General design methodology of code multi‐correlator discriminator for GNSS multi‐path mitigation

National Natural Science Foundation of China, Grant/Award Number: China, 61771272 Abstract Multi‐path becomes a main source of positioning errors in new‐generation global navigation satellite systems (GNSS). Code multi‐correlator discriminators (MCD), represented by narrow early‐minus‐late (NEML) and double delta (DD) discriminators which are designed for binary phase offset keying (BPSK) signals, are one of the main multi‐ path mitigation techniques. However, because of the complicated auto‐correlation function and intricate relationships among multiple correlators, it is hard to implement a trial and error approach, which is the conventional design method of discriminator structures, to specifically design code MCD structures for new GNSS signals which have complicated spreading waveforms. In this article, instead of a trial and error approach, a general design methodology for designing code MCD structures based on heuristic optimisation for multi‐path mitigation is proposed. The proposed method can specifically design code MCD structures for signals with various modulations and receivers with different bandwidths, indicating the adaptability of the proposed method. Multi‐path mitigation performances of such designed structures are better than those of traditional code discriminator structures with strong tracking robustness and slight degradation of thermal noise performance. Designed code MCD structures are also insensitive to multi‐path relative amplitudes and bandwidths of receivers, ensuring the practicality of the designed code MCD structures.


| INTRODUCTION
Global navigation satellite systems (GNSS) provide positioning, navigation and timing service. To promote the accuracy of positioning, on one hand, several new signals are employed in the new generation GNSS. Compared with the binary phase shift keying (BPSK) signal, which was first used in GNSS, binary offset carrier (BOC) [1] signals and multiplexed BOC (MBOC) [2] signals have more complicated autocorrelation functions (ACFs) and greater potential to improve the accuracy of positioning. On the other hand, with the development of various GNSS receiving and processing techniques, such as high precision differential techniques, performance of positioning is significantly improved. Nevertheless, there are still several non-negligible factors that reduce positioning accuracy, such as multi-path, which has become one of the main sources of positioning errors.
To alleviate the effects of multi-path, various multi-path mitigation techniques have been proposed that can be divided into three classes. The first class is the antenna-based multi-path cancellation technique, such as the choke ring [3,4], multi-pathlimiting antenna (MLA) [5,6] and phased-array antennas [7], which mitigate multi-path signals before they enter the receiver. The second class is the multi-path elimination technique based on multi-path parameter estimation, such as multi-path estimation delay lock loop (MEDLL) [8,9]. MEDLL can effectively mitigate multi-path by estimating multi-path parameters, such as the number of multi-paths, relative delay and amplitudes. However, it is hard to mitigate multi-path in real time owing to the large calculation burden, especially when the number of multi-path is unknown. In addition to MEDLL, there is also a Bayesian method implemented by particle filters to estimate multi-path parameters [10]. The third class is the multi-path suppression technique based on discriminator structure design. Several types of techniques within this class include: the early-minus-late (EML) structure, the narrow earlyminus-late (NEML) [11] structure and the enhanced traditional code discriminator structure with better multi-path mitigation performance, such as double delta (DD) correlator structures [12] and strobe correlator structures [13], which can be collectively termed as high resolution correlator (HRC) structures. HRC structures have better multi-path mitigation performance than any other traditional code discriminator structures [14].
The code multi-correlator discriminator (MCD) structure in a delay lock loop (DLL) is a general discriminator structure, of which traditional code discriminator structures are specific implementations. Since the code MCD structure has great design flexibility to fulfil diverse design requirements and potential to reach better multi-path mitigation performance, it is widely used in various applications. A specific designed code MCD structure can track the high-order BOC signal without ambiguity [15]. The code MCD structures designed by the method based on quadratic programming have better characteristics of the discriminator's output, which means that the tracking loop is less susceptible to noise and interference [16]. Furthermore, MCD structures can also be used for spoofing signal suppression [17]. According to these applications in various fields, code MCD structures have a very broad development prospect, implying that code MCD structures have great potential to improve the multi-path mitigation performance. Therefore, this article mainly focusses on the design of code MCD structures.
In the field of multi-path mitigation, the conventional method of designing code discriminator structures is the trial and error approach and traditional code discriminator structures, such as NEML structures and DD structures, are both specifically designed for BPSK signals by this method. On one hand, since new signals such as BOC signals and MBOC signals, have much more complicated ACFs than BPSK signals, it is difficult for the trial and error approach to specifically design code discriminator structures, and the existing traditional code discriminator structures are obviously suboptimal for these signals, causing that the positioning accuracy of new signals can be significantly influenced by multi-path. On the other hand, although code MCD structures have great potential to mitigate multi-path, the intricate relationships among multiple correlators make it harder to design code MCD structures by trial and error.
Corresponding to the above issues in designing code MCD structures, this article proposes a general design methodology of code MCD structures for multi-path mitigation based on heuristic optimisation. The proposed method can specifically design code MCD structures for signals with various modulations and receivers with different bandwidths, which illustrates the adaptability of the proposed method. Such code MCD structures have better multi-path mitigation performance than traditional code discriminator structures with strong tracking robustness and slight sacrifice of thermal noise performance. The proposed method is also insensitive to multi-path relative amplitudes and bandwidths of receivers. On one hand, the code MCD structure designed with one specific multi-path relative amplitude can greatly mitigate multi-path signals with other multi-path relative amplitudes. On the other hand, the code MCD structure designed with one specific bandwidth still has better anti-multi-path performance than that of traditional code discriminator structures when applied to receivers with other bandwidths.
This article is organised as follows. Section 2 formalises the satellite baseband spreading signals and analyses the ACF and power spectral density (PSD) characteristics of signals. Then the code MCD model is established to show the relationship among the multiple correlators. Section 3 introduces the multipath error optimisation model whose goal is to minimise the mean multi-path errors and to propose the general design methodology of code MCD structures for multi-path mitigation. In Section 4, the adaptability, insensitivity, thermal noise and tracking robustness analysis of the proposed design method are represented by case studies. Finally, conclusions are discussed in Section 5.

