#### 1. Introduction

In weak signal environments such as cities, canyons, forests, and other sheltered environments, the positioning accuracy of the GNSS receiver decreases drastically, leading to the development of high-precision receiver technology. The design of the tracking loop is an essential part of the GNSS receiver development process. It is vital in improving the receiver’s sensitivity performance and positioning accuracy. The carrier tracking loop is the most vulnerable link in GNSS receivers. Phase-locked loop (PLL) and frequency-locked loop (FLL) are typical carrier tracking loop implementations. Although the carrier phase measurement at the output of the PLL is entirely accurate, the phase-locked loop tends to lose lock when the noise is intense or in dynamic environments. However, the FLL has better dynamic performance and is suitable for tracking weak signals.

Traditional carrier tracking loops make it difficult to track weak GNSS signals effectively in real-time processing. A conventional strategy is to increase the integration time. However, the coherent integration time cannot be increased indefinitely due to factors including the length of the navigation message and local clock errors. In addition, long integration time requires narrow filter bandwidths, which cannot handle a dynamic signal. There are three main methods to promote the performance of the carrier tracking loop. The first approach adopts external sensors such as INS (Inertial navigation system) to assist in tracking weak satellite signals, which can effectively reduce the loop bandwidth and handle the dynamic. However, this method is limited by the accuracy of the information provided by the external sensors [1]. The second approach is the parameter estimation method, for example, maximum likelihood estimation and Kalman filter. However, this vector-based estimation method has high accuracy, but computing time is longer and more complicated. The third method is based on improving the conventional carrier tracking loop, for instance, the energy-based discriminators [2]-[4]. This discriminator type implements frequency estimation by using the integral values at three different frequency points within the main lobe of the *Sinc* function. It has higher tracking accuracy than the FFT discriminator under static conditions. The lead-lag structure discriminator [3] eliminates the effect of phase jumps by noncoherent integration. This discriminator introduces an approximation term deriving the slope near the zero frequency. The approximation process produces the estimation error in the discriminator, which is unsuitable for tracking in a weak signal environment. Guo [4] derived a new discriminator by interpolating the *Sinc* function and using the square operation to remove the effect of bit-hopping. However, the square process increases the noise impact and leads to the square loss of noncoherent integration, worsening the dynamic response of the signal tracking.

To overcome the deficiencies of the existing algorithms, this paper suggests a simple structured frequency-locked loop for tracking weak GNSS signals. In this structure, the IF signals are split into two branches, and each signal is correlated with an equally frequency-spaced local carrier. A new DFT interpolation algorithm is used as a frequency discriminator to calculate the carrier frequency deviation. The proposed new FLL structure achieves the same tracking sensitivity as the previous research [5]. The structure needs only two-point interpolation instead of three-point in the previous work, which introduces low computation load in a real-time receiver.

The rest of the paper is structured as follows: the problem is formulated in Section 2. A novel structure of the FLL is presented, and a two-point DFT interpolation frequency estimation with low computational complexity and high accuracy is applied to the design of the discriminator. Section 3 renders the performance of the frequency-locked loop. The Doppler frequency tracking sensitivity of the three discriminators is verified and analyzed. Finally, concluding remarks are given in Section 4.

#### 2. Design of FLL structure and discriminator

This section consists of three parts. Firstly, the GNSS IF signal model is introduced and analyzed. Secondly, a new frequency-locked loop structure is presented. Finally, a method based on two-point DFT interpolated frequency estimation is selected to design the discriminator in the loop by simulation comparison.

