# Joint Acquisition Estimator of Modern GNSS Tiered Signals Using Block Pre-Correlation Processing of Secondary Code

^{*}

## Abstract

**:**

## 1. Introduction

- Pilot signals can be coherently correlated over its period without navigation message bit transitions. Therefore, SC pilot signals can increase receiver sensitivity;
- Tiered signals have better cross-correlation properties in comparison with other GNSS signals;
- A tiered signal spectrum has better immunity to narrow-band jamming. The power spectral density is shaped by the sinc function but is line-like due to the primary code periodicity. These lines are more “diluted” by an additional period of the SC code.

## 2. State of the Art

## 3. Common SC Acquisition Methods Theory

_{d}), and the SC phase (n) set of received signal parameters. The n could be expressed as an integer part of the code phase in integer periods T of the primary code phase τ. This expression is beneficial according to the chosen processing of a tiered signal over blocks of primary code periods.

_{d},n] (2) between the received signal r[t] and its locally generated replica. The replica consists of the periodic primary code c[t] with the period of T samples, and the periodic SC[t] with the period of N bits. Apriori unknown navigation message could be present in some types of signals (Table 1). The symbol ($\oplus $) denotes the modulo-2 (xor) product. The star (*) is a symbol of complex conjugation. The “arg max” is the argument of the maxima operator. The “mod” is the modulo operator. The f

_{d}is usually searched sequentially, and it is not the subject of this article.

#### 3.1. Acquisition—PCS Algorithm, Zero-Padding Algorithms, and Bit Transition Problem

_{x}from the discrete signal autocorrelation function R

_{x}. The Equation (3) could be rewritten to a form (4). This form uses the FFT instead of the DFT for a speedup. The symbol (°) denotes the Hadamard product.

_{FFT}, to be equal to 2

^{a}, where a is an integer. Nevertheless, it is usually not equal to the number of samples in the primary code period T. The solution is the zero-padding [25] of the r[t] and c[t] blocks of signal samples with N

_{z}zeroes.

_{z}. An amplitude of the false parasitic peak is at the expense of the main peak. The loss is proportional to the τ [14,17,20]. The false peak has an amplitude higher than the main peak for τ > (1/2)T. This false parasitic fragment peak has an origin in a violated cyclic property of a signal, which is caused by the added zeroes. A similar negative consequence has the navigation message or SC bit transition in the Doppler shift domain.

_{FFT}where half of its result is discarded anyway. A DBZP utilization for acquisition of signals with a potential bit transition is popular. New optimizations of the algorithm exist [18,21]. The DBZP works as a single-step post-correlation SC estimator [21,23]. It is suitable for further comparisons. However, its combination with the PCA and usability for SC acquisition is limited and problematic [17]. It is discussed in Section 4.

_{d}NT) shaped lobe in the Doppler shift domain is split into two side-lobes with a pitch equal to 2/(NT) Hz and a lower amplitude, as in Figure 4. These effects in the CAF are problematic for the utilization of the PCS as the joint estimator. This phenomenon in the CAF is well described in [19], using the rectangular (REC) W-function (8).

_{d}estimation is biased. This function is illustrated in Figure 4, together with the corresponding CAF for τ = (1/2)T. The “E” means normalized CAF cross-correlation energy. The study of SC bit transition problem on a model using the W-function is a key factor for its effective suppression.

_{second}” are operators for maximum, mean, and second maximum peak value of the CAF.

#### 3.2. Acquisition—Common SC PCS Post-Correlation Methods—Results

_{FFT}equal to the 16K (2

^{14}) FFT, N

_{z}= 384, and base-band Galileo E1C signals with CS25

_{1}and 4 MHz sampling rate were used for simulation in this article.

## 4. Pre-Correlation SC Processing, PCA, mPCA

#### 4.1. PCA and mPCA

#### 4.2. PCS Algorithm Using mPCA Pre-Correlation Processing of SC—Results

## 5. Analysis of the Adverse Effects on the PCS-SBZP Algorithm CAF

_{d}NT) function shaped lobe in the frequency-Doppler shift domain. The PNR and the FSPR ratios are given only by the cross-correlation function of the used primary code and the sinc function given by a PIT time NT. However, SBZP results presented above differed. Their CAFs contained parasitic fragments that decreased the PNR and the FSPR. An analysis of bit transition effects exists [19], but the overall analysis of the effect in periodic SC has not been published yet.

