Unambiguous combined correlation functions for sine-BOC signal tracking

Abstract

Binary offset carrier (BOC) modulated signal has been widely employed in modern global navigation satellite system (GNSS). However, the multiple peaks auto-correlation function of BOC signal leads to tracking ambiguity problem. For high-order BOC signal, the above problem becomes more severe. We propose an unambiguous tracking method based on combined correlation functions for sine-BOC signal. The main feature of this technique is that two new local reference waveforms are used. Based on the cross-correlation functions between the BOC signal and these two local reference waveforms, unambiguous combined correlation functions without any main positive side-peaks are generated. To achieve unambiguous tracking for sine-BOC signal, we present two non-coherent discriminator functions, denoted as V ₁ and V ₂. The theoretical expressions of code tracking error in thermal noise are derived. Then, the impact of thermal noise and multipath is analyzed with the help of numerical simulations. Results show that V ₁ is easier to implement and has relatively better multipath mitigation performance at the expense of some performance degradation in code tracking error. V ₂ has better code tracking accuracy and provides the most robust code tracking process for high-order sine-BOC signal, but two additional complex correlators are required.

Appendix: Derivation of code tracking error variance

The non-coherent discriminator functions are used in our method, so we can set $\Delta \theta \approx 0$. In this derivation, we take some simplified hypothesis. First, we assume that filter is ideal, i.e.,

$$H\left( f \right) = H^{*} \left( { - f} \right) = \left| {H\left( f \right)} \right|$$

In the case of infinite bandwidth, it is easy to prove that

$$\begin{aligned} R_{1,2} \left( 0 \right) = 0,\;R_{1,2} \left( d \right) = R_{1,2} \left( { - d} \right) = 0 \hfill \\ R_{\text{BOC,ref1}} \left( \varepsilon \right) = R_{\text{BOC,ref2}} \left( \varepsilon \right)\quad {\text{when}}\;\left| \varepsilon \right| \le T_{\text{s}} \hfill \\ \end{aligned}$$

Thus, they are also approximately established in bandwidth limitation case, i.e.,

$$\begin{aligned} & \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref1}} (f){\text{e}}^{ - j\pi fd} {\text{d}}f} \approx \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref1}} (f){\text{e}}^{j\pi fd} {\text{d}}f} \\ & \approx \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref2}} (f){\text{e}}^{ - j\pi fd} {\text{d}}f} \approx \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref2}} (f){\text{e}}^{j\pi fd} {\text{d}}f} \\ & \approx \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref1}} \left( f \right){ \cos }\left( {\pi fd} \right){\text{d}}f} \\ \end{aligned}$$

(27)

Based on the above assumptions, we next provide a simple derivation of the approximate code tracking error variation.

Discriminator function V ₁

Ignoring the noise terms, V ₁ can be expanded to the following equation,

$$V_{1} \approx 8C\left( {\hat{R}_{\text{BOC,ref1}}^{2} \left( {\Delta \tau - \frac{d}{2}} \right) - \hat{R}_{\text{BOC,ref1}}^{2} \left( {\Delta \tau + \frac{d}{2}} \right)} \right)$$

(28)

Following (16), the CCF is converted to frequency domain, then the discriminator gain at $\Delta \tau = 0$ is

$$K_{1} = \left. {\frac{{{\text{d}}V_{1} }}{{{\text{d}}\Delta \tau }}} \right|_{\Delta \tau = 0} = 64\pi C\left( \begin{aligned} \int_{ - \infty }^{\infty } {H(f)G_{\text{BOC,ref1}} \left( f \right)\cos \left( {\pi fd} \right){\text{d}}f} \cdot \hfill \\ \int_{ - \infty }^{\infty } {f \cdot H(f)G_{\text{BOC,ref1}} \left( f \right)\sin \left( {\pi fd} \right){\text{d}}f} \hfill \\ \end{aligned} \right)$$

(29)

Noted that (29) is the discriminator gain in the case of bandwidth limitation.