##### 2.1 GNSS signals and models

After filtering, down-converting, and sampling the received satellite signal from the GNSS receiver, the obtained IF analog signal is as follows:

\[\begin{equation*} u(t)=\sqrt{2}aD(t)\cdot C(t)\cdot \cos\left(2\pi \left(f_{IF}+f_d\right)t+\phi\right) +n(t) \tag{1} \end{equation*}\] |

where \(a\) is the IF signal amplitude, \(C(t)\) represents the pseudo-random noise code modulated by the satellite signal, \(D(t)\) is the navigation bit, \(f_{IF}\) is the IF nominal frequency, \(f_d\) is the Doppler frequency, \(n(t)\) is the additive Gaussian white noise, and \(\phi\) is the random initial phase of the received signal. When the pseudocode in the IF signal \(u(t)\) is completely stripped, its in-phase and quadrature branches are mixed with the local carrier NCO, respectively. Then, the high frequency components are filtered out by a low-pass filter, and a complex signal of the following form can be obtained:

\[\begin{align} \boldsymbol{r}(t)&=aD(t)\left[\cos\left(2\pi f_et+\phi_e\right)+\mathit{jsin} \left(2\pi f_et+\phi_e\right)\right]+n(t)\notag \\ &=aD(t)e^{j\left[2\pi f_et+\phi_e\right]}+n(t) \tag{2} \end{align}\] |

where \(f_e\) and \(\phi_e\) are the carrier frequency and initial phase bias between the input signal \(u(t)\) and the local replica carrier signal. In the noise-free case, the complex signal obtained after the signal \(\boldsymbol{r}(t)\) passes through the integration and dump (I&D) circuit can be expressed as [4]:

\[\begin{equation*} \boldsymbol{r}(t)=aD(t)sinc\left(f_eT_{\mathit{coh}}\right) e^{j\left[2\pi f_e\left(t+\frac{T_{\mathit{coh}}}{2}\right)+\phi_e\right]}= Ae^{j2\pi f_et} \tag{3} \end{equation*}\] |

where \(T_{\mathit{coh}}\) represents the coherent integration time and \(\mathit{sinc}(fT)=\mathit{sin}(\pi fT)/\pi fT\), neglecting the effect of the fixed frequency error \(f_e\) on the integrated output amplitude, \(\boldsymbol{r}(n)\) can be taken as a simple BPSK signal. It’s simple to figure out \(f_e\) with the discrete Fourier transform. Because of DFT’s spectrum leakage and fence effect, refining the frequency values estimated by DFT is also necessary.

In practice, the observation of the signal \(\boldsymbol{r}(t)\) in Eq. (3) is concentrated in the period \(0\leq t\leq T_{\mathit{coh}}\). The signal’s discrete sampling and frequency domain analysis are equivalent to adding a standard rectangular window. The Fourier transform of \(\boldsymbol{r}(n)\) is:

\[\begin{equation*} R(f)=\textit{asinc}\left(f_eT_{\mathit{coh}}\right) e^{j\left[\pi f_eT_{\mathit{coh}}+\phi_e\right]}\cdot \mathit{sinc} \left(\pi T_{\mathit{coh}}\left(f-f_e\right)\right) \tag{4} \end{equation*}\] |

Since the carrier frequency changes slowly during the period \(T_{\mathit{coh}}\), \(\mathit{sinc}\left(f_eT_{\mathit{coh}}\right)\) in Eq. (4) can be considered a constant. \(\mathit{sinc}\left(\pi T_{\mathit{coh}}\left(f-f_e\right)\right)\) can be used as a reference for discriminator design.

##### 2.2 The structure of the proposed open tracking loop

Referring to the method of the delay-locked loop in code tracking and simplifying it, a new frequency-locked loop structure is constructed. The architecture is shown in Fig. 1.

In this frequency-locked loop, the IF signal is correlated with the receiver-generated C/A code to strip the pseudo-code. After stripping the pseudocode, the signal is split into two paths to be coherently integrated and sampled within the data bit by the I&D circuit. The sampled value of the complex signal \(\boldsymbol{r}(n)\) in Eq. (3) can be obtained. The local frequency NCO generates two carriers with a frequency interval \(\pm \Delta f\) to correlate and accumulate the two signals \(\boldsymbol{r}(n)\) and take the modulus to obtain \(A_f\) and \(A_s\), respectively. The carrier frequency deviation is estimated by inputting \(A_f\) and \(A_s\) into the Doppler estimator. The carrier frequency NCO controls the frequency deviation after passing through the loop filter. Thus, the satellite signal is continuously tracked.

