Performance analysis of cross-frequency Doppler-assisted carrier phase tracking

Using the Doppler frequency obtained from tracking a GNSS pilot signal to aid in tracking another signal modulated with higher rate navigation messages in a different frequency band can improve tacking robustness and lower the message demodulation threshold. Based on an analysis of received signal frequency coherence, a linearized mathematical model of the cross-frequency Doppler-assisted carrier phase tracking loop is built, a thermal noise jitter calculation equation for the assisted tracking loop is derived, and its dynamic stress response characteristics are examined. The loop design requirements for eliminating the influence of inter-frequency frequency bias are clarified, as are the cross-frequency assist signal selection criteria. Monte Carlo simulations and preliminary field tests validate the theoretical results using the B1C pilot signal-aided tracking B2b signal of the MEO satellite in the BeiDou satellite navigation system (BDS). Experimental results show that the carrier phase tracking threshold of the B2b signal can be reduced by about 4 dB.


Introduction
In recent years, satellite-based precise point positioning (PPP) technology has received a lot of attention, and several GNSS core constellations, including the Japanese Quasi-Zenith Satellite Navigation System (QZSS) (), the European satellite navigation system Galileo (Chatre and Benedicto 2020), and the Chinese BeiDou Satellite Navigation System (BDS) (Liu et al. 2020), have already started or are claiming to start providing PPP services. PPP services necessitate satellites broadcasting more correction data at faster rates than traditional satellite-based wide area differential augmentation systems like SBAS. For example, Galileo broadcasts high accuracy service (HAS) messages on E6-B signals at 1000sps (European Union 2021), and BDS broadcasts PPP-B2b messages on B2b signals at 1000sps (CSNO 2020a, b). Because of the increased broadcast rate, the coherent integration time in signal reception processing must be reduced accordingly, reducing the robustness of the carrier phase tracking and negatively impacting the demodulation of BPSK-modulated messages. Therefore, reducing the signal tracking threshold and improving the carrier phase tracking performance of signals that are modulated with higher rate navigation messages while maintaining signal transmit power has emerged as a new issue in the field of GNSS signal processing.
Researchers proposed cross-frequency Doppler-assisted carrier phase tracking techniques to improve the signal carrier phase tracking performance. In Gernot et al. (2008), the Doppler frequency obtained by the GPS L1C/A signal is used to track the GPS L2C signal during the GPS weak signal tracking process, and it is discovered that the Doppler frequency estimation error caused by the ionosphere has no effect on the assisted convergence of loop. Borio and Susi (2018) used Galileo E1C signal to assist in tracking of E6-B signal. Bolla and Borre (2019) analyzed and evaluated the tracking performance of using GPS L5 signal to assist in tracking of GPS L1 signal. Chen et al. (2020) proposed using a pilot signal to assist in tracking a BPSK-modulated signal and then assist in demodulating CSK-modulated messages on the quadrature carrier phase. In fact, as early as the early 1990s, some researchers proposed using L1 C/A signal-aided tracking L2P (Y) signal cross-frequency Doppler-assisted signal tracking technology to improve the performance of GPS dual-frequency semi-codeless receiver carrier phase measurement (Keegan 1990).
Although cross-frequency Doppler-assisted signal tracking has been a focus of research for a number of years, indepth quantitative theoretical analysis of tracking performance is lacking, and many papers rely on simulation or simple empirical equations for performance evaluation, making it difficult to reveal the operating mechanism of cross-frequency Doppler-assisted signal tracking loops and accurately guide loop design. Engineers must understand: 1) how to appropriately evaluate the assisted loop's thermal noise jitter; 2) how it behaves under dynamic stress; 3) whether or not the inter-frequency frequency bias affects the carrier phase measurement; and 4) how to select the best assist signal when designing a cross-frequency Dopplerassisted signal tracking loop.
It should be noted that the signals of different carrier frequency must be broadcasted by the same satellite regardless of which GNSS system the signal belongs to; this is required for cross-frequency Doppler-assisted signal tracking technology to be realized. The Monte Carlo method is used to simulate and verify the cross-frequency Doppler-assisted tracking theory. The applicability of the assisted signal's carrier phase thermal noise jitter equation is demonstrated. And the performance improvement effect of the tracking loop of the assisted signal is calculated. The design requirements of eliminating the influence of inter-frequency frequency bias are clarified, and the selection criteria of assisting signal are concluded. This paper focuses on the performance of cross-frequency Doppler-assisted carrier phase tracking loops in an additive white Gaussian noise (AWGN) channel; pseudocode tracking loops are not addressed. This is because most modern GNSS receivers use a carrier-assisted pseudocode tracking signal reception architecture, which eliminates the need for additional cross-frequency assistance for pseudocode tracking. Furthermore, the tracking threshold of a well-designed code tracking loop is typically much lower than the carrier phase tracking threshold (Ward 1998). We begin by describing the cross-frequency Doppler-assisted carrier phase tracking architecture. The tracking loop characteristics are then resolved by building a linear model of the cross-frequency assisted tracking loop. Following that, we present the Monte Carlo simulation validation results and the preliminary field test results. Finally, a summary and conclusions are provided.

