BOC signals with even or odd modulation orders.Figure 13
CCF ofC/N0=40dB·Hz modulated signal.Figure 14
ACF of the four BOC signals.


## 4.3. Capture Performance Analysis

Correct detection probabilityPcd and false detection probability Pfd are used to analyze capture performance. For BOC m,n signals, the phase difference between the main peak and the second peak code is fs/2fn, where fs is the signal sampling rate. The captured code phase and frequency are recorded as φc and fc, respectively; the actual code phase and frequency are recorded as φa and fa, respectively.The total number of Monte Carlo simulation isN. The number of correct detection N0 is defined as the number of times, while the φc−φa<fs/2fn and fc=fa are simultaneously satisfied in the simulation. Consequently, the correct detection probability Pcd is defined as
(15)Pcd=N0N.The false detection numberN1 is defined as the number of times φc−φa≥fs/2fn or fc≠fa in the simulation. Therefore, false detection probability Pfd is defined as
(16)Pfd=N1N.When the signal exists, if the capture statistic exceeds the capture threshold, the signal is considered to be detected, which corresponds to the detection probabilityPd. The opposite corresponds to the missed detection probability Pmd. Thus, the detection probability Pd and the missed detection probability Pmd satisfy the following relationship.
(17)Pd+Pmd=1.Considering that the correct detection probabilityPcd and the false detection probability Pfd in the paper are all defined under the condition that the signal is detected, the correct detection probability Pcd and the false detection probability Pfd satisfy the following relationship.
(18)Pcd+Pfd=Pd.The first set of simulation conditions is set as follows: for the sin-BOC (10, 2) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 30 MHz, the Doppler frequency shift is 300 Hz, the initial code phase is 905 chips, the C/N0 range is 34dB⋅Hz≤C/N0≤43dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.9 in the SEAQT method.As shown in Figure15, the correct detection probability Pcd of the SEAST method is the highest, followed by the SEAQT method. Taking 90% correct detection probability as an example, the SEAQT algorithm improves by about 1.2 dB compared with the SCPC algorithm. As C/N0≥37dB⋅Hz, the false detection probability Pfd of SEA is less than that of the Martin method or SCPC method. For the RZCC algorithm, both Pcd and Pfd are not performing well. In some cases, Pfd of RZCC exceeds 20%.Figure 15
Correct detection probability and false detection probability for sin-BOC (10, 2).The second set of simulation conditions is set as follows: for the sin-BOC (1, 1) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 40 MHz, the Doppler frequency shift is 580 Hz, the initial code phase is 1500 chips, the C/N0 range is 26dB⋅Hz≤C/N0≤35dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.7 in the SEAQT method.As shown in Figure16, the correct detection probability Pcd of SEAQT/SEAST is similar, which is slightly lower than that of the Martin method and SCPC, but the false detection probability Pfd performs better than that of the Martin method and SCPC method. The reason why SEA does not capture the BOC n,n signal well is that the second peak of the BOC n,n signal is only 1/4 of the main peak, which leads to the main peak after enfoldment is 1.5 times than the former one. In other words, the enfoldment signal may not be high enough to exceed the acquisition threshold, which results in a low detection probability. As for the RZCC algorithm, it performs even worse than SEA. When C/N0≥31dB⋅Hz, Pfd of RZCC reaches more than 25%, but Pcd is less than 60%.Figure 16
Correct detection probability and false detection probability for sin-BOC (1, 1).The third set of simulation conditions is set as follows: for the cos-BOC (15, 2.5) signal, the false alarm rate isPfa=10−4, the sampling rate is 400 MHz, the IF is 50 MHz, the Doppler frequency shift is -330 Hz, the initial code phase is 3220 chips, the C/N0 range is 36dB⋅Hz≤C/N0≤45dB⋅Hz with a 1 dB step, the frequency search step is 300 Hz, the frequency search range is −1200Hz≤f≤1200Hz, Monte Carlo simulation times are N=1000, and α=0.95 in the SEAQT method.As shown in Figure17, the correct detection probability Pcd of SEAST is the highest, followed by the SEAQT. Taking 90% correct detection probability as an example, the SEAQT algorithm improves by 2.1 dB compared with the Martin algorithm. When C/N0≥38dB⋅Hz, the false detection probability Pfd of SEA is less than that of the Martin method or SCPC method. As for the RZCC algorithm, Pfd of RZCC is smaller than SEA under certain circumstances. However, Pcd of RZCC performs worse than that of SEA. That is to say, RZCC sacrifices Pcd for the good performance of Pfd.Figure 17
Correct detection probability and false detection probability for cos-BOC (15, 2.5).The acquisition performances of the four methods are compared in Table1. The C/N0 corresponding to Pcd=90% and the average Pfd are compared. As can be seen from Table 1, when the modulation order is low (sin-BOC (1, 1)), the C/N0 required for Pcd=90% of the SEA algorithm is slightly higher than that of the Martin algorithm and SCPC algorithm, and the average Pfd is lower than that of the Martin algorithm and SCPC algorithm; when the modulation order is high, the C/N0 required for Pcd=90% and the average Pfd of the SEA algorithm are lower than those of the Martin and SCPC algorithm. By comparing comprehensively, the SEA algorithm is superior to the Martin algorithm and SCPC algorithm.Table 1
Comparison of capture performance for different algorithms.
BOC signal
Algorithm
C/N0 (Pcd=90%)
AveragePfd
sin-BOC (10, 2)
Martin
38.0 dB·Hz
3.48%
SCPC
38.0 dB·Hz
3.95%
SEAQT
36.8 dB·Hz
2.38%
SEAST
36.3 dB·Hz
3.14%
sin-BOC (1, 1)
Martin
29.4 dB·Hz
1.63%
SCPC
29.1 dB·Hz
1.82%
SEAQT
29.6 dB·Hz
1.16%
SEAST
29.6 dB·Hz
1.16%
cos-BOC (15, 2.5)
Martin
40.8 dB·Hz
6.39%
SCPC
41.0 dB·Hz
7.75%
SEAQT
38.7 dB·Hz
2.64%
SEAST
38.4 dB·Hz
3.91%