Fig. 2 normalizes the amplitudes of the output signal of the I&D circuit for coherent integration times of 10 ms and 20 ms without noise. It is observed that the longer the coherent integration time is, the smaller the discriminative frequency range is. The three red dots in Fig. 2 below represent the amplitude of the three coherent integration outputs when the frequency estimation error is zero. At this stage, P corresponds to the actual Doppler frequency, and the red dots S and F amplitudes are equal. The positions of the three green dots, s, p, and f, represent the presence of specific residuals in the frequency estimates. When the frequency estimation error is zero, we can obtain the following:

\[\begin{equation*} \left|\textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e-\Delta f\right)\right)\right| -\left| \textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e+\Delta f\right)\right)\right| =0 \tag{5} \end{equation*}\] |

Eq. (5) is the basis for the design of the two discriminators to be compared and analyzed in this paper.

##### 2.3 Proposed frequency discriminator

The methods for estimating the complex exponential signal \(\boldsymbol{r}(t)\) in Eq. (2) can be classified into two general groups: estimation algorithms in the time domain [6]-[11] and estimation algorithms in the frequency domain [12]-[22]. The latter is primarily implemented based on the DFT. Existing DFT-based frequency estimated methods of sinusoidal signals often include two steps. The first step uses the maximum amplitude for coarse estimation, and then a fine frequency estimation method is used as the second step to improve the estimated accuracy further. Most of the literature [23]-[29] using a two-step method for frequency estimation has the same coarse estimation in the first searching step. The difference lies in how the fractional frequency offset is estimated in the fine searching step. DFT coefficients interpolation commonly uses the peak magnitude and adjacent magnitudes to obtain a frequency offset. Therefore, the actual frequency of \(\boldsymbol{r}(t)\) is expressed as:

\[\begin{equation*} \boldsymbol{r}(t)=Ae^{j2\pi\left(k_m+\delta\right)\frac{f_s}{N}t} \tag{6} \end{equation*}\] |

where \(k_m\) is the frequency index, which has the peak magnitude, \(f_s\) indicates the sampling frequency, \(N\) is the number of the used samples, \(f_s/N\) represents the DFT frequency resolution. Existing methods generally depend on the three maximum DFT coefficients \(R\left[k_m\right]\) and \(R\left[k_m-1\right]\), \(R\left[k_m+1\right]\) to estimate the frequency. We selected three representative methods [30]-[32] for performance analysis and finally chose one for designing a discriminator in GNSS tracking loops.

Due to the nonlinear relationship between the frequency offset and the DFT coefficient interpolation, most direct interpolation methods inevitably introduce estimation bias [25]. The following experiments will analyze the performance of the mentioned methods in terms of the delta-bias curve in the absence of noise and the SNR-RMSE curve at different delta.

Fig. 3 shows the bias performance of different frequency estimators in the absence of noise. In the simulations, the observation samples are set to \(N=8\) and \(N=32\). And \(\delta\) varies from 0.01 to 0.49 with step size 0.01.

From Fig. 3, we can see that each frequency estimator has biases. The frequency estimation deviations become increasingly significant as the actual frequency approaches the middle of two neighboring frequency points. All the biases get smaller as observation samples increase. The NewDFT method has the least biases in the absence of noise with \(N=8\) and \(N=32\).

Frequency estimation bias performance is compared among the three methods in the case of noise. The true frequency of the signal is 42.5 Hz (\(\delta=0.3\)), \(N=8\), \(f_s=200\) Hz. The frequency estimation deviation for 1000 simulations is obtained at SNR of 40 dB and 50 dB, respectively, which is shown in Fig. 4.

Fig. 4(a) shows that the bias of Jacobsen’s method is significantly larger than those of Candan and NewDFT methods at SNR=40 dB. The frequency estimation bias of NewDFT varies in the rage \([-0.11,0.12]\), while the bias of Candan varies in the rage \([-0.25, 0.12]\). Fig. 4(b) shows the biases are further reduced at SNR=50 dB, and the NewDFT’s bias range is narrowed to \([-0.03, 0.05]\). Fig. 4 depicts that NewDFT significantly outperforms the other two methods in terms of estimation bias both at SNR=40 dB and 50 dB.