| SATELLITE SIGNALS AND CODE MCD MODEL
This section first models satellite baseband spreading signals and presents the ACF and PSD characteristics of signals. Then the code multi-correlator discriminator model is established to illustrate relationships between multiple correlators in code MCD structures.

| Satellite baseband spreading Signal
GNSS signals adopt direct sequence spread spectrum (DSSS) technology. DSSS signal, which can also be called as spreading signal, is utilised to modulate navigation data signals. Since the spreading sequence, alternatively termed as pseudo random noise (PRN) code, has much higher code rate than navigation data sequence, frequent phase reversal of spreading sequence can be used for precise ranging. According to the great orthogonal cross-correlation characteristics of spreading sequence, which is the basis of code division multiple access (CDMA) system, satellites can transmit multiple signals on the same carrier frequency, and GNSS receivers can separate different satellite signals by utilising the cross-correlation characteristic of spreading sequence [18,19].
A general form of the baseband spreading signal g(t) can be expressed as where, {c i } is a periodic spreading sequence with the code rate f c = n � f 0 , where f 0 = 1.023 MHz is the reference frequency in GNSS, T c = 1/f c is the length of one chip, and p(t) is a spreading chip waveform, which takes non-zero values for 0 ≤ t < T c and takes zero values elsewhere. It can be further expressed as where, m = {m j } is the modulation sequence of the spreading chip waveform with rate f s = m � f 0 , ψ(t) is the waveform with the rectangular pulse which takes the value one for 0 ≤ t < T s = 1/(2f s ), and zero elsewhere and K is the length of the modulation sequence. To align the spreading sequence with the modulation sequence, the ratio K = T c /T s should be a positive integer. The modulation sequence of BPSK(1) signals, sine-phased BOC(1,1) signals and cosine-phased BOC(1,1) signals are respectively. The ACF of baseband spreading signals g(t) can be calculated as where, T is the integral length. Since signals received by front-end antennas are multiplied with the local carrier and then filtered by the front-end lowpass filter (LPF), the obtained baseband spreading signals are always bandlimited. In the finite bandwidth condition, with the impulse response function of LPF h BW (t), filtered baseband spreading signals g BW (t) should be written as where ⊗ is the convolution operator. The single-side bandwidth is denoted as BW. The ACF of bandlimited signals, denoted as R BW (τ), can be calculated by substituting g(t) for g BW (t) in Equation (4). Figure 1 shows the ACFs of four kinds of modulated signals, which are BPSK(1), BOC(1,1), BOC cos (15,2.5) and CBOC(6,1,1/11,+), in infinite and finite bandwidth. Except for ACF of signals, the normalised PSD of signals is another critical characteristic, which can be calculated as where Q BOC(1,1) ( f ) and Q BOC(6,1) (f ) are the FT of BOC(1,1) and BOC(6,1) signals, respectively. Figure 2 shows the normalised PSDs of BPSK(1), BOC(1,1) and BOC cos (15,2.5) and CBOC(6,1,1/11,+) signals, which will be utilised in thermal noise analysis in Section 4.

