# A New Acquisition Algorithm with Elimination Side Peak for All BOC Signals

Authors: Fang Liu; Yongxin Feng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A new inhibition side peak acquisition (ISPA) algorithm is proposed for binary offset carrier (BOC) modulated signals, which will be utilized in global navigation satellite systems (GNSS). We eliminate all side peaks of the BOC correlation function (CF) by structuring special sequences composed of PRN code and cycle rectangular sequences. The new algorithm can be applied to both generic sine- and cosine-phased BOC signals, as well as to all modulation orders. Theoretical and simulation results demonstrate that the new algorithm can completely eliminate the ambiguity threat in the acquisition process, and it can adapt to lower SNR. In addition, this algorithm is better than the traditional algorithms in acquisition performance and inhibition side peak ability.


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## Body

## 1. Introduction

With the development and application of global navigation satellite systems (GNSS) [1], GNSS signal receiving methods have become highly valued. Because the acquisition technology is the core of receiving; therefore, it also becomes a focus problem. Thus, mass acquisition algorithms [2, 3] are proposed for GNSS signals to improve receiving performance. However, modern GNSS has provided new signals with longer PRN (pseudo random noise) codes and newer modulation methods, which aim to improve the positioning performance. Binary offset carrier (BOC) [4] modulated signals are the most widely used signal families in GNSS, and their side peak characteristics also require the highest technique complexity from GNSS receivers.BOC modulation signal acquisition techniques focus on recovering the main correlation peak or eliminating ambiguities in the form of side peaks. At present, various techniques are proposed for side peak cancellation and are built on the basis of the correlation function (CF) of the BOC signals. Thus, the side band processing method originated from BPSK-like method [5, 6], and then some improved methods [7–9] are proposed. The partial band is obtained by filtering or frequency domain processing in these kind methods, and then the main peak was estimated using similar BPSK characteristic. These kind methods can reduce the influence of subcarrier, but the energy and the necessary information are lost. Thus, the auxiliary signal methods [10, 11] are mainly through the local auxiliary signal establishment to reach the purpose of removing side peaks. These kind methods can remove the side peak, but they lack universality. Thus, some effective methods [12–14] are proposed to improve the processing performance. However, these techniques apply only to sine-phased BOC signals. Thus, in [15], a mitigating ambiguity acquisition method is proposed. This technique can counterbalance the undesired side peaks, but it applies only to cosine-phased BOC signals.In this paper, considering filter restriction and generic deficiency problems in traditional algorithms, we propose an inhibition side peak acquisition algorithm, which is applicable to all orders and to both generic sine- and cosine-phased BOC signals.


## 2. BOC Modulation Signal and Acquisition Analysis

### 2.1. BOC Modulation Signal

BOC modulation signal is obtained by the product of PRN code and the square wave. The complex form of the BOC signal is expressed as(1)St=e-iθ∑kakμεTst-k·εTs-t0CTst-t0,where ak is the modulated PRN code, CTs(t) is the subcarrier, 2Ts is the subcarrier cycle, μnTs(t) is the spread spectrum symbol, ε is the modulation order, and θ and t0, respectively, express the phase and time offset.The BOC signal is usually expressed asBOC⁡(fs,fc), the frequency of the subcarrier is fs times the benchmark frequency, and the frequency of the PRN code is fc times the benchmark frequency. The benchmark frequency is 1.023 MHz. The autocorrelation function of the BOC signal has multiple peaks and passes through zero many times. Its autocorrelation function consists of the positive peaks and the negative peaks, and the number of peaks is 2ε-1. The distance between peaks is Ts, and each peak height is (-1)l(ε-|l|)/ε, where l is the serial number of the peaks.


### 2.2. The Acquisition Analysis

From the perspective of algorithm generality, the acquisition algorithm for BOC modulation signal is usually divided into three categories, namely, the full band acquisition (FBA) algorithm [16], the peak optimization acquisition (POA) algorithm [17], and the single peak recovery acquisition (SPRA) algorithm [18]. Their principles are shown in Figures 1, 2, and 3, respectively.Figure 1
The full band acquisition algorithm principle.Figure 2
The peak optimization acquisition algorithm principle.Figure 3
The single peak recovery acquisition algorithm principle.FBA is a class of traditional algorithms, in which the correlation arithmetic is executed between the received signal and the original PRN code modulated by a square wave. POA is a class of improved algorithms, in which multiple correlations are executed to improve the main peak. SPRA is a class of new methods, in which a partial signal is separated from the received signal by the corresponding operations to inhibit the square wave.


