1. Introduction
The wide use of satellite navigation in various fields has dramatically increased the demand for more precise positioning performance. Several satellite navigation systems continue to improve their service accuracy (Thoelert et al., Reference Thoelert, Steigenberger, Montenbruck and Meurer2019; Yang et al., Reference Yang, Gao, Guo, Mao and Yang2019; Wu et al., Reference Wu, Guo and Zheng2020; Yang et al., Reference Yang, Mao and Sun2020). Most attention in recent years has been paid to the scope of modelling and evaluating the errors that cause degradation of positioning performance, derived from satellite navigation payloads and receivers. For precision positioning, the signal should be broadcast from the satellite with limited distortion and passed through the high fidelity channel of the receiver. Thus, to reduce the degradation of performance, we should model and optimise the channels during design.
Numerous signal quality evaluations for broadcast navigation signals and analyses of the impact of RF (Radio Frequency) channel distortions on signal quality have been performed (Betz, Reference Betz2002; Soellner et al., Reference Soellner, Kohl, Luetke and Erhard2002; Gu et al., Reference Guo, Kou, Zhao, Yu and Chen2014; Quan et al., Reference Quan, Lau, Roberts and Meng2015; Liu et al., Reference Liu, Chen, Yang, Pan and Ran2019). Notably, some correlation domain indicators, such as S-curve bias, slope distortion of S-curve and correlation loss, are usually selected as the targets of optimising the channel or as the indicators for evaluating the quality of the broadcast signal.
The standard deviation of the code tracking error is an important indicator for code tracking and is directly related to the positioning performance. The code tracking loop, also termed delay locked loop (DLL), is implemented by early-minus-later processing. In the processing, the discriminator curve is obtained by further combining the correlation functions calculated between the replica code and the received signal. To attain the desired loop performance, the shape of the correlation function should be as close to the ideal case as possible. Prior results from Betz (Reference Betz2002) show that the standard deviation of code tracking error is proportional to the loop bandwidth, and the loop bandwidth is inversely proportional to the discriminator gain that depends on the S-curve slope. Any distortions in the slope of the S-curve would cause the code tracking loop to deviate from the expected behaviour.
Warner and Last (Reference Warner and Last2009) presented an explanation of the discriminator curve and the sign of the tracking error. Focusing on the non-ideal distortion of the group delay from the ionosphere, Guo et al. (Reference Guo, Kou, Zhao, Yu and Chen2014) performed analysis of the S-curve bias for the Galileo E5 signal (European Union, 2013). He et al. (Reference He, Lu, Guo, Su, Wang and Wang2020) evaluated the S-curve biases of BDS (BeiDou Navigation Satellite System) signals from the sampled data using a 40 m dish antenna. B1I and B3I of BDS were selected as examples to be analysed, considering different parameter configurations of receivers. Some suggestions about choosing proper correlator spacing and bandwidth were provided. Liu et al. (Reference Liu, Yang, Chen, Pan and Ran2020) investigated the phase bias among the signal components modulated on the same centre frequency and revealed that the components having different types of modulation are more sensitive to the phase bias. Pseudo-range bias, which is directly related to S-curve bias, was evaluated using measurements extracted from the receiver (Hauschild and Montenbruck, Reference Hauschild and Montenbruck2016; Gong et al., Reference Gong, Lou, Zheng, Gu, Shi, Liu and Jing2018). These researches reveal that the bias is strongly related to the receiver configuration. The authors (Liu et al. Reference Liu, Chen, Yang, Pan and Ran2019) presented the model of the bias of the S-curve affected by the group delay. By expanding the group delay in the Taylor form (Zhu et al., Reference Zhu, Li, Yon and Zhuang2009) with different order terms, we revealed the mechanism of the bias of S-curve from the group delay. Based on these previous works, we can see that the bias of the S-curve is affected by the receiver's different configurations. It would bring unpredicted biases into the pseudo-range measurements for different types of receivers.