Let

$$\begin{aligned} X &= \left( {\left| {{\text{IE}}_{1} } \right| + \left| {{\text{IE}}_{2} } \right| - \left| {{\text{IE}}_{1} - {\text{IE}}_{2} } \right|} \right) \ge 0 \hfill \\ Y &= \left( {\left| {{\text{IL}}_{1} } \right| + \left| {{\text{IL}}_{2} } \right| - \left| {{\text{IL}}_{1} - {\text{IL}}_{2} } \right|} \right) \ge 0 \hfill \\ Z &= \left( {\left| {{\text{QE}}_{1} } \right| + \left| {{\text{QE}}_{2} } \right| - \left| {{\text{QE}}_{1} - {\text{QE}}_{2} } \right|} \right) \ge 0 \hfill \\ W &= \left( {\left| {{\text{QL}}_{1} } \right| + \left| {{\text{QL}}_{2} } \right| - \left| {{\text{QL}}_{1} - {\text{QL}}_{2} } \right|} \right) \ge 0 \hfill \\ \end{aligned}$$

(30)

then $V_{1} = \left( {X^{2} - Y^{2} } \right) + \left( {Z^{2} - W^{2} } \right)$.$E\left[ {V_{1} } \right] = 0$. Now, we discuss the distribution of X. For different values of IE₁ and IE₂, X is calculated by

$$X = \left\{ {\begin{array}{*{20}l} {2{\text{IE}}_{1} ,} \hfill & {0 < {\text{IE}}_{1} < {\text{IE}}_{2} \, } \hfill \\ {{ - }2{\text{IE}}_{1} ,} \hfill & {{\text{IE}}_{2} < {\text{IE}}_{1} < 0\;{\text{or}}} \hfill \\ { 0 ,} \hfill & {{\text{IE}}_{1} < 0 \le {\text{IE}}_{2} } \hfill \\ {2{\text{IE}}_{2} ,} \hfill & {0 < {\text{IE}}_{2} < {\text{IE}}_{1} \;{\text{or}}} \hfill \\ { - }{{\text{IE}}_{2} ,} \hfill & {{\text{IE}}_{1} < {\text{IE}}_{2} < 0\;{\text{or}}} \hfill \\ { 0 ,} \hfill & {{\text{IE}}_{2} < 0 \le {\text{IE}}_{1} } \hfill \\ \end{array} } \right.$$

(31)

According to (20) and (21), IE₁ and IE₂ can be seen as the independent and identically distributed Gaussian random variables

$$N\left( {\sqrt {2C} \hat{R}_{\text{BOC,ref1}} \left( {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}} \right),\frac{{N_{0} }}{{T_{\text{P}} }}\hat{R}_{1,1} \left( 0 \right)} \right)$$

Let $f_{X} \left( x \right)$ be the probability density function of $X$, according to (31), we derive out

$$f_{X} \left( x \right) = \varPi \left( {\sqrt {2C} \hat{R}_{\text{BOC,ref1}} \left( {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}} \right),\frac{{N_{0} }}{{T_{\text{P}} }}\hat{R}_{1,1} \left( 0 \right),x} \right)\quad ,x > 0$$

(32)

where

$$\varPi \left( {\mu ,\sigma^{2} ,x} \right) = \frac{1}{{\sqrt {2\pi } \sigma }}{\text{e}}^{{ - \frac{{\left( {\frac{x}{2} - \mu } \right)^{2} }}{{2\sigma^{2} }}}} Q\left( {\frac{{\frac{x}{2} - \mu }}{\sigma }} \right) + \frac{1}{{\sqrt {2\pi } \sigma }}{\text{e}}^{{ - \frac{{\left( {\frac{x}{2} + \mu } \right)^{2} }}{{2\sigma^{2} }}}} Q\left( {\frac{{\frac{x}{2} + \mu }}{\sigma }} \right)$$

(33)

Similarly, the probability density functions of $Y, \, Z$ and $W$ are

$$\begin{aligned} f_{Y} (y) & = \varPi \left( {\sqrt {2\tilde{C}} \hat{R}_{\text{BOC,ref1}} \left( {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}} \right),\frac{{N_{0} }}{{T_{\text{P}} }}\hat{R}_{1,1} (0),y} \right)\quad ,y > 0 \\ f_{Z} (z) & = \varPi \left( {0,\frac{{N_{0} }}{{T_{\text{P}} }}\hat{R}_{1,1} (0),z} \right),\quad z > 0 \\ f_{W} (w) & = \varPi \left( {0,\frac{{N_{0} }}{{T_{\text{P}} }}\hat{R}_{1,1} (0),w} \right),\quad w > 0 \\ \end{aligned}$$