## 2.1. BOC Modulation Signal

BOC modulation signal is obtained by the product of PRN code and the square wave. The complex form of the BOC signal is expressed as(1)St=e-iθ∑kakμεTst-k·εTs-t0CTst-t0,where ak is the modulated PRN code, CTs(t) is the subcarrier, 2Ts is the subcarrier cycle, μnTs(t) is the spread spectrum symbol, ε is the modulation order, and θ and t0, respectively, express the phase and time offset.The BOC signal is usually expressed asBOC⁡(fs,fc), the frequency of the subcarrier is fs times the benchmark frequency, and the frequency of the PRN code is fc times the benchmark frequency. The benchmark frequency is 1.023 MHz. The autocorrelation function of the BOC signal has multiple peaks and passes through zero many times. Its autocorrelation function consists of the positive peaks and the negative peaks, and the number of peaks is 2ε-1. The distance between peaks is Ts, and each peak height is (-1)l(ε-|l|)/ε, where l is the serial number of the peaks.


## 2.2. The Acquisition Analysis

From the perspective of algorithm generality, the acquisition algorithm for BOC modulation signal is usually divided into three categories, namely, the full band acquisition (FBA) algorithm [16], the peak optimization acquisition (POA) algorithm [17], and the single peak recovery acquisition (SPRA) algorithm [18]. Their principles are shown in Figures 1, 2, and 3, respectively.Figure 1
The full band acquisition algorithm principle.Figure 2
The peak optimization acquisition algorithm principle.Figure 3
The single peak recovery acquisition algorithm principle.FBA is a class of traditional algorithms, in which the correlation arithmetic is executed between the received signal and the original PRN code modulated by a square wave. POA is a class of improved algorithms, in which multiple correlations are executed to improve the main peak. SPRA is a class of new methods, in which a partial signal is separated from the received signal by the corresponding operations to inhibit the square wave.


