Introduction

For spectrum compatibility, performance enhancement, and payload sharing, the new generation of Global Navigation Satellite Systems (GNSS) introduces the subcarrier modulated signals in addition to the traditional Binary Phase Shift Keying (BPSK) modulated signals and adopts the highly efficient Constant Envelope Multiplexing (CEM) technique to combine multiple signal components in the same frequency band into a single wideband composite signal for broadcasting (Yao and Lu, 2021). The phase of the subcarriers used in each subcarrier modulated signal component maintains a fixed relative relationship with that of the spread spectrum code. However, there is no limitation on the number, polarity, combined mode and real or complex nature of the subcarriers used in a signal component, which brings a great freedom to the signal designers, thus diversifying the subcarrier modulated signals used in the new generation of GNSS.

According to the features of the waveform used in the subcarrier, the existing subcarrier modulated signals in the new generation of GNSS can be classified into three categories. The first category adopts a single bipolar real square wave as the subcarrier, which is also known as Binary Offset Carrier (BOC) modulation (Betz, 2001), currently adopted by the modernized Military (M) code signal of Global Positioning System (GPS), Public Regulated Service (PRS) signals of Galileo satellite navigation system (Galileo), and authorization signals of BeiDou Navigation Satellite System (BDS), etc. The second category uses a combination of multiple bipolar real square waves as the subcarrier, which is also known as Multiplexed Binary Offset Carrier (MBOC) modulation (Hein et al., 2006), and is currently used by GPS, Galileo, and BDS compatible and interoperable signals in the L1/E1/B1 band. The third category uses complex waveforms as subcarriers for single-sideband modulation, such as the Alternative BOC (AltBOC) modulation (Lestarquit et al., 2008) used by Galileo in the E5 band and the Asymmetric Constant Envelope BOC (ACE-BOC) modulation (Yao et al., 2016) used by BDS in the B2 band. In addition, BDS uses a Single-sideband Complex-value Binary Offset Carrier (SCBOC) in the B1 band to combine the legacy B1I signal with the modernized B1C signal (Lu et al., 2019).

These diverse subcarrier modulated signals present new characteristics in both time domain and frequency domain compared to the BPSK modulation commonly used in traditional GNSS. First, unlike the traditional BPSK modulated signals whose main energy of the spectrum is gathered near the carrier center, the introduction of the subcarrier shifts the signal power spectrum main lobe to the upper and lower sidebands away from the carrier center. Second, the Auto-Correlation Function (ACF) of the subcarrier modulated signal is no longer triangular as in the traditional BPSK modulated signal, but appears as multiple side peaks, whose number and position are related to the spread spectrum code rate and the subcarrier frequency, phase, and waveform shape. These new features bring new opportunities and challenges to receiver processing. The opportunities are that the introduction of subcarriers increases the high-frequency component of the signal, which can help reduce the code phase tracking error and thus improve the ranging performance of the signal. It has been shown that BOC modulated signals generally have better thermal noise immunity as well as multipath resistance compared to BPSK modulated signals with the same spread spectrum rate (Yao and Lu, 2021). However, the subcarrier also poses some challenges to the receiver processing. First, the multi-peak ACF makes it possible for tracking to lock onto the side peaks instead of the main peak, which affects the stability of the code loop tracking. Second, different subcarrier modulations often correspond to different ACF shapes, making it even more difficult to find a unified receiver architecture that handles the new generation of GNSS signals as well as legacy BPSK modulated signals.

To address these issues, the signal processing techniques used in traditional GNSS receivers are no longer applicable, and it is necessary and critical to design appropriate acquisition and tracking as well as observation extraction techniques for the subcarrier modulated signals. How to utilize the high ranging accuracy potential brought by the spectral splitting characteristics of subcarrier modulated signals while avoiding the ambiguity threat is a common concern in both academia and industry. For two decades, research on subcarrier modulated GNSS signal processing techniques has been going on and has become a classical topic. A few years ago, some detailed reviews (Yao, 2012) and surveys (Lohan et al., 2017) for BOC modulated signal unambiguous processing have been presented. However, new and important advances in subcarrier modulated signal processing have emerged recently. On the one hand, the unique composite subcarrier modulated signal structures in BDS B1 and B2 bands and Galileo E5 band have been gradually emphasized by academics, and a variety of high-precision processing techniques for such wideband composite signals have been proposed. On the other hand, the multidimensional tracking loop-based unambiguous processing of subcarrier modulated signals has flourished, in which the additional distance-dependent observations offered by the subcarrier tracking loop provide a new way for solving the ambiguity problem. Therefore, this paper provides a timely overview of cutting-edge advances and evolutionary patterns in the field of subcarrier modulated signal processing, based on in-depth studies of unambiguous solutions. It endeavors to serve as an entry point for the researchers embarking on a research journey in this field. In addition, it seeks to provide an exhaustive overview of the latest major developments in the field, linking the reader to the most fascinating and successful milestones in this field of research. New progress and remaining challenges with prospects are also given.

This paper first describes the types of subcarrier modulated signals and their ambiguity threat in receiver processing. Then, we present a classification of unambiguous processing techniques for subcarrier modulated signals. Next, the two categories of unambiguous processing techniques are reviewed in detail and the performance improvement brought by subcarrier observations for positioning is demonstrated. Remaining challenges and possible investigative directions to promote further research are also given. Finally, the conclusions are presented.

Subcarrier modulated signals

At the satellite transmitter side, a general subcarrier modulated signal can be modeled as

$$\begin{aligned} s_{\textrm{RF}}\left( t\right) ={\text {Re}}\left\{ \sqrt{P}c\left( t\right) g\left( t\right) {\textrm{e}}^{{\textrm{j}}\left( 2\pi f_{{\textrm{RF}} }t\right) }\right\} \end{aligned}$$
(1)

where \(f_{\textrm{RF}}\) is the carrier frequency, and P, \(c\left( t\right) \), and \(g\left( t\right) \) are the power of the signal, the baseband spread spectrum code that modulates the navigation message or secondary code, and the subcarrier, respectively. The \(c\left( t\right) \) of all current GNSS are non-zero codes with rate \(f_{c}\), and the amplitude is constrained to be \(\pm 1\). The subcarrier

$$\begin{aligned} g\left( t\right) =\sum _{i=-\infty }^{+\infty }\tilde{g}\left( t-i2T_{s} \right) \end{aligned}$$
(2)

is a periodic waveform with period \(2T_{s}=1/f_{s}\), where \(f_{s}\) is called the subcarrier frequency. Note that \(T_{s}\) here follows the convention in BOC signal representation, which is not the subcarrier period but half of it. In (2), \(\tilde{g}\left( t\right) \) is the subcarrier waveform in one cycle and there is no strict restriction on its shape. The subcarrier \(g\left( t\right) \) can also be expressed in Fourier series form

$$\begin{aligned} g\left( t\right) =\sum _{k=-\infty }^{+\infty }a_{k}{\textrm{e}}^{{\textrm{j}}2\pi kf_{s}t} \end{aligned}$$
(3)

where \(a_{k}=(2T_{s})^{-1}\int _{0}^{2T_{s}}\tilde{g}\left( t\right) {\textrm{e}}^{{-\textrm{j}}2\pi kf_{s}t}{\textrm{d}}t\). The new generation of GNSS uses various forms of subcarriers to flexibly change the temporal and frequency characteristics of the signals to achieve diverse effects in terms of spectral compatibility, thermal noise immunity, multipath resistance, and the implementation complexity of transmitters and receivers.