(34)

Based on a similar assumption with the derivation of PUDLL (Yao et al. 2010), the correlation coefficient between $X^{2}$ and $Y^{2}$ is approximately equal to

$$\rho_{{R_{\text{E}}^{2} R_{\text{L}}^{2} }} \approx {{\hat{R}_{1,1}^{2} \left( d \right)} \mathord{\left/ {\vphantom {{\hat{R}_{1,1}^{2} \left( d \right)} {\hat{R}_{1,1}^{2} (0)}}} \right. \kern-0pt} {\hat{R}_{1,1}^{2} (0)}}$$

(35)

Thus, the variance of discriminator V ₁ is

$$\sigma_{{V_{1} }}^{2} = E\left[ {V_{1}^{2} } \right] \approx 2\left( {1 - \rho_{{R_{\text{E}}^{2} R_{\text{L}}^{2} }} } \right)\left( { \, E\left[ {X^{4} } \right] + E\left[ {Z^{4} } \right] - E^{2} \left[ {X^{2} } \right] - E^{2} \left[ {Z^{2} } \right]} \right)$$

(36)

where

$$E\left[ {X^{i} } \right] = \int_{0}^{\infty } {x^{i} f_{X} (x ) {\text{d}}x} ,\quad E\left[ {Z^{i} } \right] = \int_{0}^{\infty } {z^{i} f_{Z} (z){\text{d}}z}$$

(37)

Substituting (29) and (37) into (26), we can obtain the theoretical code tracking error variance of discriminator function V ₁.

Discriminator function V ₂

Similar to discriminator function V ₁, the gain of discriminator function V ₂ at $\Delta \tau = 0$ is

$$K_{2} = 16\pi C\left( {\left( \begin{aligned} \int_{ - \infty }^{\infty } {H(f)f} G_{\text{BOC,ref1}} (f){ \sin }\left( {\pi fd} \right){\text{d}}f \hfill \\ \int_{ - \infty }^{\infty } {H(f)} G_{\text{B}} (f){ \cos }\left( {\pi fd} \right){\text{d}}f \hfill \\ \end{aligned} \right) + \left( \begin{aligned} \int_{ - \infty }^{\infty } {H(f)} G_{{{\text{BOC,ref}}1}} \left( f \right){ \cos }\left( {\pi fd} \right){\text{d}}f \hfill \\ \int_{ - \infty }^{\infty } {H(f)fG_{\text{B}} (f){ \sin }\left( {\pi fd} \right)} {\text{d}}f \hfill \\ \end{aligned} \right)} \right)$$

(38)

where $G_{\text{B}} (f) = \mathcal{F}\left\{ {R_{\text{B}} (\varepsilon } \right\}$ is the power spectral density of the BOC signal. Equation (38) provides the discriminator gain under bandwidth limitation, which is different with the one in the case of infinite bandwidth. Let

$$\begin{aligned} X_{\text{IE}} & = {\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} + \left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right| \\ X_{\text{IL}} & = {\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} + \left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right| \\ X_{\text{QE}} & = {\text{QE}}_{\text{BOC}} \cdot {\text{QE}}_{1} + \left| {{\text{QE}}_{\text{BOC}} \cdot {\text{QE}}_{2} } \right| \\ X_{\text{QL}} & = {\text{QL}}_{\text{BOC}} \cdot {\text{QL}}_{1} + \left| {{\text{QL}}_{\text{BOC}} \cdot {\text{QL}}_{2} } \right| \\ \end{aligned}$$

(39)

then $V_{2} = \left( {X_{\text{IE}} - X_{\text{IL}} } \right) + \left( {X_{\text{QE}} - X_{\text{QL}} } \right)$. Obviously,

$$\begin{aligned} &E\left[ {V_{2} } \right] = 0 \hfill \\ &E\left[ {\left( {X_{\text{IE}} - X_{\text{IL}} } \right) \cdot \left( {X_{\text{QE}} - X_{\text{QL}} } \right)} \right] = 0 \hfill \\ &\sigma_{{V_{2} }}^{2} = E\left[ {V_{2}^{2} } \right] = E\left[ {\left( {X_{\text{IE}} - X_{\text{IL}} } \right)^{2} } \right] + E\left[ {\left( {X_{\text{QE}} - X_{\text{QL}} } \right)^{2} } \right] \hfill \\ \end{aligned}$$