## 3. ISPA Algorithm Structure

Letft be the sampling frequency of the BOC signal, and the frequency of the subcarrier and PRN code are fs times and fc times the benchmark frequency, respectively. Considering square wave modulation characteristics, the product model of the spread spectrum sequence and a series of rectangular sequences is structured, which can be approximately expressed as the BOC base-band signal model. Hence, the base-band signal may be represented by the following equation:(2)SBOC⁡n=dnCn·∑j=1εM-1j+1RNn+N-jNSΔn+λ0n,where d(n) is the message, C(n) is the PRN code, λ0(n) is the mixed noise function caused by the discarded samples, SΔ(n) is the frequency error function cause by the front processing, n is the sequence position, M is the number of chips in accumulation time, and ε is both the modulation order and the number of rectangular sequences in one chip, which is expressed as (3). N is both the number of sampling points and the rectangular sequence width, which is expressed as (4). RN(n+N-jN) is a shifting rectangular sequence, which is expressed in (5), where u(n) is the step sequence:(3)ε=2fsfc,(4)N=ft2fs,(5)RNn+N-jN=un+N-jN-un-jN.Considering the represented model of the BOC base-band signal, the local rectangular sequence model is structured to inhibit the acquisition of side peaks. The structured process is shown in Figure4, in which ε is an odd number. The square wave sequence is shown in Figure 4(a), and the two structured cycle rectangular sequences are shown in Figures 4(b) and 4(c). The cycle rectangular sequences can also be structured for an even number ε using the same principle. Further, the ith cycle of two local channel rectangular sequences can be expressed as(6)RNn+εN-iεN=un+εN-iεN-un+εN-N-iεN,RNn+N-iεN=un+N-iεN-un-iεN.Figure 4
The structured process of the rectangular sequence model.The original PRN code is, respectively, multiplied by the two-channel cycle rectangular sequences to structure the two new local channel sequences, which are expressed as(7)H1n=Cn+τ∑i=1M-1iε-εRNn+εN-iεN,H2n=Cn+τ∑i=1M-1iε-1RNn+N-iεN,where C(n+τ) is the delay PRN code and τ is the time delay.The beforehand processing received signal is executed by the correlation circumferential arithmetic withX1(n) and X2(n), respectively, which are expressed as(8)X1n=SBOC⁡n⊗H1n=dnCn∑j=1εM-1j+1RNn+N-jNSΔn+λ0n⊗Cn+τ∑i=1M-1iε-εRNn+εN-iεN≈∧TsnSΔn-∧Tsn+TsSΔn+∧Tsn+2TsSΔn-⋯+λ0n=∑i=0ε-1ε∧Tsn+iTsSΔn+λ0n,X2n=SBOC⁡n⊗H2n=dnCn∑j=1εM-1j+1RNn+N-jNSΔn+λ0n⊗Cn+τ∑i=1M-1iε-1RNn+N-iεN≈∧TsnSΔn-∧Tsn-TsSΔn+∧Tsn-2TsSΔn-⋯+λ0n=∑i=0ε-1ε∧Tsn-iTsSΔn+λ0n,where ∧Ts(n) is the trigonometry sequence of width Ts and is expressed as(9)∧Tsn=2Tsn+1,-Ts2≤n<0-2Tsn+1,0≤n≤Ts20,other.Whenε is five, the autocorrelation result of the BOC base-band signal is shown in Figure 5(a), and the two structured correlation results are shown in Figures 5(b) and 5(c), respectively. The results show that the positions of the two channel main peaks exactly coincide with the position of the autocorrelation main peak, and the numbers of peaks are the same in both channels. In addition, the positions of the two channel peaks are symmetrical about the main peak position of the autocorrelation function.Figure 5
The correlation result of BOC.
(a)
(b)
(c)In view of these characteristics and combining (8) and (9), the addition and subtraction operations are performed by using the two structured correlation results, expressed as(10)ΔX1n=X1n+X2n,ΔX2n=X1n-X2n.Thus, the new correlation function is structured to eliminate side peaks, and the processing is expressed as(11)ΔXn=ΔX1n-ΔX2n≈∑i=0ε-1ε∧Tsn+iTsSΔn+λ0nddd+∑i=0ε-1ε∧Tsn-iTsSΔn+λ0n-∑i=0ε-1ε∧Tsn+iTsSΔn+λ0nddddd-∑i=0ε-1ε∧Tsn-iTsSΔn+λ0n≈2∧TsnSΔn+λ0n+∑i=1ε-1ε∧Tsn+iTsSΔn+λ0n+∑i=1ε-1ε∧Tsn-iTsSΔn+λ0n-∑i=1ε-1ε∧Tsn+iTsSΔn+λ0n-∑i=1ε-1ε∧Tsn-iTsSΔn+λ0n≈2∧TsnSΔn+λ0n.When the impacts of the frequency error and noise function are likely to be relatively weak, the relationship of the main peak valueA1 in the ΔX(n) function and the BOC autocorrelation function value A2 is expressed as(12)A1=2εA2.To improve the peak, the result ofΔX(n) is multiplied by a coefficient of ε/2 to obtain the final expression as(13)ΔX′n=ε2·ΔXn=ε∧TsnSΔn+ε2λ0n.


## 4. Performance Analysis

TheΔX1(n) and ΔX2(n) may be approximately represented by (14)ΔX1n=SBOC⁡n⊗H1n+SBOC⁡n⊗H2n=SBOC⁡n⊗Cn+τ∑i=1Mssssssssssssssss·∑i=1M-1iε-εRNn+εN-iεNssssssssssssssssss+∑i=1M-1iε-1RNn+N-iεN,ΔX2n=SBOC⁡n⊗H1n-SBOC⁡n⊗H2n=SBOC⁡n⊗Cn+τ∑i=1Mssssssssssssssss·∑i=1M-1iε-εRNn+εN-iεNssssssssssssssssss-∑i=1M-1iε-1RNn+N-iεN.At the same time, the structured square function can be expressed as(15)∑i=1M-1iε-εRNn+εN-iεN+∑i=1M-1iε-1RNn+N-iεN=εN2,∑i=1M-1iε-εRNn+εN-iεN-∑i=1M-1iε-1RNn+N-iεN=0.Hence,ΔX1(n) satisfies a Gaussian distribution whose mean is (A/4)εN and whose variance is σ2εN/2, and ΔX2(n) satisfies Gaussian distribution whose mean is 0 and whose variance is σ2εN/2.WhereA is the signal amplitude and σ2 is the noise variance, the probability density function ΔX1(n) is expressed as(16)f1x=1πσ2εN(e-(x-A/4εN)2/σ2εN+e-(x+A/4εN)2/σ2εN)and the probability density function ΔX2(n) is expressed as(17)f2x=2πσ2εNe-(x)2/σ2εN.Thus, theΔX(n) probability density function is expressed as (18)fx=14πσ2εN·e-(x-A/4εN)2/2σ2εN+e-(x+A/4εN)2/2σ2εN.The false alarm probability of the ISPA algorithm is expressed as(19)Pfa=∫G+0012πεNσe-x2/2εNσ2dx.The acquisition detection probability of the ISPA algorithm is expressed as(20)PD=∫G+0014πσ2εNsss·e-(x-A/4εN)2/2σ2εN+e-(x+A/4εN)2/2σ2εNdx,where G is the acquisition threshold.