Aided carrier phase tracking
The basic architecture of cross-frequency Doppler-assisted carrier phase tracking is described in this section. We develop models for the local signal and the received signal while accounting for multiple errors. The frequency coherence between different frequency signals is calculated using these models. Finally, the cross-frequency Doppler-assisted carrier phase tracking structure is obtained.

Received satellite signal
In an AWGN channel, the navigation signal from a GNSS satellite received by a receiver at the nominal carrier frequency f i is: where A i is the signal amplitude, D i is the navigation message, C i is the signal spread spectrum code with period T ci , i is the signal propagation time from the satellite to the receiver, i is the signal carrier phase, n i is the thermal noise of the receiver, and: Δf sclk, i (u)du . Δf sclk, i is the carrier frequency error caused by the satellite reference clock frequency error Δf sclk . Supposing the nominal value of the satellite reference clock frequency f sclk and the nominal carrier frequency f i satisfy an N si -fold relationship, we know the following (Gardner 2005): where f dpl,i is the Doppler frequency caused by changes in the position of the satellite relative to the receiver, r(t) is the relative distance between the satellite and the receiver, and i is the carrier wavelength of the signal, so: (2) is composed of the initial carrier phase, the ionospheric wave propagation delay, the tropospheric wave propagation delay, and the carrier phase change caused by the frequency deviation Δf ssyn,i of the satellite navigation payload synthesizer. Ionospheric wave propagation delay is approximated as a constant in the following discussion because it varies slowly with time. For example, in the case of the GPS L1 C/A signal, the carrier frequency shift caused by ionospheric delay variation does not exceed 0.085 Hz (Klobuchar 1996), which is much smaller than the bandwidth of a carrier phase tracking loop. For the same reason (Spilker 1996), tropospheric wave propagation delay is also approximated as a constant. When a frequency synthesizer generates the carrier frequency from the reference clock frequency, it may not exactly satisfy the N si -fold scaling relationship, resulting in a carrier frequency deviation Δf ssyn,i , which in turn causes the carrier phase to vary.

Measured Doppler frequency
The received satellite signal is down-converted into baseband and then tracked by the carrier tracking loop, which is integrated with the local replicated spread spectrum code signal and the carrier signal. After the carrier phase tracking is locked, the phase of the local carrier signal generated by the tracking loop can be expressed as: where ̂ sclk, i , ̂ i, dpl , and φ s, i are the estimated values of sclk,i , dpl,i , and s,i , respectively. ̂ r,i is the estimate of r,i , which includes the initial local carrier phase and the phase variable caused by the frequency deviation Δf rsyn, i of the receiver's frequency synthesizer. ̂ rclk,i is the estimate of ϕ rclk,i , and rclk, i (t) = 2 ∫ t t 0 Δf rclk, i (u)du , Δf rclk,i is the carrier frequency error caused by the receiver's reference clock frequency error Δf rclk . Supposing the nominal frequency f rclk of the receiver's reference clock and the nominal carrier frequency f i satisfy an N ri -fold relationship, similarly: We can define the generalized Doppler frequency when the receiver tracks the signal of carrier frequency f i as follows: Lemma 1 clarifies the frequency coherence of mixed signals with different frequencies using the concept of generalized Doppler frequency.

Lemma 1
In an AWGN channel, the generalized Doppler frequencies f gdpl, i and f gdpl, j associated with tracking signals of carrier frequencies f i and f j , respectively, which are broadcast by the same satellite, satisfy the following relationship: where λ i and λ j are the carrier wavelengths of f i and f j .