| Code MCD model
Code multi-correlator discriminators can be divided into two classes: the non-coherent discriminator and the coherent discriminator. Compared with coherent discriminators, noncoherent discriminators have greater mean code tracking errors when the carrier of signals is steadily tracked by a phase lock loop (PLL) [18,20,21]. This article focusses on the design of coherent discriminator structures to fully utilise their advantages. Although coherent discriminators are invalid when the carrier phase is out of lock, receivers can take the strategy that non-coherent discriminators are first used to ensure locking of the carrier phase, then coherent discriminators are applied to reach better performance in the estimation of code phase and multi-path mitigation. Figure 3 is the DLL structures with multiple correlators discriminators. The output of a general coherent code MCD can be written as where, N is the number of correlators, w ¼ w 1 ; …; w N ½ � is the weights vector of correlators, d ¼ d 1 ; …; d N ½ � is the code delay vector of correlators, which can also be termed as the location vector, w i and d i are the weight and location of the ith correlator, respectively.
Traditional discriminator structures, such as EML, NEML and DD structures, are special cases of code MCD structures. EML and NEML structures in DLL usually utilise two correlators: an early correlator and a late correlator.   (d) show, can cause the zero-crossing point of the S-curve tracked by DLL, which is the closest to zero chip, to deviate from the zero-chip point by contaminating the ACF of signals. Shifted distance between the zero-chip point and the zero-crossing point is the code phase error caused by multi-path signals, which cannot be eliminated by DLL. Compared with EML structures, the code multi-path errors of NEML and DD structures are smaller, illustrating the better anti-multi-path performance.

| DESIGN METHODOLOGY
This section establishes the multi-path error optimisation model. The design methodology of code MCD structures for multi-path mitigation will then be introduced.