The RMSE performance of the estimators is compared with the SNR varying in the range [\(-5\) dB, 60 dB]. We select analyzed samples for comparison as \(N=8\), and the parameter \(\delta\) is fixed to 0.3. 1000,000 Monte Carlo runs were used to work out the RMSE value. The RMSE comparison curve is shown in Fig. 5.

In the figure, CRLB represents the Cramer Rao lower bound as an analysis reference. Fig. 5 shows each estimator’s RMSE performance versus different SNRs. We can see that all the RMSEs of each estimator follow the same trend when the SNR ranges from 5 dB to 25 dB. However, a sudden drop occurs at SNR=5 dB as the SNR decreases. Because the coarse estimated frequency value predominantly determines the performance of the estimators in the first step. From 25 dB, Jacobsen gradually becomes flat, while from 55 dB, NewDFT becomes flat. The approximation error becomes the dominant effect of error at high SNR. The RMSE curve of NewDFT almost overlaps with CRLB from 5 dB to 45 dB, and the performance significantly outperforms the other two algorithms.

Setting the range of SNR and \(\delta\), acquiring a three-dimensional figure of the RMSE performance is easy. Projecting the RMSE 3D plots of Candan and NewDFT onto the 2D plane consisting of SNR and \(\delta\), Figs. 6 and 7 can be obtained.

In Figs. 6 and 7, the color transition from light yellow to dark blue represents the RMSE performance from poor to good. In the presence of noise, Candan and NewDFT have higher frequency estimation accuracy as the value of \(|\delta|\) is close to 0, which coincides with the noiseless situation in Fig. 3. The RMSE performance of NewDFT outperformed Candan at both high and low SNR. Depending on the above experiments, the NewDFT estimator is the best of the candidate methods. Thus, the NewDFT algorithm will be chosen to design the discriminator in the carrier tracking loop.

The estimated value of \(f_e\) is deduced by taking the maximum value of amplitude in the \(\boldsymbol{r}(t)\) spectrum and its adjacent spectral lines on the left \(A_s\) and right \(A_f\):

\[\begin{equation*} \widehat{f_e}=\frac{\mathit{tan}\left(\dfrac{\pi}{2N}\right)}{\dfrac{\pi}{2N}} \frac{A_f-A_s}{A_f+A_s} \tag{7} \end{equation*}\] |

Ignoring the effect of noise, assuming no data jumps during the integration time and the pseudocode is perfectly aligned, \(A_f\), \(A_p\) and \(A_s\) in Eq. (8) can be expressed as:

\[A_f=\left|\textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e-\Delta f\right)\right)\right|\] |

\[A_p=\left|\textit{asinc}\left(\pi T_{\mathit{coh}}f_e\right)\right|\] |

\[\begin{equation*} A_s=\left|\textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e+\Delta f\right)\right)\right| \tag{8} \end{equation*}\] |

\(A_f\), \(A_p\), and \(A_s\) denote the amplitudes of three spectral lines corresponding to three carriers with different frequencies, which differ by \(\Delta f\) in turn. According to Eq. (8), a new method for the calculation of the discriminator can be expressed as:

\[\begin{equation*} \mbox{$\displaystyle \widehat{f_e}=\frac{\mathit{tan}\left(\dfrac{\pi}{2N}\right)}{\dfrac{\pi}{2N}} \frac{\left|\textit{asinc} \left(\pi T_{\mathit{coh}}\left(f_{e}-\Delta f\right)\right)\right| -\left|\textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e+\Delta f\right)\right)\right|}{ \left|\textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_e-\Delta f\right)\right)\right| +\left| \textit{asinc}\left(\pi T_{\mathit{coh}}\left(f_{e}+\Delta f\right)\right)\right| } $} \tag{9} \end{equation*}\] |

#### 3. Experimental results

The following simulation compares the tracking performance of the carrier tracking loop with the DP discriminator, NewDFT-based discriminator, and FFT discriminator. In the simulation, the structure of FLL shown in Fig. 1 is adopted. And coherent integration time is set to 20 *ms*. The tracking performance of GNSS signals is tested by a semi-simulation platform [33]. A comparison of the tracking performance of the three discriminators for weak signals without loop filters is shown in Fig. 8.