P r o o f S u b s t i t u t i n g
( 3 ) , ( 4 ) , a n d ( 6 ) i n t o ( 7 ) a n d m u l t i p l y i n g b y . Denoting the signal wavelength corresponding to the clock reference nominal frequency f sclk of the satellite navigation payload as λ s , we have λ s = i N si = i N ri . Substitute s into the preceding equation, its right side becomes independent of subscript i. From this, we can obtain be the Doppler frequency measurement output from a receiver's carrier phase tracking loop. In an AWGN channel, its mean is according to (2), (5) and (7). The mean difference of the Doppler frequency measurement between two signals compensated by the proportional relationship of the two signals' carrier frequencies is defined as As mentioned previously Δf ssyn, i , Δf rsyn, i , Δf ssyn, j and Δf rsyn, j represent carrier frequency deviations caused by the imperfections in the frequency synthesizers both in the transmitter side and in the receiver side. When these frequency deviations are not removed they may introduce a fixed frequency bias into the assist Doppler frequency and this is known as the inter-frequency frequency bias. We will discuss how the frequency bias affects the cross-frequency Doppler-assisted tracking in the next section.
Lemma 1 Shows that despite the presence of clock frequency offset and clock phase noise in the satellite navigation payload and satellite signal receiver, the generalized Doppler frequencies of carrier different signals maintain the proportional relationship of carrier frequencies. This means that cross-frequency Doppler assistance can also eliminate the effect of clock phase noise on a bandwidth narrowed carrier phase tracking loop. We will later use real data to validate this inference.

Structure of the assisted carrier phase tracking
A schematic diagram of the cross-frequency Dopplerassisted carrier phase tracking structure is shown in Fig. 1 are weighting factors for the combination of Doppler frequency measurements. For instance, B1C and B2a pilot signals are combined to aid in tracking B2b signal, or E1 and E5a pilot signals are combined to aid in tracking E6-B signal, or only using a single pilot signal to aid in tracking a data signal just for simplicity. A direct-conversion type receiver is used in Fig. 1 to make the mathematical expression of signal reception processing concise. This model is also suitable for any other type of receiver, because the first down-conversion symbol in Fig. 1 indicates total down-conversion processing in a receiver including analog RF and digital IF down-conversion and even the baseband frequency offset removal. It should be noted that signal sampling clock and all NCO modules in the receiver are driven by the same reference clock although they are not depicted in Fig. 1.
In Fig. 1, the coherent integration of the mixed signal can be expressed as follows: i is the signal amplitude after correlation integral, w i t k is the receiver's thermal noise in complex form after correlation integral, its real and imaginary parts of which are independent of each other and follow a normal (Dierendonk 1996). Here, N 0 is the single side band power spectral density of the receiver's thermal noise, and (10) If message D i in the received signal is known, a Q-normalized or four-quadrant inverse tangent phase discriminator can be used for phase discrimination. If it is unknown, I*Q normalized or two-quadrant inverse tangent phase discriminator can be used. When Δϕ i t k ≪ 1: where n d, i is a white noise sequence with zero mean and variance 2 d,i . When the Q-normalized phase discriminator is used in the carrier tracking loop, the following can be derived: when the I*Q normalized phase discriminator is used, the following can be derived: i ∕2N 0 is the effective carrier-to-noise density ratio of the signal after correlation integral, and is the signal-to-noise power ratio loss due to the nonlinear processing of the discriminator, which is customarily called the squaring loss. When the carrier tracking loop uses an inverse tangent discriminator, the discriminating noise variance 2 d, i can be approximated by (13).
In Fig. 1, the weighting factor η i for the combination of Doppler frequency measurements satisfies the constraint When each tracking loop is designed in the same, which means they have the same loop order and the same noise bandwidth, the optimal weighting factor i under the maximum likelihood criterion, referring to the Appendix can be expressed as: when only one assist signal is present or selected, N = 1 and 1 = 1.

Tracking performance analysis
This section develops a linearized mathematical model of the cross-frequency Doppler-assisted carrier phase tracking Fig. 2 Linear model of the cross-frequency Doppler-assisted carrier phase tracking loop. It reflects linearized signal transfer paths within the receiver as displayed in Fig. 1 and used to analyze its signal tracking performance. Generalized Doppler carrier phase, ϕ gdpl,N + 1 , of signal, s N + 1 (t) is the real carrier phase value of H N+1 loop in carrier cycles. ϕ gdpl,N+1 N+1 is the real range between satellite and receiver in meters and is then converted to values that are applicable for each frequency band with the λ 1 , λ 2 , … , λ N loop, calculates its thermal noise jitter, and analyzes its tracking characteristics.