## 5. Analysis and Simulation

### 5.1. Side Peak Inhibition Analysis

Equations (9) and (13) show that the final correlation result has a single peak whose main waveform is a triangular peak. Thus, the ISPA algorithm can achieve the goal of side peak inhibition. The new algorithm is then simulated using the following parameters: 10.23 MHz PRN code frequency, 15.345 MHz square wave frequency, and 122.76 MHz sampling frequency, modulation order of 3, and the sine-phased BOC signal for these parameters is expressed as sinBOC(15,10).The ISPA result forsinBOC(15,10) is shown in Figure 6. The ISPA results for sinBOC(10,2), cosBOC(10,5), and cosBOC(6,1) are shown in Figures 7, 8, and 9, respectively. The simulation results show that the ISPA algorithm can clearly recover the main peak, whose position is the same as the main peak position of the autocorrelation function. In particular, the ISPA algorithm can effectively inhibit side peaks.Figure 6
The ISPA result forsinBOC(15,10).Figure 7
The ISPA result forsinBOC(10,2).Figure 8
The ISPA result forcosBOC(10,5).Figure 9
The ISPA result forcosBOC(6,1).


### 5.2. Adaptability Analysis

The new algorithm result is influenced by the frequency error and the mixed noise, according to (13). The algorithm result approximately conforms to the cycle equation because of the frequency error function cycle characteristics. When the relationship of the frequency error and accumulation time t satisfies (21), the algorithm result error reaches its maximum. We also find that the ISPA algorithm result decreases gradually along with the increase of mixed noise, according to (21)SΔn=k2t,where k is the positive integer.Furthermore, the ISPA algorithm’s adaptability is simulated with the following parameters: 15.345 MHz square wave frequency and 122.76 MHz sampling frequency, modulation mode is sine mode, and modulation orders are 15, 10, 6, and 3, respectively.The relationship between the relative main peak and frequency error is shown in Figure10, which shows that the results satisfy (21). The relationship between the relative main peak and SNR is shown in Figure 11, revealing that the relative main peak decreases gradually with decreasing SNR. And the ISPA algorithm’s adaptability to the SNR environment is more than −25 dB, according to (13) and Figure 11.Figure 10
The relationship between the relative main peak and frequency error.Figure 11
The relationship between the relative main peak and SNR.


### 5.3. Superiority Analysis

To verify the superiority of the ISPA algorithm, this ISPA algorithm is compared with other algorithms, namely, the FBA algorithm, POA algorithm, and SPRA algorithm. The simulation parameters are as follows: 2.046 MHz PRN code frequency, and the modulation mode is sine mode.With changing modulation order, the main peak width changes and the main peak relative changes are shown in Figures12 and 13. The results show that the ISPA algorithm’s main peak width is the smallest, and its main peak relative result is the greatest, demonstrating that this algorithm’s acquisition and tracking performance is the best.Figure 12
The relationship between the main peak width and the modulation order.Figure 13
The relationship between the relative main peak and the modulation order.The side peak relative changes and the main/side peak ratio changes with changing modulation order are shown in Figures14 and 15. The results show that the ISPA algorithm side peak relative result is the smallest, and the main/side peak ratio is the greatest, demonstrating that this algorithm’s side peak inhibition ability is best.Figure 14
The relationship between the relative side peak and the modulation order.Figure 15
The relationship between the main/side peak ratio and the modulation order.The main peak relative changes with changing SNR are shown in Figure16. The results show that the adaptability of the ISPA algorithm is better than the FBA algorithm and POA algorithm, but there are no significant differences between the ISPA algorithm and the SPRA algorithm.Figure 16
The relationship between the relative main peak changes and SNR.


