# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## 6 Further Insights and Outlooks
### Possible CRB Applications  
The CRB formulation proposed in this contribution being easy to manipulate (i.e., as it is expressed in terms of the baseband signal samples), several use cases can be identified. Hereafter is a non-exhaustive list of examples that could lead to further studies, to support the discussion and show that the proposed CRBs are of broad interest.  
* As a direct extension of the results presented in the previous Section 6.1, one could look for the different parameters that can be obtained from \(\epsilon\), and calculate a new CRB dedicated to these specific parameters. In [28], the study applied to the GNSS-R altimetry problem, investigates the receiver altitude and the complex ratio between the reflected and the direct amplitudes. A generalization of this method can easily be outlined as follows. The first step is to translate either the constraints or the new variables into a transform similarly to (29), then the general expression (31) yields the corresponding FIM. If it is simple enough, a matrix inversion might end up to a closed-form expression of the CRB.
* Another application could be the assessment of the impact the secondary signal has on the main signal's parameters estimation. From the CRB expression it is indeed quite simple to extract the CRB for the time-delay estimation while there is no reflection, and the same CRB when the NLOS signals interfere with the LOS one. Such study may bring a new tool to characterize the signal robustness to multipath, being a key issue for instance in safety-critical GNSS applications, an ultimately drive the future generation of signal design.
* Again in the GNSS context, such CRBs can also be used to have an optimal performance assessment of the Carrier-to-Noise Density Ratio (\(C/N_{0}\)) estimation under multipath conditions. Notice that the \(C/N_{0}\) is an important parameter used in several GNSS applications.  
* Regarding precise positioning techniques such as real time kinematics (RTK), needed in modern applications, these CRBs may bring a valuable information on the impact that multipath may have on the final position estimate. Indeed, it is known that RTK solutions are not able to fix the carrier phase ambiguities under harsh propagation conditions such as strong multipath. This would extend the results presented in [17; 18; 35] to more realistic conditions.
* In the dual antenna GNSS-R context, the dual source CRB proposed in this contribution can be exploited in order to characterize the impact that an imperfect isolation between antennas may have on the final GNSS-R product. This may account for the reflected signal leaking into the upward antenna, or the LOS signal contaminating the downward one.
* Finally, a compact CRB expression can be exploited to do optimal signal design as suggested in [16]. As an example, we can study the case of the estimation of the receiver altitude (33). It was shown that the optimal signal can be linked to the first eigenvector of the matrix \(\mathbf{V}^{\Delta,2}(0)\). The optimal CRB would then be given by: \[\mathrm{CRB}_{h|\epsilon}\geq\mathrm{CRB}_{h|\epsilon}^{\mathrm{opt.}}=\frac{ \sigma_{n}^{2}c^{2}\left(1+\left(\rho_{1}/\rho_{0}\right)^{2}\right)}{8\rho_{1 }^{2}\sin^{2}(e)F_{s}w_{1}D_{1,1}},\] (34) where \(\pi\leq D_{1,1}\leq\pi^{2}\) is the largest eigenvalue of \(\mathbf{V}^{\Delta,2}(0)\).