Linear model of assisted carrier phase tracking loop
When Δ i t k ≪ 1 , referring to Fig. 1, the carrier phase tracking loop that is aided by combined Doppler frequencies of N carrier different signals is linearized as shown in Fig. 2.
Here, each H i block with i = 1, … , N + 1 represents the linear model of a conventional carrier phase tracking loop for signal s i (t) with a system transfer function of H i (s) = , and a loop filter transfer function of F i (s) . In Fig. 2, 0, i = s, i − r, i denotes the carrier phase variable concerned with the unknown initial carrier phase and the variable phase component caused by synthesizer frequency deviation.

Thermal noise jitter
In Fig. 2, the noise n d, i can be approximated as white noise with a power spectral density of 2 d, i ∕2B T, i within the loop noise bandwidth when it is much smaller than the noise bandwidth of n d, i . Therefore, the thermal noise variance at the loop output is the noise variance of the white noise n d, i passing through a linear system H i (s) , which can be expressed as follows (Papoulis and Pillai 2002): | 2 df is known as the single side band noise bandwidth of the tracking loop. When the Q-normalized phase discriminator is used, the thermal noise jitter of the conventional carrier phase tracking loop is , and when using the I*Q-normalized discriminator or others, its thermal noise jitter becomes Fig. 2, the system transfer function of the crossfrequency Doppler-assisted carrier phase tracking loop is derived as: and the error transfer function is: Theorem 1 For a linear system such as the one displayed in Fig. 2, the variance of thermal noise at the output port ̂ N+1 is calculated by: Here, B La, i is known as the noise bandwidth of the i-th assist tracking branch. the definition of B La, i is as follows: Proof According to the principle of linear system superposition, the noise at the output port φ N+1 of Fig. 2 equals the sum of the output noise caused by n d, is the variance of the noise caused by n d, i at φ N+1 , and 2 ϕ N+1 is the variance of the noise caused by n d, N+1 at φ N+1 . As (15) (19) and (20), we obtain: 2 As an example of Theorem 1, when a pilot signal s 1 that adopts the Q-normalized phase discriminator is used to aid in carrier phase tracking of signal s 2 that uses the I*Q normalized phase discriminator, the thermal noise jitter in the carrier phase of the aided tracking signal can be expressed as: Theorem 1 provides an equation for the assisted loop carrier phase thermal noise that lays a foundation for further studying the characteristics of cross-frequency Dopplerassisted carrier phase tracking loop.

Assisted tracking properties
The characteristics of the carrier phase tracking loop of the assisted loop are the focus of our research with crossfrequency Doppler assistance. The characteristics of the assisted loop are given and proven in this section based on two aspects of stress response and thermal noise jitter. Fig. 2, the steady-state tracking error of the assisted tracking loop responding to a given input signal ϕ gdpl,N+1 (t) will tend to zero if the HN + 1 loop satisfies the condition H N+1 (0) = 1 and the steady-state tracking error of the Hi loop, i = 1,…, N is finite. That is:

Property 1 For a linear system such as the one displayed in
Proof Since the steady-state tracking error is finite for the Property 1 demonstrates that if a conventional carrier phase tacking loop is properly designed, the dynamic stress can be removed after it accepts cross-frequency Doppler assistance. The tracking loop can meet the conditions of Property 1 using the design method taken from the literature (Ward and Betz 2006). For example, for a BDS B2b signal with a jerk of 2 g/s (g is gravity's acceleration), the steadystate error of the third-order carrier phase tracking loop with a 16 Hz loop noise bandwidth is approximately 3.35°, which can be eliminated by accepting cross-frequency Doppler assistance from the B1C signal, as described in Property 1.
When designing a signal tracking loop, dynamic response characteristics are an important factor to consider because they are related to both the steady-state and the transient An example is given here to show how the tracking error of the assisted loop varies with time. Let f 1 be the BDS B1C carrier frequency, f 2 be the BDS B2b carrier frequency, and use the B1C pilot signal to aid in the tracking of the B2b signal. The B1C signal tracking loop (H1 loop) has a noise bandwidth of 16 Hz, while the B2b signal tracking loop (H2 loop) has a noise bandwidth of 2 Hz and 0.5 Hz, respectively. Figure 3 depicts the tracking error responses to a jerk dynamic input signal of 2 g/s. Figure 4 depicts the tracking error responses to a sinusoidal waveform input signal with a peak jerk dynamic of 2 g/s and a period of 10 s.
As shown in Fig. 3, the tracking error of the H1 loop converges to a constant value after the jerk step signal is applied. In contrast, the tracking error of the H2 loop gradually converges to zero with the help of Doppler, with the convergence time being related to the loop's noise bandwidth. The longer the convergence time, the smaller the bandwidth. The tracking error of the H1 loop fluctuates within ± 3.2° after the sinusoidal signal is applied, as shown in Fig. 4, with the same period of fluctuation as the input sinusoidal signal. The tracking error of the H2 loop with a noise bandwidth of 2 Hz gradually approaches zero with Doppler assistance. In contrast, the tracking error of the same loop with a different noise bandwidth of 0.5 Hz fluctuates within ± 2.5°. These analytical results indicate that the assisted tracking loop's noise bandwidth should not be set too narrowly. Property 2 demonstrates that a reasonable loop design can eliminate the influence of the assisting signals' initial phase or the inter-frequency frequency bias between the assisting signal and the assisted signal on the assisted signal's carrier phase measurements. For example, the assisting signal's ionospheric propagation delay differs from that of the assisted signal. Still, it does not introduce any bias in the assisted signal's carrier phase measurement. The assisted signal's carrier phase measurement will not be affected while accepting Doppler assistance, even if the assist signal has a fixed frequency bias relative to the assisted signal.

Property 2 In a linear system such as the one displayed in
Equation (18)  B La is known as the equivalent noise bandwidth of the crossfrequency Doppler-assisted carrier phase tracking loop, and G a,i is known as the assist gain of the cross-frequency assisting signal. Fig. 2, it is supposed that the noise bandwidth of the conventional tracking loop is B L . If B La < B L , then the thermal noise jitter of the cross-frequency Doppler-assisted carrier phase tracking loop will be smaller than that of the conventional tracking loop.

Property 3 For a linear system such as the one displayed in
Proof Referring to (15), the thermal noise variance of the HN + 1 loop output can be expressed as 2 Equation (18) shows that the Doppler-assisted thermal noise variance at the HN + 1 loop output can be expressed as Property 3 states that the equivalent noise bandwidth can be used to determine whether and how much the carrier phase measurement accuracy can be improved when crossfrequency Doppler assistance is used. Furthermore, the assist gain defined by (24) can be used to choose the best assist signal, that is, the one with the highest assist gain among a set of signals. GNSS signals with close carrier frequencies are typically used to provide Doppler-assisted information when designing the cross-frequency Doppler-assisted signal tracking loop, but Eq. (24) shows that high-frequency signal assistance can achieve better results. For example, the carrier wavelength of the BDS MEO satellite B1C pilot signal is approximately 19.04 cm. Its minimum received power level on the ground is − 160.25dBW (CSNO 2017a, b), whereas the carrier wavelength of the BDS MEO satellite B2a pilot signal is approximately 25.5 cm, and its minimum received power level on ground is − 159 dB (CSNO 2017a, b). Assuming the same propagation attenuation, noise power spectral density, signal reception processing loss, correlation integration time, and phase discriminator type for both signals. According to equation (24), the assist gain of the B1C pilot signal is approximately 1.29 dB greater than that of the B2a pilot signal.

Simulation evaluation
This section employs Monte Carlo simulation to validate the results of the previous section's theoretical analysis. Using the Monte Carlo method, we created a simulation verification platform of a cross-frequency Doppler-assisted carrier phase tracking loop in MATLAB. The previous section's theoretical analysis results were validated by varying signal dynamics and noise. Figure 5 depicts a block diagram of the Monte Carlo simulation verification and evaluation system. Here, r(t) represents the distance between the satellite and the receiver, ϕ 0,1 and ϕ 0,2 are the initial phase of the signal carrier, A 1 and A 2 are the signal amplitudes, and w 1 and w 2 are mutually independent complex forms of receiver thermal noise with independent real and imaginary parts that follow a normal distribution of N 0, 2 w ; 2 w = N 0 ∕T s . Signal sampling interval is set to T s = 0.1ms , and the receiver thermal noise power spectral density to N 0 = −204dBW/Hz. f 1 is set to the BDS B1C signal frequency, f 2 to the BDS B2b signal frequency, and the associated integration time is 1 ms. Both signals are assumed to have the same propagation attenuation, signal reception processing loss, and thermal noise power spectral density. The minimum received power ratio of the B1C pilot signal to the B2b signal is − 0.25 dB, and the carrier-to-noise ratio of the B2b signal is used as the reference, i.e., C∕N 0 ,