Real-value subcarrier modulation

The simplest case is when the subcarrier uses a bipolar square wave

$$\begin{aligned} g_{\textrm{BOC}}\left( t\right) ={\textrm{sgn}}\left( \sin \left( 2\pi f_{s}t+\psi \right) \right) \end{aligned}$$
(4)

in which case the signal is the well-known BOC modulated signal (Betz, 2001). Here, \(\textrm{sgn}\left( \cdot \right) \) is the sign function and \(\psi \) is the subcarrier phase, whose two common values are 0 and \(\pi /2\), with which the corresponding BOC modulated signals are referred to as sine-phase BOC signal and cosine-phase BOC signal respectively.

In GNSS, just as a BPSK modulated signal with code rate \(f_{c}=n\times 1.023\) MHz is usually abbreviated as BPSK\(\left( n\right) \), a specific BOC modulation is usually abbreviated as \(\hbox {BOC}_{x}\left( m,n\right) \), where x can be taken as either s or c to denote sine-phase or cosine-phase, respectively, and m in the parentheses denotes the subcarrier frequency normalized by 1.023 MHz, i.e., \(f_{s}=m\times 1.023\) MHz, and the meaning of n is the same as in the case of BPSK, with \(m\ge n\) and the ratio \(M=2m/n\) is an integer, called the modulation order. Considering that there are fewer signals with cosine-phase subcarriers in GNSS in practice, \(\hbox {BOC}_{s}\left( m,n\right) \) simplified to BOC\(\left( m,n\right) \) in this paper where not otherwise specified.

The normalized Power Spectral Density (PSD) as well as the ACF of a BOC signal with \(m=14,n=2\) are given in Fig. 1.

Fig. 1
figure 1

a The normalized power spectral density and b the auto-correlation function of a BOC\(\left( 14,2\right) \) signal

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It can be seen that compared with the traditional BPSK signal, the main PSD lobe of the BOC signal modulated by the real square wave subcarrier is symmetrically shifted from the center frequency of the carrier, \(f_{\textrm{RF}}\), to \(f_{\textrm{RF}}\pm f_{s}\), which gives the appearance of a split spectrum. The ACF then changes from triangular to sawtooth, with multiple side peaks in addition to a main peak. For sine-phase BOC(m, n) modulated signals, the number of side peaks is \(2M-2\). In general, the larger the modulation order M is for a certain spread spectrum code rate, the narrower the main peak of the ACF is, and the corresponding signal ranging potential is higher.

Subcarrier modulation, which is slightly more complex than BOC modulation, is the case of using a combination of two square waves with different frequencies as a subcarrier. The signal in this case is called Multiplexed BOC (MBOC) modulated signal (Hein et al., 2006). The two square waves in the MBOC modulated signal are not unique in the way they are combined. For example, they can be time-division multiplexed, i.e., the BOC\(\left( m_{1},n\right) \) subcarrier alternates with the BOC\(\left( m_{2},n\right) \) subcarrier in a certain ratio of time lengths, where \(m_{1}\) is an integer multiple of \(m_{2}\), in which case the modulation is referred to as Time-division Multiplexed BOC (TMBOC) (Betz et al., 2006). Composite BOC (CBOC) modulation (Avila et al., 2007) weighted superimposes two BOC subcarriers

$$\begin{aligned} g_{\textrm{CBOC}}\left( t\right) =\sqrt{1-\gamma }g_{{\textrm{BOC}}\left( m_{1},n\right) }\pm \sqrt{\gamma }g_{{\textrm{BOC}}\left( m_{2},n\right) } \end{aligned}$$
(5)

where \(\gamma \) is used to adjust the power proportion of the two square waves. Depending on whether the two square waves are added or subtracted when superimposed, CBOC modulation may be further subdivided into two categories: in-phase CBOC and anti-phase CBOC, which are respectively represented by \(\hbox {CBOC}^{+}\) and \(\hbox {CBOC}^{-}\). Quadrature Multiplexed BOC (QMBOC) modulation (Yao et al., 2010a), on the other hand, places the two BOC subcarriers in two orthogonal phases of the carrier, i.e.

$$\begin{aligned} g_{\textrm{QMBOC}}\left( t\right) =\sqrt{1-\gamma }g_{{\textrm{BOC}}\left( m_{1},n\right) }\pm {\textrm{j}}\sqrt{\gamma }g_{{\textrm{BOC}}\left( m_{2},n\right) } \end{aligned}$$
(6)

The positive and negative signs in the above equation corresponds to the in-phase QMBOC and the anti-phase QMBOC, denoted as \(\hbox {QMBOC}^{+}\) and \(\hbox {QMBOC}^{-}\), respectively.

The normalized PSD as well as the ACF of the MBOC modulated signal with \(m_{1}=6,m_{2}=n=1\), \(\gamma =1/11\) are given in Fig. 2.

Fig. 2
figure 2

a The normalized PSD and b the ACF of a MBOC\(\left( 6,1,1/11\right) \) signal

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Since this MBOC modulated signal is a mixture of BOC(1, 1) and BOC(6, 1) components, its spectrum is also symmetric about the carrier center frequency, with a pair of lobes of the BOC(1,1) component as well as that of the BOC(6,1) component on each side of the carrier frequency. In addition, since the BOC(1, 1) component occupies the main power, the ACF of the MBOC modulated signal is based on the BOC(1, 1) result with the effect of the BOC(6, 1) component superimposed on it. The introduction of the high-frequency BOC(6, 1) component makes the main peak of the ACF of the MBOC modulated signal sharper compared to the pure BOC(1,1) case, which improves the ranging performance as well as the antimultipath performance of the signal.

Complex-value subcarrier modulation

Different from the real-value subcarrier which shifts the main lobe of the signal spectrum symmetrically to both sides of the carrier center, it is worth noting that in current GNSS, there are also the signals that use a more complicated form of complex-value subcarrier to achieve single-sided spectral shifting. These complex-value subcarriers have various waveforms, but are all essentially approximations of a simple harmonic function.

$$\begin{aligned} {\textrm{e}}^{\pm {\textrm{j}}2\pi f_{s}t}=\cos \left( 2\pi f_{s}t\right) \pm {\textrm{j}}\cos \left( 2\pi f_{s}\left( t-T_{s}/2\right) \right) \end{aligned}$$
(7)

For example, in BDS B1 band, to use a single satellite transmitter to simultaneously broadcast the legacy B1I signal at the center frequency of 1561.098 MHz and the interoperable B1C signal at the center frequency of 1575.42 MHz, the carrier frequency of the transmitter is set to \(f_{\textrm{B1}}=1575.42\) MHz, and the B1I baseband component is modulated by a single sideband complex-value subcarrier (SCBOC)

$$\begin{aligned} g_{\textrm{SCBOC}}\left( t\right) =g_{{\textrm{BOC}}_{c}\left( 14,2\right) }\left( t\right) -{\textrm{j}}g_{{\textrm{BOC}}_{c}\left( 14,2\right) }\left( t-T_{s,{\textrm{B1I}}}/2\right) \end{aligned}$$
(8)

with the frequency of \(f_{s,\textrm{B1I}}=1/T_{s,\textrm{B1I}}=14.322\) MHz, which moves its main power to 1561.098 MHz, thus realizing the asymmetrical dual-frequency multiplexing of the B1I signal and B1C signal (Lu et al., 2019). Figure 3 gives the normalized PSD of the BDS B1 wideband multiplexed signal, where the authorized signal component is not shown.