Betz and Kolodziejski (Reference Betz and Kolodziejski2009a, Reference Betz and Kolodziejski2009b) presented the analytic model of the loop performance for DLL. Their results show that the slope of the S-curve is a part of the denominator of the tracking error expression of the code tracking loop. Without including the distortion of group delay, Liu et al. (Reference Liu, Ran, Ke and Hu2012) analysed the slope of the discriminator curve in the case of only considering the band-limited effect. The slope distortion was evaluated using a numerical method (Soellner et al., Reference Soellner, Kohl, Luetke and Erhard2002; Xie et al., Reference Xie, Wang, Li and Meng2018). The slope distortion of the S-curve and other indicators of signal quality were defined and analysed in the presence of group delay distortion by Betz (Reference Betz2002). The results show that the non-ideal RF channel would change the discriminator's slope. Once the group delay's effect is included in the model, the corresponding bias of the S-curve has to be handled properly before performing further calculation of slope distortion. Focusing on this issue, we present a detailed analysis of S-curve slope distortion due to the non-ideal group delay, and analyse slope distortions for different types of group delay containing different order terms. The presented results can be used for improving the channel design for navigation payloads or receivers.
The content of the paper is as follows: first, the model of the slope distortion from the group delay is provided; the theoretical analysis is then validated by the simulation; third, the slope distortions for different cases of group delay are further analysed in detail. Finally, conclusions about the impact of group delay on slope distortion are made.
2. Model of slope distortion
Filters are important in the transmitting channels of payloads and receiving channels of receivers. For the transmitting channel, the filter is responsible for filtering the out-band part of the transmitted signal. And for the receiving channel, the filters are used for blocking the adjacent out-band signal into the channel. In engineering practice, their bandwidths are usually limited, and their amplitude and group delay usually show an inevitable fluctuation within the bandwidth. The control of the amplitude fluctuation is more accessible compared with that for group delay.
In the model, the received signal 
${c^f}(t )$ denotes the signal 
$c(t )$ filtered with the frequency response of 
$h(t )$. The equivalent baseband signal containing 
${c^f}(t )$ is processed in the DLL of the receiver using an early-minus-late power discriminator algorithm. It is noted that the method presented here can also be applied to analyse slope distortion for other types of discriminator algorithms. Then, the discriminator output is given by Liu et al. (Reference Liu, Chen, Yang, Pan and Ran2019)
\begin{equation}{\textrm{D}_{EMLP}}({\tilde{\tau }} )= \textrm{R}\left( {\tilde{\tau } - \frac{d}{2}{T_c}} \right)\ast {\textrm{R}^\ast }\left( {\tilde{\tau } - \frac{d}{2}{T_c}} \right) - \textrm{R}\left( {\tilde{\tau } + \frac{d}{2}{T_c}} \right)\ast {\textrm{R}^\ast }\left( {\tilde{\tau } + \frac{d}{2}{T_c}} \right)\end{equation}
where d is the correlator spacing; 
${T_c}$ is the pseudorandom code rate; and 
$R({\tilde{\tau }} )$ is given by
\begin{equation}R({\tilde{\tau }} )= \sqrt {2P} {e^{\textrm{i}{\varphi _0}}}\int_{ - {B_f}/2}^{{B_f}/2} {P(f )H(f ){e^{\textrm{i}2\mathrm{\pi }f\tilde{\tau }}}\textrm{d}f}\end{equation}
where 
$P(f )$ means the signal's power spectral density; 
${B_f}$ is the bandwidth limited by the front-end of the receiver; 
$H(f )$ denotes the equivalent filter of the channel and can be further written as 
${A_a}(f ){e^{\textrm{i}\ast \theta (f )}}$; and 
${\varphi _0}$ is the initial carrier phase.