## 4.4.α Value Impact Analysis

Theα value of the SEAQT algorithm is used for the first detection, which is a screening of the capture statistic to reduce the probability of false detection. When the α value approaches zero, the SEAQT will degenerate into SEAST. Considering that a detection error is more likely to occur when the modulation order is high, a larger α value can be used to minimize Pfd in this case. As the enfoldment gain of the low-order BOC signal is low, the capture statistic is difficult to exceed the threshold, which makes the false detection possibility low, and Pfd is not prone to occur. Therefore, the value of α can be smaller than the high-order BOC.Taking the sin-BOC (10, 2) signal as an example, whenPfa=10−4, C/N0=36dB⋅Hz, the range of α is between 0,1 with step 0.05, and 1000 simulations are carried out. The relationship curves between Pcd and Pfd versus α are shown in Figure 18. As can be seen from the figure, in order to obtain higher Pcd and lower Pfd, α=0.65 should be taken; in order to ensure lower Pfd, α=0.95 should be taken. With the increase in the α value especially when α>0.7, Pcd gradually decreases while the size of the α value has a little effect on Pfd.Figure 18
The relationship curves betweenPcd and Pfd versus α.


## 4.5. Complexity Analysis of Algorithms

When the local code is correlated with the received signal in the frequency domain, the frequency band number of the captured signal is recorded asnf and the coherence integration length is recorded as N0. The computational complexity of a FFT operation is counted as NFFT, the computational complexity of filtering is marked as Nfilter, the computational complexity of complex multiplication is marked as Nm, and the computational complexity of real addition is marked as Na.In addition to FFT operation, the Martin algorithm needs one filtering for theI/Q branch separately and one complex multiplication for upper and lower sidebands. Then, the complexity of the Martin algorithm is
(19)NAndia=nf4NFFT+2Nfilter+2Nm+3Na.Among them,4NFFT is the sum of the FFT computational complexity of the upper/lower sideband I/Q branch, 2Nfilter is the sum of the computational complexity of the I/Q branch filter, 2Nm is the sum of the computational complexity of the upper and lower sideband complex multiplication, 3Na is the sum of the computational complexity of the upper/lower side bands of I2 plus Q2 and the upper I2+Q2 plus lower I2+Q2 side band.Similarly, the complexity of the SCPC algorithm is(20)NSCPC=nf4NFFT+2Nfilter+3Na.In the SEAQT algorithm, the computational complexity of the first threshold decision is counted asNdetection, the computational complexity of searching three peaks and two valleys is counted as Nfind, and the computational complexity of enfoldment is counted as Nfold. Therefore, the complexity of the SEAQT algorithm is as follows:
(21)NSEAQT=nf2NFFT+Na+Ndetection+Nfind+Nfold.The complexity of the SEAST algorithm is as follows:(22)NSEAST=nf2NFFT+Na+Nfind+Nfold.Since the SEA algorithm is not computationally intensive for searching the three peaks and two valleys and the enfoldment calculation is a simple multiplication and addition, the complexity of the SEA algorithm is lower compared with that of the Martin algorithm and SCPC algorithm.


## 5. Conclusions

Aimed at the acquisition ambiguity of BOCm,n signals, this paper proposes the SEA method, which is divided into SEAQT and SEAST methods. Both methods can solve the ambiguity of BOC signals. When M≥2, the main peak of SEA is 10lg2M−12+M2/M−22dB stronger than the secondary peak, which greatly improves the peak value. It turns out that the false detection probability Pfd of the SEAQT method performs better than that of SEAST through comparative analysis.The simulation results show that the SEA can be applied to the sin-BOC signal and cos-BOC signal, with even and odd modulation order. The acquisition performances of SEA, Martin, and SCPC are analyzed and compared. Except for BOCn,n signals, the acquisition performance of the SEAQT method is better than that of the Martin method and SCPC method in other cases.Considering that the modulation order is high, the detection error is more likely to occur and a largerα value can be used to minimize Pfd in this case. When the modulation order is low, the value of α can be smaller. This paper takes the sin-BOC (10, 2) signal as an example to analyze the effect of α on Pcd and Pfd. Various α values can be selected according to different needs for Pcd and Pfd.As above, both SEAQT and SEAST algorithms are superior to the RZCC algorithms. As for cos-BOC (15, 2.5), RZCC sacrificesPcd for the good performance of Pfd. In the other two cases, SEAQT and SEAST algorithms are much better than RZCC. Compared with the Martin algorithm and SCPC algorithm, the SEA algorithm does not need to use filters or an auxiliary signal. According to the comprehensive comparison, the SEA algorithm is better than the Martin, SCPC, and RZCC algorithms.



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*Source: 4314132-2020-02-25.xml*