Fig. 3
figure 3

The normalized PSD of the BDS B1 wideband multiplexed signal

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Similar complex-subcarrier modulation is used on the BDS B2 band (Yao et al., 2016) and Galileo E5 band (Lestarquit et al., 2008), where the carrier center frequency of the transmitter is located at \(f_{\textrm{B2}}=1191.795\) MHz, and the B2a/E5a and B2b/E5b signal components are modulated by the complex-value subcarrier

$$\begin{aligned} g_{\textrm{ACE}}\left( t\right) =g_{1}\left( t\right) -{\textrm{j}} g_{1}\left( t-T_{s,{\textrm{B2}}}/2\right) \end{aligned}$$
(9)

with period \(2T_{s,\textrm{B2}}=1/(f_{s,\textrm{B2}})=1/\left( 15.345\text { MHz}\right) \) and its conjugate \(g_{\textrm{ACE}}^{*}\left( t\right) \), and thus their main powers are shifted to \(f_{\textrm{B2a}}=f_{\textrm{B2} }-f_{s,\textrm{B2}}=1176.45\) MHz and \(f_{\textrm{B2b}}=f_{\textrm{B2} }+f_{s,\textrm{B2}}=1207.14\) MHz, respectively. In (9), \(g_{1}\left( t\right) \) is a multilevel cosine-like step shape waveform whose expression can be found in (Yao et al., 2016).

For a long time, the above single-sideband complex-value subcarriers were only regarded as auxiliary components in the satellite payload to achieve constant envelope multiplexing and enable backward compatibility, allowing the receiver to treat the B1I component as BPSK(2) signals with the center frequency located at \(f_{\textrm{B1}}-f_{s,\textrm{B1I}}\), and the B2a/E5a and B2b/E5b components as BPSK(10) signals with the center frequency located at \(f_{\textrm{B2}}-f_{s,\textrm{B2}}\) and \(f_{\textrm{B2} }+f_{s,\textrm{B2}}\), respectively. However, recent research (Gao et al., 2020b) has shown that single-sideband complex-value subcarriers can also be involved in receiver pseudorange measurements, rather than just being treated as “useless” multiplexing terms. Moreover, the high frequency of these subcarriers holds great potential for range performance improvement. Several studies have focused on the processing of single-sideband complex-value subcarrier modulated signals, and positive results have been obtained. We will present them in the following sections.

Table 1 Subcarrier modulations being used in GNSS
Full size table

Table 1 lists the subcarrier modulations adopted by each GNSS. It should be noted that only a few subcarrier modulated signals that have now been practically used in GNSS are introduced in this section. The subcarrier modulations that have been proposed are not limited to those, but also include such as Binary Coded Symbol (BCS) (Hegarty et al., 2004) and Composite BCS (CBCS) (Hein et al., 2005). In (Yao and Lu, 2021), subcarrier modulations are presented in more detail.

Ambiguity threat

Although subcarrier modulated signals have a potential of higher tracking accuracy than BPSK signals with the same spreading code rate, the improvement of accuracy is at the cost of reduced reliability when using traditional acquisition and tracking techniques to time synchronize and extract observations from them.

Take the BOC signal as an example. Assuming that the receiver follows the ACF-energy-based acquisition detection statistic and an Early Minus Late Power (EMLP) based code discriminator which are classically used in BPSK signal processing, Fig. 4 compares the shapes of acquisition detection statistic and the code tracking discriminator curve for the BPSK\(\left( 2\right) \) signal and the BOC\(\left( 14,2\right) \) signal.

Fig. 4
figure 4

a Acquisition detection statistic and b code-tracking discriminator curves for the BPSK\(\left( 2\right) \) signal and the BOC\(\left( 14,2\right) \) signal

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Both of these two signals have the same spreading code rate of 2.046 MHz, but the shape of the acquisition detection statistic is quite different. The acquisition detection statistic of BOC\(\left( 14,2\right) \) signal has a much sharper main peak but multiple side peaks within \(\pm 1\) code chip, which do not differ much in height from the main peak. The discriminator curve of the BOC signal, while having a higher slope around zero delay, which implies a higher discriminator gain, has multiple false locking points in the operating range. In the acquisition stage, if a side peak of the detection statistic exceeds the decision threshold due to thermal noise, interference, multipath, dynamic stress, etc., causing a false lock, after giving its initial code phase offset to the tracking loop,it will be falsely locked to the corresponding false lock point. Moreover, if noise, dynamic stress, and short-term loss of lock caused by occlusion occur during the phase tracking, even no false acquisition, the loop may change from true locking point to false locking point, which will result in an unacceptable large deviation to the final position.

Classification of unambiguous solutions

Over the past two decades, numerous solutions have been proposed to address the ambiguity problem, which first emerged with the initial proposal of BOC modulation.

According to the way of treating the subcarriers, the existing unambiguous processing techniques are mainly categorized as those based on One Dimensional (1D) correlation structure and those based on Two Dimensional (2D) correlation structure.

Among them, the methods based on 1D correlation structure treat the subcarrier and the spreading code as a whole, looking for the ways to eliminate the side peaks of the sawtooth-shaped 1D correlation function shown in Fig. 1, or to reduce the impact of these side peaks on the acquisition and tracking. This class of methods has a relatively long research history, and has been the mainstream method of subcarrier signal unambiguous processing for a long time. These methods can be classified further according to their approach to eliminating ambiguous threats. This categorization can be achieved through two distinct approaches: false lock detection and recovery techniques, as well as unambiguous processing techniques. In more specific terms, considering the operational domain, unambiguous processing can be further classified into filtering-based processing and geometry-based processing.

The methods based on the 2D correlation structure treat the spreading code and the subcarrier as independent parts, breaking the constraint that the code phase delays and the subcarrier phase delays are identical in the receiver’s replica signals, and employing two independent loops to track them. The 2D processing methods did not receive sufficient attention in the early days. However, in recent years, some 2D processing architectures have demonstrated the advantages of high accuracy and robustness in processing composite subcarrier modulated signals, and the additional distance-related observations offered by the subcarrier loop provide new ideas for solving the ambiguity problem, which makes this structure gradually emphasized by academia researchers. Depending on the object to be processed, the methods based on the 2D correlation structure can be categorized as the methods for processing real-value subcarrier modulated signals and the methods for processing complex-value subcarrier modulated signals. The methods for processing real subcarrier signals can be further divided into Delay Locked Loop (DLL) based subcarrier processing, Phase Locked Loop (PLL) based subcarrier processing, and subcarrier processing in conjunction with carriers, depending on the differences in subcarrier discriminators. The methods for processing complex-value subcarrier signals are further classified into symmetric subcarrier tracking methods and asymmetric subcarrier tracking methods based on the difference in the selection of the carrier and subcarrier center frequency points. In addition, most of the methods based on the 2D correlation structure fix the ambiguity of the subcarrier observations either within a single satellite signal tracking channel or jointly by using the subcarrier observations of multiple satellite signals in the positioning solution stage.

Figure 5 provides the classification of the subcarrier modulated signal unambiguous processing. We will describe the various classes of methods in the following Sections.

Fig. 5
figure 5

Classification of subcarrier modulated signal unambiguous processing

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1D correlation based processing

False lock detection and recovery

False lock detection and recovery technique does not remove ambiguity but checks false lock. The Bump-Jum** (BJ) technique (Fine and Wilson, 1999) is the most representative method of detection and recovery. This technique is based on the traditional ambiguous code tracking loop and continuously monitors whether this loop locks on the main peak of BOC ACF. To this end, the BJ method uses two additional correlators located at the two highest side peaks. The two correlators are called Very Early (VE) and Very Late (VL) correlators respectively. The BJ method determines whether a false lock occurs by measuring and comparing the magnitudes of the outputs of these two correlators and the prompt (P) correlator. Ignoring the effect of noise, when the loop locks on the main peak, the magnitude of P correlator output is the greatest. And if one of the VL or VE correlation outputs is the biggest, it is a sign that the tracking may have been false locked and the loop will “jump” \(T_{s}\) in the right direction.