_{1}code and τ = (1/4)T. The side-lobes are repeated with a lower amplitude on many frequencies. These frequencies are given by a quasi-periodicity of the SC. It decreases the PNR and the FSPR performance, side-lobes cause the biased f

_{d}estimation again.

_{d}NT) lobes that are situated on multiples of 1/T frequency and are weighted by sinc(2πf

_{d}N(T−τ)) (16). The splitting of the main peak in the Doppler shift domain of the CAF is illustrated for τ = (1/4)T in Figure 13 (right). The CAF contains spectral fragments with amplitude depending on τ. Such spectral fragments decrease the FSPR too.

_{d}NT) lobe on multiplies of 1/T frequency in the Doppler shift domain. The small false peak in the code phase domain is contained too.

_{d}is biased.

_{FFT}= 16 384, N

_{z}= 384, and Galileo E1C signal with CS25

_{1}. However, the results in the Doppler shift domain were almost equal even for the case N

_{z}= 0. In this case, the false peak in the code phase domain was not present. It occurred because the false parasitic peak in the code phase domain is a product of a non-zero N

_{z}. Nevertheless, the effects of W and W’-functions are a product of the SC bit transition, and therefore a splitting remained.

## 6. Proposed PCS Algorithm with mSBZP and mPCA

#### 6.1. Modified SBZP Joint Primary and SC Estimator (mSBZP for SC Algorithm)—Theory

_{1}, W

_{2}) for the original and its time-shifted version. Their power spectra could be the same, but phase-spectra differ. Let the W

_{2}-function be complementary to the original W

_{1}-function in the time domain. Then, the coherent sum of the W

_{1}-function and the W

_{2}-function spectra is equal to zero. This is illustrated in Figure 16 using W-functions in the time and the spectral domain. Therefore, the parasitic effect on the CAF is subtracted, and the false peak is not formed and split. A similar effect is observable when two cross-correlation functions for two consecutive SC bits are combined.

_{z}samples. At last, both resulting functions are coherently combined (18). The symbol “circshift” denotes the circular shift of the signal samples vector.

_{1}and the W’

_{2}are formed according to cross-correlated parts of the signal and each replica. Then, due to the proposed coherent combining (18), the original loss (17) is equal to the mean value of the sum of both W’-functions (19) for the mSBZP case. Two partial correlations form one full correlation. The zero result of (19) indicates that the loss has been removed, and the main peak is not split.

#### 6.2. The mPCA-Based Joint Primary and SC Estimator (mPCA mSBZP for SC Algorithm)

_{z}samples. The results of both schemas are the same and follow. However, the post-correlation approach in Figure 18 requires N computations of the PCS algorithm per SC period NT, whilst its mPCA version requires only one. It is an essential computational complexity benefit.

#### 6.3. The mPCA-Based Joint Primary and SC Estimator (mPCA mSBZP for SC Algorithm)—Results

_{1}code period were depicted in Figure 21 and Figure 22. The resulting functions of the PNR and the FSPR were constant for a proper n. Thus, they were independent of the τ as was intended for the joint estimator. The proposed algorithm surmounts the post-correlation DBZP in both the PNR and the FSPR ratios. A better PNR, and a main to SC side peaks ratio, respectively, was also evident here.

_{d},n] set of tested parameters and 3000 simulation points) and the Constant-False-Alarm-Rate (CFAR, comparing PNR with fixed threshold) detection in Figure 23 at the final. The detection results of both above-compared schemas (post-DBZP and mPCA mSBZP) were similar. However, the mPCA algorithm surmounts others in lower computational complexity (number of PCS operations per SC period NT). See Section 6.4.

_{d}and the τ estimation. The remaining fragments are observable, but it has no significant effect. Its origin is in a non-zero mean value of SC.

#### 6.4. Computational Complexity and Implementation Remarks

_{FFT}is required for the same resolution of the computed cross-correlation function.