According to the theory of tracking loop (Betz, Reference Betz2002; Kaplan and Hegarty, Reference Kaplan and Hegarty2006), 
${D_{EMLP}}({\tilde{\tau }} )$ is zero when the loop locks. Assume that 
${\tau _r}$ is the delay of the maximum peak correlation between the received and replica signals, then 
${D_{EMLP}}({{\tau_r},d} )= 0$ as long as the channel is ideal. Once the non-ideal property is introduced, a correlator spacing related bias 
$\delta \tau $ would exist to satisfy 
${D_{EMLP}}({\tilde{\tau },d} )= 0$, where 
$\tilde{\tau } = {\tau _r} + \delta \tau $. Thus, the slope of the S-curve can be written as
\begin{equation}{K_{\textrm{EMLP}}}(d )= { {{{D^{\prime}}_{EMLP}}({\tilde{\tau },d} )} |_{\tilde{\tau } = {\tau _r} + \delta \tau }}\end{equation}
where 
${\mathrm{D^{\prime}}_{EMLP}}({\tilde{\tau },d} )= \textrm{d}{D_{EMLP}}({\tilde{\tau },d} )/\textrm{d}\tilde{\tau }$. Using a similar method to that in Liu et al. (Reference Liu, Chen, Yang, Pan and Ran2019), 
${\tau _r}$ can be expressed as the terms of the zeroth-order of the group delay plus a constant 
$\Delta \mathrm{\tau }$, that is
\begin{equation}{\tau _r} = {\mathrm{\tau }_g} + \Delta \mathrm{\tau }\end{equation}
where 
$\Delta \mathrm{\tau } ={-} { {{{\mathrm{R^{\prime}}}_s}({\tilde{\tau }} )/{{\mathrm{R^{\prime\prime}}}_s}({\tilde{\tau }} )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}}$ and 
${R_s}({\tilde{\tau }} )= 2P\int_{ - {B_f}/2}^{{B_f}/2} {P(f )H(f ){e^{\textrm{i}2\mathrm{\pi }f\tilde{\tau }}}df} \int_{ - {B_f}/2}^{{B_f}/2} P(f) H(f) {e^{ - \textrm{i}2\mathrm{\pi }f\tilde{\tau }}}df $.
Further 
$\delta \tau (d )$ can be calculated by
\begin{equation}\delta \tau (d )={-} \frac{{{D_{EMLP}}({{\tau_r},d} )}}{{{{D^{\prime}}_{EMLP}}({{\tau_r},d} )}}\end{equation}Substituting Equation (2) into (3) yields
\begin{align} {K_{\textrm{EMLP}}}(d )&= 2P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){A_a}(f ){e^{\textrm{i}\theta (f)}}{e^{\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } -\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f}\notag\\&\quad\times \int_{ - {B_f}/2}^{{B_f}/2} {P(f ){A_a}(f ){e^{ - \textrm{i}\theta(f )}}{e^{ - \textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } - \tfrac{d}{2}{T_c}} \right)}}\textrm{d}f} \notag\\&\quad - 2P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){A_a}(f){e^{ - \textrm{i}\theta (f )}}{e^{ -\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } -\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f}\notag\\&\quad\times \int_{ -{B_f}/2}^{{B_f}/2} {P(f ){A_a}(f ){e^{\textrm{i}\theta (f)}}{e^{\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } -\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f} \notag\\&\quad -2P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){A_a}(f){e^{\textrm{i}\theta (f )}}{e^{\textrm{i}2\mathrm{\pi}f\left( {\tilde{\tau } + \tfrac{d}{2}{T_c}}\right)}}\textrm{d}f}\notag\\&\quad\times \int_{ - {B_f}/2}^{{B_f}/2} {P(f){A_a}(f ){e^{ - \textrm{i}\theta (f )}}{e^{ -\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } +\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f} \notag\\&\quad +2P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){A_a}(f){e^{ - \textrm{i}\theta (f )}}{e^{ -\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } +\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f}\notag\\&\quad\times \int_{ -{B_f}/2}^{{B_f}/2} {P(f ){A_a}(f ){e^{\textrm{i}\theta (f)}}{e^{\textrm{i}2\mathrm{\pi }f\left( {\tilde{\tau } +\tfrac{d}{2}{T_c}} \right)}}\textrm{d}f}\end{align}
where 
$\theta (f )$ is
\begin{equation}\theta (f )= {\theta _0} - 2\mathrm{\pi }{\mathrm{\tau }_g}f - \frac{1}{2}{({2\mathrm{\pi }} )^2}{\mathrm{\tau }_{g1}}{f^2} - \frac{1}{6}{({2\mathrm{\pi }} )^3}{\mathrm{\tau }_{g2}}{f^3} - \frac{1}{{24}}{({2\mathrm{\pi }} )^4}{\mathrm{\tau }_{g3}}{f^4}\end{equation}The slope distortion of the S-curve is defined as the ratio between the distorted and ideal slopes (Betz, Reference Betz2002), and its logarithmic form is given by
\begin{equation}{K_{sd}}(d )= 20\ast \textrm{LOG}\left( {\frac{{{K_{\textrm{EMLP}}}(d )}}{{K_{\textrm{EMLP}}^{\textrm{ideal}}(d )}}} \right)\end{equation} According to the model in the references, the standard deviation of code tracking error is inversely proportional to 
${K_{\textrm{EMLP}}}(d )$ (Betz and Kolodziejski, Reference Betz and Kolodziejski2009a, Reference Betz and Kolodziejski2009b). In practice, the loop design should meet the demand for dynamics, tracking error, etc. The existence of distortion on the slope of the S-curve would cause the tracking loop to deviate from the expected behaviour.