(40)

The first part of $\sigma_{{V_{2} }}^{2}$ is expanded to

$$\begin{aligned} E\left[ {\left( {X_{\text{IE}} - X_{\text{IL}} } \right)^{2} } \right] & = E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right)^{2} } \right] + E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right)^{2} } \right] + 2E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \cdot \left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|} \right] \\ & \quad + E\left[ {\left( {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} } \right)^{2} } \right] + E\left[ {\left( {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right)^{2} } \right] + 2E\left[ {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} \left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \\ &\quad - 2\left( \begin{aligned} E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \cdot {\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} } \right] + E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \hfill \\ + E\left[ {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} \left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|} \right] + E\left[ {\left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|\left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \hfill \\ \end{aligned} \right) \\ \end{aligned}$$

(41)

Considering that the correlator outputs of I-branch follow the joint Gaussian distribution, we take the following approximation,

$$\begin{aligned} & E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right)^{2} } \right] = E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right)^{2} } \right] = E\left[ {\left( {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} } \right)^{2} } \right] = E\left[ {\left( {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right)^{2} } \right] \hfill \\ & E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] = E\left[ {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} \left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|} \right] \approx E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \cdot \left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|} \right] = E\left[ {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} \left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \hfill \\ & E\left[ {\left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|\left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \approx E\left[ {\left| {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right|} \right]E\left[ {\left| {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right|} \right] \approx \sqrt {E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right)^{2} } \right]E\left[ {\left( {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{2} } \right)^{2} } \right]} = E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{2} } \right)^{2} } \right] \hfill \\ & E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} \cdot {\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} } \right] \approx E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right]E\left[ {{\text{IL}}_{\text{BOC}} \cdot {\text{IL}}_{1} } \right] = \left( {E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right]} \right)^{2} \, \hfill \\ \end{aligned}$$

(42)

then we have

$$E\left[ {\left( {X_{\text{IE}} - X_{\text{IL}} } \right)^{2} } \right] = 2\left( {E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right)^{2} } \right] - \left( {E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right]} \right)^{2} } \right)$$

(43)

Equation (43) can be calculated by the characteristic function of joint Gaussian distribution, namely

$$\begin{aligned} E\left[ {\left( {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right)^{2} } \right] & = \left( {1 + 2\rho^{2} } \right)\sigma^{4} + \left( {\mu_{1}^{2} + \mu_{2}^{2} + 4\mu_{1} \mu_{2} \rho } \right)\sigma^{2} + \mu_{1}^{2} \mu_{2}^{2} \\ \left( {E\left[ {{\text{IE}}_{\text{BOC}} \cdot {\text{IE}}_{1} } \right]} \right)^{2} & = \left( {\mu_{1} \mu_{2} + \rho \sigma^{2} } \right)^{2} \\ E\left[ {\left( {X_{\text{IE}} - X_{\text{IL}} } \right)^{2} } \right] & = 2\left( {\left( {1 + \rho^{2} } \right)\sigma^{4} + \left( {\mu_{1}^{2} + \mu_{2}^{2} + 2\mu_{1} \mu_{2} \rho } \right)\sigma^{2} } \right) \\ \end{aligned}$$

(44)

where $\mu_{1} = \sqrt {2C} \hat{R}_{\text{BOC,ref1}} \left( \frac{d}{2} \right),\quad \, \mu_{2} = \sqrt {2C} \hat{R}_{\text{B}} \left( \frac{d}{2} \right)$, $\rho = \frac{{\hat{R}_{\text{BOC,ref1}} \left( 0 \right)}}{{\hat{R}_{1,1} \left( 0 \right)}}$, $\sigma^{2} = \frac{{N_{0} }}{{T_{\text{p}} }}\hat{R}_{1,1} \left( 0 \right)$.