Simulation evaluation system
The tracking loop filter in Fig. 5 is designed following the literature (Ward and Betz 2006), with the filter parameters determined given the loop noise bandwidth. The H1 and H2 loops are both third-order loops. The bandwidth of the H1 loop was set to 16 Hz, and a Q-normalized discriminator was used. Signal amplitude was obtained by coherent averaging of the 1 ms correlated integrated output signal across ten repeats and then by taking a moving average 1000 times with a moving window of 10 s to obtain the average signal amplitude A 1 . Signal average amplitude was used for phase discriminator normalization. The noise bandwidth of H2 loop was set to 2 Hz and a I*Q-normalized discriminator was used and normalized by using the squared signal amplitude valuation A 2 2 = A 2 1 ∕0.944. Phase errors Δ 1 = 1 −̂ 1 and Δ 2 = 2 −̂ 2 were recorded continuously at 1s intervals. To calculate the RMS (mean root square) and examine cycle slips, 1000 samples of each phase error were collected. If the number of cycle slips for either phase error exceeds 100, the loop is deemed to have lost tracking, and the C/N 0 is recorded at this point.

Fig. 5
Block diagram of the Monte Carlo simulation evaluation system. Doppler information from H1 loop is obtained to assist carrier phase track of H2 loop, r(t) is the true range between satellite and receiver in meters, given signal amplitude A 1 , A 2 , we can adjust the amplitude of noise w 1 ,w 2 to control the C/N0 of signals Fig. 6 Tracking performance in a static situation. 'H1' represents the carrier phase measurement error of the B1C pilot signal. 'aided H2' represents the B2b signal's carrier phase measurement error after accepting Doppler assistance. 'aided H2 theoretic' represents the carrier phase thermal noise error calculated by equation. 'conv. H2' represents the B2b signal carrier phase measurement error without Doppler assistance (noise bandwidth 16 Hz) The above simulation process was repeated 50 times, with the average value of the RMS when the loop kept tracking used as the carrier phase measurement error and the average value of the C/N 0 when the loop lost tracking used as the signal tracking threshold.

Tracking performance in static situations
Carrier phase measurement error is shown in Fig. 6, with r(t) = 0 and the initial phases 0,1 and 0,2 as random constants in the range (0, 2π). When the carrier phase measurement error becomes singular, the tracking loop has lost track, and the corresponding C/N 0 indicates the tracking threshold. Figure 6 shows that the simulation results of the carrier phase measurement error of the assisted loop agree well with the theoretical analysis results when the loop is kept tracking. The carrier phase tracking threshold of the B2b signal is approximately 28.2 dB-Hz without Doppler aid. After accepting the Doppler aid, the carrier phase tracking threshold of the B2b signal drops to around 23 dB-Hz, representing a 5.2 dB improvement.

Tracking performance in a dynamic situation
A dynamic situation with sinusoidally varying carrier phase is used to evaluate carrier phase tracking performance. The input signal's peak jerk was set to 2 g/s, and the signal period was set to 10 s. Distance between the satellite and the receiver is expressed as r(t) = A ϕ sin(2 t∕T) , T = 10 s , and A ϕ = 2g(T∕2 ) 3 (m). Initial phases ϕ 0,1 and ϕ 0,2 are random constants in the range (0, 2π). Figure 7 depicts a comparison of the carrier phase tracking error curves in static and dynamic situations. Figure 7 shows that the dynamic stress error in the H1 loop causes the phase measurement error and tracking threshold of the B1C signal to increase slightly in the dynamic situation. In comparison with the static situation, the phase measurement error and tracking threshold of the assisted B2b signal are almost unchanged in the dynamic situation. This implies that the dynamic stress error in the cross-frequency Dopplerassisted tracking loop is very small, which agrees with the theoretical calculation results shown in Fig. 4.