3. Validation
In this section, the theoretical analysis is validated by the simulation. The group delay in the simulation is derived from the model (Liu et al., Reference Liu, Chen, Yang, Pan and Ran2019), and the corresponding curve is plotted in Figure 1. The coefficients of different order terms in the model are given as follows: 
${\mathrm{\tau }_g} = 23 \cdot 5\textrm{E} - 9\textrm{\; }(\textrm{s} )$; 
${\mathrm{\tau }_{g1}} = 1 \cdot 8\textrm{E} - 17\textrm{\; }({{\textrm{s}^2}/\textrm{rad}} );$ 
${\mathrm{\tau }_{g2}} = 1 \cdot 75\textrm{E} - 24\textrm{\; }({{\textrm{s}^3}/\textrm{ra}{\textrm{d}^2}} )$; and 
${\mathrm{\tau }_{g3}} = 4 \cdot 2\textrm{E} - 33({{\textrm{s}^4}/\textrm{ra}{\textrm{d}^3}} )$.
Figure 1. Group delay used in the analysis and simulation
Here, we select BPSK(10) as an example to be analysed. The GPS L5 signal, the BDS B3I signal, and the Galileo E5a signal all adopt the modulation of BPSK(10) or the modulation that can be treated as BPSK(10) (European Union, 2013; China Satellite Navigation Office, 2018; Global Positioning Systems Directorate, 2018). Its power spectral density is given by (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):
\begin{equation}{P_{BPSK}}(f )= {T_c}\sin{\textrm{c}^2}({\mathrm{\pi }f{T_c}} )\end{equation}
where 
${T_c}$ is defined as 1/10⋅23E6 (s). In the simulation, BPSK(10) is generated and filtered by the model shown in Figure 1. The generated filtered signal is further correlated with the replica codes for different correlator spacings; thus, the S-curves are obtained. By using the numerical method, the biases of the S-curves and the corresponding slopes for each of the given correlator spacings are calculated.
Figure 2 presents the comparisons of the theoretical and simulation results for different correlator spacings, and the results indicate that the theoretical results are in accord with the simulation results.