In a similar way, the second part of $\sigma_{{V_{2} }}^{2}$ is

$$E\left[ {\left( {X_{\text{QE}} - X_{\text{QL}} } \right)^{2} } \right] = 2\left( {1 + \rho^{2} } \right)\sigma^{4}$$

(45)

Using (40), (44) and (45), the variance of discriminator V ₂ output can be approximately expressed as

$$\sigma_{2}^{2} = 2\left( {\left( {2 + 2\rho^{2} } \right)\sigma^{4} + \left( {\mu_{1}^{2} + \mu_{2}^{2} + 2\mu_{1} \mu_{2} \rho } \right)\sigma^{2} } \right)$$

(46)

Substituting (38) and (46) into (26), we obtain the theoretical code tracking error variance of discriminator function V ₂.

The code tracking error variance of DET

The two dimensional correlation function of DET in the case of infinite bandwidth is (Hodgart et al. 2007)

$$\chi \left( {\Delta \tau ,\Delta \tau^{ * } } \right) \approx \text{trc} \left( {\Delta \tau^{ * } } \right)\varLambda \left( {\Delta \tau } \right)$$

(47)

where $\Delta \tau$ is the estimation error of code delay in DLL, $\Delta \tau^{ * }$ is the estimation error of subcarrier delay in SLL, $\varLambda \left( \varepsilon \right) = 1 - \frac{\left| \varepsilon \right|}{{T_{\text{c}} }}$ is the ACF of BPSK signal, and $\text{trc} ()$ is a continuous triangular cosine of periodicity 2T _s. Considering the effect of bandwidth limitation, Eq. (47) is rewritten as

$$\chi \left( {\Delta \tau ,\Delta \tau^{ * } } \right) \approx \text{t} \hat{r}c\left( {\Delta \tau^{ * } } \right)\hat{\varLambda }\left( {\Delta \tau } \right)$$

(48)

where $\hat{\varLambda }\left( \varepsilon \right) = \mathcal{F}^{ - 1} \left\{ {G_{\varLambda } \left( f \right)H\left( f \right)} \right\}$, and $G_{\varLambda } \left( f \right) = \mathcal{F}\left\{ {\varLambda \left( \varepsilon \right)} \right\}$. $\text{t} \hat{r}c\left( \varepsilon \right) = \frac{8}{{\pi^{2} }}{ \cos }\left( {\frac{\pi \varepsilon }{{T_{\text{s}} }}} \right)$.

The code tracking performance of DET is mainly determined by the SLL. The E and L correlator outputs of in-phased and quadrature-phase branches in SLL are

$$\begin{aligned}{\text{IE}} \approx \sqrt {2C}{\text{t}\hat{\hbox{r}}\text{c}}\left( {\Delta \tau^{ * } - {d \left/{\vphantom {d 2}} \right. 2}} \right)\hat{\varLambda}\left( {\Delta \tau } \right){ \cos }\left( {\Delta \theta }\right) + n_{\text{IE}} \\ {\text{IL}} \approx \sqrt {2C}{\text{t}\hat{\hbox{r}}\text{c}}\left( {\Delta \tau^{ * } + {d \left/{\vphantom {d 2}} \right. } 2} \right)\hat{\varLambda}\left( {\Delta \tau } \right){ \cos }\left( {\Delta \theta }\right) + n_{\text{IL}} \hfill \\ {\text{QE}} \approx \sqrt {2C}{\text{t}\hat{\hbox{r}}\text{c}}\left( \Delta \tau^{ * } - d \left/{\vphantom {d 2}} \right. 2 \right)\hat{\varLambda}\left( {\Delta \tau } \right){ \sin }\left( {\Delta \theta }\right) + n_{\text{QE}} \hfill \\ {\text{QL}} \approx \sqrt {2C}{\text{t}\hat{\hbox{r}}\text{c}}\left( {\Delta \tau^{ * } + {d \left/{\vphantom {d 2}} \right. 2}} \right)\hat{\varLambda}\left( {\Delta \tau } \right){ \sin }\left( {\Delta \theta }\right) + n_{\text{QL}} \hfill \\ \end{aligned}$$

(49)

where these noise terms $n_{\text{IE}}$, $n_{\text{IL}}$, $n_{\text{QE}}$ and $n_{\text{QL}}$ satisfy Gaussian process with zero mean and variance $\frac{{N_{0} }}{{T_{\text{p}} }}\text{t} \hat{r}c\left( 0 \right)\hat{\varLambda }\left( 0 \right)$.$E\left[ {n_{\text{IE}} n_{\text{IL}} } \right] = E\left[ {n_{\text{QE}} n_{\text{QL}} } \right] = \frac{{N_{0} }}{{T_{\text{p}} }}\text{t} \hat{r}c\left( 0 \right)\hat{\varLambda }\left( d \right)$.

