# Joint Delay-Doppler Estimation Performance in a Dual Source Context
###### Abstract  
Evaluating the time-delay, Doppler effect and carrier phase of a received signal is a challenging estimation problem that was addressed in a large variety of remote sensing applications. This problem becomes more difficult and less understood when the signal is reflected off one or multiple surfaces and interferes with itself at the receiver stage. This phenomenon might deteriorate the overall system performance, as for the multipath effect in Global Navigation Satellite Systems (GNSS), and mitigation strategies must be accounted for. In other applications such as GNSS reflectometry (GNSS-R) it may be interesting to estimate the parameters of the reflected signal to deduce the geometry and the surface characteristics. In either case, a better understanding of this estimation problem is directly brought by the corresponding lower performance bounds. In the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramer-Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this article, we derive a new compact CRB expression for the joint time-delay and Doppler estimation in a dual source context, considering a band-limited signal and its specular reflection. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. This extends existing results in the single source context and opens the door to a plethora of usages to be discussed in the article. The proposed CRB expressions are validated in two representative navigation and radar examples. time-delay and Doppler estimation; Cramer-Rao bound; maximum likelihood; band-limited signals; dual source; GNSS multipath; GNSS remote sensing 2020 October 202020 October 202020 October 202020 October 202020 October 202020 October 20202020 October 2020202020 October 20  
In some applications the user tries to mitigate the multipath effect because it tends to degrade the quality of the LOS signals of interest, directly affecting the overall system performance. This is the case in standard GNSS where numerous techniques were developed to get rid of the multipath interference (e.g., MEDLINE [8], MMT [9], Vision Correlator [10]). Indeed, GNSS multipath remains one of the main sources of error in challenging urban environments, therefore being a key point to be addressed for safety-critical applications requiring a precise receiver position, such as in intelligent transportation systems. In other applications the user wants to collect and exploit such interference in order to extract features related to the secondary propagation medium or the reflecting surface properties. A typical example is GNSS reflectometry (GNSS-R) where the secondary signal is the main source of information. In GNSS-R, a first approach is to have two separated antennas: one pointing upward collecting the LOS signal and the other one pointing downward, collecting the reflected signal [11; 12]. A second approach is to collect both signals with a single antenna and process the resulting superimposed signals [13; 14]. Both GNSS multipath (i.e., considering one LOS signal and a main secondary path contribution) and the single antenna GNSS-R approach are typical scenarios of a dual source estimation problem, being the main problem of interest in this contribution. When it comes to the estimation of deterministic parameters (i.e., such as time-delay and Doppler in this contribution), a powerful tool to determine the best statistical performance are Cramer-Rao Bounds (CRBs). Indeed, these bounds accurately describe the ultimate achievable minimum variance that an unbiased estimator can reach for a given problem. The CRBs for the delay and delay-Doppler estimation have been investigated in several contributions for different signal models, considering signals with a finite [1; 15; 16; 17; 18] or infinite [19] bandwidth.