Figure 2. Comparisons of the theoretical and simulation results
In Section 2, the original derived model includes the amplitude distortion of the channel. Here, to illuminate its impact on the slope, we add an amplitude distortion, which varies from −1 dB to 1 dB in the bandwidth linearly, to the filter model. The corresponding results in Figure 3 indicate that the addition of the amplitude distortion to the model described by the group delay in Figure 1 causes limited impacts on the slope distortion compared with the case of only existing group delay distortion. For clarity, we only consider the group delay in the model when analysing the slope distortion in the following sections, and then the frequency response of the filter becomes
\begin{equation}H(f )= {\textrm{e}^{ - \mathrm{i\ast \theta }(\textrm{f} )}}\end{equation}
Figure 3. Comparison of slope distortion between the cases with or without amplitude distortion
4. Further analyses
4.1 Case A
In this case, 
$\theta (f )= {\theta _0} - 2\mathrm{\pi }{\mathrm{\tau }_g}f$, and it is also the ideal case. The group delay is constantly equal to 
${\mathrm{\tau }_g}$, and the filter delays all frequencies with the same amount. The signal passing through the filter is delayed by 
${\mathrm{\tau }_g}$, thus 
$\tilde{\tau }$ is equal to 
${\mathrm{\tau }_g}$. Applying 
$\tilde{\tau } = {\mathrm{\tau }_g}$ to Equation (6) yields
\begin{align}{ {{K_{\textrm{EMLP}}}(d )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}} &= 4P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){e^{ - \textrm{i}d\pi f}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f ){e^{\textrm{i}d\pi f{T_c}}}\textrm{d}f} \notag\\ &\quad - 4P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){e^{\textrm{i}d\pi f}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f ){e^{ - \textrm{i}d\pi f}}\textrm{d}f} \end{align}Further simplifying Equation (11) yields
\begin{equation}{ {{K_{\textrm{EMLP}}}(d )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}} = 8P\mathrm{\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({d\mathrm{\pi }f{T_c}} )\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({d\mathrm{\pi }f{T_c}} )\textrm{d}f}\end{equation} In this case, 
$K_{\textrm{EMLP}}^{\textrm{ideal}}(d )= {K_{\textrm{EMLP}}}(d )$ and 
${K_{sd}}(d )= 0$.
4.2 Case B
In the case, the group delay contains the zeroth-order 
${\mathrm{\tau }_g}$ and the first-order 
${\mathrm{\tau }_{g1}}$ terms, then 
$\theta (f )$ is written as
\begin{equation}\theta (f )= {\theta _0} - 2\mathrm{\pi }{\mathrm{\tau }_g}f - \frac{1}{2}{({2\mathrm{\pi }} )^2}{\mathrm{\tau }_{g1}}{f^2}\end{equation} For the properties of the odd-order terms (Liu et al., Reference Liu, Chen, Yang, Pan and Ran2019), the delay 
$\tilde{\tau }$ due to the filter is equal to 
${\mathrm{\tau }_g}$. Then, 
${ {{K_{\textrm{EMLP}}}(d )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}}$ can be written as
\begin{align} { {{K_{\textrm{EMLP}}}(d )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}} &= 2P\mathrm{i\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f ){e^{ - \textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{ - \mathrm{i\pi}fd{T_c}}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f){e^{\textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{\mathrm{i\pi}fd{T_c}}}\textrm{d}f} \notag \\ &\quad - 2P\mathrm{i\pi }\int_{ -{B_f}/2}^{{B_f}/2} {fP(f){e^{\textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{\mathrm{i\pi}fd{T_c}}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f){e^{ - \textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{ - \mathrm{i\pi}fd{T_c}}}\textrm{d}f} \notag \\ &\quad - 2P\mathrm{i\pi }\int_{ -{B_f}/2}^{{B_f}/2} {fP(f ){e^{ -\textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{\mathrm{i\pi}fd{T_c}}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f){e^{\textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{ - \mathrm{i\pi}fd{T_c}}}\textrm{d}f} \notag\\ &\quad + 2P\mathrm{i\pi }\int_{ -{B_f}/2}^{{B_f}/2} {fP(f){e^{\textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{ - \mathrm{i\pi}fd{T_c}}}\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f){e^{ - \textrm{i}\frac{1}{2}{{({2\mathrm{\pi }})}^2}{\mathrm{\tau }_{g1}}{f^2}}}{e^{\mathrm{i\pi}fd{T_c}}}\textrm{d}f}\end{align}Simplifying and rearranging Equation (14) yields
\begin{equation}{ {{K_{\textrm{EMLP}}}(d )} |_{\tilde{\tau } = {\mathrm{\tau }_g}}} = 4P\mathrm{\pi }\left[ \begin{matrix} \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\cos({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ + \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {P(f )\sin({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \end{matrix} \right]\end{equation}Substituting Equations (15) and (11) into (8) yields
\begin{equation}{K_{sd}}(d )= \frac{{\left[ \begin{array}{l} \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\cos({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ + \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\sin({2{\mathrm{\pi }^2}{\mathrm{\tau }_{g1}}{f^2}} )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \end{array} \right]}}{{2\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} }}\end{equation} From the results shown in Equation (16), we can determine that the slope distortion is related to the first-order term, unlike the case for the bias. Since the group delay containing 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g1}}$ does not change the value of 
${D_{EMLP}}({{\mathrm{\tau }_g}} )$ and does change the derivative of 
${D_{EMLP}}({{\mathrm{\tau }_g}} )$, it has different impacts on the bias and slope of the S-curve.
