# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 3 Time-Delay ML Estimation (MLE) and Cramer-Rao Bound  
Considering the signal model in Equation (1), the time-delay MLE is defined as (i.e., let \(S=span\left(\mathbf{A}\right)\), with \(\mathbf{A}\) a matrix, be the linear span of the set of its column vectors, \(S^{\perp}\) the orthogonal complement of the subspace \(S\), \(\mathbf{\Pi_{A}}=\mathbf{A}\left(\mathbf{A}^{H}\mathbf{A}\right)\mathbf{A}^{H}\) the orthogonal projection over \(S\), and \(\mathbf{\Pi_{A}^{\perp}}=\mathbf{I}-\mathbf{\Pi_{A}}\).) [19]  
\[\hat{\tau}=\arg\min_{\tau}\left\{\mathbf{x}^{H}\mathbf{\Pi_{c}^{\perp}}_{(\tau )}\mathbf{x}\right\}=\arg\max_{\tau}\left\{\frac{\left|\mathbf{c}\left(\tau \right)^{H}\mathbf{x}\right|^{2}}{\mathbf{c}\left(\tau\right)^{H}\mathbf{c} \left(\tau\right)}\right\}, \tag{3}\]  
which is useful to determine the value of \(\mathrm{SNR_{out}}\) (threshold), which allows reaching the CRB, because it is known that such estimator is asymptotically efficient (e.g., in the high SNR regime) for the conditional signal model of interest [14; 15]. In a recent contribution [16], we derived a new compact closed-form CRB for the time-delay estimation of a generic band-limited signal which only depends on the signal samples, given by the following Fisher Information (FI),  
\[F_{\tau|\xi}\left(\mathbf{\varepsilon}\right)=2\mathrm{SNR_{out}}F_{s}^{2} \left(\frac{\mathbf{c}^{H}\mathbf{V}\mathbf{c}}{\mathbf{c}^{H}\mathbf{c}}- \left|\frac{\mathbf{c}^{H}\mathbf{\Lambda}\mathbf{c}}{\mathbf{c}^{H}\mathbf{c }}\right|^{2}\right)\underset{\text{real signal}}{=}2\mathrm{SNR_{out}}F_{s}^{2} \left(\frac{\mathbf{c}^{T}\mathbf{V}\mathbf{c}}{\mathbf{c}^{T}\mathbf{c}} \right), \tag{4}\]  
where \(\mathrm{SNR_{out}}=\frac{\left|a\right|^{2}\mathbb{E}}{\left(\frac{\mathbf{c} ^{2}}{\mathbf{c}^{2}}/\mathbf{F_{s}}\right)}=\frac{\left|a\right|^{2}}{\sigma _{\mathrm{s}}^{2}}\mathbf{c}^{H}\mathbf{c}\) and \(\mathbb{E}\) is the energy of the signal. \(\mathbf{\Lambda}\) and \(\mathbf{V}\) are defined as (for \(N_{1}\leq n,n^{\prime}\leq N_{2}\))  
\[\left(\mathbf{V}\right)_{n,n^{\prime}}=\left| \begin{array}{l}n^{\prime}\
eq n :\left(-1\right)^{\left|n-n^{\prime}\right|}\frac{2}{\left(n-n^{\prime}\right) ^{2}}\\ n^{\prime}=n:\frac{\pi^{2}}{3}\end{array} \right.\text{; }\left(\mathbf{ \Lambda}\right)_{n,n^{\prime}}=\left| \begin{array}{l}n^{\prime}\
eq n: \frac{\left(-1\right)^{\left|n-n^{\prime}\right|}}{\left(n-n^{\prime}\right) }\\ n^{\prime}=n:0\end{array} \right.. \tag{5}\]  
From the FI, the time-delay CRB is written as  
\[\mathrm{CRB}_{\tau|\xi}\left(\mathbf{\varepsilon}^{0}\right)=\left(F_{\tau| \xi}\left(\mathbf{\varepsilon}^{0}\right)\right)^{-1}, \tag{6}\]  
with \(\mathbf{\varepsilon}^{0}\) a selected value of \(\mathbf{\varepsilon}\). We can further extend this CRB result to the joint time-delay and phase \(\theta^{\top}=\left(\varphi,\tau\right)\) estimation resorting to the model in Equation (2), which is useful for the discussion in Section 7.5. The complete derivation of this new CRB expressed from the signal samples is given in Appendix A,  
Figure 1: (**Left**) Multilayer GNSS signal structure. (**Right**) ACF for BPSK(1) and BOC(1,1) subcarriers. \[\mathbf{CRB}_{\theta|\underline{c}} = \left[ \begin{array}{cc}\mathrm{CRB}_{\varphi|\underline{c}}&\mathrm{ CRB}_{\tau,\varphi|\underline{c}}\\ \mathrm{CRB}_{\tau,\varphi|\underline{c}}&\mathrm{CRB}_{\tau|\underline{c}} \end{array}