The advantage of the BJ technique is that when the false locking does not occur, the tracking accuracy can reach the optimal. However, strictly speaking, this method does not belong to the real sense of unambiguous tracking, since it does not completely eliminate the false locking threat. The distortion of the correlation function shape due to multipath or filtering will have a significant influence on its performance, since it is heavily dependent upon the position of the correlation function. In addition, the amplitude comparing approach used in this technique has a high likelihood of false alarms and miss detection when noise is present for large order subcarrier modulated signals where the height of the side peaks are closer to the main peak. Although a longer sequential detection process improves the decision performance, it implies a longer response time (Wang et al., 2021). That is, once the false locking occurs, the loop cannot detect and act immediately. Therefore, the BJ technique is typically used in receivers for low-order BOC signals such as BOC\(\left( 1,1\right) \) signals. In the applications with high continuity and security requirements, as well as in low signal-to-noise ratio environments, the BJ technique is not suitable.

Filtering-based processing

This category of methods attempts to reduce as much as possible the changes introduced by the subcarrier to the signal by means of demodulation or channel equalization, so that the frequency domain and time domain characteristics of the signal can be as close as possible to the BPSK cases.

Sideband processing techniques

Sideband processing techniques can essentially be understood as demodulating \(c\left( t\right) \) from the subcarrier modulated signal before performing traditional acquisition and tracking processing on it. The differences in sideband processing techniques vary in the filtering process of the received signal and in the form of the local replica signals.

To separate the upper and lower spectrum sidebands, both the received signal and the local BOC modulation baseband signal are filtered in the earliest sideband processing techniques (Fishman et al., 2000). The correlation function of the filtered signals is similar to that of the BPSK signal. Therefore, in the acquisition and tracking process, this correlation function may be used to replace the ACF of the BOC signal. The single sideband technique uses only one sideband of the BOC modulated signal, while the double sideband technique uses the correlation values of both sidebands simultaneously by summing them noncoherently. Compared to the single sideband method, the double sideband technique results in a smaller noncoherent loss of correlation. But compared to the single sideband technique, it needs two times as many selective filters on each side band.

Another unambiguous method of sideband processing is the BPSK-like method (Martin et al., 2003). Its major difference from the method in (Fishman et al., 2000) is that it uses just one low-pass filter for receiving signals. The two main lobes of the spectrum are included in the filter bandwidth. Another difference is that the local replica signal is not a filtered BOC modulated baseband signal, but \(c\left( t\right) \) multiplied by the fundamental frequency component \(\textrm{e}^{\textrm{j}2\pi f_{s}t}\) (Martin et al., 2003) or the first 2nd-order harmonic component (Li et al., 2007) of the square-wave subcarrier (see (3)). The original BPSK-like method can only be used for even-order sine-phase BOC modulation. The improved BPSK-like method proposed by Burian et al. (2006) extended its applicability to odd-order BOC signals. Further improved work (Lohan et al., 2008) reduces the number of filters required for the sideband filtering method and also reduces the complexity.

Although the correlation functions in sideband techniques do not show any side-peaks, which means that they are completely unambiguous, the main drawback of this category of methods is that they completely discard all the potential advantages in terms of thermal noise and multipath mitigation offered by the subcarriers. In addition, the combination of two sidebands in a noncoherent mode introduces correlation losses. With respect to tracking, the sideband technique has little advantage. However, the correlation function of these methods has a wider main peak, so that longer code delay steps can be used in the acquisition stage, thus shortening the average acquisition time (Lohan, 2006).

Channel equalization techniques

Such methods regard the product of the subcarrier and the code, \(c\left( t\right) g\left( t\right) \), as the result of an ideal spread spectrum code impulse sequence \(\tilde{c}\left( t\right) =\sum \nolimits _{i=-\infty }^{+\infty }c\left( iT_{c}\right) \delta \left( t-iT_{c}\right) \) passing through a selective communication channel with impulse response \(s_{b}\left( t\right) =\sum \nolimits _{k=0}^{M/2}\tilde{g}\left( t-k2T_{s}\right) \), where \(\delta \left( t\right) \) is the Dirac delta function. Therefore, a corresponding compensator or equalizer can be employed at the receiver side to cancel out the effect of this channel on the code processing, and subsequently recover the correlation function of \(c\left( t\right) \) without side peaks. Typical equalizers include Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) equalizers (Anantharamu et al., 2009, 2011b). The advantage of this class of methods is their generality (Yang et al., 2006), while the disadvantage is the implementation complexity. In addition, a further disadvantage as with sideband processing is that the high accuracy ranging performance potential inherent in the subcarrier is discarded.

Geometry-based processing

This category of methods attempts to change the code chip waveform shape of the local replica signal as well as the combination mode of the correlator outputs to eliminate the false locking points of the discriminator curves. For a given code chip waveform \(\tilde{g}\left( t\right) \) of the received subcarrier modulated signal, varying the code chip waveform of the local replica signal can change the shape of the Cross-Correlation Function (CCF), which in turn adjusts the shape of the discriminator curve. This category can be subdivided into side-peaks cancellation methods that replace the ACF of the subcarrier modulation signal with a Pseudo-Correlation Function (PCF) without side peaks for the discriminator, and discriminator curve sha** methods that construct an unambiguous discriminator curve directly.

Side-peaks cancellation techniques

This subcategory of methods achieves unambiguous processing by introducing auxiliary signals, correlating them with the received subcarrier modulated signal, and combining the CCFs to obtain a PCF without side peaks which is used in place of the subcarrier modulated signal’s multi-peak ACF for acquisition and tracking.

Two sets of local signals are used by the Autocorrelation Side-Peak Cancellation Technique (ASPeCT) (Julien et al., 2004a, b, 2007). By using the synthesized PCF instead of ACF in the EMLP discriminator, with a wide front-end bandwidth, the resulting discriminator curve can completely eliminate the two false-locking points at \(\pm 0.5\) code chips delay while maintaining good tracking accuracy and anti-multipath performance. However, for bandlimiting case, the correlation function shape becomes rounded, and ASPeCT will still have false locking points. Moreover, this method is only dedicated to BOC(n, n) signals. The subcarrier phase cancellation method proposed by Burian et al. (2007) is applicable to all BOC signals with both sine and cosine phases.

The methods of side-peak cancellation are based on the geometry of a signal’s CCF. In most of the earlier side-peak cancellation methods, the waveforms of the auxiliary signals were obtained by trial-and-error methods, which causes most of side peak cancellation methods to have the same problem of poor generalization as ASPeCT. Yao and Lu (2011) thoroughly investigate the design methodology of unambiguous processing techniques based on side peak cancellation, and give a general design framework for the side peak cancellation method. By introducing a local signal model with a step-shaped spread spectrum code waveform, this design methodology establishes the correspondence between the shape of the CCF between the local signal and the input signal and the value of a shape vector, thus transforming the design of the PCF into a four-step process of solving a set of equations. Under the guidance of this design framework, more generalized unambiguous acquisition and tracking methods have subsequently emerged, such as the General Removing Ambiguity via Side-peak Suppression (GRASS) method (Yao et al., 2010b) for acquisition and the PCF-based Unambiguous DLL (PUDLL) (Yao et al., 2010) for tracking. Both of these methods can be used to sine-phase BOC signals with arbitrary orders. Based on PUDLL, Yao (2012) and Zhou et al. (2012) further extend its generality by replacing local waveforms to enable it to be used for cosine phase BOC signals.Ren et al. (2014) also present a side-peak cancellation method that can be applied to sine BOC, cosine BOC, and AltBOC signals under the guidance of the side-peak cancellation method design framework (Yao and Lu, 2011). Although the side-peak cancellation methods are flexible and widely applicable, when the BOC modulation order increases, the shape vector degrees of freedom of the local signals increase, thus the difficulty of the design optimization increases as well. Moreover, the resistance to thermal noise and multipath decreases significantly for most of the side-peak cancellation methods for higher-order BOC signals.