The discriminator output of SLL is

$$V\left( {\Delta \tau^{ * } } \right) = \left( {{\text{IE}}^{2} + {\text{QE}}^{2} } \right) - \left( {{\text{IL}}^{2} + {\text{QL}}^{2} } \right)$$

(50)

Assume that the code delay has been accurately estimated by the DLL, i.e. $\Delta \tau \approx 0$ and $\Delta \theta \approx 0$, then

$$\begin{aligned} V\left( {\Delta \tau^{ * } } \right) & =2C\hat{\varLambda }^{2} \left( 0 \right)\left({{\text{t}\hat{\hbox{r}}\text{c}}^{2} \left( {\Delta \tau^{ * } - {d{\left/ {\vphantom {d 2}} \right. } 2}} \right) -{\text{t}\hat{\hbox{r}}\text{c}}^{2} \left( {\Delta \tau^{ * } + {d {\left/{\vphantom {d 2}} \right. } 2}} \right)} \right) \\ & \quad + 2\sqrt{2C} \hat{\varLambda }\left( 0 \right)\left({\text{t}\hat{\hbox{r}}c}\left( {\Delta \tau^{ * } - {d {\left/ {\vphantom{d 2}} \right. } 2}} \right)n_{\text{IE}} -{\text{t}\hat{\hbox{r}\text{c}}\left( {\Delta \tau^{ * } + {d{\left/ {\vphantom {d 2}} \right. } 2}} \right)n_{\text{IL}} }\right) \\ &\quad + \left( {n_{\text{IE}}^{2} - n_{\text{IL}}^{2} }\right) + n_{\text{QE}}^{2} - n_{\text{QL}}^{2} \\ \end{aligned}$$

(51)

The discriminator gain at $\Delta \tau^{ * } = 0$ is

$$K = \left. {\frac{{{\text{d}}V}}{{{\text{d}}\Delta \tau^{ * } }}} \right|_{{\Delta \tau^{ * } = 0}} = \frac{{2^{8} C\hat{\varLambda }^{2} \left( 0 \right)}}{{\pi^{3} T_{\text{s}} }}{ \sin }\frac{\pi d}{{T_{\text{s}} }}$$

(52)

Due to $\Delta \tau^{ * } \approx 0$, the noise part in (52) is

$$n \approx 2\sqrt {2C} \hat{\varLambda }\left( 0\right){\text{t}\hat{\hbox{r}}\text{c}}\left( {{d \left/ {\vphantom {d 2}}\right. 2}} \right)\left( {n_{\text{IE}} - n_{\text{IL}} } \right)+ \left( {n_{\text{IE}}^{2} - n_{\text{IL}}^{2} } \right) +n_{\text{QE}}^{2} - n_{\text{QL}}^{2}$$

(53)

The variance of discriminator function V is given by

$$\sigma_{V}^{2} = \frac{{2^{8} CN_{0} }}{{\pi^{4} T_{\text{p}} }}{ \sin }^{2} \left( {\frac{{\pi d_{{T_{\text{s}} }} }}{{T_{\text{s}} }}} \right)\hat{\varLambda }^{2} \left( 0 \right)\left( {\frac{{16\hat{\varLambda }\left( 0 \right)}}{{\pi^{2} }} + \frac{1}{{T_{\text{p}} {C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-0pt} {N_{0} }}}}} \right)$$

(54)

Substituting (52) and (54) into (26), we have

$$\sigma_{\text{DET}}^{2} = \frac{{B_{\text{L}} \left( {1 - 0.5B_{\text{L}} T_{\text{P}} } \right)T_{\text{s}}^{2} \left( {\frac{{16\hat{\varLambda }\left( 0 \right)}}{{\pi^{2} }} + \frac{1}{{T_{\text{p}} {C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-0pt} {N_{0} }}}}} \right)}}{{\frac{{2^{7} \hat{\varLambda }^{2} \left( 0 \right)}}{{\pi^{2} }}{C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-0pt} {N_{0} }}}}$$

(55)

(55) is established based on the assumption that the estimation error of code delay in DLL is close to zero.