Due to the same structure, the tracking performance of the three methods depends mainly on the estimation accuracy of the discriminator in open-loop tracking. To achieve the best frequency estimation accuracy, the DP discriminator frequency interval is set to \(\Delta f=1/(3T)\). As shown in Fig. 8, the FFT discriminator has a tracking threshold of 30 dB-Hz, and the frequency tracking jitter is greater than 1 Hz when the sampling number is \(N=8\). DP discriminator and NewDFT discriminator can track signals with a carrier-to-noise ratio of 26 dB-Hz, which has a 4 dB-Hz improvement over the tracking threshold of the FFT discriminator. The tracking jitters of the two discriminators are less than 0.5 Hz. As the sampling number increased to 16, these discriminators’ tracking thresholds and tracking jitter were further reduced. The DP and NewDFT discriminators’ tracking threshold drops to 24 dB-Hz. The FFT discriminator has the biggest tracking jitter, approaching 0.5 Hz, while the other two discriminators are close to 0.15 Hz. The NewDFT discriminator can improve the tracking threshold over 2 dB-Hz relative to that of the FFT discriminator. The proposed discriminator algorithm is less complex than the DP discriminator, and its tracking jitter is very close to the DP with superior tracking performance.

GPS signals are generated using GPS-SDR-SIM open-source software to examine the performance of Doppler tracking of satellite signals based on FFT, DP and NewDFT discriminators. A set of IF data with decreasing \(\mathrm{C}/N_0\) is generated, which falls from 40 dB-Hz to 22 dB-Hz with a speed of 2 dB-Hz/160s. A 2^{nd} order FLL is adopted for tracking the frequency Doppler. The sampling frequency is 4 MHz, 20 *ms* signal is coherently integrated, and the bandwidth of FLL is set to 15 Hz.

Fig. 9 shows the Doppler tracking results. The black dashed line in Fig. 9 represents the Doppler reference value, and the red solid line represents \(\mathrm{C}/N_0\). Fig. 9 illustrates the FFT-based FLL can track the signal up to 600s, corresponding to \(\mathrm{C}/N_0\) about 31 dB-Hz, while the DP and NewDFT-based FLLs can track the signal up to 950s, corresponding to \(\mathrm{C}/N_0\) about 27 dB-Hz. Due to the clock bias and A/D conversion loss of GPS, the tracking threshold is consistent with the theoretical values in Fig. 8 with N=8.

#### 4. Conclusion

This paper proposes a new open tracking loop structure for GNSS carrier based on two-point DFT interpolation. The tracking sensitivity of the FLL is verified and analyzed by simulation. The results show that in open-loop tracking, the tracking threshold of this frequency-locked loop with 8 samples is 26 dB-Hz, which is 4 dB-Hz better than that of the FFT discriminator, and the tracking error jitter is less than 0.5 Hz. When the signal samples are increased to 16 points, this frequency-locked loop improves the tracking threshold by 2 dB-Hz relative to that of the FFT discriminator, and its tracking error jitter is close to 0.15 Hz. Finally, a GPS software receiver verifies the performance of the proposed FLL structure. Results show that the tracking sensitivity and the frequency accuracy are the best when the signals are attenuated.

#### Acknowledgments

This work was supported in part by the Scientific Research Project of the Education Department of Hubei Province under Grant D20223002, in part by the Scientific Research Foundation of Hubei University of Education for Talent Introduction under Grant ESRC20220025, and in part by Expert Workstation for Terahertz Technology and Advanced Energy Materials and Devices, Hubei University of Education.

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#### Authors

Li Cheng

Dept. of Physics, Mechanical and Electrical Engineering, Hubei University of Education