Discriminator curve sha** techniques

Discriminator curve sha** methods can essentially be viewed as the fitting of an unambiguous DLL discriminator curve using the outputs of multiple correlators by linear or nonlinear combinations. Admittedly, some of the side-peak cancellation methods of the previous Subsection, which first use correlator outputs to assemble a PCF and then use it to construct a discriminator curve, can be viewed as this kind of method as well in a broad sense. However, the methods presented in this Subsection mainly refer to those that construct the discriminator curve directly.

The simplest discriminator curve sha** method can be referred to as the BOC-Pseudo Range Noise (BOC-PRN) method (Dovis et al., 2005). This method generates a CCF by using the BPSK\(\left( n\right) \) signal with the received \(\hbox {BOC}_{s}\left( n,n\right) \) modulated signal and directly uses it as the DLL discriminator function. This method completely eliminates the false locking point of the discriminator curve, so that it realizes unambiguous tracking and has a very low implementation complexity. However, the method is only applicable to \(\hbox {BOC}_{s}\left( n,n\right) \) modulated signals and less resistant to thermal noise and multipath. Subsequently, Musso et al. (2006) extended the applicability of the BOC-PRN method to \(\hbox {BOC}_{s}\left( m,n\right) \) signals. Wu and Dempster (2009) further extended it to \(\hbox {BOC}_{c}\left( m,n\right) \) signals. Further studies began to use the combination of more correlation functions to construct the discriminator curves. Wu and Dempster (2007) replaced the BOC-PRN method with an improved version based on the sum of the early (E) and late (L) branches of the BOC-PRN correlator. The advantage of this method is that the effect of noise can be reduced by changing the E-L code spacing. Subsequently, Wu and Dempster (2011) enhanced the BOC-PRN tracking architecture by using strobe pulses in the local code chip to improve its resistance to multipath, especially for medium- and long-range multipath.

The use of a larger number of correlators with different delays provides greater design freedom in the discriminator curve sha** process. Although discriminator curve sha** can be achieved by the trial-and-error method (Kao and Juang, 2012), it is only applicable when the number of correlators is small, and the optimality of this method is difficult to guarantee. Actually, after determining the local waveform shape, the correlator number N, the spacing \(\left\{ d_{i}\right\} \), the combination form, and the optimization criterion, the problem is mathematically equivalent to an optimization of curve fitting. The most commonly used optimization criterion is minimizing the mean-square error of a linear combination of the correlation result \(R\left( \tau +d_{i}\right) \) with the ideal discriminator curve \(D_{\textrm{ideal} }\left( \Delta \tau \right) \), that is

$$\begin{aligned} \underset{\left\{ \alpha _{i}\right\} }{\min }\mathbb {E}\left( \sum _{i=1} ^{N}\alpha _{i}R\left( \tau +d_{i}\right) -D_{\textrm{ideal}}\left( \tau \right) \right) ^{2} \end{aligned}$$
(10)

For example, Fante (2003) fits an approximately linear discriminator output curve for the \(\hbox {BOC}_{s}\left( m,n\right) \) modulation signal using 12 to 16 correlator outputs with different spacings. Besides, Pany et al. (2005) andPaonni et al. (2008) fit more refined designs of the desired discriminator curve shape \(D_{\textrm{ideal}}\) by using the outputs of dozens of correlators.

It is not difficult to understand that for discriminator functions in coherent combination form, weighted combination of the correlator outputs is equivalent to sending a weighted linear combination of local PRN code waveforms with different delays to the correlator. Therefore, in some literatures (Pany et al., 2005; Li et al., 2017), the above idea is also presented in the form of optimizing the local replica code waveforms.

The high flexibility advantage of the discriminant curve sha** method based on optimization of the multiple correlator combination coefficients allows this idea to be easily extended to different subcarrier modulation signals, such as cosine-phase BOC (Chen et al., 2012; Shen et al., 2014), CBOC (Ren et al., 2013), QMBOC (Wang and Li, 2019), and AltBOC (Chen et al., 2014). Theoretically, arbitrary desired discriminator curve shapes can be numerically fitted using sufficiently large numbers of correlator outputs of different delays. However, the structure of implementations based on a large number of correlators makes most of such methods highly complex to implement.

2D correlation based unambiguous processing

The various unambiguous acquisition and tracking methods described in the previous section all deal with the subcarrier and the spreading code in the received signal as a whole. They try to construct a code phase discriminator function without false locking points by means of filtering, equalization, or modification of the local signal waveform, etc., to replace the discriminator function in the traditional DLL. In contrast, the idea of 2D tracking methods is quite different since these methods regard the code and the subcarrier as two independent parts, breaking the restriction that the code phase delay \(\tau _{c}\) and the subcarrier phase delay \(\tau _{s}\) are exactly the same, and adopting two sets of independent loops to track them. That is, the product of the local replica code and the subcarrier is \(c\left( t-\hat{\tau } _{c}\right) \hat{g}\left( t-\hat{\tau }_{s}\right) \). When using this replica signal to correlate with the received signal \(s\left( t-\tau \right) \), the CCF is a two-dimensional function

$$\begin{aligned} R\left( \tau -\hat{\tau }_{c},\tau -\hat{\tau }_{s}\right) =\frac{1}{T}\int _{T}s\left( t-\tau \right) c\left( t-\hat{\tau }_{c}\right) \hat{g}\left( t-\hat{\tau }_{s}\right) {\textrm{d}}t \end{aligned}$$
(11)

Using the BOC\(\left( 2,1\right) \) signal as an example, the shape of the function is shown in Fig. 6.

Fig. 6
figure 6

a Oblique drawing, b profile in \(\tau _{s}\) dimension, and c profile in \(\tau _{c}\) dimension of the 2D CCF of BOC(2, 1) signal with a front-end filter bandwidth of 10.23 MHz

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As seen in Fig. 6, the 2D CCF exhibits different characteristics in two dimensions: in the \(\tau -\hat{\tau }_{s}\), or subcarrier dimension, it has multiple peaks and expands infinitely with a period of \(2T_{s}\); in the \(\tau -\hat{\tau }_{c}\), or code dimension, it has only a single peak, with a main peak width of \(2T_{c}\). For the receivers using a 2D tracking structure, any one of the peaks of the 2D CCF can be tracked. When the loop is converged, the delay estimates in each of the two dimensions at the peak can be written as

$$\begin{aligned} \hat{\tau }_{c}\approx \tau +n_{c} \end{aligned}$$
(12)
$$\begin{aligned} \hat{\tau }_{s}\approx \tau +NT_{s}+n_{s} \end{aligned}$$
(13)

where N is an arbitrary integer representing the tracking ambiguity in the subcarrier dimension, while \(n_{s}\) and \(n_{c}\) are the tracking noise jitters in the subcarrier and code dimensions, respectively. Since the subcarrier rate is higher than the code rate, in general, the variance of \(n_{s}\) is smaller than the variance of \(n_{c}\). In this way, the loops of the 2D tracking structure obtain an unambiguous but lower precision delay estimate in the code dimension, while at the same time obtaining an ambiguous but high precision delay estimate in the subcarrier dimension. If \(n_{s}\) and \(n_{c}\) satisfy

$$\begin{aligned} \left| n_{c}-n_{s}\right| <0.5T_{s} \end{aligned}$$
(14)

then by performing the following nonlinear combination of the two delay estimates

$$\begin{aligned} \hat{\tau }=\hat{\tau }_{s}-\text{round}\left( \frac{{{\hat{\tau }}_{s}}-{{\hat{\tau }}_{c}}}{T_{s}}\right) T_{s} \end{aligned}$$
(15)

where \(\text{round}\left( \cdot \right) \) is the operation of taking the nearest integer. It is easy to verify that an unambiguous and high-precision delay estimate can be obtained.