Detailed analysis with the coefficients of 
${\mathrm{\tau }_g} = 23 \cdot 5\textrm{E} - 9\textrm{\; }(\textrm{s} )$ and 
${\mathrm{\tau }_{g1}} = 1 \cdot 8\textrm{E} - 17\textrm{\; }({{\textrm{s}^2}/\textrm{rad}} )$ is performed in this case. The variation of group delay within the bandwidth in Figure 4 is 5 ns. Using Equation (16) with the given parameters of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g1}}$, we find that the slope distortion is less than 0⋅02 dB, as shown in Figure 5.
Figure 4. Group delay within the bandwidth for the case of existing 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g1}}$ terms
Figure 5. Slope distortion versus correlator spacings for the case of existing 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g1}}$ terms
To further show the effect of the first-order term on the slope distortion in a more comparable case, we increase 
${\mathrm{\tau }_{g1}}$ to 
$5 \cdot 4\textrm{E} - 17\textrm{\; }({{\textrm{s}^2}/\textrm{rad}} )$ to obtain the similar amount of the group delay fluctuation within the bandwidth caused by the second-order term in Section 4.3. As illustrated in Figures 6 and 7, the fluctuation of the group delay within the bandwidth enlarges to 15 ns, and the slope distortion for the worse case also increases to 0⋅15 dB.
Figure 6. Group delay within the bandwidth for the case of enlarged 
${\mathrm{\tau }_{g1}}$ term
Figure 7. Slope distortion versus correlator spacings for the case of enlarged 
${\mathrm{\tau }_{g1}}$ term
4.3 Case C
For the case of the group delay composed of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g2}}$ terms, 
$\theta (f )$ becomes
\begin{equation}\theta (f )= {\theta _0} - 2\mathrm{\pi }{\mathrm{\tau }_g}f - \frac{1}{6}{({2\mathrm{\pi }} )^3}{\mathrm{\tau }_{g2}}{f^3}\end{equation} Due to the existence of the second-order terms, extra biases have been added to 
${\mathrm{\tau }_g}$. Then, 
$\tilde{\tau }$ is given by
\begin{equation}\tilde{\tau } = {\mathrm{\tau }_g} + \Delta \mathrm{\tau } + \delta \tau \end{equation} The calculation of 
$\Delta \mathrm{\tau }$ and 
$\delta \tau \; $ can be implemented by referring to Equations (4) and (5) in Section 2. Substituting Equation (18) into (3) and further simplifying the equation yields
\begin{align}{K_{\textrm{EMLP}}}(d )&= 4P\mathrm{\pi }\int_{ -{B_f}/2}^{{B_f}/2} {fP(f )\sin\left({\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau}_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta\tau - \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \notag\\&\quad \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos\left({\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau}_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta\tau - \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \notag\\&\quad -4P\mathrm{\pi }\int_{ - {B_f}/2}^{{B_f}/2} {fP(f)\sin\left( {\dfrac{1}{6}{{({2\mathrm{\pi }})}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left({\Delta \tau + \delta \tau + \dfrac{d}{2}{T_c}} \right)}\right)\textrm{d}f} \notag\\ &\quad \times \int_{ - {B_f}/2}^{{B_f}/2}{P(f )\cos\left( {\dfrac{1}{6}{{({2\mathrm{\pi }})}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left({\Delta \tau + \delta \tau + \dfrac{d}{2}{T_c}} \right)}\right)\textrm{d}f}\end{align} Thus, 
${K_{sd}}(d )$ is calculated as