_{1}. However, the algorithm was approved using the Galileo CS4

_{1}(Galileo E5b-I) and the BeiDou B1 20 bits long Neumann–Hoffman SC as well. The algorithm was designed for an existing PCS based acquisition unit with the PCS core, which is implemented in System-On-a Chip (SoC) FPGA with three 16K (16,384) FFT blocks. This unit will be usable for the acquisition of these advanced signals using this easily implementable mPCA mSBZP principle in the future.

_{d}on the code rate.

## 7. Conclusions

_{FFT}than the DBZP, the mPCA mSBZP has other beneficial properties. The algorithm is τ, and f

_{d}unbiased. The essential benefit is a decrease in the number of PCS operations that are required per a period of a tiered signal from N to one PCS operation (where N is the number of SC bits). It allows savings of PCS computational resources (hardware resources or latency).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**BeiDou-B1 (Neumann–Hoffman,

**left**) and Galileo E1C (CS25

_{1},

**right**) autocorrelation function of SC.

**Figure 4.**W-function (

**left**), the effect of SC bit transition on the CAF (

**middle**) and Doppler shift domain (

**right**), for τ = (1/2)T.

**Figure 5.**Result of the utilization of the single-step PCS post-correlation schema with SC (Galileo CS25

_{1}).

**Figure 6.**Result of the utilization of the single-step PCS post-correlation schema with SC, detail of Figure 5.

**Figure 11.**Result of mPCA pre-correlation processing of SC using DBZP, τ = (1/4)T (

**left**) and τ = (3/4)T (

**right**).

**Figure 13.**Splitting of the false peak (

**left**) and the main peak (

**right**) in the Doppler shift domain for τ = (1/4)T.

**Figure 14.**Resulting CAF of the mPCA pre-correlation processing of SC using the SBZP, for τ = (1/4)T. CAF (

**left**), Dopler shift domain (

**middle**), Code phase domain (

**right**).

**Figure 15.**Resulting CAF of the mPCA pre-correlation processing of SC using the SBZP, for τ = (3/4)T. CAF (

**left**), Dopler shift domain (

**middle**), Code phase domain (

**right**).

**Figure 16.**Description of W

_{1}and W

_{2}-function in the time domain (

**left**) and the spectral domain (

**right**) for τ = (1/2)T.

**Figure 19.**Proposed mPCA mSBZP schema for SC acquisition in the pre-correlation processing algorithm.

**Figure 24.**CAF of mPCA mSBZP for SC algorithm for τ = (1/2)T. CAF (

**left**), Dopler shift domain (

**middle**), Code phase domain (

**right**).

Signal | Signal Properties | ||||
---|---|---|---|---|---|

Primary Code N [chip]/T [ms] | Secondary Code N [chip]/T [ms] | Data [bit/s] | |||

GPS L1C-P | 10,230 | 10 | 1800 | 18,000 | - |

GPS L5-I | 10,230 | 1 | 10 | 10 | 50 |

GPS L5-Q | 10,230 | 1 | 20 | 20 | - |

Galileo E1C | 4092 | 4 | 25 | 100 | - |

Galileo E5a-I | 10,230 | 1 | 20 | 20 | 25 |

Galileo E5a-Q | 10,230 | 1 | 100 | 100 | - |

Galileo E5b-I | 10,230 | 1 | 4 | 4 | 125 |

Galileo E1b-Q | 10,230 | 1 | 100 | 100 | - |

BeiDou B1-I | 2046 | 1 | 20 | 20 | 50 |

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**MDPI and ACS Style**

Svatoň, J.; Vejražka, F.
Joint Acquisition Estimator of Modern GNSS Tiered Signals Using Block Pre-Correlation Processing of Secondary Code. *Sensors* **2020**, *20*, 2965.
https://doi.org/10.3390/s20102965

**AMA Style**

Svatoň J, Vejražka F.
Joint Acquisition Estimator of Modern GNSS Tiered Signals Using Block Pre-Correlation Processing of Secondary Code. *Sensors*. 2020; 20(10):2965.
https://doi.org/10.3390/s20102965

**Chicago/Turabian Style**

Svatoň, Jiří, and František Vejražka.
2020. "Joint Acquisition Estimator of Modern GNSS Tiered Signals Using Block Pre-Correlation Processing of Secondary Code" *Sensors* 20, no. 10: 2965.
https://doi.org/10.3390/s20102965