Two-dimensional tracking methods provide higher observation dimensions than traditional 1D methods. It is easy to prove that, the 1D correlation function is a cross section in the direction of the diagonal \(\tau _{c}=\tau _{s}\) of the 2D correlation function. As a result, the 2D tracking methods will be more flexible in receiving subcarrier modulated signals.

According to the type of subcarriers in the processed signals, 2D tracking methods can be used for both real subcarrier modulated signals, such as BOC, MBOC, etc., as well as for wideband composite signals where complex subcarriers are used. These two types of methods are described separately in the following two subsections.

2D tracking of real-value subcarrier modulated signals

Depending on the loop type used to process or extract the real-value subcarrier phases, the existing 2D loop tracking techniques can be subdivided into three categories: DLL-based subcarrier processing, PLL-based subcarrier processing, and subcarrier processing in conjunction with carriers.

DLL-based subcarrier processing

Exploiting the square-wave characteristics of real-value subcarriers, this category of methods implements the discriminator and loop structure that are similar to code tracking on real-value subcarrier tracking. For instance, Double Estimator Tracking (DET) (Hodgart and Blunt, 2007; Hodgart et al., 2007a) adopts a DLL in subcarrier tracking and can be applied to both \(\hbox {BOC}_{s} \left( m,n\right) \) and \(\hbox {BOC}_{c}\left( m,n\right) \). Based on this, Hodgart and Simons (2012) present an improved DET method to relieve the coupling effects of code and subcarrier tracking loops caused by the twisted 2D ACF. Unlike DET with “two-point” correlators in code and subcarrier tracking, the improved DET method specifically designs a “four-point” correlator structure shared for both code and subcarrier tracking. By combining “four-point” correlator values and taking proper parameters, the coupling effect of subcarrier tracking to code tracking can be removed, and thus the tracking robustness is enhanced. Furthermore, utilizing the phase coherences between code and subcarrier, Schubert et al. (2014) designed an Astrium Correlator structure, which can achieve stable tracking by implementing only a subcarrier tracking loop without a code tracking loop. When two real-value subcarriers exist in signals, such as MBOC(Hein et al., 2006) and QMBOC(Yao et al., 2010a), Hodgart et al. (2008) and Gao et al. (2019) propose the three-dimensional tracking methods by introducing an additional subcarrier carrier loop to DET. The subcarrier phase delay will be combined with two subcarrier tracking loop outputs. Since the above methods assume the real-value subcarriers are bipolar square waves, they are more suitable to track signals the with wider bandwidths.

PLL-based subcarrier processing

According to (2),the real-value subcarrier has similar periodic characteristics as the carrier. Ignoring high-order terms of Fourier series form, the real-value subcarrier with periodic square waves can be approximated as a sine or cosine signal. Therefore, this category of methods treat the real-value subcarrier as a sine or cosine wave. For instance, Double Phase Estimator (DPE) proposed in (Borio, 2014b, a) adopts a DLL in code tracking and a PLL in subcarrier tracking, which can get similar tracking results as the DLL-based tracking methods. Based on this, by spectrum shifting and decimation, Decimation DPE (DDPE) (Feng, 2016) effectively decreases the calculation burden of DPE with only slight tracking performance loss, thus enhancing the practicality of DPE. Since the above methods only utilize the first-order sine or cosine terms, they are more suitable to tracking the signals with narrow bandwidths.

Subcarrier processing in conjunction with carrier

This category of methods treats the subcarrier as a part of the carrier and extracts subcarrier phases by jointly processing both lower and upper sideband components. Improved Dual Binary-phase-shifting-keying Tracking (IDBT) methods proposed by Wang et al. (2020) regard the lower and upper sidebands as two independent BPSK modulated signals and track subcarrier phases by combining lower and upper sideband carrier discriminators. Wang et al. (2020) also derived the theoretical tracking performance of IDBT. Similarly, Feng et al. (2016) generated the local upper and lower sideband carrier replicas without local subcarrier replicas. The lower and upper sideband correlation values are used for joint discrimination to estimate carrier and subcarrier phase errors. Updated carrier and subcarrier frequencies are combined to generate local lower and upper sideband carriers. Based on this, Borio (2017), Tian et al. (2021), and Tian et al. (2022) reduced the tracking complexity by degrading sampling frequencies. Specifically, Coherent Side-Band (CSB) (Borio, 2017) implements frequency conversion on signals, and thus both lower and upper sideband signals are moved to zero-frequency. By decimation and low-frequency filtering, the sampling rates of signals can be reduced. Similarly, Low Processing Rate DBT (LPR-DBT) proposed by Tian et al. (2021, 2022) specifically designs a pre-processing unit based on spectrum conversion, decimation and filtering to track the signals with lower processing rates. Since this category of methods track subcarriers with carriers, they are more suitable for the signals with narrow bandwidth like PLL-based subcarrier methods.

2D tracking of complex-value subcarrier modulated Signals

According to the previous discussions, complex-value subcarriers as multiplexing terms not only combine different signals into a wideband multiplexed signal, but also have great high-precision ranging potential due to their high frequencies. Recently, there are existing methods focusing on complex-value subcarrier processing. Different from real-value subcarrier modulated signals, complex-value subcarrier modulated signals with different local carrier frequencies can be equivalently converted because of the existence of complex-value subcarriers. Therefore, the tracking of complex-value subcarrier modulated signals does not need to fix the local carrier frequency on the center frequency between lower and upper sideband frequencies as in the case of real-value subcarrier signal tracking, but can take any value between the lower and upper sideband frequencies. According to the used local carrier frequencies, the existing research can be divided into the symmetric subcarrier tracking and asymmetric subcarrier tracking.

Symmetric subcarrier tracking

Similar to the tracking of real-value subcarrier modulated signals, the local carrier frequency adopted by symmetric subcarrier tracking methods is the center frequency between lower and upper sidebands. Both lower and upper sideband components in equivalent baseband signal are modulated by the complex-value subcarrier whose frequency is half the difference between the lower and upper sideband frequencies. Consistent with tracking methods for real-value subcarrier modulated signals, Ren et al. (2012) combine the codes of E5a and E5b pilot components so the Galileo E5 wideband multiplexed signal is converted to a standard BOC signal. The DET method is adopted to stably track such signal. Similarly, Wideband High-Accuracy joint Tracking (WHAT) proposed by Zhang et al. (2019) treats the B1I and B1C components in BDS B1 wideband multiplexed signal as the lower and upper sidebands of standard BOC signal respectively. By combining lower and upper sideband correlations, WHAT realizes the joint tracking of signals with different codes and accurately extracts subcarrier phases. Unlike the methods proposed by Ren et al. (2012) and Zhang et al. (2019), Dual BPSK Tracking (DBT) Zhu et al. (2015) treats the E5a and E5b in AltBOC as two independent signals. By joint combining of lower and upper sideband carrier discriminators, DBT achieves the stable tracking of both carrier and subcarrier. Based on this, Wang et al. (2017) and Wang et al. (2017) proposed an ASYMmetric DBT (ASYM-DBT) so as to adapt to the BDS B1 wideband multiplexed signal with unequal power characteristics.