\begin{equation}{K_{sd}}(d )= \frac{\begin{array}{l} \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin\left( {\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta \tau - \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos\left( {\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta \tau - \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \\ - \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin\left( {\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta \tau + \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos\left( {\dfrac{1}{6}{{({2\mathrm{\pi }} )}^3}{\mathrm{\tau }_{g2}}{f^3} - 2\mathrm{\pi }f\left( {\Delta \tau + \delta \tau + \dfrac{d}{2}{T_c}} \right)} \right)\textrm{d}f} \end{array}}{{2\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({d\mathrm{\pi }f{T_c}} )\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({d\mathrm{\pi }f{T_c}} )\textrm{d}f} }}\end{equation} Figure 8 presents the plot of the group delay determined by 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g2}}$ terms with the coefficients of 
${\mathrm{\tau }_g} = 23 \cdot 5\textrm{E} - 9\textrm{\; }(\textrm{s} )$ and 
${\mathrm{\tau }_{g2}} = 1 \cdot 75\textrm{E} - 24\textrm{\; }({{\textrm{s}^3}/\textrm{ra}{\textrm{d}^2}} )$. The results in Figure 8 show that the variation of the group delay is 15 ns, and the maximum of the corresponding slope distortion in Figure 9 is 0⋅23 dB. The result is slightly larger than the slope distortion with the enlarged first-order terms case in Section 4.2.
Figure 8. Group delay within the bandwidth in the presence of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g2}}$ terms
Figure 9. Slope distortion versus correlator spacings in the presence 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g1}}$ terms
4.4 Case D
For this case, the group delay contains the zeroth-order and the third-order terms, and 
$\theta (f )= {\theta _0} - 2\mathrm{\pi }{\mathrm{\tau }_g}f - \frac{1}{{24}}{({2\mathrm{\pi }} )^4}{\mathrm{\tau }_{g3}}{f^4}$. For the characteristic of the odd-order term of group delay (Liu et al., Reference Liu, Chen, Yang, Pan and Ran2019), we can find that 
$\tilde{\tau } = {\mathrm{\tau }_g}$. Then, the simplified expression can be expressed as
\begin{equation}{K_{\textrm{EMLP}}}(d)= 4P\mathrm{\pi }\left[ \begin{matrix} \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\cos\left({\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau}_{g3}}{f^4}} \right)\sin({\mathrm{\pi }fd{T_c}})\textrm{d}f} \\ \times \displaystyle\int_{ - {B_f}/2}^{{B_f}/2} {P(f)\cos\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }})}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\cos({\mathrm{\pi}fd{T_c}} )\textrm{d}f} \\ + \displaystyle\int_{ - {B_f}/2}^{{B_f}/2}{fP(f )\sin\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }})}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\sin({\mathrm{\pi}fd{T_c}} )\textrm{d}f} \\ \times \displaystyle\int_{ -{B_f}/2}^{{B_f}/2} {P(f )\sin\left({\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau}_{g3}}{f^4}} \right)\cos({\mathrm{\pi }fd{T_c}})\textrm{d}f} \end{matrix}\right]\end{equation}Substituting Equation (21) into (8) yields
\begin{equation}{K_{sd}}(d )= \frac{{\left[ \begin{array}{c} \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\cos\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ + \int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \\ \times \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\sin\left( {\dfrac{1}{{24}}{{({2\mathrm{\pi }} )}^4}{\mathrm{\tau }_{g3}}{f^4}} \right)\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \end{array} \right]}}{{2\int_{ - {B_f}/2}^{{B_f}/2} {fP(f )\sin({\mathrm{\pi }fd{T_c}} )\textrm{d}f} \int_{ - {B_f}/2}^{{B_f}/2} {P(f )\cos({\mathrm{\pi }fd{T_c}} )\textrm{d}f} }}\end{equation} Figure 10 shows the curve of the group delay with the coefficient of 
${\mathrm{\tau }_{g3}} = 4 \cdot 2\textrm{E} - 33\textrm{\; }({{\textrm{s}^4}/\textrm{ra}{\textrm{d}^3}} )$. The variation of the group delay in this figure is 3 ns within the bandwidth, and the corresponding slope distortion is negligible, as shown in Figure 11.