| Multi-path error optimisation model
Signals received by antennas consist of two parts: the direct path signal and the multi-path signals. Compared with the direct path signal, multi-path signals have a longer travel path, decay in signal power and relative shifted phase by reflecting and diffracting. With the presence of multi-path signals, S-curve of code MCD structures can be written as where, α is the multi-path relative amplitude, ϕ is the multipath relative phase and τ m is the multi-path relative delay. Since the received signal is contaminated by multi-path signals, the symmetry of the signal's ACF is broken and the code phase  tracked by DLL has multi-path errors, which are denoted as τ d and can be expressed as Multi-path error τ d is determined by multi-path relative delay τ m and multi-path relative phase ϕ with the given multipath relative amplitude α and code MCD structure parameters w and d. Multi-path errors under different multi-path relative delays constitute the multi-path error curve. When absolute amplitudes of multi-path signals have maximal values at ϕ = 0°o r 180°, the multi-path error curve reaches its envelope. For any ϕ, multi-path error curves lie in the range of the multi-path error envelope (MEE). Figure 5 exhibits the MEE of traditional code discriminators with EML, NEML and DD structures for BPSK(1) signals. As Figure 5 shows, DD structures have the smallest MEE of these three traditional code discriminator structures, which indicates the best multi-path mitigation performance. The area enclosed by multi-path error envelopes can be defined as the mean multi-path error (MME), and can be written as where MEE + and MEE − are multi-path error curves with ϕ = 0°and ϕ = 180°, respectively. Since weights vector w and location vector d are continuous bounded parameters, a minimal value can be derived from the range of MME values, which indicates that the optimisation goal of this model is to minimise mean multi-path errors. More specifically, to obtain optimal code MCD structure which has the best multi-path mitigation performance, code MCD structure parameters w and d should be optimised. According to the symmetry of the signal's ACF, w and d have the inner relationship: Therefore, a general multi-path error optimisation model can be concluded as where the multi-path error optimisation model is optimised in one specific multi-path relative amplitude condition. Although multi-path signals with one specific multi-path relative amplitude cannot represent the complicated multi-path signals received from the real-world environment, studies derived from it can provide useful diagnostic insights. The solutions of w and d obtained by optimising the multi-path error optimisation model can construct code MCD structures that have the best anti-multi-path performance.

| Proposed design methodology of code MCD structures
The multi-path error optimisation model with the goal to minimise mean multi-path errors can be effectively solved by optimisation methods. Considering that ACFs of signals are segmented, which can be derived from Figure 1, outputs of discriminators, which are the linear combination of shifted ACFs, should have complicated segmented mathematical expressions. Owing to the segmented characteristics of S-curves, zero-crossing points of S-curves should be piecewise-solved, causing their mathematical expressions to be much more complicated [22]. Therefore, it is difficult to apply derivativebased optimisation methods to the design of code MCD structures, such as gradient descent and Newton's method. Except for derivative-based optimisation methods, the heuristic algorithm, which is inspired by natural creatures, can effectively solve optimisation problems in a swarm intelligence method, such as ant colony optimisation [23], simulated annealing [24,25], genetic algorithm (GA) [26,27] and differential evolution (DE) algorithms [28,29]. Genetic algorithms and differential evolution algorithms imitate the evolution of biological populations by natural rules such as genetic hybridisation, genetic mutation and survival of the fittest. Since binary encoding for genes of populations adopted by GA introduces the 'Hamming cliff' problem [30], which includes the instability of the algorithm; the DE algorithm directly mutates and crosses populations' genes in the real number domain. Because of the DE algorithm's ability to reach global optimal solutions and the strong algorithm robustness, the proposed method adopts it to assist the design of code MCD structures.
Algorithm 1 is the proposed design method of code MCD structures for multi-path mitigation. To implement this code MCD structure design method, hyperparameters, which can be divided into three classes: structure parameters, environment parameters and DE parameters, should be set in advance. The structure parameters, such as bandwidths of receivers and number of correlators (N) in code MCD structures, determine the direction of the optimisation. The environment parameters, such as multi-path relative amplitudes, simulate the multi-path environment in the optimisation. The DE parameters, such as sizes of weights, location populations (N d , N w ), generation sizes (G d , G w ), mutation coefficient (F m ), crossover probability (P c ) and iteration size (G), control the convergence speed and the optimal multi-path mitigation performance.
In this algorithm, w where r 1 , r 2 and r 3 are randomly selected in the range of 1 to N X , X are r 1 th, r 2 th and r 3 th population in X j i . C( ·) is the process of crossover, which simulates the reproduction of populations. A crossover population is generated by Since the two parameters, weights vector w and location vector d should be optimised, the proposed design method optimises w and d in turn. In the ith iteration, the proposed method first optimises w i with the (i − 1)th optimal location population d  can construct the code MCD structure which has the optimal multi-path mitigation performance.

