An advanced GNSS code multipath detection and estimation algorithm

Abstract

A novel maximum likelihood-based range estimation algorithm is designed to provide robustness to multipath, which is recognized as a dominant error source in DS-CDMA-based navigation systems. The detection–estimation problem is jointly solved to sequentially estimate the parameters of each individual multipath component and predict the existence of a next possible component. A comparison between contemporary maximum likelihood-based multipath estimation techniques and this new technique is provided. A selection of realistic channel simulation models is used to assess relative performance under different operating situations. A set of real GPS L1/CA data processing results are also presented to further assess the applicability of the proposed algorithm for urban navigation.

Appendices

Appendix 1

It is shown here that when the number of parameters to be estimated increases and therefore the Fisher information matrix (FIM) is expanded, the estimation variance of the original parameters decreases (related to “Maximum likelihood channel estimation” section).

Assume that ${\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{1} } \right)$ is the FIM of the original parameters. Then ${\bar{\mathbf{a}}}_{1}$ is extended to ${\bar{\mathbf{a}}}_{2}$ by adding more parameters to be estimated. For simplicity and without loss of generality, it is assumed that these new parameters have been inserted to end of ${\bar{\mathbf{a}}}_{1}$. Considering the fact that the FIM is a symmetric matrix, ${\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{2} } \right)$ can be represented in the following block matrix form:

$${\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{2} } \right) = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{1} } \right)} & {\mathbf{B}} \\ {{\mathbf{B}}^{T} } & {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)} \\ \end{array} } \right],$$

(36)

where ${\bar{\mathbf{a}}}_{3}$ is the vector including only the newly added parameters. Therefore, the estimation covariance matrix for ${\bar{\mathbf{a}}}_{2}$, shown by ${\mathbf{C}}_{{{\bar{\mathbf{a}}}_{2} }}$, can be represented as (Lu and Shiou 2002; Puntanen and Styan 2005)

$${\mathbf{C}}_{{{\mathbf{a}}_{2} }} = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}\left( {\bar{\mathbf{a}}_{1} } \right)} & {\mathbf{B}} \\ {{\mathbf{B}}^{\rm H} } & {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)} \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{1} } \right) - {\mathbf{BI}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{3} } \right){\mathbf{B}}^{\rm H} } \right)^{ - 1} } & { - {\mathbf{I}}^{ - 1} \left( {\bar{\mathbf{a}}_{1} } \right){\mathbf{B}}\left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right) - {\mathbf{B}}^{\rm H} {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{1} } \right){\mathbf{B}}} \right)^{ - 1} } \\ { - {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{3} } \right){\mathbf{B}}^{\rm H} \left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{1} } \right) - {\mathbf{BI}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{3} } \right){\mathbf{B}}^{\rm H} } \right)^{ - 1} } & {\left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right) - {\mathbf{B}}^{\rm H} {\mathbf{I}}^{ - 1} \left( {\bar{\mathbf{a}}_{1} } \right){\mathbf{B}}} \right)^{ - 1} } \\ \end{array} } \right].$$

(37)

Thus, the part of the covariance matrix that is related to the primary set of the parameters is $\left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{{\mathbf{1}}} } \right) - {\mathbf{BA}}^{ - 1} {\mathbf{B}}^{\rm H} } \right)^{ - 1}$. We refer to this matrix as ${\mathbf{C^{\prime}}}_{{{\bar{\mathbf{a}}}_{1} }}$. Furthermore, this matrix can be rewritten as

$$\begin{aligned} {\mathbf{C^{\prime}}}_{{{\mathbf{a}}_{1} }} &= \left( {{\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{1} } \right) - {\mathbf{BI}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{3} } \right){\mathbf{B}}^{\rm H} } \right)^{ - 1} \hfill \\ &= {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{1} } \right) + {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{1} } \right){\mathbf{B}}\left( {{\mathbf{B}}^{\rm H} {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{1} } \right){\mathbf{B}} + {\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)} \right)^{ - 1} {\mathbf{B}}^{\rm H} {\mathbf{I}}^{ - 1} \left( {{\bar{\mathbf{a}}}_{1} } \right) \hfill \\ &= {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} + {\mathbf{C}}_{{\bar{\mathbf{a}}_{1} }} {\mathbf{B}}\left( {{\mathbf{B}}^{\rm H} {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} {\mathbf{B}} + {\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)} \right)^{ - 1} {\mathbf{B}}^{\rm H} {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} \hfill \\ \end{aligned}$$

