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 Cramér-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.

School of Mechanical Engineering, Shanghai Jiao Tong University, 200240, Shanghai, China

“…𝐶 is the power in the data/pilot component and 𝛾 is the baseband signal with bandwidth 𝐵 including both spreading code and navigation data. This signal model implicitly implies a narrowband assumption on 𝛾 which means the effect of Doppler dilatation on the baseband signal (Lubeigt, 2020) can be neglected. Mixing in the receiver frontend, using an estimate of the received carrier frequency 𝑓 F %& , performs frequency translation and adjustment of 𝑠(𝑡) as…”

“…Finally, the problem of bounding carrier tracking and in particular, frequency tracking using modern and theoretically tight bounds such as WWB and comparison between WWB and ZZB has been overlooked, although code tracking using ZZB under Gaussian assumptions has been investigated (Emmanuele, 2012). There are asymptotic derivations for frequency estimation, for the standard case, high dynamic case (Mcphee et al, 2023a) which accounts for up to 100g acceleration long the line-of-sight (LOS), and a generic non-line-ofsight (NLOS) case (Lubeigt et al, 2020). The misspecified CRB McPhee, 2023b;Lubeigt et al, 2023) have been thoroughly investigated -misspecified because the assumed signal model unintentionally and practically differs from the received signal, resulting in bias.…”

On the asymptotic performance of time-delay and Doppler estimation with a carrier modulated by a band-limited signal

“…Note that τ is the time-delay due to the propagation path and (1 − b) is the dilatation induced by the Doppler effect. Using (4), the complex analytic signal at the output of the receiver's antenna can be expressed as [25] x R (t) = αa ′ (t; η)e jwct + n A (t),…”

“…Consequently, multiple CRB expressions have emerged over recent decades for the time-delay and Doppler estimation problem. These expressions encompass a spectrum of signal types, ranging from finite narrow-band signals [2,[14][15][16][17][18][19][20][21][22] to finite wide-band signals [14,17,20,[23][24][25][26], and even infinite bandwidth signals [27]. All these previous investigations consider that the wave transmission process introduces an unknown phase element.…”

On the Impact and Mitigation of Signal Crosstalk in Ground-Based and Low Altitude Airborne GNSS-R

“…where: (i) F θ 0 | ( ) and F θ 1 | ( ) correspond to the FIMs of the signals when they are totally decoupled, derived in [43]; and (ii) F θ 0 ,θ 1 | ( ) = F θ 1 ,θ 0 | ( ) T characterizes the interference between both signals. A compact closed-form expression of F | ( ), which depends only on the baseband signal samples, in the vein of [43], was very recently derived in [45].…”