| Implementation of the proposed method
From the perspective of algorithm implementation, the time complexity of the proposed method is O(G ⋅ G X ⋅ N X ⋅ D), which means that the time consumption of the proposed design method is positively correlated with iteration size, generation size, population size and correlator number. Table 1 exhibits the time consumption of the algorithm under different hyperparameter settings, where the computing equipment is Intel(R) Core(TM) i7-4770K CPU @ 3.50 GHz. To reach better multi-path mitigation performance of code MCD structures designed by the proposed method, generation size G X and population size N X should be large enough to ensure that the algorithm can approach its optimal solutions. Under the trade-off between time cost of the proposed design method and the anti-multi-path performance of designed code MCD structures, hyperparameters adopted in case studies in Section 4 are G = 2, G X = 50, N X = 150, D = 4. When utilising the proposed method to design code MCD structures, hyperparameters of algorithm such as the modulation and bandwidth of the signal should be set in advance, which means that the proposed method can design one code MCD structure for one signal with certain modulation and bandwidth. According to the modulation of the signal and bandwidth of the receiver, the ACF of the signal, which can be calculated by Equation (4) and Equation (5), is input into the design method. The outputs of the proposed method should be optimal weights and location vectors of the code MCD structure which is specifically designed for the signal with given modulation and bandwidth of the receiver. Although receivers equipped with designed code MCD structures have more correlators compared with traditional receivers, the implementation methods of early-late correlators in both traditional receivers and receivers with code MCD structures are consistent. [18,19] Since the type and bandwidth of signals cannot vary with time when receiving GNSS signals, the designed code MCD structure can be solidified in the corresponding receiving channel and utilised all the time. Therefore, the design time cost of several hours consumed by the proposed method is acceptable.

| CASE STUDIES
This section illustrates the characteristics of the proposed method by case studies, which will be represented in four aspects: adaptability, sensitivity, thermal noise and tracking robustness analysis.
To analyse these characteristics of the proposed method, several representative GNSS signals and bandwidths of receivers are selected in the following case studies. BPSK(1) signals are widely applied in GNSS, such as global positioning system (GPS) L1 C/A and L2C. BOC(1,1) signals are used in GPS L1C and the BeiDou navigation satellite system (BDS) B1C, as the baseline components, and BOC cos (15,2.5) signals are high order BOC signals which are adopted by Galileo E1-A. Composite BOC (CBOC) signals belong to MBOC signals and the CBOC(6,1,1/11,+) signal is a specific implementation which is used in GALILEO E1-B. Three single-side bandwidths of receivers are also adopted in the case studies: 10f c , 30f c and ∞f c . The ∞f c represents the infinite bandwidth.
Since the DD structure has great multi-path mitigation performances in traditional code discriminator structures, it is adopted as a benchmark for multi-path mitigation performance analysis of code MCD structures. Table 2 lists the hyperparameter settings of case studies in this section and Table 3 is the value range of weights vector w and location vector d. Since w i and d i have the opposite values with w N−i and d N−i , the first half designed parameters of w and d are shown and the remaining half designed parameters can be calculated according to that relationship.

| Adaptability analysis
This section discusses the adaptability of the proposed method. As long as modulation types of the signals and bandwidths of the receivers can be determined, the proposed method can specifically design code MCD structures which will have better multi-path mitigation performance than DD structures in every case.
Case studies for adaptability analysis can be divided into two classes. The first class is code MCD structures designed for various signals with one specific bandwidth of the receiver. The second class is code MCD structures designed for one specific signal with various bandwidths of receivers. Table 4 lists the designed code MCD structures for signals with four modulation types and receivers with three bandwidths, respectively.