(38)

In the second term of (38), ${\mathbf{B}}^{\text{H}} {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} {\mathbf{B}} + {\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)$ is a positive definite matrix (because both of the FIM matrix and the covariance matrix are positive definite) and so is its inverse. Therefore, the quadratic form term ${\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} {\mathbf{B}}\left( {{\mathbf{B}}^{\rm H} {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }} {\mathbf{B}} + {\mathbf{I}}\left( {{\bar{\mathbf{a}}}_{3} } \right)} \right)^{ - 1} {\mathbf{B}}^{\rm H} {\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }}$ has real positive elements on its main diagonal. Hence, the diagonal elements of ${\mathbf{C^{\prime}}}_{{{\mathbf{a}}_{1} }}$ which are the variances of estimation of the original set of the parameters are greater than corresponding diagonal elements of ${\mathbf{C}}_{{{\bar{\mathbf{a}}}_{1} }}$, and therefore, the variance of estimation of the primary parameters has been increased after inserting the new set of the parameters to be estimated.

Appendix 2

The asymptotic performance of the GLRT in (28) can be represented as (Kay 1998)

$$2\ln L\left(z \right) {\mathop{\sim}^{a}}\left\{\begin{array}{cc} \chi_{2}^{2} & {\text{under}}\quad{\rm H}_{0}\\ \chi^{\prime 2}_{2} \left(\lambda \right) & {\text{under}}\quad{\rm H}_{1}\\ \end{array}\right.$$

(39)

in which

$$\lambda = \left( {\theta_{{r_{1} }} - \theta_{{r_{0} }} } \right)^{\rm H} \left[ {{\mathbf{I}}_{{\theta_{r} \theta_{r} }} \left( {\theta_{{r_{0} }} ,\theta_{s} } \right) - {\mathbf{I}}_{{\theta_{r} \theta_{s} }} \left( {\theta_{{r_{0} }} ,\theta_{s} } \right){\mathbf{I}}_{{\theta_{s} \theta_{s} }}^{ - 1} \left( {\theta_{{r_{0} }} ,\theta_{s} } \right){\mathbf{I}}_{{\theta_{s} \theta_{r} }} \left( {\theta_{{r_{0} }} ,\theta_{s} } \right)} \right]\,\left( {\theta_{{r_{1} }} - \theta_{{r_{0} }} } \right),$$

(40)

where $\theta_{{r_{0} }}$ and $\theta_{{r_{1} }}$ are the true values of the parameter to be detected under the H₀ and H₁ hypotheses, respectively, and θ _s is a vector that includes the true values of nuisance parameters. I is the Fisher information matrix that can be represented as

$${\mathbf{I}}\left( \theta \right) = {\mathbf{I}}\left( {\theta_{r} ,\theta_{s} } \right) = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{{\theta_{r} \theta_{r} }} \left( {\theta_{r} ,\theta_{s} } \right)} & {{\mathbf{I}}_{{\theta_{r} \theta_{s} }} \left( {\theta_{r} ,\theta_{s} } \right)} \\ {{\mathbf{I}}_{{\theta_{s} \theta_{r} }} \left( {\theta_{r} ,\theta_{s} } \right)} & {{\mathbf{I}}_{{\theta_{s} \theta_{s} }} \left( {\theta_{r} ,\theta_{s} } \right)} \\ \end{array} } \right].$$

(41)

Herein, $\theta_{{r_{0} }} = 0$, $\theta_{{r_{1} }} = a_{m + 1}$ and ${\mathbf{I}}\left( \theta \right)$ is computed from (12). In Fig. 15, the asymptotic performance of detection for some different values of SNR is plotted for different values of m.

Fig. 15

Asymptotic performance of detector for different values of m

It can be noticed from the plots in Fig. 15 that the adverse effect of increasing the number of signal components to be detected is insignificant when m is larger than 2 (which is the normal case for an urban channel). Therefore, this nonlinear effect can be ignored in setting PFA for detecting new multipath components.