Asymmetric subcarrier tracking

Different from symmetric subcarrier tracking, the local carrier frequency adopted by asymmetric subcarrier tracking methods is either the lower or upper sideband frequency. Therefore, the equivalent baseband signal has the subcarrier whose frequency is the difference between upper and lower sideband frequencies. Since such subcarrier frequency is twice that of the symmetric subcarrier tracking, asymmetric subcarrier tracking can more fully utilize the high-frequency characteristics of complex-value subcarrier modulated signals.

Specifically, the Cross-Assisted Tracking (CAT) proposed by Gao et al. (2020c) uses the upper sideband frequency as the local carrier frequency. Since the lower sideband components have high-frequency subcarriers while upper sideband components have no such subcarriers, the CAT utilizes the upper sideband components for carrier tracking, and lower sideband components for code and subcarrier tracking. Moreover, Gao et al. (2020a) also proposed a Coherent Processing Tracking (CPT) method which is equivalent with CAT. However, although these methods can take advantage of subcarriers with high frequencies, they inherently cannot utilize all components of both lower and upper sideband components to jointly track subcarriers as in symmetric subcarrier tracking methods, since high-frequency subcarrier modulations only exist in single sideband components. To fully utilize multi-component characteristics of complex-value subcarrier modulated signals, Qi et al. (2024a) proposed a Dual-frequency Multi-component Tracking (DMT) method. Since there are two local carrier frequencies in DMT, the lower and upper sideband frequencies, both lower and upper sideband components in two equivalent baseband signals are modulated by high-frequency subcarriers, thus can be jointly used in subcarrier tracking. Based on this, Qi et al. (2024b) effectively reduced the calculation burden of DMT with a slight performance loss by using a specifically designed multi-stage segmented-integral structure.

Integer ambiguity resolution of subcarrier observations

According to the previous discussions, the subcarrier observations extracted by 2D tracking are high-precision but ambiguous. Accurate estimation of subcarrier ambiguity N is the key to obtain unambiguous subcarrier observations. Classic 2D tracking methods (Borio, 2014b; Hodgart and Blunt, 2007) utilizes the unambiguous code observations to provide integer ambiguities for subcarrier observations, such as (15). The observation-based method can accurately fix ambiguities for subcarriers with low frequencies. However, when the subcarrier wavelength is short or the errors of code and subcarrier observations are large, constraint (14) may not be fulfilled, causing incorrect estimation of subcarrier ambiguities.

To accurately resolve subcarrier ambiguities in various cases, the positioning-based methods attract attention. Drawing on the idea of carrier ambiguity resolution, Wendel et al. (2014, 2015) utilize unambiguous but low-precision code observations and high-precision but ambiguous subcarrier observations of multiple satellites to establish a subcarrier ambiguity resolution model and then calculate the subcarrier ambiguity float solutions in the position domain. Subsequently, the Least-square AMBiguity Decorrelation Adjustment (LAMBDA)(Teunissen et al., 1997) is implemented to obtain the integer ambiguity of the subcarrier observations. Compared with observation-based methods, positioning-based methods utilize the collaborative gain of multiple satellites and can more accurately resolve subcarrier ambiguities. To ensure subcarrier ambiguities can be robustly and accurately resolved when the number of visible satellites is small or quality of observations is poor, Qi et al. (2022) introduced the concept of carrier smoothed code observations. Unlike the above methods (Wendel et al., 2014, 2015; Qi et al., 2022) processed in a single epoch, Hameed et al. (2021) implemented the Kalman filter to utilize observations of multiple epochs to jointly resolve subcarrier ambiguities. Based on this, Qi et al. (2023b) introduced the time-differenced carrier observations with ultra-high precision, and thus the subcarrier ambiguity resolution accuracy is further improved.

Performance evaluation theory for 2D loops

In contrast to the well-established tracking performance evaluation theory for 1D loop structures (Betz and Kolodziejski, 2009), the tracking performance evaluation theory for 2D loop structures is still in the process of being refined. Some early studies were generally analyzed by simulation only (Blunt, 2007), or the two dimensions were considered to be independent of each other, thus directly applying the 1D theoretical analysis method (Ren et al., 2012). Other theoretical evaluation expressions for 2D loop structure tracking performance (Hodgart et al., 2007b; Borio, 2014b; Anantharamu et al., 2011a) ignore the important RF front-end band-limit filtering effects and the inter-dimensional coupling effects for simplicity, thus failing to correctly reflect the characteristics of the 2D loop structure tracking performance, as well as the influence of loop parameters on noise performance and multipath performance.

Yao et al. (2017) presented a general tracking performance evaluation theory for 2D loop structures, which not only clearly reveals the twisting and slanting properties of the 2D correlation function and its resultant problem of coupling between the code and the subcarrier tracking loops, but also finds that, for both DET and DPE, the selection of the E-L spacing in the code dimension has a significant impact on the ranging accuracy in the subcarrier dimension. Especially when the code E-L spacing \(\Delta _{c}\) is near an even multiple of \(T_{s}\), the tracking jitter performance deteriorates rapidly, which is known as the “parity stratification” phenomenon. Therefore, it is suggested that the value of \(\Delta _{c}\) in DET and DPE should be preferably chosen near an odd multiple of \(T_{s}\). Furthermore, Gao et al. (2018) give the DET tracking performance prediction formulas with closed-form analytic forms while preserving the effects of band-limiting filtering and coupling effects, which can help receiver designers to better understand the core characteristics of the tracking performance of 2D loop structures. This paper further gives the guidelines for optimal receiver parameter selection under certain Radio Frequency (RF) front-end bandwidth. The conclusions of these recently emerged theoretical analyses can greatly simplify the work of subcarrier modulated signal receiver designers.

Use of subcarrier observations in positioning

According to the previous discussions, the 2D tracking of subcarrier modulated signals can not only extract code and carrier observations like traditional 1D signal tracking methods, but also can extract the third observation that is the subcarrier observation. From the perspective of observation, since the subcarrier is tracked by independent tracking loops, subcarrier observation is a new measurement of the distance between the satellite and the receiver, providing additional information for positioning. Therefore, subcarrier observations have the potential to further improve positioning accuracy.

In terms of Single Point Positioning (SPP), traditional methods uses only code observations for SPP. However, since subcarrier observations have higher precision than code observations, using subcarrier observations to calculate pseudorange and SPP can be expected to have higher precision. There have been studies that have verified the effectiveness of subcarrier observations. Figure 7 exhibits SPP results (Qi et al., 2022) using B1I code observations, B1C observations and subcarrier observations of BDS B1 wideband multiplexed signal respectively and their SPP errors STandard Deviations (STD) in North (N), East (E) and Upper (U) directions are also exhibited in Table 2. Intuitively, the dispersion of SPP using subcarrier observations is much smaller than that of positioning using B1I and B1C code observations. Quantitatively, SPP errors of subcarrier observations are also obviously smaller than that of B1I and B1C code observations.

