Theoretical analysis of unambiguous 2-D tracking loop performance for band-limited BOC signals

Abstract

Code-tracking accuracy, an important attribute of receiver performance assessment, depends both on characteristics of the ranging signal being tracked and on the structure of the tracking channel. The implementation of binary offset carrier (BOC) modulated signals and the development of new tracking channel structures with a two-dimensional (2-D) loop architecture, in which both the code and subcarrier delays are estimated independently by two separate tracking loops, have resulted in the theories of 2-D loop tracking performance being of increased interest. However, the theories of tracking performance in white noise for band-limited BOC signals using the most representative and most mature 2-D tracking method are still not mature. Therefore, we present the exact expression for tracking performance prediction for limited front-end bandwidths to show how well double estimator technique (DET) could perform for given conditions. While evaluation of the exact expression requires numerical integration, a simple yet accurate closed-form analytical approximation is also provided for a more intuitive description of tracking performance. Moreover, we present a bandwidth-dependent, quasi-optimal, discriminator parameter selection rule to simplify the work of receiver designers while improving tracking performance. These results can provide further guidance for the design, parameters selection, and optimization of the receiver.

Appendix derivation of closed-form analytical approximation expression

Considering the BOC_sin($\alpha k,k$) signal, assume the normalized one-sided front-end bandwidth is wide enough to contain two main lobes of the BOC_sin($\alpha k,k$) signal spectrum, i.e., $b \ge \alpha { + }1$. Substituting $\Delta_{c} { = 1} \cdot T_{s}$ into (11) and (12), one can obtain the normalized slope matrix

$$\begin{aligned} \kappa_{cc} & = \frac{1}{\alpha \pi }\left( {2\text{Si} \left( {\frac{b}{2\alpha }\pi } \right) + \text{Si} \left( {\left( {4\alpha - 1} \right)\frac{b}{2\alpha }\pi } \right) - \text{Si} \left( {\left( {4\alpha + 1} \right)\frac{b}{2\alpha }\pi } \right)} \right) \\ \kappa_{cs} & = \kappa_{sc} \\ & = - \frac{4}{\alpha \pi }\left( {\sum\limits_{m = 1}^{2\alpha - 1} {\left( { - 1} \right)^{m + 1} \text{Si} \left( {m\frac{b}{\alpha }\pi } \right)} - \frac{1}{2} \cdot \text{Si} \left( {2b\pi } \right)} \right) \\ \kappa_{ss} & = \frac{2}{\alpha \pi }\left( {\sum\limits_{m = 1}^{2\alpha } {\left[ \begin{aligned} \left( { - 1} \right)^{m + 1} \left( {2m - 1} \right)\text{Si} \left( {\frac{b}{2\alpha }\pi \left( {D - \left( {4\alpha + 2 - 2m} \right)} \right)} \right) \hfill \\ + \, \left( { - 1} \right)^{m} \left( {2m - 1} \right)\text{Si} \left( {\frac{b}{2\alpha }\pi \left( {D + \left( {4\alpha - 2m} \right)} \right)} \right) \hfill \\ \end{aligned} \right]} } \right) \\ \end{aligned}$$

(24)

Using the approximation relation (18) then (24) can be further approximately simplified as (21). So (16) can be simplified as

$$\varUpsilon \approx \left\{ {\begin{array}{*{20}l} {\left( {\frac{\alpha }{4\alpha - 1}} \right)^{2} } \hfill & {bD \ge 3\alpha } \hfill \\ {\left( {\frac{\alpha \pi }{{2\left( {4\alpha - 1} \right)\left( { - 0.28\cos (\frac{bD}{2\alpha }\pi ) + \frac{\pi }{2}} \right)}}} \right)^{2} } \hfill & {b \ge 3\alpha ,\alpha < bD < 3\alpha } \hfill \\ {\left( {\frac{{\alpha^{2} }}{{\left( {4\alpha - 1} \right)bD}}} \right)^{2} } \hfill & {b \ge 3\alpha ,bD \le \alpha } \hfill \\ {\frac{{\tilde{\kappa }_{sc}^{2} + \tilde{\kappa }_{cc}^{2} - \tilde{\kappa }_{sc} \tilde{\kappa }_{cc} }}{{3 \cdot (\tilde{\kappa }_{cc} \tilde{\kappa }_{ss} - \tilde{\kappa }_{cs} \tilde{\kappa }_{sc} )^{2} }}} \hfill & {b < 3\alpha } \hfill \\ \end{array} } \right.$$

(25)

Similarly, for (17) there is

$$\Gamma = \frac{1}{{b\pi ^{2} }}\left\{ {\begin{array}{*{20}l} { - \left( {8\alpha + 2} \right) + 8\text{F} \left( {\frac{b}{{2\alpha }}\pi \left( {D - 1} \right)} \right)} \hfill \\ { -\, 4\text{F} \left( {\frac{b}{{2\alpha }}\pi \left( {D - \left( {4\alpha + 1} \right)} \right)} \right) - 4\text{F} \left( {\frac{b}{{2\alpha }}\pi \left( {D - \left( {4\alpha - 1} \right)} \right)} \right)} \hfill \\ { + \sum\limits_{{m = 1}}^{{4\alpha - 1}} {c_{m} \cdot \text{F} \left( {\frac{b}{\alpha }\pi \left( {D - \left( {2\alpha - m + 1} \right)} \right)} \right)} } \hfill \\ { + \sum\limits_{{m = 1}}^{{2\alpha + 1}} {a_{m} \cdot \text{F} \left( {m\frac{b}{\alpha }\pi } \right)} } \hfill \\ \end{array} } \right\}$$

(26)

where $a_{m}$ and $c_{m}$ are integer coefficient, which can be eliminated using the approximation relation (19). Substituting (19) into (26), after simplification one can obtain

$$\varGamma \approx \left\{ {\begin{array}{*{20}l} {\left( {4 - \frac{2}{\alpha }} \right)D + \frac{1}{\alpha }} \hfill & {b \ge 3\alpha ,bD \ge \alpha } \hfill \\ {\frac{{2\left( {2\alpha - 1} \right)}}{{\alpha^{2} }}bD^{2} + \frac{1}{\alpha }} \hfill & {b \ge 3\alpha ,bD < \alpha } \hfill \\ \zeta \hfill & {b < 3\alpha } \hfill \\ \end{array} } \right.$$

(27)

Substituting (25) and (27) into (15), we obtain the final closed-form analytical approximation expression (20).