Preliminary field test
The working mechanism of cross-frequency Doppler-assisted carrier phase tracking and the improvement of tracking performance are initially validated in this section using real signal data collected from BeiDou satellites. We first check to see if there is an inter-frequency frequency bias. Then, we test whether cross-frequency Doppler assistance can eliminate the effect of clock phase noise on the bandwidth narrowed tracking loop. Finally, we present preliminary results of the carrier phase tracking loop's performance improvement after using cross-frequency Doppler assistance.

Real data collection
On 2020/09/24, BDS-3 satellite signals were collected in Beijing using the LabSat 3 Wideband, a radio signal record and playback device (116.40°, 39.99°). The collected data was then post-processed using a software defined receiver, which was used to capture and track satellite signals and analyze and verify the cross-frequency Doppler-assisted Fig. 7 Tracking performance comparison between the static and dynamic situations. Conv. In the figure is short for conventional. Two curves of the same color reflect the difference between static and dynamic results under the same processing method carrier phase tracking performance. Table 1 displays the data collection and software defined receiver parameters. An oven-controlled crystal oscillator (OCXO) was used for the reference clock. A 32-bit numerically controlled oscillator (NCO) driven by the sampling signal generated the digital local oscillator signal and the baseband frequency bias removal signal. The residual frequency offset caused by the finite word length effect is less than 0.01 Hz.

Inter-frequency frequency bias
A difference between the measured Doppler frequencies at the output of the receiver carrier phase tracking loop is obtained as Δf trk,i − j =f trk, i −f trk, j * f i ∕f j , where f i and f j are the nominal values of the signal carrier frequencies. If the mean value of Δf trk,i−j is nonzero or not close to zero, there will be a frequency bias in the cross-frequency Doppler assist signal as displayed in (9).
The legend 'B1C-B2a' in Fig. 8 denotes the difference in Doppler frequency measurements between the B1C and B2a signals, and the other legends follow the same rule. A 1 ms coherent integration, 16 Hz noise bandwidth third-order tracking loop was used for the test. The B1C and B2a pilot signals were tracked using a Q-normalized discriminator, while the B2b signal was tracked using a 2-quadrant inverse tangent discriminator. The effective C/N 0 of the B1C, B2a, and B2b signals, respectively, were approximately 48 dB-Hz, 45.4 dB-Hz, and 48.2 dB-Hz. After the signal tracking had stabilized, Doppler frequency measurements were taken at 1 ms intervals and a 1s moving average was applied. Figure 8 shows that the frequency bias between B1C and B2a was approximately -0.382 Hz, and the one between B1C and B2b was approximately -0.404 Hz. These two Although the precise cause of the inter-frequency frequency bias has yet to be determined, the test results shown in Fig. 8 confirm that there may be a frequency deviation in the Doppler frequency measurements of different carrier signals that is independent of the carrier frequency difference.
In the linearized mathematical model of the cross-frequency Doppler-assisted carrier phase tracking loop shown in Fig. 2, such a frequency deviation is represented by the phase variable 0, i . Property 2 demonstrates that if the tracking loop is properly designed, the carrier phase measurement of the assisted loop will not be disrupted even if there is a fixed frequency deviation in the assist Doppler frequency. Figure 9 depicts the Lissajous figure of the 1 ms coherent integrated output of the B2b signal after signal tracking has been stabilized. Figure 9 shows the conventional tracking results with loop noise bandwidths of 16 Hz and 2 Hz, respectively, and (c) shows the assisted tracking result with a loop noise bandwidth of 2 Hz. The B1C signal with a tracking loop noise bandwidth of 16 Hz is used as the assist signal.

Effect of clock phase noise
When the loop noise bandwidth is reduced to 2 Hz, the Lissajous graph shifts from a nearly circular shape to an elongated elliptical shape in the Q-axis direction, indicating that the tracked carrier phase fluctuates and tracking performance degrades. According to the reference (Ward and Betz 2006), the carrier phase tracking error caused by the phase noise of the clock crystal oscillator increases as the loop bandwidth narrows. Furthermore, because the receiver is stationary while collecting signal data, the phase fluctuation can be attributed to the receiver's clock phase noises. The Lissajous pattern is restored and the carrier phase fluctuation disappears after B1C signal assistance, as shown in Fig. 9. According to experimental results, experimental results show that cross-frequency Doppler assistance can eliminate or reduce the effect of clock phase noise on the assisted tracking loop.