| Code MCD structures designed for various signals with one specific bandwidth of the receiver
In this class of cases, the proposed method designs code MCD structures for four signals with one specific bandwidth of the receiver. As Figure 6 (a) shows, when the bandwidth of the receiver is fixed, the mean multi-path error of the designed code MCD structures for each signal is smaller than that of the DD structures, which indicates that the designed code MCD structures have greater capability to mitigate multi-path. Compared with the mean multi-path error of DD structures, it can be seen that the mean multi-path errors relative degradation of code MCD structures designed for BOC cos (15,2.5) signals and CBOC (6,1,1/11,+) signals, is larger than that of code MCD structures designed for BPSK(1) signals and BOC(1,1) signals, indicating that the DD structures cannot effectively degrade mean multipath errors for signals with complicated ACF. Since the DD structure is designed for BPSK signals by trial and error, it is no longer suitable for signals with complicated ACFs, while the proposed design method can systematically design code MCD structures by the optimisation method. These structures have great anti-multi-path performance, especially for signals with complicated ACFs, which indicates the proposed method's adaptability to various signals. Figures 7-9 show the multi-path error envelopes of the designed code MCD structures and DD structures for different modulation types and receiver bandwidths. For most cases, the multi-path error envelopes of code MCD structures are smaller than that of the DD structures, which illustrates a smaller mean multi-path error and better multi-path mitigation performance. In Figure 7 (c), multi-path error envelopes of code MCD structures have an obvious multi-path error peak at τ m = 1 chip, which greatly exceeds the multi-path error envelopes of the DD structures. Since BOC cos (15,2.5) signals are high-order BOC signals, the Gibbs effect of BOC cos (15,2.5) signals is much more severe than that of any other signals when the single-side bandwidth is 10f c . As the bandwidth of the receiver grows wider, the Gibbs effect is attenuated and the multi-path error peak of multi-path error envelopes at τ m = 1 chip will disappear.

| Code MCD structures designed for one specific Signal with various bandwidths of receivers
Since signals received from antennas will be down-converted and filtered to become baseband signals, signals processed in the receiver always have finite bandwidths. In this class of cases, the proposed design method designs code MCD structures for one specific signal with various bandwidths of receivers. As Figure 6 (b) shows, when the modulation of the signal is fixed, the mean multi-path error of code MCD structures designed for each bandwidth is obviously smaller than that of the DD structures, showing that the proposed method has the ability to design code MCD structures for various bandwidths with great multi-path mitigation performance and has adaptability to various bandwidths of receivers.

| Sensitivity analysis
For practical applications, the sensitivity of the proposed method should be analysed. This analysis can be performed for two classes: the sensitivity to multi-path relative amplitudes and to the bandwidths of receivers. According to these cited case studies, the proposed design method is insensitive to both multi-path relative amplitudes and bandwidths of receivers, ensuring the practicality of the proposed method.

| Sensitivity to different mMulti-path relative amplitudes
Corresponding to the multi-path error optimisation model, the proposed method optimises code MCD structure parameters with only one specific multi-path relative amplitude, which is the hyperparameter set before the optimisation. In the realworld complex environment, since multi-path signals received by antennas have various multi-path relative amplitudes, the code MCD structure designed with one specific multi-path relative amplitude is expected to greatly degrade code errors caused by multi-path signals with other multi-path relative amplitudes. As Figure 10 shows, mean multi-path errors of the designed code MCD structures and DD structures are both positively correlated with multi-path relative amplitudes, which means that when the multi-path relative amplitude increases, multi-path errors will become larger. In every bandwidth condition, mean multi-path errors of code MCD structures designed with α = 0.5 are smaller than that of the DD structures when applied to multi-path signals with other multi-path relative amplitudes. This shows that the proposed design method is insensitive to multi-path relative amplitudes.