## 5.1. Side Peak Inhibition Analysis

Equations (9) and (13) show that the final correlation result has a single peak whose main waveform is a triangular peak. Thus, the ISPA algorithm can achieve the goal of side peak inhibition. The new algorithm is then simulated using the following parameters: 10.23 MHz PRN code frequency, 15.345 MHz square wave frequency, and 122.76 MHz sampling frequency, modulation order of 3, and the sine-phased BOC signal for these parameters is expressed as sinBOC(15,10).The ISPA result forsinBOC(15,10) is shown in Figure 6. The ISPA results for sinBOC(10,2), cosBOC(10,5), and cosBOC(6,1) are shown in Figures 7, 8, and 9, respectively. The simulation results show that the ISPA algorithm can clearly recover the main peak, whose position is the same as the main peak position of the autocorrelation function. In particular, the ISPA algorithm can effectively inhibit side peaks.Figure 6
The ISPA result forsinBOC(15,10).Figure 7
The ISPA result forsinBOC(10,2).Figure 8
The ISPA result forcosBOC(10,5).Figure 9
The ISPA result forcosBOC(6,1).


## 5.2. Adaptability Analysis

The new algorithm result is influenced by the frequency error and the mixed noise, according to (13). The algorithm result approximately conforms to the cycle equation because of the frequency error function cycle characteristics. When the relationship of the frequency error and accumulation time t satisfies (21), the algorithm result error reaches its maximum. We also find that the ISPA algorithm result decreases gradually along with the increase of mixed noise, according to (21)SΔn=k2t,where k is the positive integer.Furthermore, the ISPA algorithm’s adaptability is simulated with the following parameters: 15.345 MHz square wave frequency and 122.76 MHz sampling frequency, modulation mode is sine mode, and modulation orders are 15, 10, 6, and 3, respectively.The relationship between the relative main peak and frequency error is shown in Figure10, which shows that the results satisfy (21). The relationship between the relative main peak and SNR is shown in Figure 11, revealing that the relative main peak decreases gradually with decreasing SNR. And the ISPA algorithm’s adaptability to the SNR environment is more than −25 dB, according to (13) and Figure 11.Figure 10
The relationship between the relative main peak and frequency error.Figure 11
The relationship between the relative main peak and SNR.


## 5.3. Superiority Analysis

To verify the superiority of the ISPA algorithm, this ISPA algorithm is compared with other algorithms, namely, the FBA algorithm, POA algorithm, and SPRA algorithm. The simulation parameters are as follows: 2.046 MHz PRN code frequency, and the modulation mode is sine mode.With changing modulation order, the main peak width changes and the main peak relative changes are shown in Figures12 and 13. The results show that the ISPA algorithm’s main peak width is the smallest, and its main peak relative result is the greatest, demonstrating that this algorithm’s acquisition and tracking performance is the best.Figure 12
The relationship between the main peak width and the modulation order.Figure 13
The relationship between the relative main peak and the modulation order.The side peak relative changes and the main/side peak ratio changes with changing modulation order are shown in Figures14 and 15. The results show that the ISPA algorithm side peak relative result is the smallest, and the main/side peak ratio is the greatest, demonstrating that this algorithm’s side peak inhibition ability is best.Figure 14
The relationship between the relative side peak and the modulation order.Figure 15
The relationship between the main/side peak ratio and the modulation order.The main peak relative changes with changing SNR are shown in Figure16. The results show that the adaptability of the ISPA algorithm is better than the FBA algorithm and POA algorithm, but there are no significant differences between the ISPA algorithm and the SPRA algorithm.Figure 16
The relationship between the relative main peak changes and SNR.


## 6. Conclusions

In this paper, the principle and characteristics of BOC modulation signals have been studied. To implement the BOC modulated signal acquisition, effective algorithms have been studied, including the full band acquisition (FBA) algorithm, the peak optimization acquisition (POA) algorithm, and the single peak recovery acquisition (SPRA) algorithm. Considering the filter restriction and generic deficiency problems in traditional algorithms, we propose the ISPA algorithm. We eliminate all side peaks of the BOC correlation function (CF) by structuring special sequences composed of PRN code and cycle rectangular sequences. The ISPA algorithm can be applied to both generic sine- and cosine-phased BOC signals and to all modulation orders. In addition, it outperforms the traditional algorithms in acquisition, inhibition side peak ability, and adaptability to lower SNR.



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