Fig. 7
figure 7

SPP of B1I code observations, B1C code observations and subcarrier observations of BDS B1 wideband multiplexed signal. Qi et al. (2022)

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Table 2 SPP error STDs of B1I code observations, B1C code observations and subcarrier observations of BDS B1 wideband multiplexed signal in N, E, and U direction. Qi et al. (2022)
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In terms of Precise Point Positioning (PPP), classical methods utilize code observations to calculate pseudoranges and assist the resolution of carrier ambiguities, achieving centimeter-level positioning results. According to discussions by Ge et al. (2019), the quality of pesudoranges involved in PPP is one of the important factors affecting PPP convergence. Since subcarrier observations can provide higher-precision pseudoranges than code observations, PPP using subcarrier observations can effectively accelerate convergence speed of PPP. Positive results have been given for PPP using subcarrier observations (Qi et al., 2023a). Figure 8 exhibits the Three Dimensional (3D) positioning results using B1I code observations, B1C code observations and subcarrier observations of BDS B1 wideband multiplexed signal. As the results show, PPP using subcarrier observations has obviously faster convergence speed than PPP using B1I or B1C code observations, showing the effectiveness of subcarrier observations in accelerating PPP convergence. Unlike the methods that replace code observations with subcarrier observations in PPP, since code and subcarrier observations are independent measurements of the distances between satellites and the receiver, PPP using both code and subcarrier observations can be expected to have better positioning performances.

Fig. 8
figure 8

PPP of B1I code observations, B1C code observations and subcarrier observations of BDS B1 wideband multiplexed signal. Qi et al. (2023a)

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Challenges and prospects

Although the research on subcarrier modulated signal processing techniques has been going on for more than two decades since BOC signals were first proposed for GNSS, new problems and challenges are still emerging with the use of new subcarrier modulated signals and the emergence of new application scenarios. In particular, the processing techniques for complex-value subcarrier modulated signals, as well as the 2D and even higher dimensional correlation based unambiguous processing are still at an early stage of development, and extensive work is required to improve their applicability and theoretical understanding.

It should be noted that high frequency subcarriers in complex-value subcarrier modulated signals bring the potential for highly accurate ranging and positioning, but also the limitations of both higher processing complexity and more serious ambiguity threats. To receive and process high frequency subcarrier modulated signals, receivers may need to use larger front-end bandwidth and higher sampling rate, which can increase the complexity of the RF and baseband processing channels, thereby affecting the utility of subcarrier high-precision applications in mobile portable or consumer electronical devices. On the other hand, because high-frequency subcarriers have similar periodic characteristics to carriers and their wavelengths are short, subcarrier tracking loops may experience more frequent half- or full- cycle slip in tracking, which affects the quality of the extracted subcarrier observations.

Moreover, although the 2D tracking technique can provide more flexible and diverse tracking modes, the coupling effect existing between the code and subcarrier components makes the 2D ACF distorted, which can lead to the interaction between the code tracking loop and the subcarrier tracking loop, possibly bringing about problems such as longer loop convergence time, reduced stability, and degradation of performance. In addition, the analytical expression of the distorted 2D ACF is complicated, which means that it is difficult to establish a complete theoretical performance analysis framework for 2D tracking and design the optimal phase discriminator structure for 2D tracking accordingly, especially for complex-value subcarrier signals. At this stage, some characteristics of the 2D tracking loop can only be partially explored by artificial trial-and-error and numerical simulation.

In response to the above status quo and challenges, we present possible future investigative directions to promote further research.

Tracking performance evaluation of multi-dimensional loop structures

Theoretical research on tracking performance evaluation of multidimensional loop structures for complex-value subcarrier modulated signals can be carried out. Although the existing tracking performance evaluation theories consider the inherent code- and subcarrier-dimensional coupling problems in 2D loop structures, the situation becomes more complicated when the signals are modulated with complex-value subcarriers, and when the challenge of mutual coupling of carriers and subcarriers in the complex correlation function arises. Therefore, analyzing the theory of tracking performance evaluation in this complicated situation is a worthwhile research direction.

Joint discriminator design for code and subcarrier tracking

More in-depth research can be carried out on the discriminator of multi-dimensional loop structure tracking techniques. The existing discriminant methods for multi-dimensional loop structures in each dimension use independent discriminators. However, considering the inter-dimensional coupling effect existing in multi-dimensional loop structures may lead to the fact that such discriminators in a single dimension are no longer optimal. Therefore, the study of multi-dimensional joint discriminators at the level of high-dimensional correlation functions is also a direction worth exploring.

Low complexity tracking methods

High-frequency subcarriers with high sampling rate requirements can greatly increase the complexity of tracking and impose a large computational burden on the device. Therefore, to enhance the practicality of subcarrier modulated signal high-precision processing in complexity-sensitive terminals, low-complexity tracking methods for high-frequency subcarrier modulated signals are worth further investigation. Most of the existing low-complexity tracking methods focus on reducing the sampling rate by frequency shifting and extraction at Inter-Frequency (IF) or baseband, while neglecting the low-complexity optimization of the tracking structure. In fact, our recent study (Qi et al., 2024b) shows that there is potential for further reduction of tracking complexity for high-frequency subcarrier modulated signals.

Subcarrier ambiguity resolution in complex environments

To achieve unambiguous ranging and positioning, the ambiguity in the subcarrier observation needs to be resolved. The existing techniques have investigated subcarrier ambiguity restoration from both the observation domain and the positioning domain, achieving robust unambiguous ranging and positioning in benign environments. However, in complex environments such as urban canyons, the performance of the existing subcarrier ambiguity resolution techniques has yet to be verified, and their resolution performance may be degraded due to the factors such as fewer satellites, stronger multipaths, and weaker signals. More robust subcarrier ambiguity resolution techniques are yet to be studied in depth.

Moreover, theoretical research on the tracking performance evaluation of multi-dimensional loop structures under the influence of multipath can be carried out, based on which the design of corresponding multipath-resistant algorithms for multi-dimensional loop structures is also a direction worthy of in-depth research.

Positioning methods based on subcarrier observations

In the application of subcarrier observations for positioning, no in-depth study has been carried out on the error modeling of subcarrier observations, such as ionospheric delay, tropospheric delay, and hardware delay, which means that the advantages of subcarrier observations have not yet been fully exploited in SPP, Real-Time Kinematic (RTK), and PPP. In addition, the existing studies only use the high-accuracy subcarrier observations to replace the low-accuracy code observations in positioning. However, both the code observation and the subcarrier observation are independent observations of the distance from the satellite to the receiver. The simultaneous use of code, subcarrier and carrier observations in SPP, RTK and PPP can be expected to achieve better performance than using only two observations.

Conclusions

This Review provides a detailed overview of subcarrier modulated signal processing for next generation GNSS. New progress and remaining challenges with future prospects are also given. The introducing of subcarriers makes the traditional acquisition and tracking methods used for BPSK signals less reliable. Some of the earlier subcarrier modulated signal processing techniques viewed the subcarrier as a harmful component. For example, filtering-based unambiguous processing techniques viewed the subcarrier as a type of interference or channel distortion, and expected to eliminate its effects and return to a BPSK-like signal processing mode. As the high accuracy ranging potential and multipath resistance inherent in subcarriers are recognized, some geometry-based processing methods have begun to take advantage of the sharp main peaks of the subcarrier modulated signal correlation function to enhance ranging accuracy. The emergence of 2D correlation structure-based methods has provided greater design freedom for subcarrier modulated signal processing. The subcarrier tracking loop in the 2D structure provides a new distance-dependent observation in addition to the code pseudorange, which combines the properties of both the code pseudorange and the carrier pseudorange, and has a wavelength and measurement accuracy intermediate between the code pseudorange and the carrier pseudorange. The introduction of subcarrier observation redefines the boundary between carrier frequency and baseband, and becomes a bridge between code and carrier, which provides more possibilities for the way of signal processing and positioning solution. In recent years, as the composite subcarrier modulated signal structure used in the B1 and B2 bands of BDS has been gradually emphasized by academics, the research on subcarrier modulated signal processing has entered a new stage and many related issues worth further investigation.