| Sensitivity to bandwidths of receivers
To put the proposed design method into practice, the bandwidth of the receiver, which is the hyperparameter of the multipath error optimisation model, should be set before the optimisation, that is the proposed method specifically designs one code MCD structure with only one specific bandwidth of the receiver. Since receivers have different bandwidths according F I G U R E 6 Mean multi-path errors of code MCD structures designed for (a) different signals with one specific bandwidth, (b) different bandwidths with one specific signal, in multi-path relative amplitude α = 0.5 condition. BOC, binary offset carrier; BPSK, binary phase shift keying; DD, double delta; MCD, multi-correlator discriminator to their various applications, it will be complicated to design various code MCD structures for these receivers by the proposed method. Therefore, the code MCD structure designed with one specific bandwidth is expected to be able to greatly mitigate multi-path when applied to receivers with other bandwidths.
In case studies shown in this section, the proposed method designs code MCD structures for receivers with three bandwidths, respectively, and mean multi-path errors of these code MCD structures applied to receivers with other bandwidths are represented in Figure 11 (a), (b) and (c).
As Figure 11 (c) shows, the code MCD structure designed with infinite bandwidth has smaller mean multi-path errors than DD structures when applied to receivers with other bandwidths. This illustrates the insensitivity of the proposed method to bandwidths of receivers. When the bandwidth used in the design becomes narrower, as Figure 11 (a) shows, the code MCD structure designed with single-side bandwidth = 10f c has worse multi-path mitigation performance than the DD structures in several cases. Since the ACFs of signals suffer severe Gibbs effect when the bandwidth is 10f c , the proposed design method is strongly

| Thermal noise analysis
According to adaptive and sensitive analysis discussed above, designed code MCD structures have great anti-multi-path performance and abilities to adapt to various multi-path environments and signals with different bandwidths. Thermal noise performance, which is one of the important standards to evaluate the performance of discriminators, should also be analysed by observing the tracking jitter of DLL with multicorrelator discriminators. Since designed discriminators adopt multi-correlator structures, the two-correlator coherent thermal noise performance formula given by [31] should be extended, which can be written as where, σ 2 is the code delay estimation variance of DLL, which represents the thermal noise performance, B L is the single-side loop bandwidth, C/N 0 is the signal-to-noise ratios (SNR), δ i is the early-minus-late spacing for ith correlator, which equals to 2|d i |. A detailed derivation of Equation (16) is displayed in Appendix. Figure 12 exhibits tracking jitters of discriminators with NEML, DD and code MCD structures for four types of signal modulations, respectively, with single-side bandwidth 10f c , single-side loop bandwidth 1 Hz and integration time 1 ms. As Figure 12 (a), (b) and (d) show, for BPSK(1), BOC(1,1) and CBOC(6,1,1/11,+) signals, the DLL tracking error variance of code MCD structures is slightly degraded when compared with DD structures. As Figure 12 (c) shows, for BOC cos (15,2.5) signals which are high-order BOC signals, designed code MCD structures have the same tracking error variance as DD structures without any degradation in thermal noise performance. Since the multi-path mitigation performance of code MCD structures is greatly enhanced, taking both the influence of thermal noise and multi-path into account, the code phase estimation errors of code MCD structures are much smaller than that of the DD structures, especially for high-order BOC signals, illustrating the superiority of the proposed method when designing for signals with complicated spreading waveforms.

| Tracking robustness analysis
Tracking robustness can be characterised by the linear tracking region of S-curves, which is located around the zero-code phase. When inputs of discriminators fall in this region, discriminators can react accordingly and their outputs can modify the code phase, so that the code phase error can gradually converge to zero code phase error. Therefore, a wider linear tracking region can handle a wider range of discriminators' inputs, which means that the tracking loop has better robustness. Figure 13 exhibits the normalised S-curves of DD and code MCD structures for four distinct signals with BW = 10f c . As Figure 13 shows, the range of linear tracking regions of code MCD structures are the same as that of DD structures, which means that code MCD structures have the similar tracking robustness with DD structures. In Figure 13 (c), for BOC cos (15,2.5) signals, there are many side peaks in the S-curve of DD structures, which may cause that tracking loop to deviate from the linear tracking region around zero code phase and lock on to the linear region of side peaks. However, code MCD structures can effectively restrain side peaks introduced by highorder BOC signals and the probability of false lock will be then reduced. Designed code MCD structures can not only outperform DD structures in multi-path mitigation performance, but also have the similar tracking robustness with DD structures, which indicates the superiority of the proposed method