Improving the tracking threshold
By superimposing noise on the collected raw satellite signal data, the effective C/N 0 of the received signal can be changed (Liu 2022). To examine whether carrier phase tracking performance can be improved by cross-frequency Doppler assistance, noise was inserted into the received raw signal to gradually reduce the effective C/N 0 . As was the case in the 'Simulation Evaluation' subsection, carrier phase cycle slips were then used as a measure of the carrier phase tracking sensitivity. Recording the tracked carrier phase as the true signal phase and comparing this with the tracked carrier phase permitted the tracking performance to be evaluated. The tracking threshold of the carrier phase was set to the average value of the carrier-to-noise ratio when the cycle slip ratio reached 10%.
We chose the B1C pilot signal as the assist signal and set the tracking loop's noise bandwidth to 16 Hz, while the B2b signal was chosen as the aided signal and the tracking loop's noise bandwidth was set to 2 Hz. The carrier phase tracking threshold of the Doppler-assisted tracking was reduced by about 4 dB when compared to the conventional tracking method with a noise bandwidth of 16 Hz, according to test results.

Conclusions
We developed a linearized mathematical model of the crossfrequency Doppler-assisted carrier phase tracking loop, derived an equation for calculating the thermal noise jitter of the assisted tracking loop, analyzed its dynamic stress response characteristics, specified the tracking loop design requirements to eliminate the influence of inter-frequency frequency bias, and provided selection criteria. These The middle, Conventional tracking, 2 Hz; The right, Aided tracking, 2 Hz analytical results reveal the inner workings of the crossfrequency Doppler-assisted carrier phase tracking loop and provide a theoretical foundation for GNSS receiver signal tracking loop design.
Monte Carlo simulation was used to validate the theoretical analysis of the cross-frequency Doppler-assisted tracking loop. The simulation results agreed well with the theoretical calculation results, proving the validity of the theoretical analysis method. Using the carrier phase tracking of the B2b signal aided by the BDS B1C pilot signal as an example, the tracking sensitivity was improved by 1.5-3 dB and the tracking threshold was reduced by about 5 dB when the unaided tracking loop's noise bandwidth was set to 16 Hz and the aided tracking loop's noise bandwidth was set to 2 Hz.
Using collected real satellite signals and a software defined receiver, we analyzed the effect of the proposed cross-frequency Doppler-assisted carrier phase tracking loop. Even after the carrier frequency ratio transformation, preliminary results revealed an inter-frequency frequency bias between the measured B1C and B2b Doppler frequencies. However, the fixed frequency bias had no effect on B1C to B2b tracking assistance. This result was consistent with the theoretical analysis. The results of quasi-real data tests of B1C pilot signal-assisted tracking of the B2b signal showed that the carrier phase tracking threshold of the B2b signal could be reduced by approximately 4 dB.
We only looked at cross-frequency Doppler-assisted carrier phase tracking in AWGN channels; more research is needed to understand loop tracking performance in fading channels.

Appendix: optimal weighting factor for the combination of Doppler frequency measurements
In Fig. 1, Doppler frequency of each signal at its carrier phase tracking loop output is denoted byf trk,i ,f trk,i = f gdpl,i + n f ,i ,i = 1, … , N . Here, f gdpl,i is the generalized Doppler frequency of to signals i ; n f , i is the Doppler frequency measurement noise that follows a normal distribution of N 0, σ 2 f , i , which are independent of each other. Suppose the noise bandwidth from the input of the tracking loop to the output of the Doppler frequency is B Lf,i , then 2 f , i = 2 d, i B T,i B Lf ,i exists. According to Lemma 1, multiplying each generalized Doppler frequency by their respective carrier wavelength obtains the common generalized Doppler velocityv gdpl . In this way, the velocity observation vector V = v 1 , … ,v N T is obtained in which v i = v gdpl + i n f ,i . The log-likelihood function of the parameter v gdpl can be expressed as:

Its maximum likelihood estimate is
Here, the definition of i is as follows When the design of the carrier phase tracking loop is the same for each signal, their Doppler measurement noise bandwidths are also the same. Substituting B T,i = 1∕2T i into the equation above obtains: If the correlation integration time of the carrier tracking is the same for each assist signal and the Q-normalized phase discriminator is used for each of them, the above equation can be written as: We calculated the optimal weighting factor for the combination of Doppler frequency measurements using the maximum likelihood criterion in this appendix. The above equation provides a method for determining the best weighting factor.