Figure 10. Group delay within the bandwidth in the presence of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g3}}$ terms
Figure 11. Slope distortion versus correlator spacings in the presence 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g3}}$ terms
For comparisons among different cases in a similar range of variations of group delay, the third-order term 
${\mathrm{\tau }_{g3}}$ is increased to 
$2 \cdot 94\textrm{E} - 32\textrm{\; }({{\textrm{s}^4}/\textrm{ra}{\textrm{d}^3}} )$. We can see that the fluctuation of the group delay in Figure 12 reaches 25 ns, and the slope distortion for this case in Figure 13 is less than 0⋅05 dB. Compared with the similar cases in Section 4.2 and Section 4.3, we find that the slope distortion affected by the terms of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g3}}$ is much less than the other cases. This phenomenon is consistent with the shape of the group delay curve within the bandwidth. For the case of containing the terms of 
${\mathrm{\tau }_g}$ and 
${\mathrm{\tau }_{g3}}$, the variation of the group delay as frequency limited to [−10 10] MHz, which corresponds to the main lode spectrum of the signal, is still limited. Due to the impact of the third power of frequency, more considerable variations of group delay concentrate on the margin of the frequency bandwidth.
Figure 12. Group delay within the bandwidth for the case of enlarged 
${\mathrm{\tau }_{g3}}$ term
Figure 13. Slope distortion versus correlator spacings for the case of enlarged 
${\mathrm{\tau }_{g3}}$ term
4.5 Case E
To compare the combined effect of more group delay terms on slope distortion, we calculate the slope distortions for the cases of (
${\mathrm{\tau }_g}$, 
${\mathrm{\tau }_{g2}}$), (
${\mathrm{\tau }_g}$, 
${\mathrm{\tau }_{g1}}$, 
${\mathrm{\tau }_{g2}}$), and (
${\mathrm{\tau }_g}$, 
${\mathrm{\tau }_{g1}}$, 
${\mathrm{\tau }_{g2}}$, 
${\mathrm{\tau }_{g3}}$). These coefficients are derived from the model in Section 2.
By using Equation (8), the slope distortions for the three cases are calculated and plotted in Figure 14. The results indicate that the addition of the term of 
${\mathrm{\tau }_{g1}}$ to the case shown in Section 4.3 causes little difference in the slope distortions. This phenomenon is because the slope distortion from the first-order term is very small, as shown in Section 4.2. Meanwhile, for the third-order term's limited impact on the slope distortion shown in Section 4.4, the addition of the term of 
${\mathrm{\tau }_{g3}}$ to the case containing the terms of 
${\mathrm{\tau }_g}$, 
${\mathrm{\tau }_{g1}}$, and 
${\mathrm{\tau }_{g2}}$ also brings little extra slope distortion.
Figure 14. Comparisons of slope distortions for different types of group delay composed of different order terms
4.6 Discussion
The optimisation of the RF channel for payloads or receivers to minimise the distortions of group delay is always an important issue. The shape of the group delay determined by different odd-order and even-order terms is related to the choice of the filter type, order and bandwidth. From the point of view of design, the selections of proper bandwidth and amount of out-of-band rejection of filters in the channel are helpful to decrease the fluctuation of group delay within the bandwidth. Too narrow bandwidth and larger suppression of adjacent out-of-band would cause more variation of group delay. Meanwhile, an equalisation network is another effective way to decrease the fluctuation of the group delay in the channel. The group delay of the equalisation network is designed to compensate for the fluctuation of the group delay of the channel. The cascade connection of the channel with the equalisation network would then show less variation of group delay.
5. Conclusion
The slope distortion of the discriminator curve has profound impacts on the performance of DLL under noise. To reduce the slope distortion due to the group delay from the transmitting or receiving channels, the slope distortions in the presence of several types of group delay terms are modelled and analysed theoretically. Based on the presented results, we find that both the odd- and even-orders terms of group delay have impacts on the slope distortion. The amplitude distortion has limited effect on the S-curve, and the higher odd-order terms of group delay cause fewer slope distortions under the same level of group delay fluctuation. The models and corresponding analysed results present a theoretical method of calculating the slope distortions derived from the group delay.