| Summary
Corresponding to case studies in this section, the proposed method can specifically design code MCD structures for various signals and bandwidths of receivers, which illustrates the adaptability of the proposed design method. The multi-F I G U R E 1 3 S-curves of code MCD structures designed for (a) BPSK(1) signal, (b) BOC(1,1) signal, (c) BOC COS (15,2.5) signal and (d) CBOC(6,1,1/11,+) signal. BPSK, binary phase shift keying; BOC, binary offset carrier; DD, double delta; MCD, multi-correlator discriminator path mitigation performance of the designed code MCD structures is always better than that of the DD structures, especially for signals with complicated ACFs. Although the proposed design method can only design one code MCD structure with one specific multi-path relative amplitude and one specific bandwidth of the receiver, the designed code MCD structures have greater multi-path mitigation performance than DD structures when applied to other multi-path relative amplitudes and bandwidths of receivers. This indicates the insensitivity of the proposed method to these effects. Although thermal noise performance of designed code MCD structures is slightly degraded, the multi-path mitigation performance of code MCD structures is greatly improved and the code tracking loop also has strong robustness.

| CONCLUSION
This article proposes a general methodology to design code MCD structures for multi-path mitigation with the goal to minimise the mean multi-path error. To conquer the problem that the multi-path error optimisation model cannot be effectively solved by derivative-based method, the heuristic algorithm DE is adopted to assist the design of code MCD structures without the problem of derivations.
In the presented case studies, some significant characteristics of the proposed method are developed. First, the proposed method can specifically design code MCD structures for signals with various modulations and receivers with different bandwidths. Compared with traditional code discriminator structures, the designed code MCD structures always have better multi-path mitigation performance, especially for signals with complicated ACFs. This illustrates the adaptability of the proposed method.
Second, the multi-path signal relative amplitude and the bandwidth of the receiver which are the two hyperparameters of the proposed method should be set in advance of optimisation. Code MCD structures designed with one specific multipath relative amplitude can effectively mitigate multi-path signals with other multi-path relative amplitudes, and code MCD structures designed for the receiver with one specific bandwidth can be migrated effectively to receivers with other bandwidths. Therefore, the proposed method is insensitive to multi-path relative amplitudes and bandwidths of receivers, indicating the great practicality of the proposed method.
Third, although thermal noise performance of designed code MCD structures is slightly degraded, multi-path mitigation performance of code MCD structures is promoted with strong tracking robustness, so that the accuracy of positioning can be eventually improved.

APPENDIX
This appendix provides the derivation of thermal noise performance of discriminators with multiple correlators.
Taking the influence of noise into account, the relationship between output of discriminators and code estimated errors can be modelled as where K(⋅) is the relation function, N c is the complex noise introduced by code discriminators. Equation (17) can be linearised around τ = 0: where k c is the slope of discriminator at τ = 0. The variance of estimated code phase b τ can be written as varfb τg ¼ Efb where, Rf⋅g takes the real part. According to the closed-loop delay estimation formulas given in [31], the closed-loop code phase estimation variance can be written as Considering the presence of noise, baseband signals with finite bandwidth can be modelled as: where, A is the magnitude of baseband signals, Δφ is the carrier phase error, and n(t) is the zero-mean Gaussian white noise with power spectrum density N 0 . The multiple correlators discriminator function is where Therefore, EfRfN 2 c gg in Equation (19) can be expanded to where R n (t − u) is the ACF of noise, which can be further expressed as where G n ( f ) is the PSD of noise. Therefore, QI ET AL.
When stably tracking, multiple correlators discriminator function can be rewritten as where ζ is near the zero value. Therefore, the slope of discriminators is From Equation (19), Equation (20), Equation (26) and Equation (28), code phase estimation variance is where C/N 0 = A 2 /N 0 is the SNR of signals.